School:Theology

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      <text bytes="9508" xml:space="preserve">{{Portal:Humanities/SchoolHeader|Theology}}

{{cquote|''Truthfulness and rationality in religions are truths that can be substantiated by science or those that cannot be proven to be wrong.'' Kurt Kawohl}}

The '''School of Theology''' is devoted to study of religion, spirituality, and deities. Participants here at this school may use rational analysis and argument to discuss, interpret, and teach on any of a myriad of religious topics. For that reason, the school works in close participation with the [[School:Religious Studies|School of Religious Studies]].

Religious Studies is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives. While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint.

The School of Theology can be undertaken to help participants understand more truly one's own religious tradition or can be undertaken with the goal of preservation of religious traditions, reform of a particular tradition, or to apply the resources of a particular religious tradition to some present day problem, situation or need.

So, whether you are interested in Bahá'í, Islam, Buddhism, Christianity, Urantia, or any other doctrinal foundation, Wikiversity is your soundboard and learning environment to find others who think and believe as you do. If you wish to further your knowledge and practice to encompass a richer base for doing so, please, by all means, feel free.

{{RightTOC}}
----
At Wikiversity, a school is a large organizational structure which can contain various departments and divisions. The departments and divisions should be listed in the departments and divisions section. The school can contain [[Portal:Learning Projects|projects]] for developing [[Portal:Learning Materials|learning resources]].
----

''The faculties, departments, and divisions are not listed in any particular order or preference.''

===Faculty of World Religions===
:Note: ''These are religions that have historically dominated, and do currently dominate cultures and nations. They have stamped large footprints in the annals of history, and are at least a thousand years old. The variant traditions within these religions may be much newer and much less dominant.''
* '''[[Buddhist Studies|Division of Buddhist Studies]]'''

{{Christian studies}}

* '''[[Portal:confucian Studies|Division of Confucian Studies]]'''
* '''[[Portal:Hindu Studies|Division of Hindu Studies]]'''
* '''[[Portal:Islamic Studies|Division of Islamic Studies]]'''
** [[Portal:Islamic Studies|The Institute for Islamic Research and Study]]
** [[Portal:Qur'anic Studies|Department of Qur'anic Studies]] &amp;mdash; ''includes topics like''
* '''[[Portal:Jainism Studies|Division of Jainism Studies]]'''
* '''[[Portal:Jewish Studies|Division of Jewish Studies]]'''
** [[Portal:Tanakh Studies|Department of Tanakh Studies]] &amp;mdash; ''includes topics like the [[Torah]] and the [[Portal:Nevi'im|Nevi'im]] and the [[Portal:Ketuvim|Ketuvim]]''
** [[Portal:Talmudic Studies|Department of Talmudic Studies]] &amp;mdash; ''includes topics like the [[Portal:Mishnah|Mishnah]] and the [[Portal:Gemara|Gemara]]''
* '''[[Portal:Taoist Studies|Division of Taoist Studies]]'''
* '''[[Portal:Sikhism Studies|Division of Sikhism Studies]]'''
* '''[[Portal:Santmat Studies|Division of Santmat Studies]]'''
* '''[[Portal:Zoroastrian Studies|Division of Zoroastrian Studies]]'''

===Faculty of historical religions===
:::''Note: These are religions that are old.''
* [[Portal:Ancient Egyprian Religion Studies|Division of Ancient Egyptian Religion Studies]]
* [[Portal:Ancient Greek Religion Studies|Division of Ancient Greek Religion Studies]]
* [[Portal:Ancient Slavic Religion Studies|Division of Ancient Slavic Religion Studies]]
* [[Portal:Ancient Celtic Religion Studies|Division of Ancient Celtic Religion Studies]]
* [[Portal:Ancient Norse Religion Studies|Division of Ancient Norse Religion Studies]]
* [[Portal:Gnostic Studies|Division of Gnostic Studies]]

===Faculty of New Religions===
:::''Note: These are religions that have existed less than a thousand years.''
* [[Portal:Ayyavazhi|Division of Ayyavazhi Studies]]
* [[Bahá'í Faith|Division of Bahá'í Studies]]
* [[Discordianism|Division of Discordian Studies]]
* [[Meher Baba|Division of Meher Baba Studies]]
* [[Portal:Scientology|Division of Scientology Studies]]
* [[Portal:Thelema|Division of Thelemic Studies]]
* [[Portal:Theistic Satanism studies|Division of Theistic Satanism Studies]]
* [[Urantia Book|Division of Urantia Studies]]
* [[Portal:Wiccan Studies|Division of Wiccan Studies]]

===Faculty of other religions===
:::''Note: These are religions that are old, but do not dominate any particular culture or nation.''

* [[Portal:Druze Studies|Division of Druze Studies]]
* [[Portal:Zoroastrian Studies|Division of Zoroastrian Studies]]

===Faculty of Indigenous Religions===
:::''Note: These are religions that are old, but dominate only relatively small local cultures and groups.''
* [[Portal:Indigenous African religious studies|Division of Indigenous African Religion Studies]]
** [[Portal:Serer religious studies|Division of Serer Religion Studies]]
* [[Portal:Shinto studies|Division of Shinto Studies]]

===Related subjects===
* [[Peace studies|Division of Peace Studies]]
* [[Portal:Religious studies|Division of Religious Studies]] &amp;mdash; ''the study of religion as a product of society''
* [[Portal:Comparative religion|Department of Comparative Religion]]

==Learning resources and projects==
{{Col list|3|
* [[God/Nils/Angel oracle|Angel oracle]]
* [[Ásatrú]]
* [[Ayyavazhi]]
* [[b:Bahá'í Faith|Bahá'í Faith]]
* [[Beyond Theism]]
* [[Book:The Dynamic Book]]
* [[Buddha oracle]]
* [[Buddhism]]
* [[Christian Anarchism]]
* [[Christianity]]
* [[Christian Theology]]
* [[Christian mysticism]]
* [[Daoism|Taoism]] - [[I Ching oracle]]
* [[Hinduism]]
* [[Yoga]]
* [[Dodekatheon]]
* [[Islam]]
* [[Jainism]]
* [[Portal:Jewish Studies|Judaism]]
* [[Sikhism]]
* [[God/Nils/Spirituality|Spirituality]]
* [[Urantia Book]]
* [[Zoroastrianism]]

Mysticism:
*[[Esoteric Buddhism]]
*[[Gnosticism]]
*[[Kabbalah]]
*[[Sufism]]
*[[A Course in Miracles]]
*[[God/Nils/God and Enlightenment|God and Enlightenment]]
}}

* [[Ethics/Nonkilling/Spiritual/Islam|Nonkilling in Islam]]
* [[Ethics/Nonkilling/Spiritual/Hinduism|Nonkilling in Hinduism]]
* [[Ethics/Nonkilling/Spiritual/Buddhism|Nonkilling in Buddhism]]
* [[Ethics/Nonkilling/Spiritual/Jainism|Nonkilling in Jainism]]
* [[Ethics/Nonkilling/Spiritual/Voudou|Nonkilling in Voudou]]
* [[Ethics/Nonkilling/Spiritual/Humanism|Nonkilling in Humanism]]

=== [[Wikisource]] ===
* [[s:Bible|The Bible]] - several complete versions ([[s:Bible (King James)|KJV]], [[s:Bible (American Standard)|ASE]], [[s:Bible (World English)|WEB]], ...) of the whole text and [[s:Bible (Free)|another]] under development
* [[s:The Holy Qur'an|The Holy Qur'an]] - central religious text of [[Islam]]
* [[s:The Bhagavad Gita|The Bhagavad Gita]] - as it is
* [[s:A Course In Miracles|A Course In Miracles]] - by [[w:Helen Schucman|Helen Schucman]]
* [[s:Tao Te Ching (Wikisource translation)|Tao Te Ching]] - key to [[Daoism|Taoism]] (or [[Daoism]])
* [[s:The Urantia Book|The Urantia Book]] - the complete public domain text
* [[s:The Book of the Law|The Book of the Law]] - The central text of Thelema
*...''[[s:Ancient and Classical texts/Titles|many more Ancient and Classical texts at Wikisource]]''...

=== Wikibooks ===
* [[b:Wikibooks:Humanities bookshelf|Humanities bookshelf]]

=== Wikipedia ===
*[[w:Portal:Religion|Portal:Religion]] - (what's the difference?)
* [[w: Scottish Journal of Theology]]

==Active participants==
Please see [[/Participants/|participants]].

==Things you can do!==

* Clean up [[Draft:Theology]] and move it to become a subpage of a supporting main page learning project.
* Clean up [[Draft:Universals]] and move it to become a subpage of a supporting main page learning project.
* Clean up [[Draft:What is a human?]] and move it to become a subpage of a supporting main page learning project.

==School news==
* '''17 June 2004''' - School founded!
* '''6 March 2017''' - [[Does God exist?]], a WikiDebate, announced on Wikiversity Main Page News!
* '''4 July 2017''' - [[Does everything happen for a sufficient reason?]], a WikiDebate, announced on Wikiversity Main Page News!
* '''4 July 2017''' - [[Do humans have free will?]], a WikiDebate, announced on Wikiversity Main Page News!
* '''15 October 2017''' - [[Seeking True Beliefs]] announced on Wikiversity Main Page News!

== See also ==
*[[School:Relosophy|The School of Relosophy]]

==External links==
* February 2008 ''[http://www.physorg.com/news122663575.html University of Oxford researchers will spend nearly $4 million to study why mankind embraces God]''
* May 2008 ''[http://www.lep.co.uk/weirdnews/I-believeGod-is-a-man.4095585.jp I believe... God is a Man]''
* [http://www.thinkunity.com Christian Mysticism]

[[Category:Wikiversity schools]]
[[Category:Theology| ]]

[[ar:كلية علوم الأديان]]
[[de:Fachbereich Religionswissenschaften und Theologie]]
[[es:Departamento de Teología]]
[[fr:Faculté:Théologie]]
[[it:Facoltà:Teologia]]
[[ru:Факультет теологии]]</text>
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      <text bytes="9506" xml:space="preserve">{{Portal:Humanities/SchoolHeader|Theology}}

{{cquote|''Truthfulness and rationality in religions are truths that can be substantiated by science or those that cannot be proven to be wrong.'' Kurt Kawohl}}

The '''School of Theology''' is devoted to study of religion, spirituality, and deities. Participants here at this school may use rational analysis and argument to discuss, interpret, and teach on any of a myriad of religious topics. For that reason, the school works in close participation with the [[School:Religious Studies|School of Religious Studies]].

Religious Studies is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives. While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint.

The School of Theology can be undertaken to help participants understand more truly one's own religious tradition or can be undertaken with the goal of preservation of religious traditions, reform of a particular tradition, or to apply the resources of a particular religious tradition to some present day problem, situation or need.

So, whether you are interested in Bahá'í, Islam, Buddhism, Christianity, Urantia, or any other doctrinal foundation, Wikiversity is your soundboard and learning environment to find others who think and believe as you do. If you wish to further your knowledge and practice to encompass a richer base for doing so, please, by all means, feel free.

{{RightTOC}}
----
At Wikiversity, a school is a large organizational structure which can contain various departments and divisions. The departments and divisions should be listed in the departments and divisions section. The school can contain [[Portal:Learning Projects|projects]] for developing [[Portal:Learning Materials|learning resources]].
----

''The faculties, departments, and divisions are not listed in any particular order or preference.''

===Faculty of World Religions===
:Note: ''These are religions that have historically dominated, and do currently dominate cultures and nations. They have stamped large footprints in the annals of history, and are at least a thousand years old. The variant traditions within these religions may be much newer and much less dominant.''
* '''[[Buddhist Studies|Division of Buddhist Studies]]'''

{{Christian studies}}

* '''[[Portal:confucian Studies|Division of Confucian Studies]]'''
* '''[[Portal:Hindu Studies|Division of Hindu Studies]]'''
* '''[[Portal:Islamic Studies|Division of Islamic Studies]]'''
** [[Portal:Islamic Studies|The Institute for Islamic Research and Study]]
** [[Portal:Qur'anic Studies|Department of Qur'anic Studies]] &amp;mdash; ''includes topics like''
* '''[[Portal:Jainism Studies|Division of Jainism Studies]]'''
* '''[[Portal:Jewish Studies|Division of Jewish Studies]]'''
** [[Portal:Tanakh Studies|Department of Tanakh Studies]] &amp;mdash; ''includes topics like the [[Torah]] and the [[Portal:Nevi'im|Nevi'im]] and the [[Portal:Ketuvim|Ketuvim]]''
** [[Portal:Talmudic Studies|Department of Talmudic Studies]] &amp;mdash; ''includes topics like the [[Portal:Mishnah|Mishnah]] and the [[Portal:Gemara|Gemara]]''
* '''[[Portal:Taoist Studies|Division of Taoist Studies]]'''
* '''[[Portal:Sikhism Studies|Division of Sikhism Studies]]'''
* '''[[Portal:Santmat Studies|Division of Santmat Studies]]'''
* '''[[Portal:Zoroastrian Studies|Division of Zoroastrian Studies]]'''

===Faculty of historical religions===
:::''Note: These are religions that are old.''
* [[Portal:Ancient Egyprian Religion Studies|Division of Ancient Egyptian Religion Studies]]
* [[Portal:Ancient Greek Religion Studies|Division of Ancient Greek Religion Studies]]
* [[Portal:Ancient Slavic Religion Studies|Division of Ancient Slavic Religion Studies]]
* [[Portal:Ancient Celtic Religion Studies|Division of Ancient Celtic Religion Studies]]
* [[Portal:Ancient Norse Religion Studies|Division of Ancient Norse Religion Studies]]
* [[Portal:Gnostic Studies|Division of Gnostic Studies]]

===Faculty of New Religions===
:::''Note: These are religions that have existed less than a thousand years.''
* [[Portal:Ayyavazhi|Division of Ayyavazhi Studies]]
* [[Bahá'í Faith|Division of Bahá'í Studies]]
* [[Discordianism|Division of Discordian Studies]]
* [[Meher Baba|Division of Meher Baba Studies]]
* [[Portal:Scientology|Division of Scientology Studies]]
* [[Portal:Thelema|Division of Thelemic Studies]]
* [[Portal:Theistic Satanism studies|Division of Theistic Satanism Studies]]
* [[Urantia Book|Division of Urantia Studies]]
* [[Portal:Wiccan Studies|Division of Wiccan Studies]]

===Faculty of other religions===
:::''Note: These are religions that are old, but do not dominate any particular culture or nation.''

* [[Portal:Druze Studies|Division of Druze Studies]]
* [[Portal:Zoroastrian Studies|Division of Zoroastrian Studies]]

===Faculty of Indigenous Religions===
:::''Note: These are religions that are old, but dominate only relatively small local cultures and groups.''
* [[Portal:Indigenous African religious studies|Division of Indigenous African Religion Studies]]
** [[Portal:Serer religious studies|Division of Serer Religion Studies]]
* [[Portal:Shinto studies|Division of Shinto Studies]]

===Related subjects===
* [[Peace studies|Division of Peace Studies]]
* [[School:Religious_Studies|School of Religious Studies]] &amp;mdash; ''the study of religion as a product of society''
* [[Portal:Comparative religion|Department of Comparative Religion]]

==Learning resources and projects==
{{Col list|3|
* [[God/Nils/Angel oracle|Angel oracle]]
* [[Ásatrú]]
* [[Ayyavazhi]]
* [[b:Bahá'í Faith|Bahá'í Faith]]
* [[Beyond Theism]]
* [[Book:The Dynamic Book]]
* [[Buddha oracle]]
* [[Buddhism]]
* [[Christian Anarchism]]
* [[Christianity]]
* [[Christian Theology]]
* [[Christian mysticism]]
* [[Daoism|Taoism]] - [[I Ching oracle]]
* [[Hinduism]]
* [[Yoga]]
* [[Dodekatheon]]
* [[Islam]]
* [[Jainism]]
* [[Portal:Jewish Studies|Judaism]]
* [[Sikhism]]
* [[God/Nils/Spirituality|Spirituality]]
* [[Urantia Book]]
* [[Zoroastrianism]]

Mysticism:
*[[Esoteric Buddhism]]
*[[Gnosticism]]
*[[Kabbalah]]
*[[Sufism]]
*[[A Course in Miracles]]
*[[God/Nils/God and Enlightenment|God and Enlightenment]]
}}

* [[Ethics/Nonkilling/Spiritual/Islam|Nonkilling in Islam]]
* [[Ethics/Nonkilling/Spiritual/Hinduism|Nonkilling in Hinduism]]
* [[Ethics/Nonkilling/Spiritual/Buddhism|Nonkilling in Buddhism]]
* [[Ethics/Nonkilling/Spiritual/Jainism|Nonkilling in Jainism]]
* [[Ethics/Nonkilling/Spiritual/Voudou|Nonkilling in Voudou]]
* [[Ethics/Nonkilling/Spiritual/Humanism|Nonkilling in Humanism]]

=== [[Wikisource]] ===
* [[s:Bible|The Bible]] - several complete versions ([[s:Bible (King James)|KJV]], [[s:Bible (American Standard)|ASE]], [[s:Bible (World English)|WEB]], ...) of the whole text and [[s:Bible (Free)|another]] under development
* [[s:The Holy Qur'an|The Holy Qur'an]] - central religious text of [[Islam]]
* [[s:The Bhagavad Gita|The Bhagavad Gita]] - as it is
* [[s:A Course In Miracles|A Course In Miracles]] - by [[w:Helen Schucman|Helen Schucman]]
* [[s:Tao Te Ching (Wikisource translation)|Tao Te Ching]] - key to [[Daoism|Taoism]] (or [[Daoism]])
* [[s:The Urantia Book|The Urantia Book]] - the complete public domain text
* [[s:The Book of the Law|The Book of the Law]] - The central text of Thelema
*...''[[s:Ancient and Classical texts/Titles|many more Ancient and Classical texts at Wikisource]]''...

=== Wikibooks ===
* [[b:Wikibooks:Humanities bookshelf|Humanities bookshelf]]

=== Wikipedia ===
*[[w:Portal:Religion|Portal:Religion]] - (what's the difference?)
* [[w: Scottish Journal of Theology]]

==Active participants==
Please see [[/Participants/|participants]].

==Things you can do!==

* Clean up [[Draft:Theology]] and move it to become a subpage of a supporting main page learning project.
* Clean up [[Draft:Universals]] and move it to become a subpage of a supporting main page learning project.
* Clean up [[Draft:What is a human?]] and move it to become a subpage of a supporting main page learning project.

==School news==
* '''17 June 2004''' - School founded!
* '''6 March 2017''' - [[Does God exist?]], a WikiDebate, announced on Wikiversity Main Page News!
* '''4 July 2017''' - [[Does everything happen for a sufficient reason?]], a WikiDebate, announced on Wikiversity Main Page News!
* '''4 July 2017''' - [[Do humans have free will?]], a WikiDebate, announced on Wikiversity Main Page News!
* '''15 October 2017''' - [[Seeking True Beliefs]] announced on Wikiversity Main Page News!

== See also ==
*[[School:Relosophy|The School of Relosophy]]

==External links==
* February 2008 ''[http://www.physorg.com/news122663575.html University of Oxford researchers will spend nearly $4 million to study why mankind embraces God]''
* May 2008 ''[http://www.lep.co.uk/weirdnews/I-believeGod-is-a-man.4095585.jp I believe... God is a Man]''
* [http://www.thinkunity.com Christian Mysticism]

[[Category:Wikiversity schools]]
[[Category:Theology| ]]

[[ar:كلية علوم الأديان]]
[[de:Fachbereich Religionswissenschaften und Theologie]]
[[es:Departamento de Teología]]
[[fr:Faculté:Théologie]]
[[it:Facoltà:Teologia]]
[[ru:Факультет теологии]]</text>
      <sha1>qb5bk4uj0mh36lbkezv53ivluqso77j</sha1>
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    <title>Category:Religious studies</title>
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        <id>2947466</id>
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      <model>wikitext</model>
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      <text bytes="801" xml:space="preserve">'''Religious Studies''' is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While Theology attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

== See Also ==
* [[School:Theology]]
* [[School:Religious Studies]]
* [[Wikipedia: Religious Studies]]

[[Category:Religion]]
[[Category:Theology]]</text>
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      <text bytes="779" xml:space="preserve">'''Religious Studies''' is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While Theology attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

== See Also ==
* [[School:Religious Studies]]
* [[Wikipedia: Religious Studies]]

[[Category:Religion]]
[[Category:Theology]]</text>
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      <text bytes="760" xml:space="preserve">{{center top}}
[[File:Great Seal of the United States (obverse).svg|thumb|left|250px]]


----

'''[[School:Fire and Emergency Management/About Open Source Learning Project|About the Open Source Learning Project]]'''

'''[[/General|General]]'''

'''[[/Scene Examination|Scene Examination]]'''

'''[[/Documenting the Scene|Documenting the Scene]]'''

'''[[/Evidence Collection and Preservation|Evidence Collection and Preservation]]'''

'''[[/Interview|Interview]]'''

'''[[/Post Incident Investigation|Post Incident Investigation]]'''

'''[[/Presentations|Presentations]]'''

'''[[/Cognitive Applications|Cognitive Applications]]'''

'''[[/Psychomotor Applications|Psychomotor Applications]]'''
----
{{center bottom}}

[[Category:Fire and Emergency Management]]</text>
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    <ns>0</ns>
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    <revision>
      <id>2411968</id>
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      <text bytes="495" xml:space="preserve">{{center top}}

[[File:Great Seal of the United States (obverse).svg|thumb|left|250px]]


----

'''[[School:Fire and Emergency Management/About Open Source Learning Project|About the Open Source Learning Project]]'''

'''[[/Response|Response]]'''

'''[[/Fire Suppression|Fire Suppression]]'''

'''[[/Rescue|Rescue]]'''

'''[[/Additional Training Materials|Additional Training Materials]]'''

{{center bottom}}


[[Category:Fire and Emergency Management]]

[[Category:Pages moved from Wikibooks]]</text>
      <sha1>tasro62a6xgyo9y9oqofuxhf1yqpfzj</sha1>
    </revision>
  </page>
  <page>
    <title>Fire and emergency management/ARFF Driver Operator</title>
    <ns>0</ns>
    <id>6924</id>
    <revision>
      <id>2411969</id>
      <parentid>2190056</parentid>
      <timestamp>2022-08-04T22:41:50Z</timestamp>
      <contributor>
        <username>Rkt2312</username>
        <id>550330</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="482" xml:space="preserve">{{center top}}

{{center top}}
[[File:Great Seal of the United States (obverse).svg|thumb|left|250px]]


----

'''[[School:Fire and Emergency Management/About Open Source Learning Project|About the Open Source Learning Project]]'''

'''[[School:Fire and Emergency Management/ARFF Driver Operator/Additional Training Materials|Additional Training Materials]]'''

{{center bottom}}
{{center bottom}}

[[Category:Fire and Emergency Management]]

[[Category:Pages moved from Wikibooks]]</text>
      <sha1>r8fbd09cwe2zbsqott5f4sabol8tp1u</sha1>
    </revision>
  </page>
  <page>
    <title>P-18 Training</title>
    <ns>0</ns>
    <id>6925</id>
    <revision>
      <id>2411966</id>
      <parentid>2191155</parentid>
      <timestamp>2022-08-04T22:40:46Z</timestamp>
      <contributor>
        <username>Rkt2312</username>
        <id>550330</id>
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[[File:Great Seal of the United States (obverse).svg|thumb|left|250px]]


----

'''[[School:Fire and Emergency Management/About Open Source Learning Project|About the Open Source Learning Project]]'''

'''[[media:P-18_LP.pdf|Lesson Plan]]'''

'''[[media:P-18_SG.pdf|Student Guide]]'''

'''[[School:Fire and Emergency Management/Cognitive Applications|Cognitive Applications]]'''

----
{{center bottom}}

[[Category:Fire and Emergency Management]]

[[Category:Pages moved from Wikibooks]]</text>
      <sha1>bkdp9zkn6t5sc18s44if04rh75s0t8f</sha1>
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  <page>
    <title>Fire and emergency management/P-19 Training</title>
    <ns>0</ns>
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    <revision>
      <id>2411970</id>
      <parentid>2190628</parentid>
      <timestamp>2022-08-04T22:42:06Z</timestamp>
      <contributor>
        <username>Rkt2312</username>
        <id>550330</id>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="475" xml:space="preserve">{{center top}}
[[File:Great Seal of the United States (obverse).svg|thumb|left|250px]]


----

'''[[School:Fire and Emergency Management/About Open Source Learning Project|About the Open Source Learning Project]]'''

'''Lesson Plan'''

'''[[b:media:P19_SG_1.pdf|Student Guide]]'''

'''[[b:media:P19_Slides.pdf|Presentation Slides]]'''

'''Cognitive Applications'''

----


{{center bottom}}

[[Category:Fire and Emergency Management]]

[[Category:Pages moved from Wikibooks]]</text>
      <sha1>34yz3vl68ye2971mmp7p60reljhli00</sha1>
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  <page>
    <title>Fire and emergency management/Fire officer I</title>
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      <id>2411971</id>
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      <contributor>
        <username>Rkt2312</username>
        <id>550330</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1018" xml:space="preserve">Return to [[Wikiversity:School_of_Fire_and_Emergency_Management|School of Fire and Emergency Management]]


{{center top}}


[[File:Great Seal of the United States (obverse).svg|thumb|left|250px]]

----

'''[[Wikiversity:School of Fire and Emergency Management/About Open Source Learning Project|About the Open Source Learning Project]]'''

'''[[/Human Resource Management|Human Resource Management]]'''

'''[[/Community and Government Relations|Community and Government Relations]]'''

'''[[/Administration|Administration]]'''

'''[[/Inspections and Investigations|Inspections and Investigations]]'''

'''[[/Emergency Services Delivery|Emergency Services Delivery]]'''

'''[[/Health and Safety|Health and Safety]]'''

'''[[/Cognitive Applications|Cognitive Applications]]'''

'''[[/Psychomotor Applications|Psychomotor Applications]]'''

'''[[/Additional Training Materials|Additional Training Materials]]'''

----
{{center bottom}}

[[Category:Fire and Emergency Management]]

[[Category:Pages moved from Wikibooks]]</text>
      <sha1>c0jxficg19794y4bomb6ebmq3qs9ynw</sha1>
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  </page>
  <page>
    <title>Portal:Religious studies</title>
    <ns>102</ns>
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    <revision>
      <id>2411939</id>
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      <contributor>
        <username>SotanScholar</username>
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      <text bytes="3398" xml:space="preserve">{{center top}}&lt;big&gt;'''Welcome to the Wikiversity Division of Religious Studies'''&lt;/big&gt;{{center bottom}}
{{center top}}
''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''{{center bottom}}
[[Image:Religions 4x5.png|thumb|150px|Symbols of religions]]

The Division of Religious Studies is for multi-disciplinary '''and''' secular study of religion.

Within Wikiversity, the [[School:Theology|School of Theology]] can be distinguished from the Division of Religious Studies. The School of Theology is devoted to study of religion, spirituality, and deities. Studies within the School of Theology can be undertaken to help participants understand more truly one's own religious tradition or can be undertaken with the goal of preservation of religious traditions, reform of a particular tradition, or to apply the resources of a particular religious tradition to some present day problem, situation or need. In contrast, participants in the Division of Religious Studies never adopt the position of any particular religious tradition except in an attempt to understand it as a particular way of thinking.

The Wikiversity Division of Religious Studies works in close cooperation with the Wikiversity [[School:Theology|School of Theology]].

==School news==
* '''9 September 2006''' - Original Division (under Theology) founded! 
* '''4 August 2022''' - Division converted to a School!

==Subdivisions and Departments==
* [[Topic:Ayyavazhi|Department of Ayyavazhi Studies]]
* [[Bahá'í Faith|Department of Bahá'í Studies]]
* [[Buddhist Studies|Department of Buddhist Studies]]
* [[Hinduism|Department for the Study of Hinduism]]
* [[Abrahamic religions|Department of Abrahamic Science]]
** [[Bahá'í Faith|Division of Baha'i Studies]]
** [[Topic:Christian studies|Division of Christian Studies]]
** [[Portal:Jewish Studies|Division of Jewish Studies]]
** [[Portal:Islamic Studies|Division  of Islamic Studies]]
* [[Topic:Zoroastrian Studies|Division of Mazdayasna Studies]]
* [[Meher Baba|Department of Meher Baba Studies]]
* [[Topic:Interreligious conflict studies|Department of Inter-Religious Conflict Studies]]
* [[Portal:Religious studies|Department for the Study of New Age Religions]]
* [[Portal:Thelema|Department of Thelemic Studies]]
* [[Portal:Comparative religion|Department of Comparative Religion]]
* [[Topic:Geography of Religion|Department of Religious &amp; Mystical Geography]]

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this division, you can list your name here (this can help small divisions grow and the participants communicate better; for large divisions a list of active participants is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]], email:[mailto:dewnada@icloud.com dewnada@icloud.com]

===Inactive participants===

* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]

==Things you can do==


== External links ==

[[Category:{{PAGENAME}}| ]]</text>
      <sha1>pyrcaw0hkitvx9w3df9evtjrjs40kqd</sha1>
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      <text bytes="3223" xml:space="preserve">{{center top}}&lt;big&gt;'''Welcome to the Wikiversity Division of Religious Studies'''&lt;/big&gt;{{center bottom}}
{{center top}}
''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''{{center bottom}}
[[Image:Religions 4x5.png|thumb|150px|Symbols of religions]]

T'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].

==School news==
* '''9 September 2006''' - Original Division (under Theology) founded! 
* '''4 August 2022''' - Division converted to a School!

==Subdivisions and Departments==
* [[Topic:Ayyavazhi|Department of Ayyavazhi Studies]]
* [[Bahá'í Faith|Department of Bahá'í Studies]]
* [[Buddhist Studies|Department of Buddhist Studies]]
* [[Hinduism|Department for the Study of Hinduism]]
* [[Abrahamic religions|Department of Abrahamic Science]]
** [[Bahá'í Faith|Division of Baha'i Studies]]
** [[Topic:Christian studies|Division of Christian Studies]]
** [[Portal:Jewish Studies|Division of Jewish Studies]]
** [[Portal:Islamic Studies|Division  of Islamic Studies]]
* [[Topic:Zoroastrian Studies|Division of Mazdayasna Studies]]
* [[Meher Baba|Department of Meher Baba Studies]]
* [[Topic:Interreligious conflict studies|Department of Inter-Religious Conflict Studies]]
* [[Portal:Religious studies|Department for the Study of New Age Religions]]
* [[Portal:Thelema|Department of Thelemic Studies]]
* [[Portal:Comparative religion|Department of Comparative Religion]]
* [[Topic:Geography of Religion|Department of Religious &amp; Mystical Geography]]

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this division, you can list your name here (this can help small divisions grow and the participants communicate better; for large divisions a list of active participants is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]], email:[mailto:dewnada@icloud.com dewnada@icloud.com]

===Inactive participants===

* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]

==Things you can do==


== External links ==

[[Category:{{PAGENAME}}| ]]</text>
      <sha1>58nt7jxbvijmb7y52y93yjb6v3sflr5</sha1>
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    <revision>
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      <text bytes="3328" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].

==School news==
* '''9 September 2006''' - Original Division (under Theology) founded! 
* '''4 August 2022''' - Division converted to a School!

==Subdivisions and Departments==
* [[Topic:Ayyavazhi|Department of Ayyavazhi Studies]]
* [[Bahá'í Faith|Department of Bahá'í Studies]]
* [[Buddhist Studies|Department of Buddhist Studies]]
* [[Hinduism|Department for the Study of Hinduism]]
* [[Abrahamic religions|Department of Abrahamic Science]]
** [[Bahá'í Faith|Division of Baha'i Studies]]
** [[Topic:Christian studies|Division of Christian Studies]]
** [[Portal:Jewish Studies|Division of Jewish Studies]]
** [[Portal:Islamic Studies|Division  of Islamic Studies]]
* [[Topic:Zoroastrian Studies|Division of Mazdayasna Studies]]
* [[Meher Baba|Department of Meher Baba Studies]]
* [[Topic:Interreligious conflict studies|Department of Inter-Religious Conflict Studies]]
* [[Portal:Religious studies|Department for the Study of New Age Religions]]
* [[Portal:Thelema|Department of Thelemic Studies]]
* [[Portal:Comparative religion|Department of Comparative Religion]]
* [[Topic:Geography of Religion|Department of Religious &amp; Mystical Geography]]

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this division, you can list your name here (this can help small divisions grow and the participants communicate better; for large divisions a list of active participants is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]], email:[mailto:dewnada@icloud.com dewnada@icloud.com]

===Inactive participants===

* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]

==Things you can do==


== External links ==

[[Category:{{PAGENAME}}| ]]</text>
      <sha1>fn3sp05syec2o39k24f4z2dmnzmz6yo</sha1>
    </revision>
    <revision>
      <id>2411942</id>
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      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
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      <comment>/* School news */</comment>
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      <text bytes="3358" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].

==School news==
* '''9 September 2006''' - Original Division (under Theology) founded! 
* '''4 August 2022''' - Division converted to a School! -- [School:Religious Studies]

==Subdivisions and Departments==
* [[Topic:Ayyavazhi|Department of Ayyavazhi Studies]]
* [[Bahá'í Faith|Department of Bahá'í Studies]]
* [[Buddhist Studies|Department of Buddhist Studies]]
* [[Hinduism|Department for the Study of Hinduism]]
* [[Abrahamic religions|Department of Abrahamic Science]]
** [[Bahá'í Faith|Division of Baha'i Studies]]
** [[Topic:Christian studies|Division of Christian Studies]]
** [[Portal:Jewish Studies|Division of Jewish Studies]]
** [[Portal:Islamic Studies|Division  of Islamic Studies]]
* [[Topic:Zoroastrian Studies|Division of Mazdayasna Studies]]
* [[Meher Baba|Department of Meher Baba Studies]]
* [[Topic:Interreligious conflict studies|Department of Inter-Religious Conflict Studies]]
* [[Portal:Religious studies|Department for the Study of New Age Religions]]
* [[Portal:Thelema|Department of Thelemic Studies]]
* [[Portal:Comparative religion|Department of Comparative Religion]]
* [[Topic:Geography of Religion|Department of Religious &amp; Mystical Geography]]

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this division, you can list your name here (this can help small divisions grow and the participants communicate better; for large divisions a list of active participants is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]], email:[mailto:dewnada@icloud.com dewnada@icloud.com]

===Inactive participants===

* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]

==Things you can do==


== External links ==

[[Category:{{PAGENAME}}| ]]</text>
      <sha1>1zyfstubqjai57tkwzh5z0tab45tiqb</sha1>
    </revision>
    <revision>
      <id>2411943</id>
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      <text bytes="2348" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].

==School news==
* '''9 September 2006''' - Original Division (under Theology) founded! 
* '''4 August 2022''' - Division converted to a School! -- [School:Religious Studies]

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this division, you can list your name here (this can help small divisions grow and the participants communicate better; for large divisions a list of active participants is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]], email:[mailto:dewnada@icloud.com dewnada@icloud.com]

===Inactive participants===

* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]

==Things you can do==


== External links ==

[[Category:{{PAGENAME}}| ]]</text>
      <sha1>3o3dnbdd9j0oo3dp50mwqla8mg9aepo</sha1>
    </revision>
    <revision>
      <id>2411944</id>
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        <id>2947466</id>
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      <text bytes="2404" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].

==School news==
* '''9 September 2006''' - Original Division (under Theology) founded! 
* '''4 August 2022''' - Division converted to a School! -- [School:Religious Studies]

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this division, you can list your name here (this can help small divisions grow and the participants communicate better; for large divisions a list of active participants is not needed).

* [[user:SotanScholar|SotanScholar]] as of Aug 3, 2022.
* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]], email:[mailto:dewnada@icloud.com dewnada@icloud.com]

===Inactive participants===

* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]

==Things you can do==


== External links ==

[[Category:{{PAGENAME}}| ]]</text>
      <sha1>l3dkgi3cop9uxrjxilp572e3yr8avmy</sha1>
    </revision>
    <revision>
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      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* Things you can do */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2474" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].

==School news==
* '''9 September 2006''' - Original Division (under Theology) founded! 
* '''4 August 2022''' - Division converted to a School! -- [School:Religious Studies]

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this division, you can list your name here (this can help small divisions grow and the participants communicate better; for large divisions a list of active participants is not needed).

* [[user:SotanScholar|SotanScholar]] as of Aug 3, 2022.
* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]], email:[mailto:dewnada@icloud.com dewnada@icloud.com]

===Inactive participants===

* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]

==Things you can do==

Head over to the school page and help out! [School:Religious Studies]

== External links ==

[[Category:{{PAGENAME}}| ]]</text>
      <sha1>a6gp0c4tqjhe4osxxanfbn0lhcf3gbi</sha1>
    </revision>
    <revision>
      <id>2411946</id>
      <parentid>2411945</parentid>
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      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* Things you can do */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2476" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].

==School news==
* '''9 September 2006''' - Original Division (under Theology) founded! 
* '''4 August 2022''' - Division converted to a School! -- [School:Religious Studies]

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this division, you can list your name here (this can help small divisions grow and the participants communicate better; for large divisions a list of active participants is not needed).

* [[user:SotanScholar|SotanScholar]] as of Aug 3, 2022.
* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]], email:[mailto:dewnada@icloud.com dewnada@icloud.com]

===Inactive participants===

* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]

==Things you can do==

Head over to the school page and help out! [[School:Religious Studies]]

== External links ==

[[Category:{{PAGENAME}}| ]]</text>
      <sha1>to6u9ys72q2j3h4wgfyf1srij9au98i</sha1>
    </revision>
    <revision>
      <id>2411947</id>
      <parentid>2411946</parentid>
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      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* Things you can do */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2481" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].

==School news==
* '''9 September 2006''' - Original Division (under Theology) founded! 
* '''4 August 2022''' - Division converted to a School! -- [School:Religious Studies]

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this division, you can list your name here (this can help small divisions grow and the participants communicate better; for large divisions a list of active participants is not needed).

* [[user:SotanScholar|SotanScholar]] as of Aug 3, 2022.
* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]], email:[mailto:dewnada@icloud.com dewnada@icloud.com]

===Inactive participants===

* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]

==Things you can do==

Head over to the school page and help out!  --- [[School:Religious Studies]]

== External links ==

[[Category:{{PAGENAME}}| ]]</text>
      <sha1>9kwmfb9cjkzhc2kelqqqvb8d5rs36ee</sha1>
    </revision>
    <revision>
      <id>2411948</id>
      <parentid>2411947</parentid>
      <timestamp>2022-08-04T21:15:20Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* School news */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2483" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].

==School news==
* '''9 September 2006''' - Original Division (under Theology) founded! 
* '''4 August 2022''' - Division converted to a School!  --- [School:Religious Studies]

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this division, you can list your name here (this can help small divisions grow and the participants communicate better; for large divisions a list of active participants is not needed).

* [[user:SotanScholar|SotanScholar]] as of Aug 3, 2022.
* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]], email:[mailto:dewnada@icloud.com dewnada@icloud.com]

===Inactive participants===

* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]

==Things you can do==

Head over to the school page and help out!  --- [[School:Religious Studies]]

== External links ==

[[Category:{{PAGENAME}}| ]]</text>
      <sha1>5mywjj51xgsiypo2thqfb6mwg723sff</sha1>
    </revision>
    <revision>
      <id>2411949</id>
      <parentid>2411948</parentid>
      <timestamp>2022-08-04T21:15:35Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
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      <comment>/* School news */</comment>
      <model>wikitext</model>
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      <text bytes="2485" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].

==School news==
* '''9 September 2006''' - Original Division (under Theology) founded! 
* '''4 August 2022''' - Division converted to a School!  --- [[School:Religious Studies]]

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this division, you can list your name here (this can help small divisions grow and the participants communicate better; for large divisions a list of active participants is not needed).

* [[user:SotanScholar|SotanScholar]] as of Aug 3, 2022.
* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]], email:[mailto:dewnada@icloud.com dewnada@icloud.com]

===Inactive participants===

* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]

==Things you can do==

Head over to the school page and help out!  --- [[School:Religious Studies]]

== External links ==

[[Category:{{PAGENAME}}| ]]</text>
      <sha1>npum83g6cjhz0s2i0xcvewvhkzu4qyb</sha1>
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  </page>
  <page>
    <title>Calculus II</title>
    <ns>0</ns>
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      <minor />
      <comment>Added a somewhat detailed description of what the Calculus II course covers. Added: &quot;Calculus II covers integral calculus of functions of one variable with applications, specific methods of integration, convergence of numerical and power series, parametric equations and polar coordinates, and separable differential equations.&quot;</comment>
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      <text bytes="1876" xml:space="preserve">'''Calculus II''' is the second course involving calculus, after [[Introduction to Calculus]]. Because of this, you are expected to know derivatives inside and out, and also know basic integrals. Calculus II covers integral calculus of functions of one variable with applications, specific methods of integration, convergence of numerical and power series, parametric equations and polar coordinates, and separable differential equations. In this course, we will cover series, calculus in more than one variable, and vectors. Because you should know basic calculus, we'll just start where you left off.

==Tests==
&lt;big&gt;'''[[Wright State University Lake Campus/2017-1/MTH2310]]'''&lt;/big&gt;

* The four midterm tests are on 2/2/17, 3/9/17, 3/30/17, and 4/20/17 (all Thursdays) from 11:30 - 12:50 pm.   
* The final exam is on Monday 4/24/17, at 3:15-5:15 pm.
[[/Test 1/]] {{spaces|3}}
[[/Test 2/]] {{spaces|3}}
[[/Test 3/]] {{spaces|3}}
[[/Test 4/]] {{spaces|3}}
==Content==
*Chapter 1 - [[Vectors]]
*Chapter 2 - [[/Vector Geometry Operation/]]
*Chapter 3 - [[/Vector Algebra Operation/]]
*Chapter 2 - [[/Vector operations/]]

==Lessons==
*[[More ways to use the derivative]]
*[[Partial derivative]]
*[[Applying definite integrals]]
*[[Integrals of solids and planes]]
*[[Physics equations/Integrals for Maxwell's equations]] (Under construction)
*[[Sequences and Series]]
*[[Taylor's series]]
*[[/Moments/]]
*[[Complex Numbers]]
*[[Models]]

*[[Vector derivatives]]
*[[Integrals of a line]]
*[[Important theorems of the vector]]

==External resources/References==
̈* [[Wikipedia:Vector calculus identities|Vector calculus identities]]
* K. Stein, Sherman. Calculus and Analytic Geometry. 2nd ed. New York: Mc-Graw Hill Book Company, 1973.
* [http://www.emathhelp.net/calculators/calculus-2/integral-calculator/ Integral Calculator]

[[Category:Calculus]]
[[Category:Calculus II]]</text>
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  <page>
    <title>Portal:Comparative religion</title>
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      <text bytes="13957" xml:space="preserve">&lt;div style=&quot;margin-top:-3px; margin-bottom:0.3em; text-align: center; font-size: 98%;&quot;&gt; 
{{center top}}'''[[School:Theology|School of Theology]]'''{{center bottom}}
[[Portal:Anthropology|Anthropology]] • [[Communication]] • [[School:Development Science| Development Science]] • [[School:Economics|Economics]] • [[School:Geography|Geography]] • [[School:History|History]] • [[School:Linguistics|Linguistics]] • [[School:Media studies|Media Studies]] • [[School:Political science|Political Science]] • [[School:Psychology|Psychology]] • [[School:Public Policy|Public Policy]] • [[Social work|Social Work]] • [[School:Sociology|Sociology]] • [[School:Theology|Theology]] • [[Women's studies|Women's Studies]] &lt;/div&gt;


{{center top}}
{|style=&quot;width:100%;margin-top:+.7em;background-color:#fcfcfc;border:1px solid #ccc&quot;
|style=&quot;width:50%;color:#000&quot;|
{|style=&quot;width:280px;border:solid 0px;background:none&quot;
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|style=&quot;width:280px;text-align:center;white-space:nowrap;color:#000&quot; |
&lt;div style=&quot;font-size:133%;border:none;margin: 0;padding:.1em;color:#000&quot;&gt;'''Welcome to the Department of Comparative Religion,'''&lt;/div&gt;

|}
&lt;div style=&quot;top:+0.2em;font-size: 95%&quot;&gt;part of the School for [[School:Theology|Theology]].&lt;/div&gt;
|style=&quot;width:18%;font-size:95%;color:#000&quot;|
*[[Topic:Jesus Christ|Jesus Christ]]
*[[Topic:Siddhārtha Gautama|Siddhārtha Gautama]]
|style=&quot;width:22%;font-size:95%&quot;;color:#000&quot;|
*[[Topic:Muhammad|Muhammad]]
*[[Topic:Bahaullah|Bahá'u'lláh]]
*[[Topic:Aleister Crowley|Aleister Crowley]]
|style=&quot;width:10%;font-size:95%&quot;;color:#000&quot;|
|}
{{center bottom}}

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:99%; background-color:#f9f9ff;margin-bottom:10px;margin-top:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Comparative Religion&lt;/h2 &gt;
[[Image:91px-Fedorovskaya.jpg|right|100px|Fedorovskaya]] Hello and welcome to the Wikiversity Department of Comparative Religion. Thank you for your interest and please join in '''[[Wikiversity:Adding content|editing]]''' the Comparative Religion pages! Prior to doing so, we recommend you check out our page '''[[/Templates|Templates]]''' for use in the Comparative Religion Department. The '''[[/Templates|Templates]]''' page also details the basic organization of this department.  This department is dedicated to unifying the diverse religious community here on Wikiversity so that we may come together to study and expand each others understandings of theology.  You can help by continuing to grow and expand this page with accurate and informative articles and information regarding your particular cosmology.
&lt;/div&gt;

&lt;div style=&quot;display:block;width:99%;float:left&quot;&gt;
&lt;div style=&quot;width:49%;display:block;float:left;&quot;&gt;
&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Current Learning Projects&lt;/h2 &gt;
[[Image:Nuvola apps date.png|right|44px|News]]
The development of large scale projects for group learning and future pedagogical use is one of the goals of this school. Some of the ongoing learning projects are:

==Comparative mythology/metaphysics==
===Creation===
* [[2 creation stories of Genesis]]
* [[Taoist]]
* [[Hindu myth of Purusha]]
* [[Gnostic 'demiurge']]
===consciousness science===
* [[7 planes and sheaths]]
* [[7 spirits before the throne]]
===Eschatology (End of the world)===
====Judgment====
* [[Jewish Eschatology|Judaism]]
* [[Apocalypse of John|Christianity]]
* [[Qiyama|Islam]]
* [[Egyptian Eschatology|Egyptian]]
* [[Hindu Eschatology|Hindu]]

====Wheel of Rebirth====
* [[Reincarnation in Judaism|Judaist]]
* [[Reincarnation in Hinduism|Hindu]]
* [[Rebirth and Buddhism|Buddhist]]
* [[Reincarnation in Jainism|Jainist]]
* [[Reincarnation in Taoism|Taoist]]

==Comparative ethics==

===Laws of right conduct===
* [[Ten Commandments|10 Commandments (613 in total) of Judaism]]
* [[Sila of Buddhism]]
* [[Yama &amp; Niyama of Yoga]]
* [[True Will|The 'Do what thou wilt shall be the whole of the Law' of Thelema]]
&lt;br&gt;
&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Department History&lt;/h2 &gt;
*The Department of Comparative Religion was founded within the Wikiversity's School of Theology on 28 February, 2008.
&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;School news&lt;/h2 &gt;
*The [[Portal:Comparative religion|Department of Comparative Religion]] was created at the request of discussion on the [[School:Theology|School of Theology]] talk page. It is not yet finished. Do you want to expand this ? ''from 28th February 2008''
&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Related Schools&lt;/h2 &gt;
{| style=&quot;width:100%;border:solid 0px #FFFFFF;background-color:transparent;align:center;padding:0px;padding-top:1em;&quot; cellspacing=&quot;0px;&quot;
|-
| style=&quot;width:50%;align:center;&quot; |
&lt;div align=&quot;center&quot;&gt;[[Image:91px-Fedorovskaya.jpg|right|55px|Fedorovskaya]]&lt;/div&gt;
| style=&quot;width:50%;align:center;&quot; |
&lt;div align=&quot;center&quot;&gt;[[Image:Codex Manesse Heinrich von Rugge.jpg|55px|History]]&lt;/div&gt;
|-
! style=&quot;text-align:center;width:33%&quot;|'''[[School:Theology|Theology]]'''
! style=&quot;text-align:center;width:33%&quot;|'''[[School:History|History]]'''
|-
| style=&quot;width:50%;align:center;&quot; |
&lt;div align=&quot;center&quot;&gt;[[Image:Nuvola apps kuser.svg|55px|Humanities]]&lt;/div&gt;
| style=&quot;width:50%;align:center;&quot; |
&lt;div align=&quot;center&quot;&gt;[[Image:Nuvola_filesystems_www.png|55px|Social Sciences]]&lt;/div&gt;
|-
! style=&quot;text-align:center;width:33%&quot;|'''[[Portal:Humanities|Humanities]]'''
! style=&quot;text-align:center;width:33%&quot;|'''[[Portal:Social Sciences|Social Sciences]]'''
|}
&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Selected Research Topics&lt;/h2 &gt;
Please feel free to add a topic or contribute your hunting and gathering time (see [[Other Free Learning Resources|Hunter-gatherers_project]]) to expanding the resources in an existing one!
{{{{FULLPAGENAME}}/Topics}}
&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Quote&lt;/h2 &gt;
[[Image:Nuvola apps edu languages.png|right|44px|Quotation]]
&quot;Magick is the science and art of causing change to occur in conformity with will&quot; 
&lt;br&gt;&lt;br&gt;[[w:Aleister_Crowley]], ''[http://www.gutenberg.org/files/14040/14040.txt]''&lt;br&gt;
British Mystic (1875-1947 CE)
&lt;/div&gt;

&lt;/div &gt;
&lt;div style=&quot;width:40%;display:block;float:right;&quot;&gt;
&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Self Paced Learning Program: Theology&lt;/h2 &gt;
The intent of this School is to offer  a program, not unlike a degree program, in Comparative Religion. Given this objective, a series of courses are listed below so that students will be able to fulfill their goal of undertaking a comprehensive and university-style Undergraduate degree program. The program is structured over a hypothetical four-year period, in which a student takes ten classes per year. The core classes must be completed and should be augmented by selections from the electives. Strategic Studies is a multidisciplinary study. It is therefore encouraged that students select courses from other Schools that have application in light of the direction they wish to take their studies. Students taking a Christian focus should feel free to round out their general understanding of Christianity by taking dominantly Christian courses that perhaps have a relation to the student's ethnic areas of interest from '''[[School: Theology]]'''. Students wishing to emphasize religion from a socialogical standpoint in their studies are encouraged to take general courses from '''[[School:Religious Studies]]'''. Students may notice that Comparative Religion has a strong overlap with '''[[School:Theology|Theology]]''', and Courses from that Department may make up part of a student's Course load.&lt;p&gt;
Guidelines: for definitions of Courses, Classes, and Projects, please see the [[/Templates|Templates]] page. Students should complete ten Courses per &quot;year.&quot; All Core Courses must be completed. The student may round out their Course load with appropriate Electives. Appropriate would imply that they come from the &quot;year&quot; in which the student is studying at that time. While there is nothing to stop you from taking courses in any order you wish, it is recommended that courses be taken in some semblance of order. Upon completion of the full degree program... pat yourself on the back. Education is its own reward.&lt;/p&gt;
&lt;/div &gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Core Courses&lt;/h2 &gt;
'''First Year'''
*CORE 100 - [[Introduction to Comparative Religion]]
*CORE 110 - [[Comparative Mythology/Introduction|Introduction to Comparative Mythology]]
*CORE 120 - [[Comparative Axiology: Aesthetics &amp; Ethics]]
*CORE 130 - [[Epistemology: Agnosticism or Gnosticism?]]
*CORE 140 - [[Introduction to Metaphysics: Cosmology &amp; Ontology]]
*CORE 142 - [[Creation Myths Around The World]]
*CORE 150 - [[A Study in Eschatology]] 
*CORE 152 - [[A Study on The Wheel of Rebirth]] 
'''Second Year'''
*CORE 210 - [[Varieties and Unity of Theism]]
*CORE 256 - [[Worldwide Sacred Texts Temurah And Introduction to The Seven Keys]]
'''Third Year'''
*CORE 310 - [[Masters, Saviours, Teachers: Muktis, Buddhas, Messiahs]]
'''Fourth Year'''
*CORE 410 - [[&quot;Do Even Greater Things:&quot; The Mukti, Bodhisattva, Service Path]]
&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Elective Courses&lt;/h2 &gt;
[[Image:Nuvola apps kfind.png|right|70px|Electives]]
'''&lt;big&gt;Electives&lt;/big&gt;'''&lt;br&gt;
'''First Year'''&lt;br&gt;
*CORE 132 - [[Introduction to Alchemy]]
*CORE 150 - [[Introduction to Shamanism]]
*CORE 151 - [[Comparative Afterlifes]]
*CORE 154 - [[Introduction to Mysticism]]
*CORE 155 - [[The place of Allegory in Religion]]
*CORE 156 - [[Introduction to Esoterism]]
*CORE 160 - [[Esoteric Mysticism of The Pagan Religions]]
*CORE 170 - [[Esoteric Mysticism of The Great Religious Founders]]
*CORE 180 - [[Esoteric Mysticism of The Perennial Philosophers]]
'''Second Year'''
*CORE 260 - [[Comparative Gnosticism]]
'''Third Year'''
*CORE 360 - [[Comparative Yogas And Hermetism]]
'''Fourth Year'''
*CORE 450 - [[Advanced Theosophy]]
&lt;/div&gt;
&lt;/div&gt;

&lt;div style=&quot;display:block;float:left;width:100%;&quot;&gt;
&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Associated Wikimedia for Comparative Religious Topics&lt;/h2 &gt;
{| class=&quot;noprint&quot; align=&quot;center&quot; style=&quot;clear:both; background: transparent; text-align:center; font-size:91%&quot; cellspacing=&quot;3&quot; cellpadding=&quot;3&quot; rules=&quot;none&quot; border=&quot;0&quot;
|-
|'''[[w:Religion|Religion on Wikipedia]]''' &amp;nbsp; &amp;nbsp;
|'''[[n:Religion|Religion on Wikinews]]''' &amp;nbsp; &amp;nbsp;
|'''[[q:Religion|Religion on Wikiquote]]''' &amp;nbsp; &amp;nbsp;
|'''[[b:Religion|Religion on Wikibooks]]''' &amp;nbsp; &amp;nbsp;
|'''[[s:Religion|Religion on Wikisource]]''' &amp;nbsp; &amp;nbsp;
|'''[[wikt:Religion|Religion on Wiktionary]]'''
|}
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;

__NOEDITSECTION__
__NOTOC__

[[Category:Comparative Religion]]</text>
      <sha1>j5xejgrmgdcprdmoy3wmthqfoma09q1</sha1>
    </revision>
    <revision>
      <id>2411917</id>
      <parentid>2411916</parentid>
      <timestamp>2022-08-04T20:38:32Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13983" xml:space="preserve">&lt;div style=&quot;margin-top:-3px; margin-bottom:0.3em; text-align: center; font-size: 98%;&quot;&gt; 
{{center top}}'''[[School:Theology|School of Theology]]'''{{center bottom}}
[[Portal:Anthropology|Anthropology]] • [[Communication]] • [[School:Development Science| Development Science]] • [[School:Economics|Economics]] • [[School:Geography|Geography]] • [[School:History|History]] • [[School:Linguistics|Linguistics]] • [[School:Media studies|Media Studies]] • [[School:Political science|Political Science]] • [[School:Psychology|Psychology]] • [[School:Public Policy|Public Policy]] • [[Social work|Social Work]] • [[School:Sociology|Sociology]] • [[School:Theology|Theology]] • [[Women's studies|Women's Studies]] &lt;/div&gt;


{{center top}}
{|style=&quot;width:100%;margin-top:+.7em;background-color:#fcfcfc;border:1px solid #ccc&quot;
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{|style=&quot;width:280px;border:solid 0px;background:none&quot;
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|style=&quot;width:280px;text-align:center;white-space:nowrap;color:#000&quot; |
&lt;div style=&quot;font-size:133%;border:none;margin: 0;padding:.1em;color:#000&quot;&gt;'''Welcome to the Department of Comparative Religion,'''&lt;/div&gt;

|}
&lt;div style=&quot;top:+0.2em;font-size: 95%&quot;&gt;part of the School for [[School:Theology|Theology]].&lt;/div&gt;
|style=&quot;width:18%;font-size:95%;color:#000&quot;|
*[[Topic:Jesus Christ|Jesus Christ]]
*[[Topic:Siddhārtha Gautama|Siddhārtha Gautama]]
|style=&quot;width:22%;font-size:95%&quot;;color:#000&quot;|
*[[Topic:Muhammad|Muhammad]]
*[[Topic:Bahaullah|Bahá'u'lláh]]
*[[Topic:Aleister Crowley|Aleister Crowley]]
|style=&quot;width:10%;font-size:95%&quot;;color:#000&quot;|
|}
{{center bottom}}

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:99%; background-color:#f9f9ff;margin-bottom:10px;margin-top:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Comparative Religion&lt;/h2 &gt;
[[Image:91px-Fedorovskaya.jpg|right|100px|Fedorovskaya]] Hello and welcome to the Wikiversity Department of Comparative Religion. Thank you for your interest and please join in '''[[Wikiversity:Adding content|editing]]''' the Comparative Religion pages! Prior to doing so, we recommend you check out our page '''[[/Templates|Templates]]''' for use in the Comparative Religion Department. The '''[[/Templates|Templates]]''' page also details the basic organization of this department.  This department is dedicated to unifying the diverse religious community here on Wikiversity so that we may come together to study and expand each others understandings of theology.  You can help by continuing to grow and expand this page with accurate and informative articles and information regarding your particular cosmology.
&lt;/div&gt;

&lt;div style=&quot;display:block;width:99%;float:left&quot;&gt;
&lt;div style=&quot;width:49%;display:block;float:left;&quot;&gt;
&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Current Learning Projects&lt;/h2 &gt;
[[Image:Nuvola apps date.png|right|44px|News]]
The development of large scale projects for group learning and future pedagogical use is one of the goals of this school. Some of the ongoing learning projects are:

==Comparative mythology/metaphysics==
===Creation===
* [[2 creation stories of Genesis]]
* [[Taoist]]
* [[Hindu myth of Purusha]]
* [[Gnostic 'demiurge']]
===consciousness science===
* [[7 planes and sheaths]]
* [[7 spirits before the throne]]
===Eschatology (End of the world)===
====Judgment====
* [[Jewish Eschatology|Judaism]]
* [[Apocalypse of John|Christianity]]
* [[Qiyama|Islam]]
* [[Egyptian Eschatology|Egyptian]]
* [[Hindu Eschatology|Hindu]]

====Wheel of Rebirth====
* [[Reincarnation in Judaism|Judaist]]
* [[Reincarnation in Hinduism|Hindu]]
* [[Rebirth and Buddhism|Buddhist]]
* [[Reincarnation in Jainism|Jainist]]
* [[Reincarnation in Taoism|Taoist]]

==Comparative ethics==

===Laws of right conduct===
* [[Ten Commandments|10 Commandments (613 in total) of Judaism]]
* [[Sila of Buddhism]]
* [[Yama &amp; Niyama of Yoga]]
* [[True Will|The 'Do what thou wilt shall be the whole of the Law' of Thelema]]
&lt;br&gt;
&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Department History&lt;/h2 &gt;
*The Department of Comparative Religion was founded within the Wikiversity's School of Theology on 28 February, 2008.
&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;School news&lt;/h2 &gt;
*The [[Portal:Comparative religion|Department of Comparative Religion]] was created at the request of discussion on the [[School:Theology|School of Theology]] talk page. It is not yet finished. Do you want to expand this ? ''from 28th February 2008''
&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Related Schools&lt;/h2 &gt;
{| style=&quot;width:100%;border:solid 0px #FFFFFF;background-color:transparent;align:center;padding:0px;padding-top:1em;&quot; cellspacing=&quot;0px;&quot;
|-
| style=&quot;width:50%;align:center;&quot; |
&lt;div align=&quot;center&quot;&gt;[[Image:91px-Fedorovskaya.jpg|right|55px|Fedorovskaya]]&lt;/div&gt;
| style=&quot;width:50%;align:center;&quot; |
&lt;div align=&quot;center&quot;&gt;[[Image:Codex Manesse Heinrich von Rugge.jpg|55px|History]]&lt;/div&gt;
|-
! style=&quot;text-align:center;width:33%&quot;|'''[[School:Theology|Theology]]'''
! style=&quot;text-align:center;width:33%&quot;|'''[[School:History|History]]'''
|-
| style=&quot;width:50%;align:center;&quot; |
&lt;div align=&quot;center&quot;&gt;[[Image:Nuvola apps kuser.svg|55px|Humanities]]&lt;/div&gt;
| style=&quot;width:50%;align:center;&quot; |
&lt;div align=&quot;center&quot;&gt;[[Image:Nuvola_filesystems_www.png|55px|Social Sciences]]&lt;/div&gt;
|-
! style=&quot;text-align:center;width:33%&quot;|'''[[Portal:Humanities|Humanities]]'''
! style=&quot;text-align:center;width:33%&quot;|'''[[Portal:Social Sciences|Social Sciences]]'''
|}
&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Selected Research Topics&lt;/h2 &gt;
Please feel free to add a topic or contribute your hunting and gathering time (see [[Other Free Learning Resources|Hunter-gatherers_project]]) to expanding the resources in an existing one!
{{{{FULLPAGENAME}}/Topics}}
&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Quote&lt;/h2 &gt;
[[Image:Nuvola apps edu languages.png|right|44px|Quotation]]
&quot;Magick is the science and art of causing change to occur in conformity with will&quot; 
&lt;br&gt;&lt;br&gt;[[w:Aleister_Crowley]], ''[http://www.gutenberg.org/files/14040/14040.txt]''&lt;br&gt;
British Mystic (1875-1947 CE)
&lt;/div&gt;

&lt;/div &gt;
&lt;div style=&quot;width:40%;display:block;float:right;&quot;&gt;
&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Self Paced Learning Program: Theology&lt;/h2 &gt;
The intent of this School is to offer  a program, not unlike a degree program, in Comparative Religion. Given this objective, a series of courses are listed below so that students will be able to fulfill their goal of undertaking a comprehensive and university-style Undergraduate degree program. The program is structured over a hypothetical four-year period, in which a student takes ten classes per year. The core classes must be completed and should be augmented by selections from the electives. Strategic Studies is a multidisciplinary study. It is therefore encouraged that students select courses from other Schools that have application in light of the direction they wish to take their studies. Students taking a Christian focus should feel free to round out their general understanding of Christianity by taking dominantly Christian courses that perhaps have a relation to the student's ethnic areas of interest from '''[[School: Theology]]'''. Students wishing to emphasize religion from a socialogical standpoint in their studies are encouraged to take general courses from '''[[School:Religious Studies|School: Religious Studies]]'''. Students may notice that Comparative Religion has a strong overlap with '''[[School:Theology|Theology]]''', and Courses from that Department may make up part of a student's Course load.&lt;p&gt;
Guidelines: for definitions of Courses, Classes, and Projects, please see the [[/Templates|Templates]] page. Students should complete ten Courses per &quot;year.&quot; All Core Courses must be completed. The student may round out their Course load with appropriate Electives. Appropriate would imply that they come from the &quot;year&quot; in which the student is studying at that time. While there is nothing to stop you from taking courses in any order you wish, it is recommended that courses be taken in some semblance of order. Upon completion of the full degree program... pat yourself on the back. Education is its own reward.&lt;/p&gt;
&lt;/div &gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Core Courses&lt;/h2 &gt;
'''First Year'''
*CORE 100 - [[Introduction to Comparative Religion]]
*CORE 110 - [[Comparative Mythology/Introduction|Introduction to Comparative Mythology]]
*CORE 120 - [[Comparative Axiology: Aesthetics &amp; Ethics]]
*CORE 130 - [[Epistemology: Agnosticism or Gnosticism?]]
*CORE 140 - [[Introduction to Metaphysics: Cosmology &amp; Ontology]]
*CORE 142 - [[Creation Myths Around The World]]
*CORE 150 - [[A Study in Eschatology]] 
*CORE 152 - [[A Study on The Wheel of Rebirth]] 
'''Second Year'''
*CORE 210 - [[Varieties and Unity of Theism]]
*CORE 256 - [[Worldwide Sacred Texts Temurah And Introduction to The Seven Keys]]
'''Third Year'''
*CORE 310 - [[Masters, Saviours, Teachers: Muktis, Buddhas, Messiahs]]
'''Fourth Year'''
*CORE 410 - [[&quot;Do Even Greater Things:&quot; The Mukti, Bodhisattva, Service Path]]
&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Elective Courses&lt;/h2 &gt;
[[Image:Nuvola apps kfind.png|right|70px|Electives]]
'''&lt;big&gt;Electives&lt;/big&gt;'''&lt;br&gt;
'''First Year'''&lt;br&gt;
*CORE 132 - [[Introduction to Alchemy]]
*CORE 150 - [[Introduction to Shamanism]]
*CORE 151 - [[Comparative Afterlifes]]
*CORE 154 - [[Introduction to Mysticism]]
*CORE 155 - [[The place of Allegory in Religion]]
*CORE 156 - [[Introduction to Esoterism]]
*CORE 160 - [[Esoteric Mysticism of The Pagan Religions]]
*CORE 170 - [[Esoteric Mysticism of The Great Religious Founders]]
*CORE 180 - [[Esoteric Mysticism of The Perennial Philosophers]]
'''Second Year'''
*CORE 260 - [[Comparative Gnosticism]]
'''Third Year'''
*CORE 360 - [[Comparative Yogas And Hermetism]]
'''Fourth Year'''
*CORE 450 - [[Advanced Theosophy]]
&lt;/div&gt;
&lt;/div&gt;

&lt;div style=&quot;display:block;float:left;width:100%;&quot;&gt;
&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top; background-color:#f9f9ff;margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Associated Wikimedia for Comparative Religious Topics&lt;/h2 &gt;
{| class=&quot;noprint&quot; align=&quot;center&quot; style=&quot;clear:both; background: transparent; text-align:center; font-size:91%&quot; cellspacing=&quot;3&quot; cellpadding=&quot;3&quot; rules=&quot;none&quot; border=&quot;0&quot;
|-
|'''[[w:Religion|Religion on Wikipedia]]''' &amp;nbsp; &amp;nbsp;
|'''[[n:Religion|Religion on Wikinews]]''' &amp;nbsp; &amp;nbsp;
|'''[[q:Religion|Religion on Wikiquote]]''' &amp;nbsp; &amp;nbsp;
|'''[[b:Religion|Religion on Wikibooks]]''' &amp;nbsp; &amp;nbsp;
|'''[[s:Religion|Religion on Wikisource]]''' &amp;nbsp; &amp;nbsp;
|'''[[wikt:Religion|Religion on Wiktionary]]'''
|}
&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;

__NOEDITSECTION__
__NOTOC__

[[Category:Comparative Religion]]</text>
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  </page>
  <page>
    <title>Bible/King James/Documentary Hypothesis/Numbers</title>
    <ns>0</ns>
    <id>54718</id>
    <revision>
      <id>2411829</id>
      <parentid>2411396</parentid>
      <timestamp>2022-08-04T13:36:01Z</timestamp>
      <contributor>
        <username>Huz and Buz</username>
        <id>2928717</id>
      </contributor>
      <comment>/* Chapter 24 */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="261633" xml:space="preserve">{{RightTOC}}
&lt;[[Bible, English, King James, According to the documentary hypothesis]]

According to the [[w:documentary hypothesis|documentary hypothesis]], [[w:Numbers|Numbers]] is composed from a number of originally independant sources joined by a [[w:redaction|redactor]].

There follows the text of Numbers in the [[w:King James Version|King James Version]], with sources highlighted according to the documentary hypothesis
*The [[w:Priestly source|&quot;Priestly source&quot;]] is highlighted in {{font|color=#888800|olive yellow}} &lt;small&gt;{{font|color=#0000FF|([[Bible, English, King James, Documentary Hypothesis, Priestly source|view in isolation]])}}&lt;/small&gt;
*The [[w:Jahwist|&quot;Jahwist source&quot;]] is highlighted in {{font|color=#000088|navy blue}} &lt;small&gt;{{font|color=#0000FF|([[Bible, English, King James, Documentary Hypothesis, JE source, Jahwist source|view in isolation]])}}&lt;/small&gt;
*The [[w:Elohist|&quot;Elohist source&quot;]] is highlighted in {{font|color=#008888|teal blueish grey}} &lt;small&gt;{{font|color=#0000FF|([[Bible, English, King James, Documentary Hypothesis, JE source, Elohist source|view in isolation]])}}&lt;/small&gt;
*The [[w:Torah redactor|&quot;Additions by the redactor&quot;]] are highlighted in {{font|color=#880000|maroon red}}

==Chapter 1==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses in the wilderness of Sinai, in the tabernacle of the congregation, on the first day of the second month, in the second year after they were come out of the land of Egypt, saying,}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| Take ye the sum of all the congregation of the children of Israel, after their families, by the house of their fathers, with the number of their names, every male by their polls;}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| From twenty years old and upward, all that are able to go forth to war in Israel: thou and Aaron shall number them by their armies.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And with you there shall be a man of every tribe; every one head of the house of his fathers.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And these are the names of the men that shall stand with you: of the tribe of Reuben; Elizur the son of Shedeur.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| Of Simeon; Shelumiel the son of Zurishaddai.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| Of Judah; Nahshon the son of Amminadab.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| Of Issachar; Nethaneel the son of Zuar.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| Of Zebulun; Eliab the son of Helon.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| Of the children of Joseph: of Ephraim; Elishama the son of Ammihud: of Manasseh; Gamaliel the son of Pedahzur.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| Of Benjamin; Abidan the son of Gideoni.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| Of Dan; Ahiezer the son of Ammishaddai.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| Of Asher; Pagiel the son of Ocran.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| Of Gad; Eliasaph the son of Deuel.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| Of Naphtali; Ahira the son of Enan.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| These were the renowned of the congregation, princes of the tribes of their fathers, heads of thousands in Israel.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| And Moses and Aaron took these men which are expressed by their names:}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| And they assembled all the congregation together on the first day of the second month, and they declared their pedigrees after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, by their polls.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| As the LORD commanded Moses, so he numbered them in the wilderness of Sinai.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| And the children of Reuben, Israel's eldest son, by their generations, after their families, by the house of their fathers, according to the number of the names, by their polls, every male from twenty years old and upward, all that were able to go forth to war;}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| Those that were numbered of them, even of the tribe of Reuben, were forty and six thousand and five hundred.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| Of the children of Simeon, by their generations, after their families, by the house of their fathers, those that were numbered of them, according to the number of the names, by their polls, every male from twenty years old and upward, all that were able to go forth to war;}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| Those that were numbered of them, even of the tribe of Simeon, were fifty and nine thousand and three hundred.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| Of the children of Gad, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| Those that were numbered of them, even of the tribe of Gad, were forty and five thousand six hundred and fifty.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| Of the children of Judah, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| Those that were numbered of them, even of the tribe of Judah, were threescore and fourteen thousand and six hundred. }}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| Of the children of Issachar, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#888800| Those that were numbered of them, even of the tribe of Issachar, were fifty and four thousand and four hundred.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#888800| Of the children of Zebulun, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#888800| Those that were numbered of them, even of the tribe of Zebulun, were fifty and seven thousand and four hundred.}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#888800| Of the children of Joseph, namely, of the children of Ephraim, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#888800| Those that were numbered of them, even of the tribe of Ephraim, were forty thousand and five hundred.}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#888800| Of the children of Manasseh, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#888800| Those that were numbered of them, even of the tribe of Manasseh, were thirty and two thousand and two hundred.}}

&lt;small&gt;{{font|color=#0000FF|36}}&lt;/small&gt;{{font|color=#888800| Of the children of Benjamin, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}}

&lt;small&gt;{{font|color=#0000FF|37}}&lt;/small&gt;{{font|color=#888800| Those that were numbered of them, even of the tribe of Benjamin, were thirty and five thousand and four hundred.}}

&lt;small&gt;{{font|color=#0000FF|38}}&lt;/small&gt;{{font|color=#888800| Of the children of Dan, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}}

&lt;small&gt;{{font|color=#0000FF|39}}&lt;/small&gt;{{font|color=#888800| Those that were numbered of them, even of the tribe of Dan, were threescore and two thousand and seven hundred.}}

&lt;small&gt;{{font|color=#0000FF|40}}&lt;/small&gt;{{font|color=#888800| Of the children of Asher, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}}

&lt;small&gt;{{font|color=#0000FF|41}}&lt;/small&gt;{{font|color=#888800| Those that were numbered of them, even of the tribe of Asher, were forty and one thousand and five hundred.}}

&lt;small&gt;{{font|color=#0000FF|42}}&lt;/small&gt;{{font|color=#888800| Of the children of Naphtali, throughout their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}}

&lt;small&gt;{{font|color=#0000FF|43}}&lt;/small&gt;{{font|color=#888800| Those that were numbered of them, even of the tribe of Naphtali, were fifty and three thousand and four hundred.}}

&lt;small&gt;{{font|color=#0000FF|44}}&lt;/small&gt;{{font|color=#888800| These are those that were numbered, which Moses and Aaron numbered, and the princes of Israel, being twelve men: each one was for the house of his fathers.}}

&lt;small&gt;{{font|color=#0000FF|45}}&lt;/small&gt;{{font|color=#888800| So were all those that were numbered of the children of Israel, by the house of their fathers, from twenty years old and upward, all that were able to go forth to war in Israel; }}

&lt;small&gt;{{font|color=#0000FF|46}}&lt;/small&gt;{{font|color=#888800| Even all they that were numbered were six hundred thousand and three thousand and five hundred and fifty.}}

&lt;small&gt;{{font|color=#0000FF|47}}&lt;/small&gt;{{font|color=#888800| But the Levites after the tribe of their fathers were not numbered among them.}}

&lt;small&gt;{{font|color=#0000FF|48}}&lt;/small&gt;{{font|color=#888800| For the LORD had spoken unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|49}}&lt;/small&gt;{{font|color=#888800| Only thou shalt not number the tribe of Levi, neither take the sum of them among the children of Israel:}}

&lt;small&gt;{{font|color=#0000FF|50}}&lt;/small&gt;{{font|color=#888800| But thou shalt appoint the Levites over the tabernacle of testimony, and over all the vessels thereof, and over all things that belong to it: they shall bear the tabernacle, and all the vessels thereof; and they shall minister unto it, and shall encamp round about the tabernacle.}}

&lt;small&gt;{{font|color=#0000FF|51}}&lt;/small&gt;{{font|color=#888800| And when the tabernacle setteth forward, the Levites shall take it down: and when the tabernacle is to be pitched, the Levites shall set it up: and the stranger that cometh nigh shall be put to death.}}

&lt;small&gt;{{font|color=#0000FF|52}}&lt;/small&gt;{{font|color=#888800| And the children of Israel shall pitch their tents, every man by his own camp, and every man by his own standard, throughout their hosts.}}

&lt;small&gt;{{font|color=#0000FF|53}}&lt;/small&gt;{{font|color=#888800| But the Levites shall pitch round about the tabernacle of testimony, that there be no wrath upon the congregation of the children of Israel: and the Levites shall keep the charge of the tabernacle of testimony.}}

&lt;small&gt;{{font|color=#0000FF|54}}&lt;/small&gt;{{font|color=#888800| And the children of Israel did according to all that the LORD commanded Moses, so did they.}}

==Chapter 2==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses and unto Aaron, saying,}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| Every man of the children of Israel shall pitch by his own standard, with the ensign of their father's house: far off about the tabernacle of the congregation shall they pitch.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| And on the east side toward the rising of the sun shall they of the standard of the camp of Judah pitch throughout their armies: and Nahshon the son of Amminadab shall be captain of the children of Judah.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And his host, and those that were numbered of them, were threescore and fourteen thousand and six hundred.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And those that do pitch next unto him shall be the tribe of Issachar: and Nethaneel the son of Zuar shall be captain of the children of Issachar.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| And his host, and those that were numbered thereof, were fifty and four thousand and four hundred.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| Then the tribe of Zebulun: and Eliab the son of Helon shall be captain of the children of Zebulun.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And his host, and those that were numbered thereof, were fifty and seven thousand and four hundred.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| All that were numbered in the camp of Judah were an hundred thousand and fourscore thousand and six thousand and four hundred, throughout their armies. These shall first set forth.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| On the south side shall be the standard of the camp of Reuben according to their armies: and the captain of the children of Reuben shall be Elizur the son of Shedeur.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And his host, and those that were numbered thereof, were forty and six thousand and five hundred.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| And those which pitch by him shall be the tribe of Simeon: and the captain of the children of Simeon shall be Shelumiel the son of Zurishaddai.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| And his host, and those that were numbered of them, were fifty and nine thousand and three hundred.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| Then the tribe of Gad: and the captain of the sons of Gad shall be Eliasaph the son of Reuel.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| And his host, and those that were numbered of them, were forty and five thousand and six hundred and fifty.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| All that were numbered in the camp of Reuben were an hundred thousand and fifty and one thousand and four hundred and fifty, throughout their armies. And they shall set forth in the second rank.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| Then the tabernacle of the congregation shall set forward with the camp of the Levites in the midst of the camp: as they encamp, so shall they set forward, every man in his place by their standards.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| On the west side shall be the standard of the camp of Ephraim according to their armies: and the captain of the sons of Ephraim shall be Elishama the son of Ammihud.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| And his host, and those that were numbered of them, were forty thousand and five hundred.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| And by him shall be the tribe of Manasseh: and the captain of the children of Manasseh shall be Gamaliel the son of Pedahzur.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| And his host, and those that were numbered of them, were thirty and two thousand and two hundred.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| Then the tribe of Benjamin: and the captain of the sons of Benjamin shall be Abidan the son of Gideoni.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| And his host, and those that were numbered of them, were thirty and five thousand and four hundred.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| All that were numbered of the camp of Ephraim were an hundred thousand and eight thousand and an hundred, throughout their armies. And they shall go forward in the third rank.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| The standard of the camp of Dan shall be on the north side by their armies: and the captain of the children of Dan shall be Ahiezer the son of Ammishaddai.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| And his host, and those that were numbered of them, were threescore and two thousand and seven hundred.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| And those that encamp by him shall be the tribe of Asher: and the captain of the children of Asher shall be Pagiel the son of Ocran.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| And his host, and those that were numbered of them, were forty and one thousand and five hundred.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#888800| Then the tribe of Naphtali: and the captain of the children of Naphtali shall be Ahira the son of Enan.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#888800| And his host, and those that were numbered of them, were fifty and three thousand and four hundred.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#888800| All they that were numbered in the camp of Dan were an hundred thousand and fifty and seven thousand and six hundred. They shall go hindmost with their standards.}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#888800| These are those which were numbered of the children of Israel by the house of their fathers: all those that were numbered of the camps throughout their hosts were six hundred thousand and three thousand and five hundred and fifty.}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#888800| But the Levites were not numbered among the children of Israel; as the LORD commanded Moses.}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#888800| And the children of Israel did according to all that the LORD commanded Moses: so they pitched by their standards, and so they set forward, every one after their families, according to the house of their fathers.}}

==Chapter 3==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| These also are the generations of Aaron and Moses in the day that the LORD spake with Moses in mount Sinai.}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| And these are the names of the sons of Aaron; Nadab the firstborn, and Abihu, Eleazar, and Ithamar.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| These are the names of the sons of Aaron, the priests which were anointed, whom he consecrated to minister in the priest's office.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And Nadab and Abihu died before the LORD, when they offered strange fire before the LORD, in the wilderness of Sinai, and they had no children: and Eleazar and Ithamar ministered in the priest's office in the sight of Aaron their father.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| Bring the tribe of Levi near, and present them before Aaron the priest, that they may minister unto him.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| And they shall keep his charge, and the charge of the whole congregation before the tabernacle of the congregation, to do the service of the tabernacle.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And they shall keep all the instruments of the tabernacle of the congregation, and the charge of the children of Israel, to do the service of the tabernacle.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| And thou shalt give the Levites unto Aaron and to his sons: they are wholly given unto him out of the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And thou shalt appoint Aaron and his sons, and they shall wait on their priest's office: and the stranger that cometh nigh shall be put to death.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying, }}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| And I, behold, I have taken the Levites from among the children of Israel instead of all the firstborn that openeth the matrix among the children of Israel: therefore the Levites shall be mine;}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| Because all the firstborn are mine; for on the day that I smote all the firstborn in the land of Egypt I hallowed unto me all the firstborn in Israel, both man and beast: mine shall they be: I am the LORD.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses in the wilderness of Sinai, saying,}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| Number the children of Levi after the house of their fathers, by their families: every male from a month old and upward shalt thou number them.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| And Moses numbered them according to the word of the LORD, as he was commanded.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| And these were the sons of Levi by their names; Gershon, and Kohath, and Merari.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| And these are the names of the sons of Gershon by their families; Libni, and Shimei.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| And the sons of Kohath by their families; Amram, and Izehar, Hebron, and Uzziel.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| And the sons of Merari by their families; Mahli, and Mushi. These are the families of the Levites according to the house of their fathers.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| Of Gershon was the family of the Libnites, and the family of the Shimites: these are the families of the Gershonites.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| Those that were numbered of them, according to the number of all the males, from a month old and upward, even those that were numbered of them were seven thousand and five hundred.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| The families of the Gershonites shall pitch behind the tabernacle westward.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| And the chief of the house of the father of the Gershonites shall be Eliasaph the son of Lael.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| And the charge of the sons of Gershon in the tabernacle of the congregation shall be the tabernacle, and the tent, the covering thereof, and the hanging for the door of the tabernacle of the congregation,}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| And the hangings of the court, and the curtain for the door of the court, which is by the tabernacle, and by the altar round about, and the cords of it for all the service thereof.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| And of Kohath was the family of the Amramites, and the family of the Izeharites, and the family of the Hebronites, and the family of the Uzzielites: these are the families of the Kohathites.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| In the number of all the males, from a month old and upward, were eight thousand and six hundred, keeping the charge of the sanctuary.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#888800| The families of the sons of Kohath shall pitch on the side of the tabernacle southward.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#888800| And the chief of the house of the father of the families of the Kohathites shall be Elizaphan the son of Uzziel.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#888800| And their charge shall be the ark, and the table, and the candlestick, and the altars, and the vessels of the sanctuary wherewith they minister, and the hanging, and all the service thereof.}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#888800| And Eleazar the son of Aaron the priest shall be chief over the chief of the Levites, and have the oversight of them that keep the charge of the sanctuary.}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#888800| Of Merari was the family of the Mahlites, and the family of the Mushites: these are the families of Merari.}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#888800| And those that were numbered of them, according to the number of all the males, from a month old and upward, were six thousand and two hundred.}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#888800| And the chief of the house of the father of the families of Merari was Zuriel the son of Abihail: these shall pitch on the side of the tabernacle northward.}}

&lt;small&gt;{{font|color=#0000FF|36}}&lt;/small&gt;{{font|color=#888800| And under the custody and charge of the sons of Merari shall be the boards of the tabernacle, and the bars thereof, and the pillars thereof, and the sockets thereof, and all the vessels thereof, and all that serveth thereto,}}

&lt;small&gt;{{font|color=#0000FF|37}}&lt;/small&gt;{{font|color=#888800| And the pillars of the court round about, and their sockets, and their pins, and their cords.}}

&lt;small&gt;{{font|color=#0000FF|38}}&lt;/small&gt;{{font|color=#888800| But those that encamp before the tabernacle toward the east, even before the tabernacle of the congregation eastward, shall be Moses, and Aaron and his sons, keeping the charge of the sanctuary for the charge of the children of Israel; and the stranger that cometh nigh shall be put to death.}}

&lt;small&gt;{{font|color=#0000FF|39}}&lt;/small&gt;{{font|color=#888800| All that were numbered of the Levites, which Moses and Aaron numbered at the commandment of the LORD, throughout their families, all the males from a month old and upward, were twenty and two thousand.}}

&lt;small&gt;{{font|color=#0000FF|40}}&lt;/small&gt;{{font|color=#888800| And the LORD said unto Moses, Number all the firstborn of the males of the children of Israel from a month old and upward, and take the number of their names.}}

&lt;small&gt;{{font|color=#0000FF|41}}&lt;/small&gt;{{font|color=#888800| And thou shalt take the Levites for me (I am the LORD) instead of all the firstborn among the children of Israel; and the cattle of the Levites instead of all the firstlings among the cattle of the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|42}}&lt;/small&gt;{{font|color=#888800| And Moses numbered, as the LORD commanded him, all the firstborn among the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|43}}&lt;/small&gt;{{font|color=#888800| And all the firstborn males by the number of names, from a month old and upward, of those that were numbered of them, were twenty and two thousand two hundred and threescore and thirteen.}}

&lt;small&gt;{{font|color=#0000FF|44}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying, }}

&lt;small&gt;{{font|color=#0000FF|45}}&lt;/small&gt;{{font|color=#888800| Take the Levites instead of all the firstborn among the children of Israel, and the cattle of the Levites instead of their cattle; and the Levites shall be mine: I am the LORD.}}

&lt;small&gt;{{font|color=#0000FF|46}}&lt;/small&gt;{{font|color=#888800| And for those that are to be redeemed of the two hundred and threescore and thirteen of the firstborn of the children of Israel, which are more than the Levites;}}

&lt;small&gt;{{font|color=#0000FF|47}}&lt;/small&gt;{{font|color=#888800| Thou shalt even take five shekels apiece by the poll, after the shekel of the sanctuary shalt thou take them: (the shekel is twenty gerahs:)}}

&lt;small&gt;{{font|color=#0000FF|48}}&lt;/small&gt;{{font|color=#888800| And thou shalt give the money, wherewith the odd number of them is to be redeemed, unto Aaron and to his sons.}}

&lt;small&gt;{{font|color=#0000FF|49}}&lt;/small&gt;{{font|color=#888800| And Moses took the redemption money of them that were over and above them that were redeemed by the Levites:}}

&lt;small&gt;{{font|color=#0000FF|50}}&lt;/small&gt;{{font|color=#888800| Of the firstborn of the children of Israel took he the money; a thousand three hundred and threescore and five shekels, after the shekel of the sanctuary:}}

&lt;small&gt;{{font|color=#0000FF|51}}&lt;/small&gt;{{font|color=#888800| And Moses gave the money of them that were redeemed unto Aaron and to his sons, according to the word of the LORD, as the LORD commanded Moses.}}

==Chapter 4==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses and unto Aaron, saying, }}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| Take the sum of the sons of Kohath from among the sons of Levi, after their families, by the house of their fathers,}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| From thirty years old and upward even until fifty years old, all that enter into the host, to do the work in the tabernacle of the congregation.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| This shall be the service of the sons of Kohath in the tabernacle of the congregation, about the most holy things:}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And when the camp setteth forward, Aaron shall come, and his sons, and they shall take down the covering vail, and cover the ark of testimony with it:}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| And shall put thereon the covering of badgers' skins, and shall spread over it a cloth wholly of blue, and shall put in the staves thereof.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| And upon the table of shewbread they shall spread a cloth of blue, and put thereon the dishes, and the spoons, and the bowls, and covers to cover withal: and the continual bread shall be thereon:}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And they shall spread upon them a cloth of scarlet, and cover the same with a covering of badgers' skins, and shall put in the staves thereof.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| And they shall take a cloth of blue, and cover the candlestick of the light, and his lamps, and his tongs, and his snuffdishes, and all the oil vessels thereof, wherewith they minister unto it:}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And they shall put it and all the vessels thereof within a covering of badgers' skins, and shall put it upon a bar.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And upon the golden altar they shall spread a cloth of blue, and cover it with a covering of badgers' skins, and shall put to the staves thereof:}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| And they shall take all the instruments of ministry, wherewith they minister in the sanctuary, and put them in a cloth of blue, and cover them with a covering of badgers' skins, and shall put them on a bar:}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| And they shall take away the ashes from the altar, and spread a purple cloth thereon:}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| And they shall put upon it all the vessels thereof, wherewith they minister about it, even the censers, the fleshhooks, and the shovels, and the basons, all the vessels of the altar; and they shall spread upon it a covering of badgers' skins, and put to the staves of it.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| And when Aaron and his sons have made an end of covering the sanctuary, and all the vessels of the sanctuary, as the camp is to set forward; after that, the sons of Kohath shall come to bear it: but they shall not touch any holy thing, lest they die. These things are the burden of the sons of Kohath in the tabernacle of the congregation.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| And to the office of Eleazar the son of Aaron the priest pertaineth the oil for the light, and the sweet incense, and the daily meat offering, and the anointing oil, and the oversight of all the tabernacle, and of all that therein is, in the sanctuary, and in the vessels thereof.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses and unto Aaron saying,}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| Cut ye not off the tribe of the families of the Kohathites from among the Levites:}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| But thus do unto them, that they may live, and not die, when they approach unto the most holy things: Aaron and his sons shall go in, and appoint them every one to his service and to his burden:}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| But they shall not go in to see when the holy things are covered, lest they die.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| Take also the sum of the sons of Gershon, throughout the houses of their fathers, by their families;}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| From thirty years old and upward until fifty years old shalt thou number them; all that enter in to perform the service, to do the work in the tabernacle of the congregation.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| This is the service of the families of the Gershonites, to serve, and for burdens:}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| And they shall bear the curtains of the tabernacle, and the tabernacle of the congregation, his covering, and the covering of the badgers' skins that is above upon it, and the hanging for the door of the tabernacle of the congregation,}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| And the hangings of the court, and the hanging for the door of the gate of the court, which is by the tabernacle and by the altar round about, and their cords, and all the instruments of their service, and all that is made for them: so shall they serve.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| At the appointment of Aaron and his sons shall be all the service of the sons of the Gershonites, in all their burdens, and in all their service: and ye shall appoint unto them in charge all their burdens.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| This is the service of the families of the sons of Gershon in the tabernacle of the congregation: and their charge shall be under the hand of Ithamar the son of Aaron the priest.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#888800| As for the sons of Merari, thou shalt number them after their families, by the house of their fathers;}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#888800| From thirty years old and upward even unto fifty years old shalt thou number them, every one that entereth into the service, to do the work of the tabernacle of the congregation.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#888800| And this is the charge of their burden, according to all their service in the tabernacle of the congregation; the boards of the tabernacle, and the bars thereof, and the pillars thereof, and sockets thereof,}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#888800| And the pillars of the court round about, and their sockets, and their pins, and their cords, with all their instruments, and with all their service: and by name ye shall reckon the instruments of the charge of their burden.}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#888800| This is the service of the families of the sons of Merari, according to all their service, in the tabernacle of the congregation, under the hand of Ithamar the son of Aaron the priest.}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#888800| And Moses and Aaron and the chief of the congregation numbered the sons of the Kohathites after their families, and after the house of their fathers,}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#888800| From thirty years old and upward even unto fifty years old, every one that entereth into the service, for the work in the tabernacle of the congregation:}}

&lt;small&gt;{{font|color=#0000FF|36}}&lt;/small&gt;{{font|color=#888800| And those that were numbered of them by their families were two thousand seven hundred and fifty.}}

&lt;small&gt;{{font|color=#0000FF|37}}&lt;/small&gt;{{font|color=#888800| These were they that were numbered of the families of the Kohathites, all that might do service in the tabernacle of the congregation, which Moses and Aaron did number according to the commandment of the LORD by the hand of Moses.}}

&lt;small&gt;{{font|color=#0000FF|38}}&lt;/small&gt;{{font|color=#888800| And those that were numbered of the sons of Gershon, throughout their families, and by the house of their fathers,}}

&lt;small&gt;{{font|color=#0000FF|39}}&lt;/small&gt;{{font|color=#888800| From thirty years old and upward even unto fifty years old, every one that entereth into the service, for the work in the tabernacle of the congregation,}}

&lt;small&gt;{{font|color=#0000FF|40}}&lt;/small&gt;{{font|color=#888800| Even those that were numbered of them, throughout their families, by the house of their fathers, were two thousand and six hundred and thirty.}}

&lt;small&gt;{{font|color=#0000FF|41}}&lt;/small&gt;{{font|color=#888800| These are they that were numbered of the families of the sons of Gershon, of all that might do service in the tabernacle of the congregation, whom Moses and Aaron did number according to the commandment of the LORD.}}

&lt;small&gt;{{font|color=#0000FF|42}}&lt;/small&gt;{{font|color=#888800| And those that were numbered of the families of the sons of Merari, throughout their families, by the house of their fathers,}}

&lt;small&gt;{{font|color=#0000FF|43}}&lt;/small&gt;{{font|color=#888800| From thirty years old and upward even unto fifty years old, every one that entereth into the service, for the work in the tabernacle of the congregation,}}

&lt;small&gt;{{font|color=#0000FF|44}}&lt;/small&gt;{{font|color=#888800| Even those that were numbered of them after their families, were three thousand and two hundred.}}

&lt;small&gt;{{font|color=#0000FF|45}}&lt;/small&gt;{{font|color=#888800| These be those that were numbered of the families of the sons  of Merari, whom Moses and Aaron numbered according to the word of the LORD by the hand of Moses.}}

&lt;small&gt;{{font|color=#0000FF|46}}&lt;/small&gt;{{font|color=#888800| All those that were numbered of the Levites, whom Moses and Aaron and the chief of Israel numbered, after their families, and after the house of their fathers,}}

&lt;small&gt;{{font|color=#0000FF|47}}&lt;/small&gt;{{font|color=#888800| From thirty years old and upward even unto fifty years old, every one that came to do the service of the ministry, and the service of the burden in the tabernacle of the congregation.}}

&lt;small&gt;{{font|color=#0000FF|48}}&lt;/small&gt;{{font|color=#888800| Even those that were numbered of them, were eight thousand and five hundred and fourscore,}}

&lt;small&gt;{{font|color=#0000FF|49}}&lt;/small&gt;{{font|color=#888800| According to the commandment of the LORD they were numbered by the hand of Moses, every one according to his service, and according to his burden: thus were they numbered of him, as the LORD commanded Moses.}}

==Chapter 5==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| Command the children of Israel, that they put out of the camp every leper, and every one that hath an issue, and whosoever is defiled by the dead:}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| Both male and female shall ye put out, without the camp shall ye put them; that they defile not their camps, in the midst whereof I dwell.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And the children of Israel did so, and put them out without the camp: as the LORD spake unto Moses, so did the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying, }}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| Speak unto the children of Israel, When a man or woman shall commit any sin that men commit, to do a trespass against the LORD, and that person be guilty;}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| Then they shall confess their sin which they have done: and he shall recompense his trespass with the principal thereof, and add unto it the fifth part thereof, and give it unto him against whom he hath trespassed.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| But if the man have no kinsman to recompense the trespass unto, let the trespass be recompensed unto the LORD, even to the priest; beside the ram of the atonement, whereby an atonement shall be made for him.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| And every offering of all the holy things of the children of Israel, which they bring unto the priest, shall be his.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And every man's hallowed things shall be his: whatsoever any man giveth the priest, it shall be his.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| Speak unto the children of Israel, and say unto them, If any man's wife go aside, and commit a trespass against him,}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| And a man lie with her carnally, and it be hid from the eyes of her husband, and be kept close, and she be defiled, and there be no witness against her, neither she be taken with the manner;}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| And the spirit of jealousy come upon him, and he be jealous of his wife, and she be defiled: or if the spirit of jealousy come upon him, and he be jealous of his wife, and she be not defiled:}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| Then shall the man bring his wife unto the priest, and he shall bring her offering for her, the tenth part of an ephah of barley meal; he shall pour no oil upon it, nor put frankincense thereon; for it is an offering of jealousy, an offering of memorial, bringing iniquity to remembrance.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| And the priest shall bring her near, and set her before the LORD:}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| And the priest shall take holy water in an earthen vessel; and of the dust that is in the floor of the tabernacle the priest shall take, and put it into the water:}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| And the priest shall set the woman before the LORD, and uncover the woman's head, and put the offering of memorial in her hands, which is the jealousy offering: and the priest shall have in his hand the bitter water that causeth the curse:}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| And the priest shall charge her by an oath, and say unto the woman, If no man have lain with thee, and if thou hast not gone aside to uncleanness with another instead of thy husband, be thou free from this bitter water that causeth the curse:}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| But if thou hast gone aside to another instead of thy husband, and if thou be defiled, and some man have lain with thee beside thine husband:}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| Then the priest shall charge the woman with an oath of cursing, and the priest shall say unto the woman, The LORD make thee a curse and an oath among thy people, when the LORD doth make thy thigh to rot, and thy belly to swell;}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| And this water that causeth the curse shall go into thy bowels, to make thy belly to swell, and thy thigh to rot: And the woman shall say, Amen, amen.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| And the priest shall write these curses in a book, and he shall blot them out with the bitter water:}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| And he shall cause the woman to drink the bitter water that causeth the curse: and the water that causeth the curse shall enter into her, and become bitter.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| Then the priest shall take the jealousy offering out of the woman's hand, and shall wave the offering before the LORD, and offer it upon the altar:}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| And the priest shall take an handful of the offering, even the memorial thereof, and burn it upon the altar, and afterward shall cause the woman to drink the water.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| And when he hath made her to drink the water, then it shall come to pass, that, if she be defiled, and have done trespass against her husband, that the water that causeth the curse shall enter into her, and become bitter, and her belly shall swell, and her thigh shall rot: and the woman shall be a curse among her people.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| And if the woman be not defiled, but be clean; then she shall be free, and shall conceive seed.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#888800| This is the law of jealousies, when a wife goeth aside to another instead of her husband, and is defiled;}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#888800| Or when the spirit of jealousy cometh upon him, and he be jealous over his wife, and shall set the woman before the LORD, and the priest shall execute upon her all this law.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#888800| Then shall the man be guiltless from iniquity, and this woman shall bear her iniquity.}}

==Chapter 6==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| Speak unto the children of Israel, and say unto them, When either man or woman shall separate themselves to vow a vow of a Nazarite, to separate themselves unto the LORD:}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| He shall separate himself from wine and strong drink, and shall drink no vinegar of wine, or vinegar of strong drink, neither shall he drink any liquor of grapes, nor eat moist grapes, or dried.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| All the days of his separation shall he eat nothing that is made of the vine tree, from the kernels even to the husk.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| All the days of the vow of his separation there shall no razor come upon his head: until the days be fulfilled, in the which he separateth himself unto the LORD, he shall be holy, and shall let the locks of the hair of his head grow.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| All the days that he separateth himself unto the LORD he shall come at no dead body.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| He shall not make himself unclean for his father, or for his mother, for his brother, or for his sister, when they die: because the consecration of his God is upon his head.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| All the days of his separation he is holy unto the LORD.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| And if any man die very suddenly by him, and he hath defiled the head of his consecration; then he shall shave his head in the day of his cleansing, on the seventh day shall he shave it.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And on the eighth day he shall bring two turtles, or two young pigeons, to the priest, to the door of the tabernacle of the congregation:}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And the priest shall offer the one for a sin offering, and the other for a burnt offering, and make an atonement for him, for that he sinned by the dead, and shall hallow his head that same day.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| And he shall consecrate unto the LORD the days of his separation, and shall bring a lamb of the first year for a trespass offering: but the days that were before shall be lost, because his separation was defiled.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| And this is the law of the Nazarite, when the days of his separation are fulfilled: he shall be brought unto the door of the tabernacle of the congregation:}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| And he shall offer his offering unto the LORD, one he lamb of the first year without blemish for a burnt offering, and one ewe lamb of the first year without blemish for a sin offering, and one ram without blemish for peace offerings,}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| And a basket of unleavened bread, cakes of fine flour mingled with oil, and wafers of unleavened bread anointed with oil, and their meat offering, and their drink offerings.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| And the priest shall bring them before the LORD, and shall offer his sin offering, and his burnt offering:}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| And he shall offer the ram for a sacrifice of peace offerings unto the LORD, with the basket of unleavened bread: the priest shall offer also his meat offering, and his drink offering.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| And the Nazarite shall shave the head of his separation at the door of the tabernacle of the congregation, and shall take the hair of the head of his separation, and put it in the fire which is under the sacrifice of the peace offerings.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| And the priest shall take the sodden shoulder of the ram, and one unleavened cake out of the basket, and one unleavened wafer, and shall put them upon the hands of the Nazarite, after the hair of his separation is shaven:}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| And the priest shall wave them for a wave offering before the LORD: this is holy for the priest, with the wave breast and heave shoulder: and after that the Nazarite may drink wine.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| This is the law of the Nazarite who hath vowed, and of his offering unto the LORD for his separation, beside that that his hand shall get: according to the vow which he vowed, so he must do after the law of his separation.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| Speak unto Aaron and unto his sons, saying, On this wise ye shall bless the children of Israel, saying unto them,}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| The LORD bless thee, and keep thee:}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| The LORD make his face shine upon thee, and be gracious unto thee:}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| The LORD lift up his countenance upon thee, and give thee peace.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| And they shall put my name upon the children of Israel, and I will bless them.}}

==Chapter 7==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And it came to pass on the day that Moses had fully set up the tabernacle, and had anointed it, and sanctified it, and all the instruments thereof, both the altar and all the vessels thereof, and had anointed them, and sanctified them;}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| That the princes of Israel, heads of the house of their fathers, who were the princes of the tribes, and were over them that were numbered, offered:}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| And they brought their offering before the LORD, six covered wagons, and twelve oxen; a wagon for two of the princes, and for each one an ox: and they brought them before the tabernacle.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| Take it of them, that they may be to do the service of the tabernacle of the congregation; and thou shalt give them unto the Levites, to every man according to his service.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| And Moses took the wagons and the oxen, and gave them unto the Levites.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| Two wagons and four oxen he gave unto the sons of Gershon, according to their service:}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And four wagons and eight oxen he gave unto the sons of Merari, according unto their service, under the hand of Ithamar the son of Aaron the priest.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| But unto the sons of Kohath he gave none: because the service of the sanctuary belonging unto them was that they should bear upon their shoulders.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And the princes offered for dedicating of the altar in the day that it was anointed, even the princes offered their offering before the altar.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And the LORD said unto Moses, They shall offer their offering, each prince on his day, for the dedicating of the altar.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| And he that offered his offering the first day was Nahshon the son of Amminadab, of the tribe of Judah:}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| And his offering was one silver charger, the weight thereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them were full of fine flour mingled with oil for a meat offering:}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| One spoon of ten shekels of gold, full of incense:}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| One kid of the goats for a sin offering:}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Nahshon the son of Amminadab.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| On the second day Nethaneel the son of Zuar, prince of Issachar, did offer:}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| He offered for his offering one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| One spoon of gold of ten shekels, full of incense:}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| One kid of the goats for a sin offering:}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Nethaneel the son of Zuar.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| On the third day Eliab the son of Helon, prince of the children of Zebulun, did offer:}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| One golden spoon of ten shekels, full of incense:}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| One kid of the goats for a sin offering:}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Eliab the son of Helon.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#888800| On the fourth day Elizur the son of Shedeur, prince of the children of Reuben, did offer:}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#888800| His offering was one silver charger of the weight of an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#888800| One golden spoon of ten shekels, full of incense:}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#888800| One kid of the goats for a sin offering:}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Elizur the son of Shedeur.}}

&lt;small&gt;{{font|color=#0000FF|36}}&lt;/small&gt;{{font|color=#888800| On the fifth day Shelumiel the son of Zurishaddai, prince of the children of Simeon, did offer:}}

&lt;small&gt;{{font|color=#0000FF|37}}&lt;/small&gt;{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}}

&lt;small&gt;{{font|color=#0000FF|38}}&lt;/small&gt;{{font|color=#888800| One golden spoon of ten shekels, full of incense:}}

&lt;small&gt;{{font|color=#0000FF|39}}&lt;/small&gt;{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}}

&lt;small&gt;{{font|color=#0000FF|40}}&lt;/small&gt;{{font|color=#888800| One kid of the goats for a sin offering:}}

&lt;small&gt;{{font|color=#0000FF|41}}&lt;/small&gt;{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Shelumiel the son of Zurishaddai.}}

&lt;small&gt;{{font|color=#0000FF|42}}&lt;/small&gt;{{font|color=#888800| On the sixth day Eliasaph the son of Deuel, prince of the children of Gad, offered:}}

&lt;small&gt;{{font|color=#0000FF|43}}&lt;/small&gt;{{font|color=#888800| His offering was one silver charger of the weight of an hundred and thirty shekels, a silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}}

&lt;small&gt;{{font|color=#0000FF|44}}&lt;/small&gt;{{font|color=#888800| One golden spoon of ten shekels, full of incense:}}

&lt;small&gt;{{font|color=#0000FF|45}}&lt;/small&gt;{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}}

&lt;small&gt;{{font|color=#0000FF|46}}&lt;/small&gt;{{font|color=#888800| One kid of the goats for a sin offering:}}

&lt;small&gt;{{font|color=#0000FF|47}}&lt;/small&gt;{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Eliasaph the son of Deuel.}}

&lt;small&gt;{{font|color=#0000FF|48}}&lt;/small&gt;{{font|color=#888800| On the seventh day Elishama the son of Ammihud, prince of the children of Ephraim, offered:}}

&lt;small&gt;{{font|color=#0000FF|49}}&lt;/small&gt;{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}}

&lt;small&gt;{{font|color=#0000FF|50}}&lt;/small&gt;{{font|color=#888800| One golden spoon of ten shekels, full of incense:}}

&lt;small&gt;{{font|color=#0000FF|51}}&lt;/small&gt;{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}}

&lt;small&gt;{{font|color=#0000FF|52}}&lt;/small&gt;{{font|color=#888800| One kid of the goats for a sin offering:}}

&lt;small&gt;{{font|color=#0000FF|53}}&lt;/small&gt;{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Elishama the son of Ammihud.}}

&lt;small&gt;{{font|color=#0000FF|54}}&lt;/small&gt;{{font|color=#888800| On the eighth day offered Gamaliel the son of Pedahzur, prince of the children of Manasseh:}}

&lt;small&gt;{{font|color=#0000FF|55}}&lt;/small&gt;{{font|color=#888800| His offering was one silver charger of the weight of an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}}

&lt;small&gt;{{font|color=#0000FF|56}}&lt;/small&gt;{{font|color=#888800| One golden spoon of ten shekels, full of incense:}}

&lt;small&gt;{{font|color=#0000FF|57}}&lt;/small&gt;{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}}

&lt;small&gt;{{font|color=#0000FF|58}}&lt;/small&gt;{{font|color=#888800| One kid of the goats for a sin offering:}}

&lt;small&gt;{{font|color=#0000FF|59}}&lt;/small&gt;{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Gamaliel the son of Pedahzur.}}

&lt;small&gt;{{font|color=#0000FF|60}}&lt;/small&gt;{{font|color=#888800| On the ninth day Abidan the son of Gideoni, prince of the children of Benjamin, offered:}}

&lt;small&gt;{{font|color=#0000FF|61}}&lt;/small&gt;{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}}

&lt;small&gt;{{font|color=#0000FF|62}}&lt;/small&gt;{{font|color=#888800| One golden spoon of ten shekels, full of incense:}}

&lt;small&gt;{{font|color=#0000FF|63}}&lt;/small&gt;{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}}

&lt;small&gt;{{font|color=#0000FF|64}}&lt;/small&gt;{{font|color=#888800| One kid of the goats for a sin offering:}}

&lt;small&gt;{{font|color=#0000FF|65}}&lt;/small&gt;{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Abidan the son of Gideoni.}}

&lt;small&gt;{{font|color=#0000FF|66}}&lt;/small&gt;{{font|color=#888800| On the tenth day Ahiezer the son of Ammishaddai, prince of the children of Dan, offered:}}

&lt;small&gt;{{font|color=#0000FF|67}}&lt;/small&gt;{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}}

&lt;small&gt;{{font|color=#0000FF|68}}&lt;/small&gt;{{font|color=#888800| One golden spoon of ten shekels, full of incense:}}

&lt;small&gt;{{font|color=#0000FF|69}}&lt;/small&gt;{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}}

&lt;small&gt;{{font|color=#0000FF|70}}&lt;/small&gt;{{font|color=#888800| One kid of the goats for a sin offering:}}

&lt;small&gt;{{font|color=#0000FF|71}}&lt;/small&gt;{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Ahiezer the son of Ammishaddai.}}

&lt;small&gt;{{font|color=#0000FF|72}}&lt;/small&gt;{{font|color=#888800| On the eleventh day Pagiel the son of Ocran, prince of the children of Asher, offered:}}

&lt;small&gt;{{font|color=#0000FF|73}}&lt;/small&gt;{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}}

&lt;small&gt;{{font|color=#0000FF|74}}&lt;/small&gt;{{font|color=#888800| One golden spoon of ten shekels, full of incense:}}

&lt;small&gt;{{font|color=#0000FF|75}}&lt;/small&gt;{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}}

&lt;small&gt;{{font|color=#0000FF|76}}&lt;/small&gt;{{font|color=#888800| One kid of the goats for a sin offering:}}

&lt;small&gt;{{font|color=#0000FF|77}}&lt;/small&gt;{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Pagiel the son of Ocran.}}

&lt;small&gt;{{font|color=#0000FF|78}}&lt;/small&gt;{{font|color=#888800| On the twelfth day Ahira the son of Enan, prince of the children of Naphtali, offered:}}

&lt;small&gt;{{font|color=#0000FF|79}}&lt;/small&gt;{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}}

&lt;small&gt;{{font|color=#0000FF|80}}&lt;/small&gt;{{font|color=#888800| One golden spoon of ten shekels, full of incense:}}

&lt;small&gt;{{font|color=#0000FF|81}}&lt;/small&gt;{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}}

&lt;small&gt;{{font|color=#0000FF|82}}&lt;/small&gt;{{font|color=#888800| One kid of the goats for a sin offering:}}

&lt;small&gt;{{font|color=#0000FF|83}}&lt;/small&gt;{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Ahira the son of Enan.}}

&lt;small&gt;{{font|color=#0000FF|84}}&lt;/small&gt;{{font|color=#888800| This was the dedication of the altar, in the day when it was anointed, by the princes of Israel: twelve chargers of silver, twelve silver bowls, twelve spoons of gold:}}

&lt;small&gt;{{font|color=#0000FF|85}}&lt;/small&gt;{{font|color=#888800| Each charger of silver weighing an hundred and thirty shekels, each bowl seventy: all the silver vessels weighed two thousand and four hundred shekels, after the shekel of the sanctuary:}}

&lt;small&gt;{{font|color=#0000FF|86}}&lt;/small&gt;{{font|color=#888800| The golden spoons were twelve, full of incense, weighing ten shekels apiece, after the shekel of the sanctuary: all the gold of the spoons was an hundred and twenty shekels.}}

&lt;small&gt;{{font|color=#0000FF|87}}&lt;/small&gt;{{font|color=#888800| All the oxen for the burnt offering were twelve bullocks, the rams twelve, the lambs of the first year twelve, with their meat offering: and the kids of the goats for sin offering twelve.}}

&lt;small&gt;{{font|color=#0000FF|88}}&lt;/small&gt;{{font|color=#888800| And all the oxen for the sacrifice of the peace offerings were twenty and four bullocks, the rams sixty, the he goats sixty, the lambs of the first year sixty. This was the dedication of the altar, after that it was anointed.}}

&lt;small&gt;{{font|color=#0000FF|89}}&lt;/small&gt;{{font|color=#888800| And when Moses was gone into the tabernacle of the congregation to speak with him, then he heard the voice of one speaking unto him from off the mercy seat that was upon the ark of testimony, from between the two cherubims: and he spake unto him.}}

==Chapter 8==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| Speak unto Aaron and say unto him, When thou lightest the lamps, the seven lamps shall give light over against the candlestick.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| And Aaron did so; he lighted the lamps thereof over against the candlestick, as the LORD commanded Moses.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And this work of the candlestick was of beaten gold, unto the shaft thereof, unto the flowers thereof, was beaten work: according unto the pattern which the LORD had shewed Moses, so he made the candlestick.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| Take the Levites from among the children of Israel, and cleanse them.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| And thus shalt thou do unto them, to cleanse them: Sprinkle water of purifying upon them, and let them shave all their flesh, and let them wash their clothes, and so make themselves clean.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| Then let them take a young bullock with his meat offering, even fine flour mingled with oil, and another young bullock shalt thou take for a sin offering.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| And thou shalt bring the Levites before the tabernacle of the congregation: and thou shalt gather the whole assembly of the children of Israel together:}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And thou shalt bring the Levites before the LORD: and the children of Israel shall put their hands upon the Levites:}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And Aaron shall offer the Levites before the LORD for an offering of the children of Israel, that they may execute the service of the LORD.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| And the Levites shall lay their hands upon the heads of the bullocks: and thou shalt offer the one for a sin offering, and the other for a burnt offering, unto the LORD, to make an atonement for the Levites.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| And thou shalt set the Levites before Aaron, and before his sons, and offer them for an offering unto the LORD.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| Thus shalt thou separate the Levites from among the children of Israel: and the Levites shall be mine.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| And after that shall the Levites go in to do the service of the tabernacle of the congregation: and thou shalt cleanse them, and offer them for an offering.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| For they are wholly given unto me from among the children of Israel; instead of such as open every womb, even instead of the firstborn of all the children of Israel, have I taken them unto me.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| For all the firstborn of the children of Israel are mine, both man and beast: on the day that I smote every firstborn in the land of Egypt I sanctified them for myself.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| And I have taken the Levites for all the firstborn of the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| And I have given the Levites as a gift to Aaron and to his sons from among the children of Israel, to do the service of the children of Israel in the tabernacle of the congregation, and to make an atonement for the children of Israel: that there be no plague among the children of Israel, when the children of Israel come nigh unto the sanctuary.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| And Moses, and Aaron, and all the congregation of the children of Israel, did to the Levites according unto all that the LORD commanded Moses concerning the Levites, so did the children of Israel unto them.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| And the Levites were purified, and they washed their clothes; and Aaron offered them as an offering before the LORD; and Aaron made an atonement for them to cleanse them.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| And after that went the Levites in to do their service in the tabernacle of the congregation before Aaron, and before his sons: as the LORD had commanded Moses concerning the Levites, so did they unto them.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| This is it that belongeth unto the Levites: from twenty and five years old and upward they shall go in to wait upon the service of the tabernacle of the congregation:}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| And from the age of fifty years they shall cease waiting upon the service thereof, and shall serve no more:}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| But shall minister with their brethren in the tabernacle of the congregation, to keep the charge, and shall do no service. Thus shalt thou do unto the Levites touching their charge.}}

==Chapter 9==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses in the wilderness of Sinai, in the first month of the second year after they were come out of the land of Egypt, saying,}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| Let the children of Israel also keep the passover at his appointed season.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| In the fourteenth day of this month, at even, ye shall keep it in his appointed season: according to all the rites of it, and according to all the ceremonies thereof, shall ye keep it.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And Moses spake unto the children of Israel, that they should keep the passover.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And they kept the passover on the fourteenth day of the first month at even in the wilderness of Sinai: according to all that the LORD commanded Moses, so did the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| And there were certain men, who were defiled by the dead body of a man, that they could not keep the passover on that day: and they came before Moses and before Aaron on that day:}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| And those men said unto him, We are defiled by the dead body of a man: wherefore are we kept back, that we may not offer an offering of the LORD in his appointed season among the children of Israel?}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And Moses said unto them, Stand still, and I will hear what the LORD will command concerning you.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| Speak unto the children of Israel, saying, If any man of you or of your posterity shall be unclean by reason of a dead body, or be in a journey afar off, yet he shall keep the passover unto the LORD.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| The fourteenth day of the second month at even they shall keep it, and eat it with unleavened bread and bitter herbs.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| They shall leave none of it unto the morning, nor break any bone of it: according to all the ordinances of the passover they shall keep it.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| But the man that is clean, and is not in a journey, and forbeareth to keep the passover, even the same soul shall be cut off from among his people: because he brought not the offering of the LORD in his appointed season, that man shall bear his sin.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| And if a stranger shall sojourn among you, and will keep the passover unto the LORD; according to the ordinance of the passover, and according to the manner thereof, so shall he do: ye shall have one ordinance, both for the stranger, and for him that was born in the land.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| And on the day that the tabernacle was reared up the cloud covered the tabernacle, namely, the tent of the testimony: and at even there was upon the tabernacle as it were the appearance of fire, until the morning.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| So it was alway: the cloud covered it by day, and the appearance of fire by night.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| And when the cloud was taken up from the tabernacle, then after that the children of Israel journeyed: and in the place where the cloud abode, there the children of Israel pitched their tents.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| At the commandment of the LORD the children of Israel journeyed, and at the commandment of the LORD they pitched: as long as the cloud abode upon the tabernacle they rested in their tents.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| And when the cloud tarried long upon the tabernacle many days, then the children of Israel kept the charge of the LORD, and journeyed not.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| And so it was, when the cloud was a few days upon the tabernacle; according to the commandment of the LORD they abode in their tents, and according to the commandment of the LORD they journeyed.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| And so it was, when the cloud abode from even unto the morning, and that the cloud was taken up in the morning, then they journeyed: whether it was by day or by night that the cloud was taken up, they journeyed.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| Or whether it were two days, or a month, or a year, that the cloud tarried upon the tabernacle, remaining thereon, the children of Israel abode in their tents, and journeyed not: but when it was taken up, they journeyed.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| At the commandment of the LORD they rested in the tents, and at the commandment of the LORD they journeyed: they kept the charge of the LORD, at the commandment of the LORD by the hand of Moses.}}

==Chapter 10==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| Make thee two trumpets of silver; of a whole piece shalt thou make them: that thou mayest use them for the calling of the assembly, and for the journeying of the camps.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| And when they shall blow with them, all the assembly shall assemble themselves to thee at the door of the tabernacle of the congregation.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And if they blow but with one trumpet, then the princes, which are heads of the thousands of Israel, shall gather themselves unto thee.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| When ye blow an alarm, then the camps that lie on the east parts shall go forward.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| When ye blow an alarm the second time, then the camps that lie on the south side shall take their journey: they shall blow an alarm for their journeys.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| But when the congregation is to be gathered together, ye shall blow, but ye shall not sound an alarm.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And the sons of Aaron, the priests, shall blow with the trumpets; and they shall be to you for an ordinance for ever throughout your generations.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| And if ye go to war in your land against the enemy that oppresseth you, then ye shall blow an alarm with the trumpets; and ye shall be remembered before the LORD your God, and ye shall be saved from your enemies.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| Also in the day of your gladness, and in your solemn days, and in the beginnings of your months, ye shall blow with the trumpets over your burnt offerings, and over the sacrifices of your peace offerings; that they may be to you for a memorial before your God: I am the LORD your God.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And it came to pass on the twentieth day of the second month, in the second year, that the cloud was taken up from off the tabernacle of the testimony.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| And the children of Israel took their journeys out of the wilderness of Sinai; and the cloud rested in the wilderness of Paran.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| And they first took their journey according to the commandment of the LORD by the hand of Moses.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| In the first place went the standard of the camp of the children of Judah according to their armies: and over his host was Nahshon the son of Amminadab.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| And over the host of the tribe of the children of Issachar was Nethaneel the son of Zuar.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| And over the host of the tribe of the children of Zebulun was Eliab the son of Helon.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| And the tabernacle was taken down; and the sons of Gershon and the sons of Merari set forward, bearing the tabernacle.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| And the standard of the camp of Reuben set forward according to their armies: and over his host was Elizur the son of Shedeur.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| And over the host of the tribe of the children of Simeon was Shelumiel the son of Zurishaddai.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| And over the host of the tribe of the children of Gad was Eliasaph the son of Deuel.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| And the Kohathites set forward, bearing the sanctuary: and the other did set up the tabernacle against they came.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| And the standard of the camp of the children of Ephraim set forward according to their armies: and over his host was Elishama the son of Ammihud.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| And over the host of the tribe of the children of Manasseh was Gamaliel the son of Pedahzur.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| And over the host of the tribe of the children of Benjamin was Abidan the son of Gideoni.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| And the standard of the camp of the children of Dan set forward, which was the rereward of all the camps throughout their hosts: and over his host was Ahiezer the son of Ammishaddai.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| And over the host of the tribe of the children of Asher was Pagiel the son of Ocran.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| And over the host of the tribe of the children of Naphtali was Ahira the son of Enan.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| Thus were the journeyings of the children of Israel according to their armies, when they set forward.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#000088| And Moses said unto Hobab, the son of Raguel the Midianite, Moses' father in law, We are journeying unto the place of which the LORD said, I will give it you: come thou with us, and we will do thee good: for the LORD hath spoken good concerning Israel.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#000088| And he said unto him, I will not go; but I will depart to mine own land, and to my kindred.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#000088| And he said, Leave us not, I pray thee; forasmuch as thou knowest how we are to encamp in the wilderness, and thou mayest be to us instead of eyes.}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#000088| And it shall be, if thou go with us, yea, it shall be, that what goodness the LORD shall do unto us, the same will we do unto thee.}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#000088| And they departed from the mount of the LORD three days' journey: and the ark of the covenant of the LORD went before them in the three days' journey, to search out a resting place for them.}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#000088| And the cloud of the LORD was upon them by day, when they went out of the camp.}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#000088| And it came to pass, when the ark set forward, that Moses said, Rise up, LORD, and let thine enemies be scattered; and let them that hate thee flee before thee.}}

&lt;small&gt;{{font|color=#0000FF|36}}&lt;/small&gt;{{font|color=#000088| And when it rested, he said, Return, O LORD, unto the many thousands of Israel.}}

==Chapter 11==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#000088| And when the people complained, it displeased the LORD: and the LORD heard it; and his anger was kindled; and the fire of the LORD burnt among them, and consumed them that were in the uttermost parts of the camp.}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#000088| And the people cried unto Moses; and when Moses prayed unto the LORD, the fire was quenched.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#000088| And he called the name of the place Taberah: because the fire of the LORD burnt among them.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#000088| And the mixt multitude that was among them fell a lusting: and the children of Israel also wept again, and said, Who shall give us flesh to eat?}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#000088| We remember the fish, which we did eat in Egypt freely; the cucumbers, and the melons, and the leeks, and the onions, and the garlick:}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#000088| But now our soul is dried away: there is nothing at all, beside this manna, before our eyes.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#880000| And the manna was as coriander seed, and the colour thereof as the colour of bdellium.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#880000| And the people went about, and gathered it, and ground it in mills, or beat it in a mortar, and baked it in pans, and made cakes of it: and the taste of it was as the taste of fresh oil.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#880000| And when the dew fell upon the camp in the night, the manna fell upon it.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#000088| Then Moses heard the people weep throughout their families, every man in the door of his tent: and the anger of the LORD was kindled greatly; Moses also was displeased.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#008888| And Moses said unto the LORD, Wherefore hast thou afflicted thy servant? and wherefore have I not found favour in thy sight, that thou layest the burden of all this people upon me?}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#008888| Have I conceived all this people? have I begotten them, that thou shouldest say unto me, Carry them in thy bosom, as a nursing father beareth the sucking child, unto the land which thou swarest unto their fathers?}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#000088| Whence should I have flesh to give unto all this people? for they weep unto me, saying, Give us flesh, that we may eat.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#008888| I am not able to bear all this people alone, because it is too heavy for me.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#008888| And if thou deal thus with me, kill me, I pray thee, out of hand, if I have found favour in thy sight; and let me not see my wretchedness.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#008888| And the LORD said unto Moses, Gather unto me seventy men of the elders of Israel, whom thou knowest to be the elders of the people, and officers over them; and bring them unto the tabernacle of the congregation, that they may stand there with thee.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#008888| And I will come down and talk with thee there: and I will take of the spirit which is upon thee, and will put it upon them; and they shall bear the burden of the people with thee, that thou bear it not thyself alone.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#000088| And say thou unto the people, Sanctify yourselves against to morrow, and ye shall eat flesh: for ye have wept in the ears of the LORD, saying, Who shall give us flesh to eat? for it was well with us in Egypt: therefore the LORD will give you flesh, and ye shall eat.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#000088| Ye shall not eat one day, nor two days, nor five days, neither ten days, nor twenty days;}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#000088| But even a whole month, until it come out at your nostrils, and it be loathsome unto you: because that ye have despised the LORD which is among you, and have wept before him, saying, Why came we forth out of Egypt?}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#000088| And Moses said, The people, among whom I am, are six hundred thousand footmen; and thou hast said, I will give them flesh, that they may eat a whole month.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#000088| Shall the flocks and the herds be slain for them, to suffice them? or shall all the fish of the sea be gathered together for them, to suffice them?}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#000088| And the LORD said unto Moses, Is the LORD's hand waxed short? thou shalt see now whether my word shall come to pass unto thee or not.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#000088| And Moses went out, and told the people the words of the LORD, }} {{font|color=#880000|and}} {{font|color=#008888|gathered the seventy men of the elders of the people, and set them round about the tabernacle.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#008888| And the LORD came down in a cloud, and spake unto him, and took of the spirit that was upon him, and gave it unto the seventy elders: and it came to pass, that, when the spirit rested upon them, they prophesied, and did not cease.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#008888| But there remained two of the men in the camp, the name of the one was Eldad, and the name of the other Medad: and the spirit rested upon them; and they were of them that were written, but went not out unto the tabernacle: and they prophesied in the camp.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#008888| And there ran a young man, and told Moses, and said, Eldad and Medad do prophesy in the camp.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#008888| And Joshua the son of Nun, the servant of Moses, one of his young men, answered and said, My lord Moses, forbid them.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#008888| And Moses said unto him, Enviest thou for my sake? would God that all the LORD's people were prophets, and that the LORD would put his spirit upon them!}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#008888| And Moses gat him into the camp, he and the elders of Israel.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#000088| And there went forth a wind from the LORD, and brought quails from the sea, and let them fall by the camp, as it were a day's journey on this side, and as it were a day's journey on the other side, round about the camp, and as it were two cubits high upon the face of the earth.}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#000088| And the people stood up all that day, and all that night, and all the next day, and they gathered the quails: he that gathered least gathered ten homers: and they spread them all abroad for themselves round about the camp.}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#000088| And while the flesh was yet between their teeth, ere it was chewed, the wrath of the LORD was kindled against the people, and the LORD smote the people with a very great plague.}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#000088| And he called the name of that place Kibrothhattaavah: because there they buried the people that lusted.}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#880000| And the people journeyed from Kibrothhattaavah unto Hazeroth; and abode at Hazeroth.}}

==Chapter 12==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#008888| And Miriam and Aaron spake against Moses}} {{font|color=#880000| because of the Ethiopian woman whom he had married: for he had married an Ethiopian woman.}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#008888| And they said, Hath the LORD indeed spoken only by Moses? hath he not spoken also by us? And the LORD heard it.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#008888| (Now the man Moses was very meek, above all the men which were upon the face of the earth.)}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#008888| And the LORD spake suddenly unto Moses, and unto Aaron, and unto Miriam, Come out ye three unto the tabernacle of the congregation. And they three came out.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#008888| And the LORD came down in the pillar of the cloud, and stood in the door of the tabernacle, and called Aaron and Miriam: and they both came forth.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#008888| And he said, Hear now my words: If there be a prophet among you, I the LORD will make myself known unto him in a vision, and will speak unto him in a dream.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#008888| My servant Moses is not so, who is faithful in all mine house. }}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#008888| With him will I speak mouth to mouth, even apparently, and not in dark speeches; and the similitude of the LORD shall he behold: wherefore then were ye not afraid to speak against my servant Moses?}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#008888| And the anger of the LORD was kindled against them; and he departed.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#008888| And the cloud departed from off the tabernacle; and, behold, Miriam became leprous, white as snow: and Aaron looked upon Miriam, and, behold, she was leprous.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#008888| And Aaron said unto Moses, Alas, my lord, I beseech thee, lay not the sin upon us, wherein we have done foolishly, and wherein we have sinned.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#008888| Let her not be as one dead, of whom the flesh is half consumed when he cometh out of his mother's womb.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#008888| And Moses cried unto the LORD, saying, Heal her now, O God, I beseech thee.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#008888| And the LORD said unto Moses, If her father had but spit in her face, should she not be ashamed seven days? let her be shut out from the camp seven days, and after that let her be received in again.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#008888| And Miriam was shut out from the camp seven days: and the people journeyed not till Miriam was brought in again.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#008888| And afterward the people removed from Hazeroth, and pitched in the wilderness of Paran.}}

==Chapter 13==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| Send thou men, that they may search the land of Canaan, which I give unto the children of Israel: of every tribe of their fathers shall ye send a man, every one a ruler among them.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| And Moses by the commandment of the LORD sent them from the wilderness of Paran: all those men were heads of the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And these were their names: of the tribe of Reuben, Shammua the son of Zaccur.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| Of the tribe of Simeon, Shaphat the son of Hori.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| Of the tribe of Judah, Caleb the son of Jephunneh.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| Of the tribe of Issachar, Igal the son of Joseph.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| Of the tribe of Ephraim, Oshea the son of Nun.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| Of the tribe of Benjamin, Palti the son of Raphu.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| Of the tribe of Zebulun, Gaddiel the son of Sodi.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| Of the tribe of Joseph, namely, of the tribe of Manasseh, Gaddi the son of Susi.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| Of the tribe of Dan, Ammiel the son of Gemalli.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| Of the tribe of Asher, Sethur the son of Michael.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| Of the tribe of Naphtali, Nahbi the son of Vophsi.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| Of the tribe of Gad, Geuel the son of Machi.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| These are the names of the men which Moses sent to spy out the land. And Moses called Oshea the son of Nun Jehoshua.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| And Moses sent them to spy out the land of Canaan, }} {{font|color=#000088|and said unto them, Get you up this way southward, and go up into the mountain:}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#000088|And see the land, what it is, and the people that dwelleth therein, whether they be strong or weak, few or many;}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#000088| And what the land is that they dwell in, whether it be good or bad; and what cities they be that they dwell in, whether in tents, or in strong holds;}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#000088| And what the land is, whether it be fat or lean, whether there be wood therein, or not. And be ye of good courage, and bring of the fruit of the land. Now the time was the time of the firstripe grapes.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800|So they went up, and searched the land from the wilderness of Zin unto Rehob, as men come to Hamath.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#000088| And they ascended by the south, and came unto Hebron; where Ahiman, Sheshai, and Talmai, the children of Anak, were. (Now Hebron was built seven years before Zoan in Egypt.)}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#000088| And they came unto the brook of Eshcol, and cut down from thence a branch with one cluster of grapes, and they bare it between two upon a staff; and they brought of the pomegranates, and of the figs.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#000088| The place was called the brook Eshcol, because of the cluster of grapes which the children of Israel cut down from thence.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| And they returned from searching of the land after forty days.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| And they went and came to Moses, and to Aaron, and to all the congregation of the children of Israel, unto the wilderness of Paran,}} {{font|color=#000088|to Kadesh; and brought back word unto them, and unto all the congregation, and shewed them the fruit of the land.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#000088| And they told him, and said, We came unto the land whither thou sentest us, and surely it floweth with milk and honey; and this is the fruit of it.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#000088| Nevertheless the people be strong that dwell in the land, and the cities are walled, and very great: and moreover we saw the children of Anak there.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#000088| The Amalekites dwell in the land of the south: and the Hittites, and the Jebusites, and the Amorites, dwell in the mountains: and the Canaanites dwell by the sea, and by the coast of Jordan.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#000088| And Caleb stilled the people before Moses, and said, Let us go up at once, and possess it; for we are well able to overcome it.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#000088| But the men that went up with him said, We be not able to go up against the people; for they are stronger than we.}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#888800| And they brought up an evil report of the land which they had searched unto the children of Israel, saying, The land, through which we have gone to search it, is a land that eateth up the inhabitants thereof; and all the people that we saw in it are men of a great stature.}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#000088| And there we saw the giants, the sons of Anak, which come of the giants: and we were in our own sight as grasshoppers, and so we were in their sight.}}

==Chapter 14==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And all the congregation lifted up their voice, and cried;}} {{font|color=#000088| and the people wept that night.}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| And all the children of Israel murmured against Moses and against Aaron: and the whole congregation said unto them, Would God that we had died in the land of Egypt! or would God we had died in this wilderness!}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| And wherefore hath the LORD brought us unto this land, to fall by the sword, that our wives and our children should be a prey? were it not better for us to return into Egypt?}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And they said one to another, Let us make a captain, and let us return into Egypt.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| Then Moses and Aaron fell on their faces before all the assembly of the congregation of the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| And Joshua the son of Nun, and Caleb the son of Jephunneh, which were of them that searched the land, rent their clothes:}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800|And they spake unto all the company of the children of Israel, saying, The land, which we passed through to search it, is an exceeding good land.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| If the LORD delight in us, then he will bring us into this land, and give it us; a land which floweth with milk and honey.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| Only rebel not ye against the LORD, neither fear ye the people of the land; for they are bread for us: their defence is departed from them, and the LORD is with us: fear them not.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| But all the congregation bade stone them with stones. And the glory of the LORD appeared in the tabernacle of the congregation before all the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#000088| And the LORD said unto Moses, How long will this people provoke me? and how long will it be ere they believe me, for all the signs which I have shewed among them?}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#000088| I will smite them with the pestilence, and disinherit them, and will make of thee a greater nation and mightier than they.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#000088| And Moses said unto the LORD, Then the Egyptians shall hear it, (for thou broughtest up this people in thy might from among them;)}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#000088| And they will tell it to the inhabitants of this land: for they have heard that thou LORD art among this people, that thou LORD art seen face to face, and that thy cloud standeth over them, and that thou goest before them, by day time in a pillar of a cloud, and in a pillar of fire by night.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#000088| Now if thou shalt kill all this people as one man, then the nations which have heard the fame of thee will speak, saying,}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#000088| Because the LORD was not able to bring this people into the land which he sware unto them, therefore he hath slain them in the wilderness.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#000088| And now, I beseech thee, let the power of my Lord be great, according as thou hast spoken, saying,}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#000088| The LORD is longsuffering, and of great mercy, forgiving iniquity and transgression, and by no means clearing the guilty, visiting the iniquity of the fathers upon the children unto the third and fourth generation.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#000088| Pardon, I beseech thee, the iniquity of this people according unto the greatness of thy mercy, and as thou hast forgiven this people, from Egypt even until now.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#000088| And the LORD said, I have pardoned according to thy word: }}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#000088| But as truly as I live, all the earth shall be filled with the glory of the LORD.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#000088| Because all those men which have seen my glory, and my miracles, which I did in Egypt and in the wilderness, and have tempted me now these ten times, and have not hearkened to my voice;}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#000088| Surely they shall not see the land which I sware unto their fathers, neither shall any of them that provoked me see it:}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#000088| But my servant Caleb, because he had another spirit with him, and hath followed me fully, him will I bring into the land whereinto he went; and his seed shall possess it.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#000088| (Now the Amalekites and the Canaanites dwelt in the valley.) Tomorrow turn you, and get you into the wilderness by the way of the Red sea.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses and unto Aaron, saying, }}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| How long shall I bear with this evil congregation, which murmur against me? I have heard the murmurings of the children of Israel, which they murmur against me.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| Say unto them, As truly as I live, saith the LORD, as ye have spoken in mine ears, so will I do to you:}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#888800| Your carcases shall fall in this wilderness; and all that were numbered of you, according to your whole number, from twenty years old and upward which have murmured against me.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#888800| Doubtless ye shall not come into the land, concerning which I sware to make you dwell therein, save Caleb the son of Jephunneh, and Joshua the son of Nun.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#888800| But your little ones, which ye said should be a prey, them will I bring in, and they shall know the land which ye have despised.}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#888800| But as for you, your carcases, they shall fall in this wilderness.}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#888800| And your children shall wander in the wilderness forty years, and bear your whoredoms, until your carcases be wasted in the wilderness.}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#888800| After the number of the days in which ye searched the land, even forty days, each day for a year, shall ye bear your iniquities, even forty years, and ye shall know my breach of promise.}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#888800| I the LORD have said, I will surely do it unto all this evil congregation, that are gathered together against me: in this wilderness they shall be consumed, and there they shall die.}}

&lt;small&gt;{{font|color=#0000FF|36}}&lt;/small&gt;{{font|color=#888800| And the men, which Moses sent to search the land, who returned, and made all the congregation to murmur against him, by bringing up a slander upon the land,}}

&lt;small&gt;{{font|color=#0000FF|37}}&lt;/small&gt;{{font|color=#888800| Even those men that did bring up the evil report upon the land, died by the plague before the LORD.}}

&lt;small&gt;{{font|color=#0000FF|38}}&lt;/small&gt;{{font|color=#888800| But Joshua the son of Nun, and Caleb the son of Jephunneh, which were of the men that went to search the land, lived still.}}

&lt;small&gt;{{font|color=#0000FF|39}}&lt;/small&gt;{{font|color=#000088| And Moses told these sayings unto all the children of Israel: and the people mourned greatly.}}

&lt;small&gt;{{font|color=#0000FF|40}}&lt;/small&gt;{{font|color=#000088| And they rose up early in the morning, and gat them up into the top of the mountain, saying, Lo, we be here, and will go up unto the place which the LORD hath promised: for we have sinned.}}

&lt;small&gt;{{font|color=#0000FF|41}}&lt;/small&gt;{{font|color=#000088| And Moses said, Wherefore now do ye transgress the commandment of the LORD? but it shall not prosper.}}

&lt;small&gt;{{font|color=#0000FF|42}}&lt;/small&gt;{{font|color=#000088| Go not up, for the LORD is not among you; that ye be not smitten before your enemies.}}

&lt;small&gt;{{font|color=#0000FF|43}}&lt;/small&gt;{{font|color=#000088| For the Amalekites and the Canaanites are there before you, and ye shall fall by the sword: because ye are turned away from the LORD, therefore the LORD will not be with you.}}

&lt;small&gt;{{font|color=#0000FF|44}}&lt;/small&gt;{{font|color=#000088| But they presumed to go up unto the hill top: nevertheless the ark of the covenant of the LORD, and Moses, departed not out of the camp.}}

&lt;small&gt;{{font|color=#0000FF|45}}&lt;/small&gt;{{font|color=#000088| Then the Amalekites came down, and the Canaanites which dwelt in that hill, and smote them, and discomfited them, even unto Hormah.}}

==Chapter 15==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| Speak unto the children of Israel, and say unto them, When ye be come into the land of your habitations, which I give unto you,}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| And will make an offering by fire unto the LORD, a burnt offering, or a sacrifice in performing a vow, or in a freewill offering, or in your solemn feasts, to make a sweet savour unto the LORD, of the herd or of the flock:}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| Then shall he that offereth his offering unto the LORD bring a meat offering of a tenth deal of flour mingled with the fourth part of an hin of oil.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And the fourth part of an hin of wine for a drink offering shalt thou prepare with the burnt offering or sacrifice, for one lamb.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| Or for a ram, thou shalt prepare for a meat offering two tenth deals of flour mingled with the third part of an hin of oil.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| And for a drink offering thou shalt offer the third part of an hin of wine, for a sweet savour unto the LORD.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And when thou preparest a bullock for a burnt offering, or for a sacrifice in performing a vow, or peace offerings unto the LORD:}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| Then shall he bring with a bullock a meat offering of three tenth deals of flour mingled with half an hin of oil.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And thou shalt bring for a drink offering half an hin of wine, for an offering made by fire, of a sweet savour unto the LORD.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| Thus shall it be done for one bullock, or for one ram, or for a lamb, or a kid.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| According to the number that ye shall prepare, so shall ye do to every one according to their number.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| All that are born of the country shall do these things after this manner, in offering an offering made by fire, of a sweet savour unto the LORD.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| And if a stranger sojourn with you, or whosoever be among you in your generations, and will offer an offering made by fire, of a sweet savour unto the LORD; as ye do, so he shall do.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| One ordinance shall be both for you of the congregation, and also for the stranger that sojourneth with you, an ordinance for ever in your generations: as ye are, so shall the stranger be before the LORD.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| One law and one manner shall be for you, and for the stranger that sojourneth with you.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| Speak unto the children of Israel, and say unto them, When ye come into the land whither I bring you,}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| Then it shall be, that, when ye eat of the bread of the land, ye shall offer up an heave offering unto the LORD.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| Ye shall offer up a cake of the first of your dough for an heave offering: as ye do the heave offering of the threshingfloor, so shall ye heave it.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| Of the first of your dough ye shall give unto the LORD an heave offering in your generations.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| And if ye have erred, and not observed all these commandments, which the LORD hath spoken unto Moses,}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| Even all that the LORD hath commanded you by the hand of Moses, from the day that the LORD commanded Moses, and henceforward among your generations;}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| Then it shall be, if ought be committed by ignorance without the knowledge of the congregation, that all the congregation shall offer one young bullock for a burnt offering, for a sweet savour unto the LORD, with his meat offering, and his drink offering, according to the manner, and one kid of the goats for a sin offering.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| And the priest shall make an atonement for all the congregation of the children of Israel, and it shall be forgiven them; for it is ignorance: and they shall bring their offering, a sacrifice made by fire unto the LORD, and their sin offering before the LORD, for their ignorance:}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| And it shall be forgiven all the congregation of the children of Israel, and the stranger that sojourneth among them; seeing all the people were in ignorance.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| And if any soul sin through ignorance, then he shall bring a she goat of the first year for a sin offering.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| And the priest shall make an atonement for the soul that sinneth ignorantly, when he sinneth by ignorance before the LORD, to make an atonement for him; and it shall be forgiven him.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#888800| Ye shall have one law for him that sinneth through ignorance, both for him that is born among the children of Israel, and for the stranger that sojourneth among them.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#888800| But the soul that doeth ought presumptuously, whether he be born in the land, or a stranger, the same reproacheth the LORD; and that soul shall be cut off from among his people.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#888800| Because he hath despised the word of the LORD, and hath broken his commandment, that soul shall utterly be cut off; his iniquity shall be upon him.}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#888800| And while the children of Israel were in the wilderness, they found a man that gathered sticks upon the sabbath day.}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#888800| And they that found him gathering sticks brought him unto Moses and Aaron, and unto all the congregation.}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#888800| And they put him in ward, because it was not declared what should be done to him.}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#888800| And the LORD said unto Moses, The man shall be surely put to death: all the congregation shall stone him with stones without the camp.}}

&lt;small&gt;{{font|color=#0000FF|36}}&lt;/small&gt;{{font|color=#888800| And all the congregation brought him without the camp, and stoned him with stones, and he died; as the LORD commanded Moses.}}

&lt;small&gt;{{font|color=#0000FF|37}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|38}}&lt;/small&gt;{{font|color=#888800| Speak unto the children of Israel, and bid them that they make them fringes in the borders of their garments throughout their generations, and that they put upon the fringe of the borders a ribband of blue:}}

&lt;small&gt;{{font|color=#0000FF|39}}&lt;/small&gt;{{font|color=#888800| And it shall be unto you for a fringe, that ye may look upon it, and remember all the commandments of the LORD, and do them; and that ye seek not after your own heart and your own eyes, after which ye use to go a whoring:}}

&lt;small&gt;{{font|color=#0000FF|40}}&lt;/small&gt;{{font|color=#888800| That ye may remember, and do all my commandments, and be holy unto your God.}}

&lt;small&gt;{{font|color=#0000FF|41}}&lt;/small&gt;{{font|color=#888800| I am the LORD your God, which brought you out of the land of Egypt, to be your God: I am the LORD your God.}}

==Chapter 16==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| Now Korah, the son of Izhar, the son of Kohath, the son of Levi,}} {{font|color=#008888| and Dathan and Abiram, the sons of Eliab,}} {{font|color=#880000|and On, the son of Peleth,}} {{font|color=#008888| son}}{{font|color=#888800|s}} {{font|color=#008888|of Reuben, took men:}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| And they rose up before Moses, with certain of the children of Israel, two hundred and fifty princes of the assembly, famous in the congregation, men of renown:}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| And they gathered themselves together against Moses and against Aaron, and said unto them, Ye take too much upon you, seeing all the congregation are holy, every one of them, and the LORD is among them: wherefore then lift ye up yourselves above the congregation of the LORD?}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And when Moses heard it, he fell upon his face:}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And he spake unto Korah and unto all his company, saying, Even to morrow the LORD will shew who are his, and who is holy; and will cause him to come near unto him: even him whom he hath chosen will he cause to come near unto him.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| This do; Take you censers, Korah, and all his company;}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| And put fire therein, and put incense in them before the LORD to morrow: and it shall be that the man whom the LORD doth choose, he shall be holy: ye take too much upon you, ye sons of Levi.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And Moses said unto Korah, Hear, I pray you, ye sons of Levi:}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| Seemeth it but a small thing unto you, that the God of Israel hath separated you from the congregation of Israel, to bring you near to himself to do the service of the tabernacle of the LORD, and to stand before the congregation to minister unto them?}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And he hath brought thee near to him, and all thy brethren the sons of Levi with thee: and seek ye the priesthood also?}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| For which cause both thou and all thy company are gathered together against the LORD: and what is Aaron, that ye murmur against him?}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#008888| And Moses sent to call Dathan and Abiram, the sons of Eliab: which said, We will not come up:}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#008888| Is it a small thing that thou hast brought us up out of a land that floweth with milk and honey, to kill us in the wilderness, except thou make thyself altogether a prince over us?}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#008888| Moreover thou hast not brought us into a land that floweth with milk and honey, or given us inheritance of fields and vineyards: wilt thou put out the eyes of these men? we will not come up.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#008888| And Moses was very wroth, and said unto the LORD, Respect not thou their offering: I have not taken one ass from them, neither have I hurt one of them.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| And Moses said unto Korah, Be thou and all thy company before the LORD, thou, and they, and Aaron, to morrow:}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| And take every man his censer, and put incense in them, and bring ye before the LORD every man his censer, two hundred and fifty censers; thou also, and Aaron, each of you his censer.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| And they took every man his censer, and put fire in them, and laid incense thereon, and stood in the door of the tabernacle of the congregation with Moses and Aaron.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| And Korah gathered all the congregation against them unto the door of the tabernacle of the congregation: and the glory of the LORD appeared unto all the congregation.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses and unto Aaron, saying, }}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| Separate yourselves from among this congregation, that I may consume them in a moment.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| And they fell upon their faces, and said, O God, the God of the spirits of all flesh, shall one man sin, and wilt thou be wroth with all the congregation?}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| Speak unto the congregation, saying, Get you up from about the tabernacle}} {{font|color=#880000|of Korah, Dathan, and Abiram.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#008888| And Moses rose up and went unto Dathan and Abiram; and the elders of Israel followed him.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| And he spake unto the congregation,}} {{font|color=#008888| saying, Depart, I pray you, from the tents of these wicked men, and touch nothing of their's, lest ye be consumed in all their sins.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| So they gat up from the tabernacle }}  {{font|color=#880000| of Korah, Dathan, and Abiram,}} {{font|color=#888800| on every side:}} {{font|color=#008888| and Dathan and Abiram came out, and stood in the door of their tents, and their wives, and their sons, and their little children.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#008888| And Moses said, Hereby ye shall know that the LORD hath sent me to do all these works; for I have not done them of mine own mind.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#008888| If these men die the common death of all men, or if they be visited after the visitation of all men; then the LORD hath not sent me.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#008888| But if the LORD make a new thing, and the earth open her mouth, and swallow them up, with all that appertain unto them, and they go down quick into the pit; then ye shall understand that these men have provoked the LORD.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#008888| And it came to pass, as he had made an end of speaking all these words, that the ground clave asunder that was under them:}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#008888| And the earth opened her mouth, and swallowed them up, and their houses,}} {{font|color=#880000| and all the men that appertained unto Korah,}} {{font|color=#008888| and all their goods.}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#008888| They, and all that appertained to them, went down alive into the pit, and the earth closed upon them: and they perished from among the congregation.}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#008888| And all Israel that were round about them fled at the cry of them: for they said, Lest the earth swallow us up also.}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#888800| And there came out a fire from the LORD, and consumed the two hundred and fifty men that offered incense.}}

&lt;small&gt;{{font|color=#0000FF|36}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying, }}

&lt;small&gt;{{font|color=#0000FF|37}}&lt;/small&gt;{{font|color=#888800| Speak unto Eleazar the son of Aaron the priest, that he take up the censers out of the burning, and scatter thou the fire yonder; for they are hallowed.}}

&lt;small&gt;{{font|color=#0000FF|38}}&lt;/small&gt;{{font|color=#888800| The censers of these sinners against their own souls, let them make them broad plates for a covering of the altar: for they offered them before the LORD, therefore they are hallowed: and they shall be a sign unto the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|39}}&lt;/small&gt;{{font|color=#888800| And Eleazar the priest took the brasen censers, wherewith they that were burnt had offered; and they were made broad plates for a covering of the altar:}}

&lt;small&gt;{{font|color=#0000FF|40}}&lt;/small&gt;{{font|color=#888800| To be a memorial unto the children of Israel, that no stranger, which is not of the seed of Aaron, come near to offer incense before the LORD; that he be not as Korah, and as his company: as the LORD said to him by the hand of Moses.}}

&lt;small&gt;{{font|color=#0000FF|41}}&lt;/small&gt;{{font|color=#888800| But on the morrow all the congregation of the children of Israel murmured against Moses and against Aaron, saying, Ye have killed the people of the LORD.}}

&lt;small&gt;{{font|color=#0000FF|42}}&lt;/small&gt;{{font|color=#888800| And it came to pass, when the congregation was gathered against Moses and against Aaron, that they looked toward the tabernacle of the congregation: and, behold, the cloud covered it, and the glory of the LORD appeared.}}

&lt;small&gt;{{font|color=#0000FF|43}}&lt;/small&gt;{{font|color=#888800| And Moses and Aaron came before the tabernacle of the congregation.}}

&lt;small&gt;{{font|color=#0000FF|44}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|45}}&lt;/small&gt;{{font|color=#888800| Get you up from among this congregation, that I may consume them as in a moment. And they fell upon their faces. }}

&lt;small&gt;{{font|color=#0000FF|46}}&lt;/small&gt;{{font|color=#888800| And Moses said unto Aaron, Take a censer, and put fire therein from off the altar, and put on incense, and go quickly unto the congregation, and make an atonement for them: for there is wrath gone out from the LORD; the plague is begun.}}

&lt;small&gt;{{font|color=#0000FF|47}}&lt;/small&gt;{{font|color=#888800| And Aaron took as Moses commanded, and ran into the midst of the congregation; and, behold, the plague was begun among the people: and he put on incense, and made an atonement for the people.}}

&lt;small&gt;{{font|color=#0000FF|48}}&lt;/small&gt;{{font|color=#888800| And he stood between the dead and the living; and the plague was stayed.}}

&lt;small&gt;{{font|color=#0000FF|49}}&lt;/small&gt;{{font|color=#888800| Now they that died in the plague were fourteen thousand and seven hundred, beside them that died about the matter of Korah.}}

&lt;small&gt;{{font|color=#0000FF|50}}&lt;/small&gt;{{font|color=#888800| And Aaron returned unto Moses unto the door of the tabernacle of the congregation: and the plague was stayed.}}

==Chapter 17==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| Speak unto the children of Israel, and take of every one of them a rod according to the house of their fathers, of all their princes according to the house of their fathers twelve rods: write thou every man's name upon his rod.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| And thou shalt write Aaron's name upon the rod of Levi: for one rod shall be for the head of the house of their fathers.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And thou shalt lay them up in the tabernacle of the congregation before the testimony, where I will meet with you.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And it shall come to pass, that the man's rod, whom I shall choose, shall blossom: and I will make to cease from me the murmurings of the children of Israel, whereby they murmur against you.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| And Moses spake unto the children of Israel, and every one of their princes gave him a rod apiece, for each prince one, according to their fathers' houses, even twelve rods: and the rod of Aaron was among their rods.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| And Moses laid up the rods before the LORD in the tabernacle of witness.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And it came to pass, that on the morrow Moses went into the tabernacle of witness; and, behold, the rod of Aaron for the house of Levi was budded, and brought forth buds, and bloomed blossoms, and yielded almonds.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| And Moses brought out all the rods from before the LORD unto all the children of Israel: and they looked, and took every man his rod.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And the LORD said unto Moses, Bring Aaron's rod again before the testimony, to be kept for a token against the rebels; and thou shalt quite take away their murmurings from me, that they die not.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And Moses did so: as the LORD commanded him, so did he. }}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| And the children of Israel spake unto Moses, saying, Behold, we die, we perish, we all perish.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| Whosoever cometh any thing near unto the tabernacle of the LORD shall die: shall we be consumed with dying?}}

==Chapter 18==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD said unto Aaron, Thou and thy sons and thy father's house with thee shall bear the iniquity of the sanctuary: and thou and thy sons with thee shall bear the iniquity of your priesthood.}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| And thy brethren also of the tribe of Levi, the tribe of thy father, bring thou with thee, that they may be joined unto thee, and minister unto thee: but thou and thy sons with thee shall minister before the tabernacle of witness.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| And they shall keep thy charge, and the charge of all the tabernacle: only they shall not come nigh the vessels of the sanctuary and the altar, that neither they, nor ye also, die.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And they shall be joined unto thee, and keep the charge of the tabernacle of the congregation, for all the service of the tabernacle: and a stranger shall not come nigh unto you.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And ye shall keep the charge of the sanctuary, and the charge of the altar: that there be no wrath any more upon the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| And I, behold, I have taken your brethren the Levites from among the children of Israel: to you they are given as a gift for the LORD, to do the service of the tabernacle of the congregation.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| Therefore thou and thy sons with thee shall keep your priest's office for everything of the altar, and within the vail; and ye shall serve: I have given your priest's office unto you as a service of gift: and the stranger that cometh nigh shall be put to death.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Aaron, Behold, I also have given thee the charge of mine heave offerings of all the hallowed things of the children of Israel; unto thee have I given them by reason of the anointing, and to thy sons, by an ordinance for ever.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| This shall be thine of the most holy things, reserved from the fire: every oblation of their's, every meat offering of their's, and every sin offering of their's, and every trespass offering of their's which they shall render unto me, shall be most holy for thee and for thy sons.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| In the most holy place shalt thou eat it; every male shall eat it: it shall be holy unto thee.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And this is thine; the heave offering of their gift, with all the wave offerings of the children of Israel: I have given them unto thee, and to thy sons and to thy daughters with thee, by a statute for ever: every one that is clean in thy house shall eat of it.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| All the best of the oil, and all the best of the wine, and of the wheat, the firstfruits of them which they shall offer unto the LORD, them have I given thee.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| And whatsoever is first ripe in the land, which they shall bring unto the LORD, shall be thine; every one that is clean in thine house shall eat of it.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| Every thing devoted in Israel shall be thine.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| Every thing that openeth the matrix in all flesh, which they bring unto the LORD, whether it be of men or beasts, shall be thine: nevertheless the firstborn of man shalt thou surely redeem, and the firstling of unclean beasts shalt thou redeem.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| And those that are to be redeemed from a month old shalt thou redeem, according to thine estimation, for the money of five shekels, after the shekel of the sanctuary, which is twenty gerahs.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| But the firstling of a cow, or the firstling of a sheep, or the firstling of a goat, thou shalt not redeem; they are holy: thou shalt sprinkle their blood upon the altar, and shalt burn their fat for an offering made by fire, for a sweet savour unto the LORD.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| And the flesh of them shall be thine, as the wave breast and as the right shoulder are thine.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| All the heave offerings of the holy things, which the children of Israel offer unto the LORD, have I given thee, and thy sons and thy daughters with thee, by a statute for ever: it is a covenant of salt for ever before the LORD unto thee and to thy seed with thee.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Aaron, Thou shalt have no inheritance in their land, neither shalt thou have any part among them: I am thy part and thine inheritance among the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| And, behold, I have given the children of Levi all the tenth in Israel for an inheritance, for their service which they serve, even the service of the tabernacle of the congregation.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| Neither must the children of Israel henceforth come nigh the tabernacle of the congregation, lest they bear sin, and die.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| But the Levites shall do the service of the tabernacle of the congregation, and they shall bear their iniquity: it shall be a statute for ever throughout your generations, that among the children of Israel they have no inheritance.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| But the tithes of the children of Israel, which they offer as an heave offering unto the LORD, I have given to the Levites to inherit: therefore I have said unto them, Among the children of Israel they shall have no inheritance.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| Thus speak unto the Levites, and say unto them, When ye take of the children of Israel the tithes which I have given you from them for your inheritance, then ye shall offer up an heave offering of it for the LORD, even a tenth part of the tithe.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| And this your heave offering shall be reckoned unto you, as though it were the corn of the threshingfloor, and as the fulness of the winepress.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| Thus ye also shall offer an heave offering unto the LORD of all your tithes, which ye receive of the children of Israel; and ye shall give thereof the LORD's heave offering to Aaron the priest.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#888800| Out of all your gifts ye shall offer every heave offering of the LORD, of all the best thereof, even the hallowed part thereof out of it.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#888800| Therefore thou shalt say unto them, When ye have heaved the best thereof from it, then it shall be counted unto the Levites as the increase of the threshingfloor, and as the increase of the winepress.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#888800| And ye shall eat it in every place, ye and your households: for it is your reward for your service in the tabernacle of the congregation.}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#888800| And ye shall bear no sin by reason of it, when ye have heaved from it the best of it: neither shall ye pollute the holy things of the children of Israel, lest ye die.}}

==Chapter 19==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses and unto Aaron, saying, }}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| This is the ordinance of the law which the LORD hath commanded, saying, Speak unto the children of Israel, that they bring thee a red heifer without spot, wherein is no blemish, and upon which never came yoke:}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| And ye shall give her unto Eleazar the priest, that he may bring her forth without the camp, and one shall slay her before his face:}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And Eleazar the priest shall take of her blood with his finger, and sprinkle of her blood directly before the tabernacle of the congregation seven times:}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And one shall burn the heifer in his sight; her skin, and her flesh, and her blood, with her dung, shall he burn:}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| And the priest shall take cedar wood, and hyssop, and scarlet, and cast it into the midst of the burning of the heifer.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| Then the priest shall wash his clothes, and he shall bathe his flesh in water, and afterward he shall come into the camp, and the priest shall be unclean until the even.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And he that burneth her shall wash his clothes in water, and bathe his flesh in water, and shall be unclean until the even.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| And a man that is clean shall gather up the ashes of the heifer, and lay them up without the camp in a clean place, and it shall be kept for the congregation of the children of Israel for a water of separation: it is a purification for sin.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And he that gathereth the ashes of the heifer shall wash his clothes, and be unclean until the even: and it shall be unto the children of Israel, and unto the stranger that sojourneth among them, for a statute for ever.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| He that toucheth the dead body of any man shall be unclean seven days.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| He shall purify himself with it on the third day, and on the seventh day he shall be clean: but if he purify not himself the third day, then the seventh day he shall not be clean.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| Whosoever toucheth the dead body of any man that is dead, and purifieth not himself, defileth the tabernacle of the LORD; and that soul shall be cut off from Israel: because the water of separation was not sprinkled upon him, he shall be unclean; his uncleanness is yet upon him.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| This is the law, when a man dieth in a tent: all that come into the tent, and all that is in the tent, shall be unclean seven days.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| And every open vessel, which hath no covering bound upon it, is unclean.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| And whosoever toucheth one that is slain with a sword in the open fields, or a dead body, or a bone of a man, or a grave, shall be unclean seven days.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| And for an unclean person they shall take of the ashes of the burnt heifer of purification for sin, and running water shall be put thereto in a vessel:}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| And a clean person shall take hyssop, and dip it in the water, and sprinkle it upon the tent, and upon all the vessels, and upon the persons that were there, and upon him that touched a bone, or one slain, or one dead, or a grave:}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| And the clean person shall sprinkle upon the unclean on the third day, and on the seventh day: and on the seventh day he shall purify himself, and wash his clothes, and bathe himself in water, and shall be clean at even.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| But the man that shall be unclean, and shall not purify himself, that soul shall be cut off from among the congregation, because he hath defiled the sanctuary of the LORD: the water of separation hath not been sprinkled upon him; he is unclean.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| And it shall be a perpetual statute unto them, that he that sprinkleth the water of separation shall wash his clothes; and he that toucheth the water of separation shall be unclean until even.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| And whatsoever the unclean person toucheth shall be unclean; and the soul that toucheth it shall be unclean until even.}}

==Chapter 20==
 
&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| Then came the children of Israel, even the whole congregation, into the desert of Zin in the first month:}} {{font|color=#008888| and the people abode in Kadesh; and Miriam died there, and was buried there.}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| And there was no water for the congregation: and they gathered themselves together against Moses and against Aaron.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#008888| And the people chode with Moses, and spake, }} {{font|color=#888800|saying, Would God that we had died when our brethren died before the LORD!}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And why have ye brought up the congregation of the LORD into this wilderness, that we and our cattle should die there?}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#008888| And wherefore have ye made us to come up out of Egypt, to bring us in unto this evil place? it is no place of seed, or of figs, or of vines, or of pomegranates; neither is there any water to drink.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| And Moses and Aaron went from the presence of the assembly unto the door of the tabernacle of the congregation, and they fell upon their faces: and the glory of the LORD appeared unto them.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| Take the rod, and gather thou the assembly together, thou, and Aaron thy brother, and speak ye unto the rock before their eyes; and it shall give forth his water, and thou shalt bring forth to them water out of the rock: so thou shalt give the congregation and their beasts drink.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| And Moses took the rod from before the LORD, as he commanded him.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And Moses and Aaron gathered the congregation together before the rock, and he said unto them, Hear now, ye rebels; must we fetch you water out of this rock?}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And Moses lifted up his hand, and with his rod he smote the rock twice: and the water came out abundantly, and the congregation drank, and their beasts also.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses and Aaron, Because ye believed me not, to sanctify me in the eyes of the children of Israel, therefore ye shall not bring this congregation into the land which I have given them.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| This is the water of Meribah; because the children of Israel strove with the LORD, and he was sanctified in them.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#008888| And Moses sent messengers from Kadesh unto the king of Edom, Thus saith thy brother Israel, Thou knowest all the travail that hath befallen us:}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#008888| How our fathers went down into Egypt, and we have dwelt in Egypt a long time; and the Egyptians vexed us, and our fathers:}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#008888| And when we cried unto the LORD, he heard our voice, and sent an angel, and hath brought us forth out of Egypt: and, behold, we are in Kadesh, a city in the uttermost of thy border:}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#008888| Let us pass, I pray thee, through thy country: we will not pass through the fields, or through the vineyards, neither will we drink of the water of the wells: we will go by the king's high way, we will not turn to the right hand nor to the left, until we have passed thy borders.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#008888| And Edom said unto him, Thou shalt not pass by me, lest I come out against thee with the sword.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#008888| And the children of Israel said unto him, We will go by the high way: and if I and my cattle drink of thy water, then I will pay for it: I will only, without doing anything else, go through on my feet.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#008888| And he said, Thou shalt not go through. And Edom came out against him with much people, and with a strong hand.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#008888| Thus Edom refused to give Israel passage through his border: wherefore Israel turned away from him.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| And the children of Israel, even the whole congregation,}} {{font|color=#008888|journeyed from Kadesh, and}} {{font|color=#888800|came unto mount Hor.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses and Aaron in mount Hor, by the coast of the land of Edom, saying,}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| Aaron shall be gathered unto his people: for he shall not enter into the land which I have given unto the children of Israel, because ye rebelled against my word at the water of Meribah.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| Take Aaron and Eleazar his son, and bring them up unto mount Hor:}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| And strip Aaron of his garments, and put them upon Eleazar his son: and Aaron shall be gathered unto his people, and shall die there.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| And Moses did as the LORD commanded: and they went up into mount Hor in the sight of all the congregation.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| And Moses stripped Aaron of his garments, and put them upon Eleazar his son; and Aaron died there in the top of the mount: and Moses and Eleazar came down from the mount.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#888800| And when all the congregation saw that Aaron was dead, they mourned for Aaron thirty days, even all the house of Israel.}}

==Chapter 21==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#000088| And when king Arad the Canaanite, which dwelt in the south, heard tell that Israel came by the way of the spies; then he fought against Israel, and took some of them prisoners.}}
 
&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#000088| And Israel vowed a vow unto the LORD, and said, If thou wilt indeed deliver this people into my hand, then I will utterly destroy their cities.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#000088| And the LORD hearkened to the voice of Israel, and delivered up the Canaanites; and they utterly destroyed them and their cities: and he called the name of the place Hormah.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And they journeyed from mount Hor by the way of the Red sea,}} {{font|color=#008888| to compass the land of Edom: and the soul of the people was much discouraged because of the way.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#008888| And the people spake against God, and against Moses, Wherefore have ye brought us up out of Egypt to die in the wilderness? for there is no bread, neither is there any water;}} {{font|color=#880000|and our soul loatheth this light bread.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#008888| And the LORD sent fiery serpents among the people, and they bit the people; and much people of Israel died.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#008888| Therefore the people came to Moses, and said, We have sinned, for we have spoken against the LORD, and against thee; pray unto the LORD, that he take away the serpents from us. And Moses prayed for the people.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#008888| And the LORD said unto Moses, Make thee a fiery serpent, and set it upon a pole: and it shall come to pass, that every one that is bitten, when he looketh upon it, shall live.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#008888| And Moses made a serpent of brass, and put it upon a pole, and it came to pass, that if a serpent had bitten any man, when he beheld the serpent of brass, he lived.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And the children of Israel set forward, and pitched in Oboth. }}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And they journeyed from Oboth, and pitched at Ijeabarim, in the wilderness which is before Moab, toward the sunrising.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#008888| From thence they removed, and pitched in the valley of Zared.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#008888| From thence they removed, and pitched on the other side of Arnon, which is in the wilderness that cometh out of the coasts of the Amorites: for Arnon is the border of Moab, between Moab and the Amorites.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#008888| Wherefore it is said in the book of the wars of the LORD, What he did in the Red sea, and in the brooks of Arnon,}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#008888| And at the stream of the brooks that goeth down to the dwelling of Ar, and lieth upon the border of Moab.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#000088|And from thence they went to Beer: that is the well whereof the LORD spake unto Moses, Gather the people together, and I will give them water.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#000088| Then Israel sang this song, Spring up, O well; sing ye unto it:}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#000088|The princes digged the well, the nobles of the people digged it, by the direction of the lawgiver, with their staves. And from the wilderness they went to Mattanah: }}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#000088| And from Mattanah to Nahaliel: and from Nahaliel to Bamoth:}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#000088| And from Bamoth in the valley, that is in the country of Moab, to the top of Pisgah, which looketh toward Jeshimon.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#008888| And Israel sent messengers unto Sihon king of the Amorites, saying,}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#008888| Let me pass through thy land: we will not turn into the fields, or into the vineyards; we will not drink of the waters of the well: but we will go along by the king's high way, until we be past thy borders.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#008888| And Sihon would not suffer Israel to pass through his border: but Sihon gathered all his people together, and went out against Israel into the wilderness: and he came to Jahaz, and fought against Israel.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#008888| And Israel smote him with the edge of the sword, and possessed his land from Arnon unto Jabbok, even unto the children of Ammon: for the border of the children of Ammon was strong.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#008888| And Israel took all these cities: and Israel dwelt in all the cities of the Amorites, in Heshbon, and in all the villages thereof.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#008888| For Heshbon was the city of Sihon the king of the Amorites, who had fought against the former king of Moab, and taken all his land out of his hand, even unto Arnon.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#008888| Wherefore they that speak in proverbs say, Come into Heshbon, let the city of Sihon be built and prepared:}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#008888| For there is a fire gone out of Heshbon, a flame from the city of Sihon: it hath consumed Ar of Moab, and the lords of the high places of Arnon.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#008888| Woe to thee, Moab! thou art undone, O people of Chemosh: he hath given his sons that escaped, and his daughters, into captivity unto Sihon king of the Amorites.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#008888| We have shot at them; Heshbon is perished even unto Dibon, and we have laid them waste even unto Nophah, which reacheth unto Medeba.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#008888| Thus Israel dwelt in the land of the Amorites. }}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#008888| And Moses sent to spy out Jaazer, and they took the villages thereof, and drove out the Amorites that were there.}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#880000| And they turned and went up by the way of Bashan: and Og the king of Bashan went out against them, he, and all his people, to the battle at Edrei.}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#880000| And the LORD said unto Moses, Fear him not: for I have delivered him into thy hand, and all his people, and his land; and thou shalt do to him as thou didst unto Sihon king of the Amorites, which dwelt at Heshbon.}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#880000| So they smote him, and his sons, and all his people, until there was none left him alive: and they possessed his land.}}

==Chapter 22==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the children of Israel set forward, and pitched in the plains of Moab on this side Jordan by Jericho.}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#880000| And Balak the son of Zippor saw all that Israel had done to the Amorites.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#000088|And Moab was sore afraid of the people, because they were many: and Moab was distressed because of the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#000088|And Moab said}} {{font|color=#880000| unto the elders of Midian,}} {{font|color=#000088|Now shall this company lick up all that are round about us, as the ox licketh up the grass of the field. And Balak the son of Zippor was king of the Moabites at that time.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#000088| He sent messengers therefore unto Balaam the son of Beor to Pethor, which is by the river of the land of the children of his people, to call him, saying, Behold, there is a people come out from Egypt: behold, they cover the face of the earth, and they abide over against me:}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#000088| Come now therefore, I pray thee, curse me this people; for they are too mighty for me: peradventure I shall prevail, that we may smite them, and that I may drive them out of the land: for I wot that he whom thou blessest is blessed, and he whom thou cursest is cursed.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#000088| And the elders of Moab}} {{font|color=#880000|and the elders of Midian}} {{font|color=#000088|departed with the rewards of divination in their hand; and they came unto Balaam, and spake unto him the words of Balak.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#000088| And he said unto them, Lodge here this night, and I will bring you word again, as the LORD shall speak unto me: and the princes of Moab abode with Balaam.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#000088| And God came unto Balaam, and said, What men are these with thee?}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#000088| And Balaam said unto God, Balak the son of Zippor, king of Moab, hath sent unto me saying,}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#000088| Behold, there is a people come out of Egypt, which covereth the face of the earth: come now, curse me them; peradventure I shall be able to overcome them, and drive them out.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#000088| And God said unto Balaam, Thou shalt not go with them; thou shalt not curse the people: for they are blessed.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#000088| And Balaam rose up in the morning, and said unto the princes of Balak, Get you into your land: for the LORD refuseth to give me leave to go with you.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#000088| And the princes of Moab rose up, and they went unto Balak, and said, Balaam refuseth to come with us.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#000088| And Balak sent yet again princes, more, and more honourable than they.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#000088| And they came to Balaam, and said to him, Thus saith Balak the son of Zippor, Let nothing, I pray thee, hinder thee from coming unto me:}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#000088| For I will promote thee unto very great honour, and I will do whatsoever thou sayest unto me: come therefore, I pray thee, curse me this people.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#000088| And Balaam answered and said unto the servants of Balak, If Balak would give me his house full of silver and gold, I cannot go beyond the word of the LORD my God, to do less or more.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#000088| Now therefore, I pray you, tarry ye also here this night, that I may know what the LORD will say unto me more.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#000088| And God came unto Balaam at night, and said unto him, If the men come to call thee, rise up, and go with them; but yet the word which I shall say unto thee, that shalt thou do.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#000088| And Balaam rose up in the morning, and saddled his ass, and went with the princes of Moab.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#880000| And God's anger was kindled because he went:}} {{font|color=#000088|and the angel of the LORD stood in the way for an adversary against him. Now he was riding upon his ass, and his two servants were with him.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#000088| And the ass saw the angel of the LORD standing in the way, and his sword drawn in his hand: and the ass turned aside out of the way, and went into the field: and Balaam smote the ass, to turn her into the way.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#000088| But the angel of the LORD stood in a path of the vineyards, a wall being on this side, and a wall on that side.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#000088| And when the ass saw the angel of the LORD, she thrust herself unto the wall, and crushed Balaam's foot against the wall: and he smote her again.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#000088| And the angel of the LORD went further, and stood in a narrow place, where was no way to turn either to the right hand or to the left.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#000088| And when the ass saw the angel of the LORD, she fell down under Balaam: and Balaam's anger was kindled, and he smote the ass with a staff.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#000088| And the LORD opened the mouth of the ass, and she said unto Balaam, What have I done unto thee, that thou hast smitten me these three times?}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#000088| And Balaam said unto the ass, Because thou hast mocked me: I would there were a sword in mine hand, for now would I kill thee.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#000088| And the ass said unto Balaam, Am not I thine ass, upon which thou hast ridden ever since I was thine unto this day? was I ever wont to do so unto thee? And he said, Nay.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#000088| Then the LORD opened the eyes of Balaam, and he saw the angel of the LORD standing in the way, and his sword drawn in his hand: and he bowed down his head, and fell flat on his face.}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#000088| And the angel of the LORD said unto him, Wherefore hast thou smitten thine ass these three times? behold, I went out to withstand thee, because thy way is perverse before me:}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#000088| And the ass saw me, and turned from me these three times: unless she had turned from me, surely now also I had slain thee, and saved her alive.}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#000088| And Balaam said unto the angel of the LORD, I have sinned; for I knew not that thou stoodest in the way against me: now therefore, if it displease thee, I will get me back again.}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#000088| And the angel of the LORD said unto Balaam, Go with the men: but only the word that I shall speak unto thee, that thou shalt speak. So Balaam went with the princes of Balak.}}

&lt;small&gt;{{font|color=#0000FF|36}}&lt;/small&gt;{{font|color=#000088| And when Balak heard that Balaam was come, he went out to meet him unto a city of Moab, which is in the border of Arnon, which is in the utmost coast.}}

&lt;small&gt;{{font|color=#0000FF|37}}&lt;/small&gt;{{font|color=#000088| And Balak said unto Balaam, Did I not earnestly send unto thee to call thee? wherefore camest thou not unto me? am I not able indeed to promote thee to honour?}}

&lt;small&gt;{{font|color=#0000FF|38}}&lt;/small&gt;{{font|color=#000088| And Balaam said unto Balak, Lo, I am come unto thee: have I now any power at all to say any thing? the word that God putteth in my mouth, that shall I speak.}}

&lt;small&gt;{{font|color=#0000FF|39}}&lt;/small&gt;{{font|color=#000088| And Balaam went with Balak, and they came unto Kirjathhuzoth.}}

&lt;small&gt;{{font|color=#0000FF|40}}&lt;/small&gt;{{font|color=#000088| And Balak offered oxen and sheep, and sent to Balaam, and to the princes that were with him.}}

&lt;small&gt;{{font|color=#0000FF|41}}&lt;/small&gt;{{font|color=#000088| And it came to pass on the morrow, that Balak took Balaam, and brought him up into the high places of Baal, that thence he might see the utmost part of the people.}}

==Chapter 23==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#000088| And Balaam said unto Balak, Build me here seven altars, and prepare me here seven oxen and seven rams.}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#000088| And Balak did as Balaam had spoken; and Balak and Balaam offered on every altar a bullock and a ram.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#000088| And Balaam said unto Balak, Stand by thy burnt offering, and I will go: peradventure the LORD will come to meet me: and whatsoever he sheweth me I will tell thee. And he went to an high place.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#000088| And God met Balaam: and he said unto him, I have prepared seven altars, and I have offered upon every altar a bullock and a ram.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#000088| And the LORD put a word in Balaam's mouth, and said, Return unto Balak, and thus thou shalt speak.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#000088| And he returned unto him, and, lo, he stood by his burnt sacrifice, he, and all the princes of Moab.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#000088| And he took up his parable, and said, Balak the king of Moab hath brought me from Aram, out of the mountains of the east, saying, Come, curse me Jacob, and come, defy Israel.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#000088| How shall I curse, whom God hath not cursed? or how shall I defy, whom the LORD hath not defied?}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#000088| For from the top of the rocks I see him, and from the hills I behold him: lo, the people shall dwell alone, and shall not be reckoned among the nations.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#000088| Who can count the dust of Jacob, and the number of the fourth part of Israel? Let me die the death of the righteous, and let my last end be like his!}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#000088| And Balak said unto Balaam, What hast thou done unto me? I took thee to curse mine enemies, and, behold, thou hast blessed them altogether.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#000088| And he answered and said, Must I not take heed to speak that which the LORD hath put in my mouth?}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#000088| And Balak said unto him, Come, I pray thee, with me unto another place, from whence thou mayest see them: thou shalt see but the utmost part of them, and shalt not see them all: and curse me them from thence.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#000088| And he brought him into the field of Zophim, to the top of Pisgah, and built seven altars, and offered a bullock and a ram on every altar.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#000088| And he said unto Balak, Stand here by thy burnt offering, while I meet the LORD yonder.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#000088| And the LORD met Balaam, and put a word in his mouth, and said, Go again unto Balak, and say thus.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#000088| And when he came to him, behold, he stood by his burnt offering, and the princes of Moab with him. And Balak said unto him, What hath the LORD spoken?}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#000088| And he took up his parable, and said, Rise up, Balak, and hear; hearken unto me, thou son of Zippor:}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#000088| God is not a man, that he should lie; neither the son of man, that he should repent: hath he said, and shall he not do it? or hath he spoken, and shall he not make it good?}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#000088| Behold, I have received commandment to bless: and he hath blessed; and I cannot reverse it.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#000088| He hath not beheld iniquity in Jacob, neither hath he seen perverseness in Israel: the LORD his God is with him, and the shout of a king is among them.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#000088| God brought them out of Egypt; he hath as it were the strength of an unicorn.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#000088| Surely there is no enchantment against Jacob, neither is there any divination against Israel: according to this time it shall be said of Jacob and of Israel, What hath God wrought!}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#000088| Behold, the people shall rise up as a great lion, and lift up himself as a young lion: he shall not lie down until he eat of the prey, and drink the blood of the slain.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#000088| And Balak said unto Balaam, Neither curse them at all, nor bless them at all.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#000088| But Balaam answered and said unto Balak, Told not I thee, saying, All that the LORD speaketh, that I must do?}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#000088| And Balak said unto Balaam, Come, I pray thee, I will bring thee unto another place; peradventure it will please God that thou mayest curse me them from thence.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#000088| And Balak brought Balaam unto the top of Peor, that looketh toward Jeshimon.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#000088| And Balaam said unto Balak, Build me here seven altars, and prepare me here seven bullocks and seven rams.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#000088| And Balak did as Balaam had said, and offered a bullock and a ram on every altar.}}

==Chapter 24==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#000088| And when Balaam saw that it pleased the LORD to bless Israel, he went not, as at other times, to seek for enchantments, but he set his face toward the wilderness.}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#000088| And Balaam lifted up his eyes, and he saw Israel abiding in his tents according to their tribes; and the spirit of God came upon him.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#000088| And he took up his parable, and said, Balaam the son of Beor hath said, and the man whose eyes are open hath said:}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#000088| He hath said, which heard the words of God, which saw the vision of the Almighty, falling into a trance, but having his eyes open:}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#000088| How goodly are thy tents, O Jacob, and thy tabernacles, O Israel!}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#000088| As the valleys are they spread forth, as gardens by the river's side, as the trees of lign aloes which the LORD hath planted, and as cedar trees beside the waters.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#000088| He shall pour the water out of his buckets, and his seed shall be in many waters, and his king shall be higher than Agag, and his kingdom shall be exalted.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#000088| God brought him forth out of Egypt; he hath as it were the strength of an unicorn: he shall eat up the nations his enemies, and shall break their bones, and pierce them through with his arrows.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#000088| He couched, he lay down as a lion, and as a great lion: who shall stir him up? Blessed is he that blesseth thee, and cursed is he that curseth thee.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#000088| And Balak's anger was kindled against Balaam, and he smote his hands together: and Balak said unto Balaam, I called thee to curse mine enemies, and, behold, thou hast altogether blessed them these three times.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#000088| Therefore now flee thou to thy place: I thought to promote thee unto great honour; but, lo, the LORD hath kept thee back from honour.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#000088| And Balaam said unto Balak, Spake I not also to thy messengers which thou sentest unto me, saying,}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#000088| If Balak would give me his house full of silver and gold, I cannot go beyond the commandment of the LORD, to do either good or bad of mine own mind; but what the LORD saith, that will I speak?}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#000088| And now, behold, I go unto my people: come therefore, and I will advertise thee what this people shall do to thy people in the latter days.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#000088| And he took up his parable, and said, Balaam the son of Beor hath said, and the man whose eyes are open hath said:}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#000088| He hath said, which heard the words of God, and knew the knowledge of the most High, which saw the vision of the Almighty, falling into a trance, but having his eyes open:}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#000088| I shall see him, but not now: I shall behold him, but not nigh: there shall come a Star out of Jacob, and a Sceptre shall rise out of Israel, and shall smite the corners of Moab, and destroy all the children of Sheth.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#000088| And Edom shall be a possession, Seir also shall be a possession for his enemies; and Israel shall do valiantly.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#000088| Out of Jacob shall come he that shall have dominion, and shall destroy him that remaineth of the city.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#000088| And when he looked on Amalek, he took up his parable, and said, Amalek was the first of the nations; but his latter end shall be that he perish for ever.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#000088| And he looked on the Kenites, and took up his parable, and said, Strong is thy dwellingplace, and thou puttest thy nest in a rock.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#000088| Nevertheless the Kenite shall be wasted, until Asshur shall carry thee away captive.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#000088| And he took up his parable, and said, Alas, who shall live when God doeth this!}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#000088| And ships shall come from the coast of Chittim, and shall afflict Asshur, and shall afflict Eber, and he also shall perish for ever.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#000088| And Balaam rose up, and went and returned to his place: and Balak also went his way.}}

==Chapter 25==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#008888| And Israel abode in Shittim, }}{{font|color=#000088|and the people began to commit whoredom with the daughters of Moab.}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#000088| And they called the people unto the sacrifices of their gods: and the people did eat, and bowed down to their gods.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#008888| And Israel joined himself unto Baalpeor: }}{{font|color=#000088|and the anger of the LORD was kindled against Israel.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#000088| And the LORD said unto Moses, Take all the heads of the people, and hang them up before the LORD against the sun, that the fierce anger of the LORD may be turned away from Israel.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#008888| And Moses said unto the judges of Israel, Slay ye every one his men that were joined unto Baalpeor.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| And, behold, one of the children of Israel came and brought unto his brethren a Midianitish woman in the sight of Moses, and in the sight of all the congregation of the children of Israel, who were weeping before the door of the tabernacle of the congregation.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| And when Phinehas, the son of Eleazar, the son of Aaron the priest, saw it, he rose up from among the congregation, and took a javelin in his hand;}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And he went after the man of Israel into the tent, and thrust both of them through, the man of Israel, and the woman through her belly. So the plague was stayed from the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| And those that died in the plague were twenty and four thousand.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| Phinehas, the son of Eleazar, the son of Aaron the priest, hath turned my wrath away from the children of Israel, while he was zealous for my sake among them, that I consumed not the children of Israel in my jealousy.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| Wherefore say, Behold, I give unto him my covenant of peace:}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| And he shall have it, and his seed after him, even the covenant of an everlasting priesthood; because he was zealous for his God, and made an atonement for the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| Now the name of the Israelite that was slain, even that was slain with the Midianitish woman, was Zimri, the son of Salu, a prince of a chief house among the Simeonites.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| And the name of the Midianitish woman that was slain was Cozbi, the daughter of Zur; he was head over a people, and of a chief house in Midian.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| Vex the Midianites, and smite them:}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| For they vex you with their wiles, wherewith they have beguiled you in the matter of Peor, and in the matter of Cozbi, the daughter of a prince of Midian, their sister, which was slain in the day of the plague for Peor's sake.}}

==Chapter 26==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And it came to pass after the plague, that the LORD spake unto Moses and unto Eleazar the son of Aaron the priest, saying, }}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| Take the sum of all the congregation of the children of Israel, from twenty years old and upward, throughout their fathers' house, all that are able to go to war in Israel.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| And Moses and Eleazar the priest spake with them in the plains of Moab by Jordan near Jericho, saying,}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| Take the sum of the people, from twenty years old and upward; as the LORD commanded Moses and the children of Israel, which went forth out of the land of Egypt.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| Reuben, the eldest son of Israel: the children of Reuben; Hanoch, of whom cometh the family of the Hanochites: of Pallu, the family of the Palluites:}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| Of Hezron, the family of the Hezronites: of Carmi, the family of the Carmites.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| These are the families of the Reubenites: and they that were numbered of them were forty and three thousand and seven hundred and thirty.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And the sons of Pallu; Eliab.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#880000| And the sons of Eliab; Nemuel, and Dathan, and Abiram. This is that Dathan and Abiram, which were famous in the congregation, who strove against Moses and against Aaron in the company of Korah, when they strove against the LORD:}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#880000| And the earth opened her mouth, and swallowed them up together with Korah, when that company died, what time the fire devoured two hundred and fifty men: and they became a sign.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#880000| Notwithstanding the children of Korah died not.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| The sons of Simeon after their families: of Nemuel, the family of the Nemuelites: of Jamin, the family of the Jaminites: of Jachin, the family of the Jachinites:}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| Of Zerah, the family of the Zarhites: of Shaul, the family of the Shaulites.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| These are the families of the Simeonites, twenty and two thousand and two hundred.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| The children of Gad after their families: of Zephon, the family of the Zephonites: of Haggi, the family of the Haggites: of Shuni, the family of the Shunites:}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| Of Ozni, the family of the Oznites: of Eri, the family of the Erites:}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| Of Arod, the family of the Arodites: of Areli, the family of the Arelites.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| These are the families of the children of Gad according to those that were numbered of them, forty thousand and five hundred.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| The sons of Judah were Er and Onan: and Er and Onan died in the land of Canaan.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| And the sons of Judah after their families were; of Shelah, the family of the Shelanites: of Pharez, the family of the Pharzites: of Zerah, the family of the Zarhites.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| And the sons of Pharez were; of Hezron, the family of the Hezronites: of Hamul, the family of the Hamulites.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| These are the families of Judah according to those that were numbered of them, threescore and sixteen thousand and five hundred.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| Of the sons of Issachar after their families: of Tola, the family of the Tolaites: of Pua, the family of the Punites:}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| Of Jashub, the family of the Jashubites: of Shimron, the family of the Shimronites.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| These are the families of Issachar according to those that were numbered of them, threescore and four thousand and three hundred.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| Of the sons of Zebulun after their families: of Sered, the family of the Sardites: of Elon, the family of the Elonites: of Jahleel, the family of the Jahleelites.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| These are the families of the Zebulunites according to those that were numbered of them, threescore thousand and five hundred.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| The sons of Joseph after their families were Manasseh and Ephraim.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#888800| Of the sons of Manasseh: of Machir, the family of the Machirites: and Machir begat Gilead: of Gilead come the family of the Gileadites.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#888800| These are the sons of Gilead: of Jeezer, the family of the Jeezerites: of Helek, the family of the Helekites:}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#888800| And of Asriel, the family of the Asrielites: and of Shechem, the family of the Shechemites:}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#888800| And of Shemida, the family of the Shemidaites: and of Hepher, the family of the Hepherites.}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#888800| And Zelophehad the son of Hepher had no sons, but daughters: and the names of the daughters of Zelophehad were Mahlah, and Noah, Hoglah, Milcah, and Tirzah.}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#888800| These are the families of Manasseh, and those that were numbered of them, fifty and two thousand and seven hundred.}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#888800| These are the sons of Ephraim after their families: of Shuthelah, the family of the Shuthalhites: of Becher, the family of the Bachrites: of Tahan, the family of the Tahanites.}}

&lt;small&gt;{{font|color=#0000FF|36}}&lt;/small&gt;{{font|color=#888800| And these are the sons of Shuthelah: of Eran, the family of the Eranites.}}

&lt;small&gt;{{font|color=#0000FF|37}}&lt;/small&gt;{{font|color=#888800| These are the families of the sons of Ephraim according to those that were numbered of them, thirty and two thousand and five hundred. These are the sons of Joseph after their families.}}

&lt;small&gt;{{font|color=#0000FF|38}}&lt;/small&gt;{{font|color=#888800| The sons of Benjamin after their families: of Bela, the family of the Belaites: of Ashbel, the family of the Ashbelites: of Ahiram, the family of the Ahiramites:}}

&lt;small&gt;{{font|color=#0000FF|39}}&lt;/small&gt;{{font|color=#888800| Of Shupham, the family of the Shuphamites: of Hupham, the family of the Huphamites.}}

&lt;small&gt;{{font|color=#0000FF|40}}&lt;/small&gt;{{font|color=#888800| And the sons of Bela were Ard and Naaman: of Ard, the family of the Ardites: and of Naaman, the family of the Naamites.}}

&lt;small&gt;{{font|color=#0000FF|41}}&lt;/small&gt;{{font|color=#888800| These are the sons of Benjamin after their families: and they that were numbered of them were forty and five thousand and six hundred.}}

&lt;small&gt;{{font|color=#0000FF|42}}&lt;/small&gt;{{font|color=#888800| These are the sons of Dan after their families: of Shuham, the family of the Shuhamites. These are the families of Dan after their families.}}

&lt;small&gt;{{font|color=#0000FF|43}}&lt;/small&gt;{{font|color=#888800| All the families of the Shuhamites, according to those that were numbered of them, were threescore and four thousand and four hundred.}}

&lt;small&gt;{{font|color=#0000FF|44}}&lt;/small&gt;{{font|color=#888800| Of the children of Asher after their families: of Jimna, the family of the Jimnites: of Jesui, the family of the Jesuites: of Beriah, the family of the Beriites.}}

&lt;small&gt;{{font|color=#0000FF|45}}&lt;/small&gt;{{font|color=#888800| Of the sons of Beriah: of Heber, the family of the Heberites: of Malchiel, the family of the Malchielites.}}

&lt;small&gt;{{font|color=#0000FF|46}}&lt;/small&gt;{{font|color=#888800| And the name of the daughter of Asher was Sarah.}}

&lt;small&gt;{{font|color=#0000FF|47}}&lt;/small&gt;{{font|color=#888800| These are the families of the sons of Asher according to those that were numbered of them; who were fifty and three thousand and four hundred.}}

&lt;small&gt;{{font|color=#0000FF|48}}&lt;/small&gt;{{font|color=#888800| Of the sons of Naphtali after their families: of Jahzeel, the family of the Jahzeelites: of Guni, the family of the Gunites:}}

&lt;small&gt;{{font|color=#0000FF|49}}&lt;/small&gt;{{font|color=#888800| Of Jezer, the family of the Jezerites: of Shillem, the family of the Shillemites.}}

&lt;small&gt;{{font|color=#0000FF|50}}&lt;/small&gt;{{font|color=#888800| These are the families of Naphtali according to their families: and they that were numbered of them were forty and five thousand and four hundred.}}

&lt;small&gt;{{font|color=#0000FF|51}}&lt;/small&gt;{{font|color=#888800| These were the numbered of the children of Israel, six hundred thousand and a thousand seven hundred and thirty.}}

&lt;small&gt;{{font|color=#0000FF|52}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|53}}&lt;/small&gt;{{font|color=#888800| Unto these the land shall be divided for an inheritance according to the number of names.}}

&lt;small&gt;{{font|color=#0000FF|54}}&lt;/small&gt;{{font|color=#888800| To many thou shalt give the more inheritance, and to few thou shalt give the less inheritance: to every one shall his inheritance be given according to those that were numbered of him.}}

&lt;small&gt;{{font|color=#0000FF|55}}&lt;/small&gt;{{font|color=#888800| Notwithstanding the land shall be divided by lot: according to the names of the tribes of their fathers they shall inherit.}}

&lt;small&gt;{{font|color=#0000FF|56}}&lt;/small&gt;{{font|color=#888800| According to the lot shall the possession thereof be divided between many and few.}}

&lt;small&gt;{{font|color=#0000FF|57}}&lt;/small&gt;{{font|color=#888800| And these are they that were numbered of the Levites after their families: of Gershon, the family of the Gershonites: of Kohath, the family of the Kohathites: of Merari, the family of the Merarites.}}

&lt;small&gt;{{font|color=#0000FF|58}}&lt;/small&gt;{{font|color=#888800| These are the families of the Levites: the family of the Libnites, the family of the Hebronites, the family of the Mahlites, the family of the Mushites, the family of the Korathites. And Kohath begat Amram.}}

&lt;small&gt;{{font|color=#0000FF|59}}&lt;/small&gt;{{font|color=#888800| And the name of Amram's wife was Jochebed, the daughter of Levi, whom her mother bare to Levi in Egypt: and she bare unto Amram Aaron and Moses, and Miriam their sister.}}

&lt;small&gt;{{font|color=#0000FF|60}}&lt;/small&gt;{{font|color=#888800| And unto Aaron was born Nadab, and Abihu, Eleazar, and Ithamar.}}

&lt;small&gt;{{font|color=#0000FF|61}}&lt;/small&gt;{{font|color=#888800| And Nadab and Abihu died, when they offered strange fire before the LORD.}}

&lt;small&gt;{{font|color=#0000FF|62}}&lt;/small&gt;{{font|color=#888800| And those that were numbered of them were twenty and three thousand, all males from a month old and upward: for they were not numbered among the children of Israel, because there was no inheritance given them among the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|63}}&lt;/small&gt;{{font|color=#888800| These are they that were numbered by Moses and Eleazar the priest, who numbered the children of Israel in the plains of Moab by Jordan near Jericho.}}

&lt;small&gt;{{font|color=#0000FF|64}}&lt;/small&gt;{{font|color=#888800| But among these there was not a man of them whom Moses and Aaron the priest numbered, when they numbered the children of Israel in the wilderness of Sinai.}}

&lt;small&gt;{{font|color=#0000FF|65}}&lt;/small&gt;{{font|color=#888800| For the LORD had said of them, They shall surely die in the wilderness. And there was not left a man of them, save Caleb the son of Jephunneh, and Joshua the son of Nun.}}

==Chapter 27==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| Then came the daughters of Zelophehad, the son of Hepher, the son of Gilead, the son of Machir, the son of Manasseh, of the families of Manasseh the son of Joseph: and these are the names of his daughters; Mahlah, Noah, and Hoglah, and Milcah, and Tirzah.}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| And they stood before Moses, and before Eleazar the priest, and before the princes and all the congregation, by the door of the tabernacle of the congregation, saying,}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| Our father died in the wilderness, and he was not in the company of them that gathered themselves together against the LORD in the company of Korah; but died in his own sin, and had no sons.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| Why should the name of our father be done away from among his family, because he hath no son? Give unto us therefore a possession among the brethren of our father.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And Moses brought their cause before the LORD.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| The daughters of Zelophehad speak right: thou shalt surely give them a possession of an inheritance among their father's brethren; and thou shalt cause the inheritance of their father to pass unto them.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And thou shalt speak unto the children of Israel, saying, If a man die, and have no son, then ye shall cause his inheritance to pass unto his daughter.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| And if he have no daughter, then ye shall give his inheritance unto his brethren.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And if he have no brethren, then ye shall give his inheritance unto his father's brethren.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And if his father have no brethren, then ye shall give his inheritance unto his kinsman that is next to him of his family, and he shall possess it: and it shall be unto the children of Israel a statute of judgment, as the LORD commanded Moses.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| And the LORD said unto Moses, Get thee up into this mount Abarim, and see the land which I have given unto the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| And when thou hast seen it, thou also shalt be gathered unto thy people, as Aaron thy brother was gathered.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| For ye rebelled against my commandment in the desert of Zin, in the strife of the congregation, to sanctify me at the water before their eyes: that is the water of Meribah in Kadesh in the wilderness of Zin.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| And Moses spake unto the LORD, saying,}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| Let the LORD, the God of the spirits of all flesh, set a man over the congregation,}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| Which may go out before them, and which may go in before them, and which may lead them out, and which may bring them in; that the congregation of the LORD be not as sheep which have no shepherd.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| And the LORD said unto Moses, Take thee Joshua the son of Nun, a man in whom is the spirit, and lay thine hand upon him;}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| And set him before Eleazar the priest, and before all the congregation; and give him a charge in their sight.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| And thou shalt put some of thine honour upon him, that all the congregation of the children of Israel may be obedient.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| And he shall stand before Eleazar the priest, who shall ask counsel for him after the judgment of Urim before the LORD: at his word shall they go out, and at his word they shall come in, both he, and all the children of Israel with him, even all the congregation.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| And Moses did as the LORD commanded him: and he took Joshua, and set him before Eleazar the priest, and before all the congregation:}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| And he laid his hands upon him, and gave him a charge, as the LORD commanded by the hand of Moses.}}

==Chapter 28==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#880088| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#880088| Command the children of Israel, and say unto them, My offering, and my bread for my sacrifices made by fire, for a sweet savour unto me, shall ye observe to offer unto me in their due season.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#880088| And thou shalt say unto them, This is the offering made by fire which ye shall offer unto the LORD; two lambs of the first year without spot day by day, for a continual burnt offering.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#880088| The one lamb shalt thou offer in the morning, and the other lamb shalt thou offer at even;}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#880088| And a tenth part of an ephah of flour for a meat offering, mingled with the fourth part of an hin of beaten oil.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#880088| It is a continual burnt offering, which was ordained in mount Sinai for a sweet savour, a sacrifice made by fire unto the LORD.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#880088| And the drink offering thereof shall be the fourth part of an hin for the one lamb: in the holy place shalt thou cause the strong wine to be poured unto the LORD for a drink offering.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#880088| And the other lamb shalt thou offer at even: as the meat offering of the morning, and as the drink offering thereof, thou shalt offer it, a sacrifice made by fire, of a sweet savour unto the LORD.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#880088| And on the sabbath day two lambs of the first year without spot, and two tenth deals of flour for a meat offering, mingled with oil, and the drink offering thereof:}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#880088| This is the burnt offering of every sabbath, beside the continual burnt offering, and his drink offering.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#880088| And in the beginnings of your months ye shall offer a burnt offering unto the LORD; two young bullocks, and one ram, seven lambs of the first year without spot;}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#880088| And three tenth deals of flour for a meat offering, mingled with oil, for one bullock; and two tenth deals of flour for a meat offering, mingled with oil, for one ram;}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#880088| And a several tenth deal of flour mingled with oil for a meat offering unto one lamb; for a burnt offering of a sweet savour, a sacrifice made by fire unto the LORD.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#880088| And their drink offerings shall be half an hin of wine unto a bullock, and the third part of an hin unto a ram, and a fourth part of an hin unto a lamb: this is the burnt offering of every month throughout the months of the year.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#880088| And one kid of the goats for a sin offering unto the LORD shall be offered, beside the continual burnt offering, and his drink offering.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#880088| And in the fourteenth day of the first month is the passover of the LORD.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#880088| And in the fifteenth day of this month is the feast: seven days shall unleavened bread be eaten.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#880088| In the first day shall be an holy convocation; ye shall do no manner of servile work therein:}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#880088| But ye shall offer a sacrifice made by fire for a burnt offering unto the LORD; two young bullocks, and one ram, and seven lambs of the first year: they shall be unto you without blemish:}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#880088| And their meat offering shall be of flour mingled with oil: three tenth deals shall ye offer for a bullock, and two tenth deals for a ram;}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#880088| A several tenth deal shalt thou offer for every lamb, throughout the seven lambs:}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#880088| And one goat for a sin offering, to make an atonement for you. }}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#880088| Ye shall offer these beside the burnt offering in the morning, which is for a continual burnt offering.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#880088| After this manner ye shall offer daily, throughout the seven days, the meat of the sacrifice made by fire, of a sweet savour unto the LORD: it shall be offered beside the continual burnt offering, and his drink offering.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#880088| And on the seventh day ye shall have an holy convocation; ye shall do no servile work.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#880088| Also in the day of the firstfruits, when ye bring a new meat offering unto the LORD, after your weeks be out, ye shall have an holy convocation; ye shall do no servile work:}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#880088| But ye shall offer the burnt offering for a sweet savour unto the LORD; two young bullocks, one ram, seven lambs of the first year;}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#880088| And their meat offering of flour mingled with oil, three tenth deals unto one bullock, two tenth deals unto one ram,}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#880088| A several tenth deal unto one lamb, throughout the seven lambs;}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#880088| And one kid of the goats, to make an atonement for you. }}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#880088| Ye shall offer them beside the continual burnt offering, and his meat offering, (they shall be unto you without blemish) and their drink offerings.}}

==Chapter 29==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#880088| And in the seventh month, on the first day of the month, ye shall have an holy convocation; ye shall do no servile work it is a day of blowing the trumpets unto you.}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#880088| And ye shall offer a burnt offering for a sweet savour unto the LORD; one young bullock, one ram, and seven lambs of the first year without blemish:}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#880088| And their meat offering shall be of flour mingled with oil, three tenth deals for a bullock, and two tenth deals for a ram,}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#880088| And one tenth deal for one lamb, throughout the seven lambs:}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#880088| And one kid of the goats for a sin offering, to make an atonement for you:}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#880088| Beside the burnt offering of the month, and his meat offering, and the daily burnt offering, and his meat offering, and their drink offerings, according unto their manner, for a sweet savour, a sacrifice made by fire unto the LORD.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#880088| And ye shall have on the tenth day of this seventh month an holy convocation; and ye shall afflict your souls: ye shall not do any work therein:}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#880088| But ye shall offer a burnt offering unto the LORD for a sweet savour; one young bullock, one ram, and seven lambs of the first year; they shall be unto you without blemish:}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#880088| And their meat offering shall be of flour mingled with oil, three tenth deals to a bullock, and two tenth deals to one ram,}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#880088| A several tenth deal for one lamb, throughout the seven lambs:}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#880088| One kid of the goats for a sin offering; beside the sin offering of atonement, and the continual burnt offering, and the meat offering of it, and their drink offerings.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#880088| And on the fifteenth day of the seventh month ye shall have an holy convocation; ye shall do no servile work, and ye shall keep a feast unto the LORD seven days:}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#880088| And ye shall offer a burnt offering, a sacrifice made by fire, of a sweet savour unto the LORD; thirteen young bullocks, two rams, and fourteen lambs of the first year; they shall be without blemish:}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#880088| And their meat offering shall be of flour mingled with oil, three tenth deals unto every bullock of the thirteen bullocks, two tenth deals to each ram of the two rams,}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#880088| And a several tenth deal to each lamb of the fourteen lambs:}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#880088| And one kid of the goats for a sin offering; beside the continual burnt offering, his meat offering, and his drink offering.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#880088| And on the second day ye shall offer twelve young bullocks, two rams, fourteen lambs of the first year without spot:}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#880088| And their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#880088| And one kid of the goats for a sin offering; beside the continual burnt offering, and the meat offering thereof, and their drink offerings.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#880088| And on the third day eleven bullocks, two rams, fourteen lambs of the first year without blemish;}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#880088| And their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#880088| And one goat for a sin offering; beside the continual burnt offering, and his meat offering, and his drink offering.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#880088| And on the fourth day ten bullocks, two rams, and fourteen lambs of the first year without blemish:}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#880088| Their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#880088| And one kid of the goats for a sin offering; beside the continual burnt offering, his meat offering, and his drink offering.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#880088| And on the fifth day nine bullocks, two rams, and fourteen lambs of the first year without spot:}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#880088| And their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#880088| And one goat for a sin offering; beside the continual burnt offering, and his meat offering, and his drink offering.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#880088| And on the sixth day eight bullocks, two rams, and fourteen lambs of the first year without blemish:}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#880088| And their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#880088| And one goat for a sin offering; beside the continual burnt offering, his meat offering, and his drink offering.}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#880088| And on the seventh day seven bullocks, two rams, and fourteen lambs of the first year without blemish:}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#880088| And their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#880088| And one goat for a sin offering; beside the continual burnt offering, his meat offering, and his drink offering.}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#880088| On the eighth day ye shall have a solemn assembly: ye shall do no servile work therein:}}

&lt;small&gt;{{font|color=#0000FF|36}}&lt;/small&gt;{{font|color=#880088| But ye shall offer a burnt offering, a sacrifice made by fire, of a sweet savour unto the LORD: one bullock, one ram, seven lambs of the first year without blemish:}}

&lt;small&gt;{{font|color=#0000FF|37}}&lt;/small&gt;{{font|color=#880088| Their meat offering and their drink offerings for the bullock, for the ram, and for the lambs, shall be according to their number, after the manner:}}

&lt;small&gt;{{font|color=#0000FF|38}}&lt;/small&gt;{{font|color=#880088| And one goat for a sin offering; beside the continual burnt offering, and his meat offering, and his drink offering.}}

&lt;small&gt;{{font|color=#0000FF|39}}&lt;/small&gt;{{font|color=#880088| These things ye shall do unto the LORD in your set feasts, beside your vows, and your freewill offerings, for your burnt offerings, and for your meat offerings, and for your drink offerings, and for your peace offerings.}}

&lt;small&gt;{{font|color=#0000FF|40}}&lt;/small&gt;{{font|color=#880088| And Moses told the children of Israel according to all that the LORD commanded Moses.}}

==Chapter 30==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And Moses spake unto the heads of the tribes concerning the children of Israel, saying, This is the thing which the LORD hath commanded.}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| If a man vow a vow unto the LORD, or swear an oath to bind his soul with a bond; he shall not break his word, he shall do according to all that proceedeth out of his mouth.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| If a woman also vow a vow unto the LORD, and bind herself by a bond, being in her father's house in her youth;}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And her father hear her vow, and her bond wherewith she hath bound her soul, and her father shall hold his peace at her; then all her vows shall stand, and every bond wherewith she hath bound her soul shall stand.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| But if her father disallow her in the day that he heareth; not any of her vows, or of her bonds wherewith she hath bound her soul, shall stand: and the LORD shall forgive her, because her father disallowed her.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| And if she had at all an husband, when she vowed, or uttered ought out of her lips, wherewith she bound her soul;}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| And her husband heard it, and held his peace at her in the day that he heard it: then her vows shall stand, and her bonds wherewith she bound her soul shall stand.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| But if her husband disallowed her on the day that he heard it; then he shall make her vow which she vowed, and that which she uttered with her lips, wherewith she bound her soul, of none effect: and the LORD shall forgive her.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| But every vow of a widow, and of her that is divorced, wherewith they have bound their souls, shall stand against her.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And if she vowed in her husband's house, or bound her soul by a bond with an oath;}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And her husband heard it, and held his peace at her, and disallowed her not: then all her vows shall stand, and every bond wherewith she bound her soul shall stand.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| But if her husband hath utterly made them void on the day he heard them; then whatsoever proceeded out of her lips concerning her vows, or concerning the bond of her soul, shall not stand: her husband hath made them void; and the LORD shall forgive her.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| Every vow, and every binding oath to afflict the soul, her husband may establish it, or her husband may make it void.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| But if her husband altogether hold his peace at her from day to day; then he establisheth all her vows, or all her bonds, which are upon her: he confirmeth them, because he held his peace at her in the day that he heard them.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| But if he shall any ways make them void after that he hath heard them; then he shall bear her iniquity.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| These are the statutes, which the LORD commanded Moses, between a man and his wife, between the father and his daughter, being yet in her youth in her father's house.}}

==Chapter 31==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| Avenge the children of Israel of the Midianites: afterward shalt thou be gathered unto thy people.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| And Moses spake unto the people, saying, Arm some of yourselves unto the war, and let them go against the Midianites, and avenge the LORD of Midian.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| Of every tribe a thousand, throughout all the tribes of Israel, shall ye send to the war.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| So there were delivered out of the thousands of Israel, a thousand of every tribe, twelve thousand armed for war.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| And Moses sent them to the war, a thousand of every tribe, them and Phinehas the son of Eleazar the priest, to the war, with the holy instruments, and the trumpets to blow in his hand.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| And they warred against the Midianites, as the LORD commanded Moses; and they slew all the males.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And they slew the kings of Midian, beside the rest of them that were slain; namely, Evi, and Rekem, and Zur, and Hur, and Reba, five kings of Midian: Balaam also the son of Beor they slew with the sword.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| And the children of Israel took all the women of Midian captives, and their little ones, and took the spoil of all their cattle, and all their flocks, and all their goods.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And they burnt all their cities wherein they dwelt, and all their goodly castles, with fire.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And they took all the spoil, and all the prey, both of men and of beasts.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| And they brought the captives, and the prey, and the spoil, unto Moses, and Eleazar the priest, and unto the congregation of the children of Israel, unto the camp at the plains of Moab, which are by Jordan near Jericho.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| And Moses, and Eleazar the priest, and all the princes of the congregation, went forth to meet them without the camp.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| And Moses was wroth with the officers of the host, with the captains over thousands, and captains over hundreds, which came from the battle.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| And Moses said unto them, Have ye saved all the women alive?}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| Behold, these caused the children of Israel, through the counsel of Balaam, to commit trespass against the LORD in the matter of Peor, and there was a plague among the congregation of the LORD.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| Now therefore kill every male among the little ones, and kill every woman that hath known man by lying with him.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| But all the women children, that have not known a man by lying with him, keep alive for yourselves.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| And do ye abide without the camp seven days: whosoever hath killed any person, and whosoever hath touched any slain, purify both yourselves and your captives on the third day, and on the seventh day.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| And purify all your raiment, and all that is made of skins, and all work of goats' hair, and all things made of wood.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| And Eleazar the priest said unto the men of war which went to the battle, This is the ordinance of the law which the LORD commanded Moses;}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| Only the gold, and the silver, the brass, the iron, the tin, and the lead,}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| Every thing that may abide the fire, ye shall make it go through the fire, and it shall be clean: nevertheless it shall be purified with the water of separation: and all that abideth not the fire ye shall make go through the water.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| And ye shall wash your clothes on the seventh day, and ye shall be clean, and afterward ye shall come into the camp.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| Take the sum of the prey that was taken, both of man and of beast, thou, and Eleazar the priest, and the chief fathers of the congregation:}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| And divide the prey into two parts; between them that took the war upon them, who went out to battle, and between all the congregation:}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| And levy a tribute unto the LORD of the men of war which went out to battle: one soul of five hundred, both of the persons, and of the beeves, and of the asses, and of the sheep:}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#888800| Take it of their half, and give it unto Eleazar the priest, for an heave offering of the LORD.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#888800| And of the children of Israel's half, thou shalt take one portion of fifty, of the persons, of the beeves, of the asses, and of the flocks, of all manner of beasts, and give them unto the Levites, which keep the charge of the tabernacle of the LORD.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#888800| And Moses and Eleazar the priest did as the LORD commanded Moses.}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#888800| And the booty, being the rest of the prey which the men of war had caught, was six hundred thousand and seventy thousand and five thousand sheep,}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#888800| And threescore and twelve thousand beeves,}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#888800| And threescore and one thousand asses,}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#888800| And thirty and two thousand persons in all, of women that had not known man by lying with him.}}

&lt;small&gt;{{font|color=#0000FF|36}}&lt;/small&gt;{{font|color=#888800| And the half, which was the portion of them that went out to war, was in number three hundred thousand and seven and thirty thousand and five hundred sheep:}}

&lt;small&gt;{{font|color=#0000FF|37}}&lt;/small&gt;{{font|color=#888800| And the LORD's tribute of the sheep was six hundred and threescore and fifteen.}}

&lt;small&gt;{{font|color=#0000FF|38}}&lt;/small&gt;{{font|color=#888800| And the beeves were thirty and six thousand; of which the LORD's tribute was threescore and twelve.}}

&lt;small&gt;{{font|color=#0000FF|39}}&lt;/small&gt;{{font|color=#888800| And the asses were thirty thousand and five hundred; of which the LORD's tribute was threescore and one.}}

&lt;small&gt;{{font|color=#0000FF|40}}&lt;/small&gt;{{font|color=#888800| And the persons were sixteen thousand; of which the LORD's tribute was thirty and two persons.}}

&lt;small&gt;{{font|color=#0000FF|41}}&lt;/small&gt;{{font|color=#888800| And Moses gave the tribute, which was the LORD's heave offering, unto Eleazar the priest, as the LORD commanded Moses.}}

&lt;small&gt;{{font|color=#0000FF|42}}&lt;/small&gt;{{font|color=#888800| And of the children of Israel's half, which Moses divided from the men that warred,}}

&lt;small&gt;{{font|color=#0000FF|43}}&lt;/small&gt;{{font|color=#888800| (Now the half that pertained unto the congregation was three hundred thousand and thirty thousand and seven thousand and five hundred sheep,}}

&lt;small&gt;{{font|color=#0000FF|44}}&lt;/small&gt;{{font|color=#888800| And thirty and six thousand beeves,}}

&lt;small&gt;{{font|color=#0000FF|45}}&lt;/small&gt;{{font|color=#888800| And thirty thousand asses and five hundred,}}

&lt;small&gt;{{font|color=#0000FF|46}}&lt;/small&gt;{{font|color=#888800| And sixteen thousand persons;)}}

&lt;small&gt;{{font|color=#0000FF|47}}&lt;/small&gt;{{font|color=#888800| Even of the children of Israel's half, Moses took one portion of fifty, both of man and of beast, and gave them unto the Levites, which kept the charge of the tabernacle of the LORD; as the LORD commanded Moses.}}

&lt;small&gt;{{font|color=#0000FF|48}}&lt;/small&gt;{{font|color=#888800| And the officers which were over thousands of the host, the captains of thousands, and captains of hundreds, came near unto Moses:}}

&lt;small&gt;{{font|color=#0000FF|49}}&lt;/small&gt;{{font|color=#888800| And they said unto Moses, Thy servants have taken the sum of the men of war which are under our charge, and there lacketh not one man of us.}}

&lt;small&gt;{{font|color=#0000FF|50}}&lt;/small&gt;{{font|color=#888800| We have therefore brought an oblation for the LORD, what every man hath gotten, of jewels of gold, chains, and bracelets, rings, earrings, and tablets, to make an atonement for our souls before the LORD.}}

&lt;small&gt;{{font|color=#0000FF|51}}&lt;/small&gt;{{font|color=#888800| And Moses and Eleazar the priest took the gold of them, even all wrought jewels.}}

&lt;small&gt;{{font|color=#0000FF|52}}&lt;/small&gt;{{font|color=#888800| And all the gold of the offering that they offered up to the LORD, of the captains of thousands, and of the captains of hundreds, was sixteen thousand seven hundred and fifty shekels.}}

&lt;small&gt;{{font|color=#0000FF|53}}&lt;/small&gt;{{font|color=#888800| (For the men of war had taken spoil, every man for himself.)}}

&lt;small&gt;{{font|color=#0000FF|54}}&lt;/small&gt;{{font|color=#888800| And Moses and Eleazar the priest took the gold of the captains of thousands and of hundreds, and brought it into the tabernacle of the congregation, for a memorial for the children of Israel before the LORD.}}

==Chapter 32==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#008888| Now the children of Reuben and the children of Gad had a very great multitude of cattle: }}{{font|color=#000088|and when they saw the land of Jazer, and the land of Gilead, that, behold, the place was a place for cattle;}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| The children of Gad and the children of Reuben came and spake unto Moses, and to Eleazar the priest, and unto the princes of the congregation, saying,}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| Ataroth, and Dibon, and Jazer, and Nimrah, and Heshbon, and Elealeh, and Shebam, and Nebo, and Beon,}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| Even the country which the LORD smote before the congregation of Israel, is a land for cattle, and thy servants have cattle:}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#000088| Wherefore, said they, if we have found grace in thy sight, let this land be given unto thy servants for a possession, and bring us not over Jordan.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#000088| And Moses said unto the children of Gad and to the children of Reuben, Shall your brethren go to war, and shall ye sit here?}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#000088| And wherefore discourage ye the heart of the children of Israel from going over into the land which the LORD hath given them?}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#000088| Thus did your fathers, when I sent them from Kadeshbarnea to see the land.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#000088| For when they went up unto the valley of Eshcol, and saw the land, they discouraged the heart of the children of Israel, that they should not go into the land which the LORD had given them.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#000088| And the LORD's anger was kindled the same time, and he sware, saying,}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#000088| Surely none of the men that came up out of Egypt, from twenty years old and upward, shall see the land which I sware unto Abraham, unto Isaac, and unto Jacob; because they have not wholly followed me:}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#000088| Save Caleb the son of Jephunneh the Kenezite, and Joshua the son of Nun: for they have wholly followed the LORD.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#000088| And the LORD's anger was kindled against Israel, and he made them wander in the wilderness forty years, until all the generation, that had done evil in the sight of the LORD, was consumed.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#000088| And, behold, ye are risen up in your fathers' stead, an increase of sinful men, to augment yet the fierce anger of the LORD toward Israel.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#000088| For if ye turn away from after him, he will yet again leave them in the wilderness; and ye shall destroy all this people.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#008888| And they came near unto him, and said, We will build sheepfolds here for our cattle, and cities for our little ones:}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#008888| But we ourselves will go ready armed before the children of Israel, until we have brought them unto their place: and our little ones shall dwell in the fenced cities because of the inhabitants of the land.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#008888| We will not return unto our houses, until the children of Israel have inherited every man his inheritance.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#008888| For we will not inherit with them on yonder side Jordan, or forward; because our inheritance is fallen to us on this side Jordan eastward.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#008888| And Moses said unto them,}}{{font|color=#000088| If ye will do this thing, if ye will go armed before the LORD to war,}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#000088| And will go all of you armed over Jordan before the LORD, until he hath driven out his enemies from before him,}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#000088| And the land be subdued before the LORD: then afterward ye shall return, and be guiltless before the LORD, and before Israel; and this land shall be your possession before the LORD.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#000088| But if ye will not do so, behold, ye have sinned against the LORD: and be sure your sin will find you out.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#008888| Build you cities for your little ones, and folds for your sheep; and do that which hath proceeded out of your mouth.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#000088| And the children of Gad and the children of Reuben spake unto Moses, saying, Thy servants will do as my lord commandeth.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#000088| Our little ones, our wives, our flocks, and all our cattle, shall be there in the cities of Gilead:}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#000088| But thy servants will pass over, every man armed for war, before the Lord to battle, as my lord saith.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| So concerning them Moses commanded Eleazar the priest, and Joshua the son of Nun, and the chief fathers of the tribes of the children of Israel:}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#888800| And Moses said unto them, If the children of Gad and the children of Reuben will pass with you over Jordan, every man armed to battle, before the LORD, and the land shall be subdued before you; then ye shall give them the land of Gilead for a possession:}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#888800| But if they will not pass over with you armed, they shall have possessions among you in the land of Canaan.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#888800| And the children of Gad and the children of Reuben answered, saying, As the LORD hath said unto thy servants, so will we do.}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#888800| We will pass over armed before the LORD into the land of Canaan, that the possession of our inheritance on this side Jordan may be our's.}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#888800| And Moses gave unto them, even to the children of Gad, and to the children of Reuben, and unto half the tribe of Manasseh the son of Joseph, the kingdom of Sihon king of the Amorites, and the kingdom of Og king of Bashan, the land, with the cities thereof in the coasts, even the cities of the country round about.}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#008888| And the children of Gad built Dibon, and Ataroth, and Aroer,}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#008888| And Atroth, Shophan, and Jaazer, and Jogbehah,}}

&lt;small&gt;{{font|color=#0000FF|36}}&lt;/small&gt;{{font|color=#008888| And Bethnimrah, and Bethharan, fenced cities: and folds for sheep.}}

&lt;small&gt;{{font|color=#0000FF|37}}&lt;/small&gt;{{font|color=#008888| And the children of Reuben built Heshbon, and Elealeh, and Kirjathaim,}}

&lt;small&gt;{{font|color=#0000FF|38}}&lt;/small&gt;{{font|color=#008888| And Nebo, and Baalmeon, (their names being changed,) and Shibmah: and gave other names unto the cities which they builded.}}

&lt;small&gt;{{font|color=#0000FF|39}}&lt;/small&gt;{{font|color=#000088| And the children of Machir the son of Manasseh went to Gilead, and took it, and dispossessed the Amorite which was in it.}}

&lt;small&gt;{{font|color=#0000FF|40}}&lt;/small&gt;{{font|color=#008888| And Moses gave Gilead unto Machir the son of Manasseh; and he dwelt therein.}}

&lt;small&gt;{{font|color=#0000FF|41}}&lt;/small&gt;{{font|color=#000088| And Jair the son of Manasseh went and took the small towns thereof, and called them Havothjair.}}

&lt;small&gt;{{font|color=#0000FF|42}}&lt;/small&gt;{{font|color=#000088| And Nobah went and took Kenath, and the villages thereof, and called it Nobah, after his own name.}}

==Chapter 33==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#FF8800| These are the journeys of the children of Israel, which went forth out of the land of Egypt with their armies under the hand of Moses and Aaron.}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#FF8800| And Moses wrote their goings out according to their journeys by the commandment of the LORD: and these are their journeys according to their goings out.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Rameses in the first month, on the fifteenth day of the first month; on the morrow after the passover the children of Israel went out with an high hand in the sight of all the Egyptians.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#FF8800| For the Egyptians buried all their firstborn, which the LORD had smitten among them: upon their gods also the LORD executed judgments.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#FF8800| And the children of Israel removed from Rameses, and pitched in Succoth.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Succoth, and pitched in Etham, which is in the edge of the wilderness.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#FF8800| And they removed from Etham, and turned again unto Pihahiroth, which is before Baalzephon: and they pitched before Migdol.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#FF8800| And they departed from before Pihahiroth, and passed through the midst of the sea into the wilderness, and went three days' journey in the wilderness of Etham, and pitched in Marah.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#FF8800| And they removed from Marah, and came unto Elim: and in Elim were twelve fountains of water, and threescore and ten palm trees; and they pitched there.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#FF8800| And they removed from Elim, and encamped by the Red sea.}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#FF8800| And they removed from the Red sea, and encamped in the wilderness of Sin.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#FF8800| And they took their journey out of the wilderness of Sin, and encamped in Dophkah.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Dophkah, and encamped in Alush. }}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#FF8800| And they removed from Alush, and encamped at Rephidim, where was no water for the people to drink.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Rephidim, and pitched in the wilderness of Sinai.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#FF8800| And they removed from the desert of Sinai, and pitched at Kibrothhattaavah.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Kibrothhattaavah, and encamped at Hazeroth.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Hazeroth, and pitched in Rithmah.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Rithmah, and pitched at Rimmonparez.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Rimmonparez, and pitched in Libnah.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#FF8800| And they removed from Libnah, and pitched at Rissah.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#FF8800| And they journeyed from Rissah, and pitched in Kehelathah.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#FF8800| And they went from Kehelathah, and pitched in mount Shapher.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#FF8800| And they removed from mount Shapher, and encamped in Haradah.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#FF8800| And they removed from Haradah, and pitched in Makheloth.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#FF8800| And they removed from Makheloth, and encamped at Tahath.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Tahath, and pitched at Tarah.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#FF8800| And they removed from Tarah, and pitched in Mithcah.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#FF8800| And they went from Mithcah, and pitched in Hashmonah.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Hashmonah, and encamped at Moseroth.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Moseroth, and pitched in Benejaakan.}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#FF8800| And they removed from Benejaakan, and encamped at Horhagidgad.}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#FF8800| And they went from Horhagidgad, and pitched in Jotbathah.}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#FF8800| And they removed from Jotbathah, and encamped at Ebronah.}}

&lt;small&gt;{{font|color=#0000FF|35}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Ebronah, and encamped at Eziongaber.}}

&lt;small&gt;{{font|color=#0000FF|36}}&lt;/small&gt;{{font|color=#FF8800| And they removed from Eziongaber, and pitched in the wilderness of Zin, which is Kadesh.}}

&lt;small&gt;{{font|color=#0000FF|37}}&lt;/small&gt;{{font|color=#FF8800| And they removed from Kadesh, and pitched in mount Hor, in the edge of the land of Edom.}}

&lt;small&gt;{{font|color=#0000FF|38}}&lt;/small&gt;{{font|color=#FF0000| And Aaron the priest went up into mount Hor at the commandment of the LORD, and died there, in the fortieth year after the children of Israel were come out of the land of Egypt, in the first day of the fifth month.}}

&lt;small&gt;{{font|color=#0000FF|39}}&lt;/small&gt;{{font|color=#FF0000| And Aaron was an hundred and twenty and three years old when he died in mount Hor.}}

&lt;small&gt;{{font|color=#0000FF|40}}&lt;/small&gt;{{font|color=#FF8800| And king Arad the Canaanite, which dwelt in the south in the land of Canaan, heard of the coming of the children of Israel.}}

&lt;small&gt;{{font|color=#0000FF|41}}&lt;/small&gt;{{font|color=#FF8800| And they departed from mount Hor, and pitched in Zalmonah.}}

&lt;small&gt;{{font|color=#0000FF|42}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Zalmonah, and pitched in Punon.}}

&lt;small&gt;{{font|color=#0000FF|43}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Punon, and pitched in Oboth.}}

&lt;small&gt;{{font|color=#0000FF|44}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Oboth, and pitched in Ijeabarim, in the border of Moab.}}

&lt;small&gt;{{font|color=#0000FF|45}}&lt;/small&gt;{{font|color=#FF8800| And they departed from Iim, and pitched in Dibongad. }}

&lt;small&gt;{{font|color=#0000FF|46}}&lt;/small&gt;{{font|color=#FF8800| And they removed from Dibongad, and encamped in Almondiblathaim.}}

&lt;small&gt;{{font|color=#0000FF|47}}&lt;/small&gt;{{font|color=#FF8800| And they removed from Almondiblathaim, and pitched in the mountains of Abarim, before Nebo.}}

&lt;small&gt;{{font|color=#0000FF|48}}&lt;/small&gt;{{font|color=#FF8800| And they departed from the mountains of Abarim, and pitched in the plains of Moab by Jordan near Jericho.}}

&lt;small&gt;{{font|color=#0000FF|49}}&lt;/small&gt;{{font|color=#FF8800| And they pitched by Jordan, from Bethjesimoth even unto Abelshittim in the plains of Moab.}}

&lt;small&gt;{{font|color=#0000FF|50}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses in the plains of Moab by Jordan near Jericho, saying,}}

&lt;small&gt;{{font|color=#0000FF|51}}&lt;/small&gt;{{font|color=#888800| Speak unto the children of Israel, and say unto them, When ye are passed over Jordan into the land of Canaan;}}

&lt;small&gt;{{font|color=#0000FF|52}}&lt;/small&gt;{{font|color=#888800| Then ye shall drive out all the inhabitants of the land from before you, and destroy all their pictures, and destroy all their molten images, and quite pluck down all their high places:}}

&lt;small&gt;{{font|color=#0000FF|53}}&lt;/small&gt;{{font|color=#888800| And ye shall dispossess the inhabitants of the land, and dwell therein: for I have given you the land to possess it.}}

&lt;small&gt;{{font|color=#0000FF|54}}&lt;/small&gt;{{font|color=#888800| And ye shall divide the land by lot for an inheritance among your families: and to the more ye shall give the more inheritance, and to the fewer ye shall give the less inheritance: every man's inheritance shall be in the place where his lot falleth; according to the tribes of your fathers ye shall inherit.}}

&lt;small&gt;{{font|color=#0000FF|55}}&lt;/small&gt;{{font|color=#888800| But if ye will not drive out the inhabitants of the land from before you; then it shall come to pass, that those which ye let remain of them shall be pricks in your eyes, and thorns in your sides, and shall vex you in the land wherein ye dwell.}}

&lt;small&gt;{{font|color=#0000FF|56}}&lt;/small&gt;{{font|color=#888800| Moreover it shall come to pass, that I shall do unto you, as I thought to do unto them.}}

==Chapter 34==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| Command the children of Israel, and say unto them, When ye come into the land of Canaan; (this is the land that shall fall unto you for an inheritance, even the land of Canaan with the coasts thereof:)}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| Then your south quarter shall be from the wilderness of Zin along by the coast of Edom, and your south border shall be the outmost coast of the salt sea eastward:}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And your border shall turn from the south to the ascent of Akrabbim, and pass on to Zin: and the going forth thereof shall be from the south to Kadeshbarnea, and shall go on to Hazaraddar, and pass on to Azmon:}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And the border shall fetch a compass from Azmon unto the river of Egypt, and the goings out of it shall be at the sea.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| And as for the western border, ye shall even have the great sea for a border: this shall be your west border. }}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| And this shall be your north border: from the great sea ye shall point out for you mount Hor:}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| From mount Hor ye shall point out your border unto the entrance of Hamath; and the goings forth of the border shall be to Zedad:}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| And the border shall go on to Ziphron, and the goings out of it shall be at Hazarenan: this shall be your north border.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| And ye shall point out your east border from Hazarenan to Shepham:}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| And the coast shall go down from Shepham to Riblah, on the east side of Ain; and the border shall descend, and shall reach unto the side of the sea of Chinnereth eastward:}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| And the border shall go down to Jordan, and the goings out of it shall be at the salt sea: this shall be your land with the coasts thereof round about.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| And Moses commanded the children of Israel, saying, This is the land which ye shall inherit by lot, which the LORD commanded to give unto the nine tribes, and to the half tribe:}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| For the tribe of the children of Reuben according to the house of their fathers, and the tribe of the children of Gad according to the house of their fathers, have received their inheritance; and half the tribe of Manasseh have received their inheritance:}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| The two tribes and the half tribe have received their inheritance on this side Jordan near Jericho eastward, toward the sunrising.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| These are the names of the men which shall divide the land unto you: Eleazar the priest, and Joshua the son of Nun.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| And ye shall take one prince of every tribe, to divide the land by inheritance.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| And the names of the men are these: Of the tribe of Judah, Caleb the son of Jephunneh.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| And of the tribe of the children of Simeon, Shemuel the son of Ammihud.}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| Of the tribe of Benjamin, Elidad the son of Chislon.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| And the prince of the tribe of the children of Dan, Bukki the son of Jogli.}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| The prince of the children of Joseph, for the tribe of the children of Manasseh, Hanniel the son of Ephod.}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| And the prince of the tribe of the children of Ephraim, Kemuel the son of Shiphtan.}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| And the prince of the tribe of the children of Zebulun, Elizaphan the son of Parnach.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| And the prince of the tribe of the children of Issachar, Paltiel the son of Azzan.}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| And the prince of the tribe of the children of Asher, Ahihud the son of Shelomi.}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| And the prince of the tribe of the children of Naphtali, Pedahel the son of Ammihud.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#888800| These are they whom the LORD commanded to divide the inheritance unto the children of Israel in the land of Canaan.}}

==Chapter 35==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses in the plains of Moab by Jordan near Jericho, saying,}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| Command the children of Israel, that they give unto the Levites of the inheritance of their possession cities to dwell in; and ye shall give also unto the Levites suburbs for the cities round about them.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| And the cities shall they have to dwell in; and the suburbs of them shall be for their cattle, and for their goods, and for all their beasts.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And the suburbs of the cities, which ye shall give unto the Levites, shall reach from the wall of the city and outward a thousand cubits round about.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And ye shall measure from without the city on the east side two thousand cubits, and on the south side two thousand cubits, and on the west side two thousand cubits, and on the north side two thousand cubits; and the city shall be in the midst: this shall be to them the suburbs of the cities.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| And among the cities which ye shall give unto the Levites there shall be six cities for refuge, which ye shall appoint for the manslayer, that he may flee thither: and to them ye shall add forty and two cities.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| So all the cities which ye shall give to the Levites shall be forty and eight cities: them shall ye give with their suburbs.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And the cities which ye shall give shall be of the possession of the children of Israel: from them that have many ye shall give many; but from them that have few ye shall give few: every one shall give of his cities unto the Levites according to his inheritance which he inheriteth.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| And the LORD spake unto Moses, saying,}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| Speak unto the children of Israel, and say unto them, When ye be come over Jordan into the land of Canaan;}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| Then ye shall appoint you cities to be cities of refuge for you; that the slayer may flee thither, which killeth any person at unawares.}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| And they shall be unto you cities for refuge from the avenger; that the manslayer die not, until he stand before the congregation in judgment.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| And of these cities which ye shall give six cities shall ye have for refuge.}}

&lt;small&gt;{{font|color=#0000FF|14}}&lt;/small&gt;{{font|color=#888800| Ye shall give three cities on this side Jordan, and three cities shall ye give in the land of Canaan, which shall be cities of refuge.}}

&lt;small&gt;{{font|color=#0000FF|15}}&lt;/small&gt;{{font|color=#888800| These six cities shall be a refuge, both for the children of Israel, and for the stranger, and for the sojourner among them: that every one that killeth any person unawares may flee thither.}}

&lt;small&gt;{{font|color=#0000FF|16}}&lt;/small&gt;{{font|color=#888800| And if he smite him with an instrument of iron, so that he die, he is a murderer: the murderer shall surely be put to death.}}

&lt;small&gt;{{font|color=#0000FF|17}}&lt;/small&gt;{{font|color=#888800| And if he smite him with throwing a stone, wherewith he may die, and he die, he is a murderer: the murderer shall surely be put to death.}}

&lt;small&gt;{{font|color=#0000FF|18}}&lt;/small&gt;{{font|color=#888800| Or if he smite him with an hand weapon of wood, wherewith he may die, and he die, he is a murderer: the murderer shall surely be put to death.}}

&lt;small&gt;{{font|color=#0000FF|19}}&lt;/small&gt;{{font|color=#888800| The revenger of blood himself shall slay the murderer: when he meeteth him, he shall slay him.}}

&lt;small&gt;{{font|color=#0000FF|20}}&lt;/small&gt;{{font|color=#888800| But if he thrust him of hatred, or hurl at him by laying of wait, that he die;}}

&lt;small&gt;{{font|color=#0000FF|21}}&lt;/small&gt;{{font|color=#888800| Or in enmity smite him with his hand, that he die: he that smote him shall surely be put to death; for he is a murderer: the revenger of blood shall slay the murderer, when he meeteth him.}}

&lt;small&gt;{{font|color=#0000FF|22}}&lt;/small&gt;{{font|color=#888800| But if he thrust him suddenly without enmity, or have cast upon him any thing without laying of wait,}}

&lt;small&gt;{{font|color=#0000FF|23}}&lt;/small&gt;{{font|color=#888800| Or with any stone, wherewith a man may die, seeing him not, and cast it upon him, that he die, and was not his enemy, neither sought his harm:}}

&lt;small&gt;{{font|color=#0000FF|24}}&lt;/small&gt;{{font|color=#888800| Then the congregation shall judge between the slayer and the revenger of blood according to these judgments:}}

&lt;small&gt;{{font|color=#0000FF|25}}&lt;/small&gt;{{font|color=#888800| And the congregation shall deliver the slayer out of the hand of the revenger of blood, and the congregation shall restore him to the city of his refuge, whither he was fled: and he shall abide in it unto the death of the high priest, which was anointed with the holy oil.}}

&lt;small&gt;{{font|color=#0000FF|26}}&lt;/small&gt;{{font|color=#888800| But if the slayer shall at any time come without the border of the city of his refuge, whither he was fled;}}

&lt;small&gt;{{font|color=#0000FF|27}}&lt;/small&gt;{{font|color=#888800| And the revenger of blood find him without the borders of the city of his refuge, and the revenger of blood kill the slayer; he shall not be guilty of blood:}}

&lt;small&gt;{{font|color=#0000FF|28}}&lt;/small&gt;{{font|color=#888800| Because he should have remained in the city of his refuge until the death of the high priest: but after the death of the high priest the slayer shall return into the land of his possession.}}

&lt;small&gt;{{font|color=#0000FF|29}}&lt;/small&gt;{{font|color=#888800| So these things shall be for a statute of judgment unto you throughout your generations in all your dwellings.}}

&lt;small&gt;{{font|color=#0000FF|30}}&lt;/small&gt;{{font|color=#888800| Whoso killeth any person, the murderer shall be put to death by the mouth of witnesses: but one witness shall not testify against any person to cause him to die.}}

&lt;small&gt;{{font|color=#0000FF|31}}&lt;/small&gt;{{font|color=#888800| Moreover ye shall take no satisfaction for the life of a murderer, which is guilty of death: but he shall be surely put to death.}}

&lt;small&gt;{{font|color=#0000FF|32}}&lt;/small&gt;{{font|color=#888800| And ye shall take no satisfaction for him that is fled to the city of his refuge, that he should come again to dwell in the land, until the death of the priest.}}

&lt;small&gt;{{font|color=#0000FF|33}}&lt;/small&gt;{{font|color=#888800| So ye shall not pollute the land wherein ye are: for blood it defileth the land: and the land cannot be cleansed of the blood that is shed therein, but by the blood of him that shed it.}}

&lt;small&gt;{{font|color=#0000FF|34}}&lt;/small&gt;{{font|color=#888800| Defile not therefore the land which ye shall inhabit, wherein I dwell: for I the LORD dwell among the children of Israel.}}

==Chapter 36==

&lt;small&gt;{{font|color=#0000FF|1}}&lt;/small&gt;{{font|color=#888800| And the chief fathers of the families of the children of Gilead, the son of Machir, the son of Manasseh, of the families of the sons of Joseph, came near, and spake before Moses, and before the princes, the chief fathers of the children of Israel:}}

&lt;small&gt;{{font|color=#0000FF|2}}&lt;/small&gt;{{font|color=#888800| And they said, The LORD commanded my lord to give the land for an inheritance by lot to the children of Israel: and my lord was commanded by the LORD to give the inheritance of Zelophehad our brother unto his daughters.}}

&lt;small&gt;{{font|color=#0000FF|3}}&lt;/small&gt;{{font|color=#888800| And if they be married to any of the sons of the other tribes of the children of Israel, then shall their inheritance be taken from the inheritance of our fathers, and shall be put to the inheritance of the tribe whereunto they are received: so shall it be taken from the lot of our inheritance.}}

&lt;small&gt;{{font|color=#0000FF|4}}&lt;/small&gt;{{font|color=#888800| And when the jubile of the children of Israel shall be, then shall their inheritance be put unto the inheritance of the tribe whereunto they are received: so shall their inheritance be taken away from the inheritance of the tribe of our fathers.}}

&lt;small&gt;{{font|color=#0000FF|5}}&lt;/small&gt;{{font|color=#888800| And Moses commanded the children of Israel according to the word of the LORD, saying, The tribe of the sons of Joseph hath said well.}}

&lt;small&gt;{{font|color=#0000FF|6}}&lt;/small&gt;{{font|color=#888800| This is the thing which the LORD doth command concerning the daughters of Zelophehad, saying, Let them marry to whom they think best; only to the family of the tribe of their father shall they marry.}}

&lt;small&gt;{{font|color=#0000FF|7}}&lt;/small&gt;{{font|color=#888800| So shall not the inheritance of the children of Israel remove from tribe to tribe: for every one of the children of Israel shall keep himself to the inheritance of the tribe of his fathers.}}

&lt;small&gt;{{font|color=#0000FF|8}}&lt;/small&gt;{{font|color=#888800| And every daughter, that possesseth an inheritance in any tribe of the children of Israel, shall be wife unto one of the family of the tribe of her father, that the children of Israel may enjoy every man the inheritance of his fathers.}}

&lt;small&gt;{{font|color=#0000FF|9}}&lt;/small&gt;{{font|color=#888800| Neither shall the inheritance remove from one tribe to another tribe; but every one of the tribes of the children of Israel shall keep himself to his own inheritance.}}

&lt;small&gt;{{font|color=#0000FF|10}}&lt;/small&gt;{{font|color=#888800| Even as the LORD commanded Moses, so did the daughters of Zelophehad:}}

&lt;small&gt;{{font|color=#0000FF|11}}&lt;/small&gt;{{font|color=#888800| For Mahlah, Tirzah, and Hoglah, and Milcah, and Noah, the daughters of Zelophehad, were married unto their father's brothers' sons:}}

&lt;small&gt;{{font|color=#0000FF|12}}&lt;/small&gt;{{font|color=#888800| And they were married into the families of the sons of Manasseh the son of Joseph, and their inheritance remained in the tribe of the family of their father.}}

&lt;small&gt;{{font|color=#0000FF|13}}&lt;/small&gt;{{font|color=#888800| These are the commandments and the judgments, which the LORD commanded by the hand of Moses unto the children of Israel in the plains of Moab by Jordan near Jericho.}}

[[Category:Documentary hypothesis]]</text>
      <sha1>aj3hiu6247l6o0gi38nwhuyqt6xriag</sha1>
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  </page>
  <page>
    <title>Talk:Educational Media Awareness Campaign</title>
    <ns>1</ns>
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      <id>2411863</id>
      <parentid>2038291</parentid>
      <timestamp>2022-08-04T16:38:38Z</timestamp>
      <contributor>
        <ip>111.119.190.67</ip>
      </contributor>
      <comment>/* Chemistry */ new section</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3242" xml:space="preserve">==Social sciences==
I realise this is targetted at primary/secondary, but what do you think about a section for [[Portal:Social Sciences|Social Science]] educational pictures? e.g., I'm thinking of images such as [[:Image:Maslow's hierarchy of needs.png]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]]&lt;/small&gt; 07:15, 4 May 2008 (UTC)
: Also, I recommend [[:Image:Structural-Iceberg.svg]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:44, 4 October 2008 (UTC)
::I'm curious:  why? {{unsigned|Jade Knight|16:03, 4 October 2008}}
::: Mainly because it illustrates some commonly-taught concepts in introductory psychology and it's clearly drawn in svg format. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:11, 4 October 2008 (UTC)
::::I see.  You are a Psychology professor, if I recall, so if you think it's a good depiction of basic Psychology, I'll take your word for it.  [[user:Jade Knight|The Jade Knight]] &lt;sup&gt;[[User talk:Jade Knight|(d'viser)]]&lt;/sup&gt; 06:50, 4 October 2008 (UTC)

== Who? ==

Who takes care of all this?  I mean, it's a great idea, but where did it come from?  Who rotates the pictures?  How are they chosen?  [[User:Jade Knight|The Jade Knight]] 06:01, 3 September 2008 (UTC)

: It's all automated; [[User:McCormack]] pretty much created it, I gather. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:23, 3 September 2008 (UTC)

== &quot;20th century&quot; ==
Why do you say &quot;works of art created in the 20th century&quot; -- what's the actual date of creation after which this becomes relevant?  Isn't this also &quot;created since 1923&quot;? [[User:Sj|Sj]] 01:08, 3 November 2009 (UTC)

==Actually being factual about use, especially fair use==
I left this comment on the [[The Great Wave off Kanagawa]] page that was featured as an &quot;educational picture of they day.&quot;  My concern is [[wikipedia:creep | instruction creep]] of the worst sort: fascism.--[[User:John Bessa|John]][[Image:bessa66.png|12px]][[User:John Bessa|Bessa]]&lt;sup&gt;[[User talk:John Bessa|talk]]&lt;/sup&gt; 13:54, 12 April 2010 (UTC)

:The blurb says &quot;''For works of art created in the 20th century, normally you cannot reuse any copies.''&quot;  This statement wholly ignores fair use, as is hence a lie.  

:One can use a copy of a work to discuss the work; I believe that this statement is not so much misconception but part of an orchestrated attempt across the wm to strengthen property, as in intellectual property but also as in real estate, in opposition to the ideas of the public domain that go to social sharing, which is how humanity evolved.

:The wm commons refuses to support fair use, and not for any legal reasons; this fact supports my above statement.

== Chemistry: Absorption/spectroscopy ==

This may be of interest to the rotation boxes.  Does wikuniversity have annotations enabled?
[[File:Spectroscopy_overview.svg|thumb|upright=2]]
[[User:Jkwchui|Jkwchui]] 23:30, 16 January 2011 (UTC)

== Chemistry ==

What is chemistry [[Special:Contributions/111.119.190.67|111.119.190.67]] ([[User talk:111.119.190.67|discuss]]) 16:38, 4 August 2022 (UTC)</text>
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  <page>
    <title>Wikiversity:Portal reform</title>
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      <comment>/* Reform proposals */</comment>
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      <text bytes="19710" xml:space="preserve">== Introduction ==

If you try to list [[Wikiversity:Portal|Wikiversity portals]] like [http://en.wikiversity.org/w/index.php?title=Special%3AAllPages&amp;from=&amp;namespace=102 this], it isn't very helpful. The following is a list of current portals on Wikiversity, together with comments and reform proposals. (Date: 12th May 2008).

== Reform proposals ==

{| class=&quot;wikitable sortable&quot;
 ! Name !! Notes !! Reform proposal
 |-
 | [[Portal:Advanced General Studies]]
 | 
 | Implement old proposal to merge with [[School:Advanced General Studies]] (leave a redirect here)
 |-
 | [[Portal:Applied sciences]]
 | Needs finishing.
 | 
 |-
 | [[Portal:Art and design]]
 | Needs portal-izing.
 | 
 |-
 | [[Portal:Arts]]
 | Needs finishing.
 | 
 |-
 | [[Portal:Asteroid Surveys]]
 | Redirects to mainspace article.
 | Delete the redirect.
 |-
 | [[Portal:Biology]]
 | 
 | 
 |-
 | [[Portal:Browse]]
 | Redirects to [[Wikiversity:Major portals]].
 | Redirect to [[Wikiversity:Browse]].
 |-
 | [[Portal:Chemistry]]
 | 
 | 
 |-
 | [[Portal:Christianity]]
 | Mainly a list of links; not portal-ized.
 | Move to Topic or main namespace, leaving a redirect.
 |-
 | [[Portal:Communities]]
 | Redirects to [[Portal:Learning Projects]].
 | 
 |-
 | [[Portal:Community]]
 | Almost no content; an idea which never took off.
 | Redirect to [[Portal:Learning Projects]].
 |-
 | [[Portal:Computer Science]]
 | 
 | 
 |-
 | [[Portal:Construction]]
 | 
 | Implement existing proposal to merge with [[School:Construction]]; replace with a redirect when done.
 |-
 | [[Portal:Cultural studies]]
 | Underdeveloped; needs portal-izing.
 | Perhaps move to another namespace?
 |-
 | [[Portal:Education]]
 | 
 | 
 |-
 | [[Portal:Engineering and Technology]]
 | 
 | 
 |-
 | [[Portal:English]]
 | 
 | 
 |-
 | [[Portal:FLOSS4Science]]
 | Includes a very large number of subpages with real content.
 | The entire project needs moving to the main namespace.
 |-
 | [[Portal:Fine Arts]]
 | Not portal-ized, but well-developed.
 | Merge with [[Portal:Arts]] leaving redirect.
 |-
 | [[Portal:First Year Medical School Program]]
 | Very new. Left message with sole author offering to move to main namespace.
 | 
 |-
 | [[Portal:French]]
 | Redirects to [[Topic:French]].
 | 
 |-
 | [[Portal:Game design]]
 | Underdeveloped.
 | Merge with [[School:Game Design]] and leave redirect here.
 |-
 | [[Portal:Geography]]
 | 
 | 
 |-
 | [[Portal:German]]
 | 
 | Merge with [[Topic:German]] and leave redirect here.
 |-
 | [[Portal:History]]
 | 
 | 
 |-
 | [[Portal:Humanities]]
 | Well developed.
 | Could do with a facelift.
 |-
 | [[Portal:Information sciences]]
 | Needs finishing.
 | 
 |-
 | [[Portal:Interdisciplinary Studies]]
 | 
 | 
 |-
 | [[Portal:Languages]]
 | 
 | 
 |-
 | [[Portal:Law]]
 | Very underdeveloped. Needs portal-izing.
 |
 |-
 | [[Portal:Learning Materials]]
 | 
 | 
 |-
 | [[Portal:Learning projects]]
 | 
 | 
 |-
 | [[Portal:Life Sciences]]
 | 
 | 
 |-
 | [[Portal:Linguistics]]
 | Already tagged for cleanup.
 | Merge with [[School:Linguistics]] and leave redirect here.
 |-
 | [[Portal:Mathematics]]
 | 
 | The problem here is that this subject is massively important at pre-tertiary level, but the portal is very tertiary-orientated. Suggestion: move current portal to [[Portal:Mathematical sciences]] and then re-create [[Portal:Mathematics]] as a pre-tertiary portal. Coordinate between the two.
 |-
 | [[Portal:Media]]
 | 
 | 
 |-
 | [[Portal:Medical School Program]]
 | Freshly created. Could be a speedy delete, but need to ask [[User:Pnstarr]] first.
 | 
 |-
 | [[Portal:Music]]
 | Originally this had a lot of real content, but most of it was moved to the main namespace. This is still quite a mess.
 | Move the remaining real content to the main namespace (watch especially for subpages). Then redesign the portal.
 |-
 | [[Portal:Non-formal Education]]
 | 
 | 
 |-
 | [[Portal:Participants]]
 | 
 | 
 |-
 | [[Portal:Physical Sciences]]
 | 
 | 
 |-
 | [[Portal:Physics]]
 | (redirects to Physics and Astronomy)
 | 
 |-
 | [[Portal:Physics and Astronomy]]
 | 
 | 
 |-
 | [[Portal:Plant Sciences]]
 | OK, but a redesign wouldn't hurt.
 | 
 |-
 | [[Portal:Portals]]
 | A guide to all portals. Not portal-ized. Could be redesigned and improved. 
 | Merge with [[Wikiversity:Portals]]?
 |-
 | [[Portal:Practical Arts and Sciences]]
 | 
 | This is a poorly named miscellany of subject areas. It is suggested that the individual subject areas should be moved out to different and more conventional portals (e.g. vocational studies; applied sciences), and that the portal should then be deleted.
 |-
 | [[Portal:Pre-school Education]]
 | 
 | 
 |-
 | [[Portal:Primary Education]]
 | 
 | 
 |-
 | [[Portal:Professional and vocational studies]]
 | Needs finishing.
 | 
 |-
 | [[Portal:Professions]]
 | 
 | Merge with [[Portal:Professional and vocational studies]]. Some of the areas covered are either already in more suitably named portals, or would deserve their own portal (e.g. [[Portal:medicine]]).
 |-
 | [[Portal:Psychology]]
 | Needs finishing.
 | see also: [[Talk:Psychology#Trying_to_understand_the_psychology_pages.27_hierarchy|here]]
 |-
 | [[Portal:Public Engagement]]
 | Only content is a link to [[Portal:Public Engagement and Civic Innovation]].
 | Proposed for deletion.
 |-
 | [[Portal:Public Engagement and Civic Innovation]]
 | Skeleton only.
 | Proposed for deletion.
 |-
 | [[Portal:Reading clubs]]
 | Redirects to [[Portal:Reading groups]]
 | 
 |-
 | [[Portal:Reading groups]]
 | 
 | 
 |-
 | [[Portal:Research]]
 | 
 | 
 |-
 | [[Portal:School of History]]
 | Redirects to [[School:history]].
 | 
 |-
 | [[Portal:Science]]
 | 
 | &quot;Science&quot; is far too wide ranging for this to be helpful. Replace with a &quot;disambiguation portal&quot; which quickly and clearly guides users to more useful and slight narrower portals such as Bioloy, Physics, Chemistry, Applied Sciences, Life Sciences, Information Sciences, etc.
 |-
 | [[Portal:Secondary Education]]
 | 
 | 
 |-
 | &lt;s&gt;[[Portal:Slavistics]]&lt;/s&gt;
 | Was empty. Already moved to [[School:Slavic studies]].
 | 
 |-
 | [[Portal:Social Entrepreneurship Network]]
 | Redirects to [[Portal:Social entrepreneurship]].
 | Delete the redirect.
 |-
 | [[Portal:Social Research]]
 | Underdeveloped and forgotten. Was created together with [[Portal:Social Theory]].
 | Move to main namespace.
 |-
 | [[Portal:Social Sciences]]
 | 
 | 
 |-
 | [[Portal:Social Theory]]
 | Underdeveloped and forgotten. Was created together with [[Portal:Social Research]].
 |  Move to main namespace and merge with [[Portal:Social Research]].
 |-
 | [[Portal:Social entrepreneurship]]
 | Loads of real content here.
 | Move the entire project to the main namespace.
 |-
 | [[Portal:Sociology]]
 | Redirects to [[School:Sociology]].
 | 
 |-
 | [[Portal:Tertiary Education]]
 | 
 | 
 |-
 | [[Portal:Test]]
 | 
 | Delete. We don't need this any more.
 |-
 | &lt;s&gt;[[Portal:TestPortal]]&lt;/s&gt;
 | A one-time experiment, now deleted.
 | 
 |-
 | [[Portal:Test preparation]]
 | Underdeveloped.
 | Move to main namespace.
 |-
 | [[Portal:Theology and philosophy]]
 | Needs separation. Two different schools.
 | 
 |-
 | [[Portal:Thought]]
 | Redirects to [[Topic:thought]].
 | 
 |-
 | [[Portal:Wiki]]
 | 
 | Merge with [[Portal:Wikimedia Studies]] and leave redirect here.
 |-
 | [[Portal:Wikimedia Studies]]
 | Unfinished.
 | 
 |-
 | [[Portal:Wikiversity]]
 | 
 | Is its purpose justified? Or is this what the main page should be doing? The lack of activity here suggests it hasn't really worked. Redirect to the main page. 
 |-
 | [[Portal:Writing Center]]
 | One of the most active portals.
 | Move to the Topic namespace. Clear up the double redirects. Note: loads of redirects are built up around the various English language pages - lots of tidying needed here.
 |-
 |}

== Voting and comment ==

* Merge [[Portal:Plant Sciences]] with [[School:Plant sciences]] --[[User:Gbaor|Gbaor]] 12:26, 12 May 2008 (UTC)
* Move [[Portal:Life Sciences]] to ''Portal:Life '''s'''ciences'' - naming conventions. The existing redirect should be deleted first. --[[User:Gbaor|Gbaor]] 12:26, 12 May 2008 (UTC)
* Pages linking to [[Portal:Sociology]] and [[Portal:Thought]] have been tidied up, thus these redirects could be deleted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]]&lt;/small&gt; 05:24, 13 May 2008 (UTC)
* Note that [[Portal:Social entrepreneurship]] has a rather subterranean subpage structure which follows... -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]]&lt;/small&gt; 14:24, 13 May 2008 (UTC)
** Yes - I'm aware. But I reckon it will have to be moved - its main namespace stuff. --[[User:McCormack|McCormack]] 17:03, 13 May 2008 (UTC)
* merge [[Portal:Advanced General Studies]] into [[School:Advanced General Studies]]: support, ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] &lt;small&gt;uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])&lt;/small&gt; 16:40, 13 May 2008 (UTC)
* Just noting that on the talk page, [[User:Ktucker|Kim Tucker]], who is mainly (solely?) responsible for [[Portal:FLOSS4Science]] and [[Portal:Social Entrepreneurship]], has expressed his opposition to a move of these portals to the main namespace on the grounds that they contain primarily organisational material rather than &quot;real content&quot;. I will therefore suggest to him that the Topic namespace might also be a good alternative for his purposes. --[[User:McCormack|McCormack]] 17:10, 13 May 2008 (UTC)
** Agree re moving to Topic namespace; content appears to largely consist of links to external resources, with several subpages. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]]&lt;/small&gt; 14:04, 15 May 2008 (UTC)
*** Mind you, if the links are mainly external, it ought really to be main namespace stuff. Organisational namespaces like Portal and Topic are for organising ''wikiversity'' material. --[[User:McCormack|McCormack]] 15:35, 15 May 2008 (UTC)
** Not convinced: [[User:Ktucker|Ktucker]] 12:21, 16 May 2008 (UTC) &lt;small&gt;''[Note: reasons given just below.]''&lt;/small&gt;
*** I thought the purpose of the &quot;Portal&quot; namespace was to identify/cluster areas of wikiversity which serve as &quot;portals&quot;. The definition of &quot;portal&quot; has been contentious. But to me ''in this context'', a portal is a site which 
**** helps people find out what educational initiatives related to a topic are out there (including those on already on Wikiversity), 
**** facilitates/catalyses collaboration among parties to co-create learning resources - preferably on Wikiversity - the challenge for the broader community is towards interoperability (see for example [http://cosl.usu.edu/events/oerinterop2007 OER Interoperability meeting] and [http://cosl.usu.edu/events/oerinteropsprint2008 OER Interoperability Sprint])
**** serves as a venue for showcasing learning resources, educators, student projects, etc.
**** The actual learning resources developed should be in the general default name space. [[User:Ktucker|Kim]] 10:24, 16 May 2008 (UTC)
*** Re: 'Topic:' name space: 
**** [[Portal:Social entrepreneurship|Social Entrepreneurship]] is an example of a growing number of multidisciplinary fields which do not fit naturally in traditional hierarchies (e.g. Social Science? Business Science? Humanities?). In the [[Portal:Social entrepreneurship|Social entrepreneurship portal]] we have a &quot;Topics&quot; box which is really just links to some of the top level [[Portal:Social entrepreneurship/Guidelines for Editors/List of Categories|categories we have created]] - a custom hierarchy for the [[Portal:Social entrepreneurship|Social Entrepreneurship portal]]. The pages found when browsing categories includes pages in various name spaces (as required :-). The categories do not correspond to the Wikiversity concept of &quot;[[Wikiversity:Topics|Topics]]&quot; (traditional university hierarchies - &quot;divisions&quot;, &quot;departments&quot;, &quot;centers&quot;, &quot;programs&quot;). 
**** [[Portal:FLOSS4Science|FLOSS4Science]] is intended as a portal for any user of Wikiversity from any field of science to be able to find FLOSS to use, find people to share experiences and co-create learning resources for their field which use the software, and network with (or among) developers of FLOSS for science. So, the portal itself does not fit in the standard &quot;topic&quot; hierarchies, though the learning resources which appear on Wikiversity and linked to from this portal probably will (though they should be in the default main name space and structured in Wikiversity via Categories). [[User:Ktucker|Kim]] 10:24, 16 May 2008 (UTC)
*** So, I am still not convinced that these portals should be moved to the Topics name space. [[User:Ktucker|Kim]] 10:24, 16 May 2008 (UTC) 
* [[Portal:Portals]] could be merged somehow with: [[Wikiversity:Portal]], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] &lt;small&gt;uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])&lt;/small&gt; 17:50, 13 May 2008 (UTC)
*[[Portal:Science]]. I do not understand the idea that &quot;'Science' is far too wide ranging for this to be helpful.&quot; I believe that, &quot;[[Wikiversity:Portal|Portals can be created to serve the needs of specific topic areas, whether large or small]]&quot;. --[[User:JWSchmidt|JWSchmidt]] 01:29, 16 May 2008 (UTC)
* [[Wikiversity:Major portals]] - another portal related page in the ''Wikiversity'' namespace. It seems to be useful, but I think it a little bit out of date. Shouldn't be also &quot;reformed&quot;? --[[User:Gbaor|Gbaor]] 12:47, 16 May 2008 (UTC)
::Wikipedia has a short list of links to major portals at the upper right corner of the main page. In addition to links to portals for different educational levels (primary, secondary, etc) I think Wikiversity should return to the practice of including a group of links from our main page to our most important major subject portals. --[[User:JWSchmidt|JWSchmidt]] 16:02, 16 May 2008 (UTC)
{{Template:Main Page navigation}}
*** To the unsigned editor (above): the main page ''does'' include links to the main subject portals in a tree form; at the current time not all of these links are in the portal namespace (some are schools and some are categories or topics) because things need overhauling. Once the portal reform is implemented, the explorer tree can be updated to link more coherently to the major portals. Don't forget that the main page needs to have a certain aesthetic coherence - one should perhaps avoid throwing in a non-matching template. There is already an alternative template to the one you gave which coordinates aesthetically. --[[User:McCormack|McCormack]] 15:06, 16 May 2008 (UTC)
::::This particular template &quot;Main Page navigation&quot; is just one that was available as an example. The key point is that many people who come to Wikiversity are familiar with navigating Wikipedia. Wikiversity can help itself by not abandoning the idea of providing new visitors with navigation aids that are familiar to them. --[[User:JWSchmidt|JWSchmidt]] 16:02, 16 May 2008 (UTC) 
* [[School:Law]] is really portal-type content and should be merged into the undeveloped [[Portal:Law]] page. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]]&lt;/small&gt; 11:43, 17 May 2008 (UTC)

----


===questions===
====merging and version history====
*What should we do when merging pages and both have talk pages with their version history  ? ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] &lt;small&gt;uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])&lt;/small&gt; 16:40, 13 May 2008 (UTC)
** Edit manually. ;-) --[[User:McCormack|McCormack]] 17:03, 13 May 2008 (UTC)
***Well, both version histories can be kept with moving one talk page (e.g. talk page2) as subpage of the talk page1 (e.g. talk page1/Archive) and then make a link on talk page1 to it. But I was asking because there is a [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=merge&amp;user=&amp;page=&amp;year=&amp;month=-1 merge log] available - I wonder if there is also a tool available which allows merge of pages ? I will digg later into this, ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] &lt;small&gt;uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])&lt;/small&gt; 17:22, 13 May 2008 (UTC)
**** Many of these pages don't have much of a history. On the other hand, you raise a good point in the context of some mergers. For example, to keep the history, [[Portal:Fine Arts]] should probably be moved to [[Portal:Arts]] before the integration is started. --[[User:McCormack|McCormack]] 17:28, 13 May 2008 (UTC)

* This is an important point for us in the work we are doing. We are supporting content development by incentivising and rewarding people (students and educators) who edit pages. If we lose the history of the page, this approach and the research around uptake and usage among learners and educators will become really difficult. [[User:Ktucker|Kim]] 10:28, 16 May 2008 (UTC)

====I like boilerplates====
At some of the portals (e.g. [[Portal:History]]) I have seen that it is possible to create pages with boilerplates. I like this idea (e.g. because it helps newcomers easily to understand the syntax). About how to improve the [[Portal:Biology/New page boilerplate|boilerplate]] we can talk later.

Unfortunately this is used only at about 6 portals :-( Which is the best place to note this and more important to use this in the portal reforms ? I thought about here [[Portal talk:Portals]], but well seems there is not so much action :-) ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] &lt;small&gt;uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])&lt;/small&gt; 17:39, 13 May 2008 (UTC)
:probably somewhere at [[Wikiversity:Portal]], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] &lt;small&gt;uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])&lt;/small&gt; 17:50, 13 May 2008 (UTC)
:: Yes - I had been intending to expand the use of these boilerplate page creation facilities. It's all a matter of time availability! One suggestion: most of these portals already have a related [[Help:project boxes|project box]], and the boilerplate could include a project box. --[[User:McCormack|McCormack]] 04:32, 14 May 2008 (UTC)

====Odd portal names====
* On hitting [[Special:Random/Portal|random portal]], I got some weird ones: [[Portal:Learning_Materialsnew]] and [[Portal:Researchnew]]. What are these? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]]&lt;/small&gt; 14:07, 15 May 2008 (UTC)
** Things I deliberately left out because they were once work in progress. They can be deleted - &lt;s&gt;just ''be careful''&lt;/s&gt; (well, perhaps not - just zap 'em)! --[[User:McCormack|McCormack]] 15:29, 15 May 2008 (UTC)
:: Or at least made into subpages, and perhaps changed to &quot;/Concept&quot; - which I think would make it uniform with the mainpage. But... --[[User:Remi|Remi]] 01:38, 16 May 2008 (UTC)
*** The biggest problem with all these &quot;concept&quot; pages in the portal namespace is that they confuse the listings for anyone trying to analyse the namespace - there are perhaps 100 or so such pages. And currently they contain nothing other than copies of the portals they are supposed to belong to. It would be helpful in future if portal &quot;concept&quot; pages (or draft revisions) were created for the short time they were needed and then deleted afterwards. It's just basic housekeeping. --[[User:McCormack|McCormack]] 04:01, 16 May 2008 (UTC)
*** Perhaps encourage users to do this sort of thing in a &quot;Sandbox&quot; &lt;nowiki&gt;[[User:JBloggs/MySandBox]]&lt;/nowiki&gt; [[User:Ktucker|Kim]] 10:31, 16 May 2008 (UTC)

==See also==
* [[Wikiversity:Portal]]


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The Kingdom of Cambodia is a country in South East Asia with a population of over 3 million people. Cambodia is the successor state of the once-powerful Hindu and Buddhist Khmer Empire, which ruled most of the Indochinese Peninsula between the eleventh and fourteenth centuries. Today, Cambodia is one of the greatest developing countries. Yet, the country is still having some issues such as its educational system. 

== Land ==
[[Image:Cb-map.png|left]]
Cambodia has an area of 181,035 square kilometers, and it borders [[Thailand]] to its west and northwest, Laos to its northeast, and [[Vietnam]] to its east and southeast. Cambodia gained independence from [[France]] on November 9, 1953. Cambodia's main industries are garments, tourism, and construction. As for its language, more than 90% of its population is of Khmer origin, therefore, speaks the Khmer language, the country's official language. The remainder includes Chinese, Vietnamese, and Indians. As for the religion, most Cambodians are Buddhists of Khmer extraction, but the country also has a substantial number of Muslim Cham, as well as small animist tribes.　In addition, according to the report of The World Bank, “Cambodia's economy is showing resilience in spite of the challenging international economic environment. Economic growth in 2001 was estimated at 6.3 percent, driven by an expanding tourism sector and robust garment exports. In 2001, there was nearly zero inflation. In 2002 inflation continued to be low (World Bank)”. That is, Cambodia’s economic growth has been moderately strong over the past decade. However, this growth has been concentrated in urban areas (The World Bank). Therefore the rural-urban income gap is widening these days. And around 40 percent of the population still remains under the national poverty line. 

== History ==
[[Image:TaPhrom_Cambodia.jpg|right|Ta Phrom by Berglind R]]
# 1863 - Cambodia becomes a protectorate of France. French colonial rule lasts for 90 years. 
# 1941 - Prince Norodom Sihanouk becomes king. Cambodia is occupied by Japan during World War II. 
# 1945 - The Japanese occupation ends. 
# 1946 - France re-imposes its protectorate. Communist guerrillas begin an armed campaign against the French. 
# 1953 - Cambodia wins its independence from France. Under King Sihanouk, it becomes the Kingdom of Cambodia. 
# Early 1970s - Cambodian army faces two enemies: the North Vietnamese and communist Khmer Rouge. Gradually, the army loses territory. 
# 1975 - Lon Nol is overthrown as the Khmer Rouge led by Pol Pot occupied Phnom Penh. Sihanouk briefly becomes head of state, the country is re-named Kampuchea. Basic freedoms are curtailed and religion is banned. Hundreds of thousands of the educated middle-classes are tortured and executed in special centres. Others starve, or die from disease or exhaustion. 
# 1976 - The country is re-named Democratic Kampuchea. Sihanouk resigns, Khieu Samphan becomes head of state, Pol Pot is prime minister. 
# 1977 - Fighting breaks out with Vietnam. 
# 1979 January - The Vietnamese take Phnom Penh. Pol Pot and Khmer Rouge forces flee to the border region with Thailand. The People's Republic of Kampuchea is established. Many elements of life before the Khmer Rouge take-over started to be re-established. 
# 1981 - The pro-Vietnamese Kampuchean People's Revolutionary Party wins the elections to the National Assembly. 
# 1989 - Vietnamese troops withdraw. The country is re-named the State of Cambodia. Buddhism is re-established as the state religion. 
# 1991 - A peace agreement is signed in Paris. A UN transitional authority shares power temporarily with representatives of the various factions in Cambodia. Sihanouk becomes head of state. 
# 1993 - General election sees the royalist Funcinpec party win the most seats followed by Hun Sen's Cambodian People's Party (CPP). A three-party coalition is formed with Funcinpec's Prince Norodom Ranariddh as prime minister and Hun Sen as deputy prime minister. The monarchy is restored, Sihanouk becomes king again. The country is re-named the Kingdom of Cambodia.
# 1994 - Thousands of Khmer Rouge guerrillas surrender in government amnesty. 
# 1997 - Hun Sen stages a coup against the prime ministar, Prince Ranariddh, and replaces him with Ung Huot. The coup attracts international condemnation and Cambodia's membership of the Association of Southeast Asian Nations (ASEAN) is delayed. The Khmer Rouge put Pol Pot on trial and sentence him to life imprisonment.
# 1998 - Pol Pot dies in his jungle hideout. Elections in July are won by Hun Sen's CPP, amid allegations of harassment. A coalition is formed between the CPP and Funcinpec. Hun Sen becomes prime minister, Ranariddh is president of the National Assembly. 
# 2001 - Senate approves a law to create a tribunal to bring genocide charges against Khmer Rouge leaders. 
# 2001 June - International donors, encouraged by Cambodia's reform efforts, pledge $560 million in aid at a donor conference in Tokyo. 
# 2001 June - Five Cambodians, three of them US citizens, are sentenced to life for involvement in armed attack by US-based Cambodian Freedom Fighters (CFF) on government buildings in Phnom Penh in 2000. CFF says it will continue campaign to overthrow Prime Minister Hun Sen. 
# 2001 December - First bridge across the Mekong River opens, linking the east and west of the country. The 1.36km bridge cost $56 million. 
# 2002 February - First multi-party local elections; ruling Cambodian People's Party wins in all but 23 out of 1,620 communes. 
# 2004 August - Parliament ratifies kingdom's entry into World Trade Organization (WTO). 
# 2004 October - King Sihanouk abdicates and is succeeded by his son Norodom Sihamoni. 
# 2005 April - Tribunal to try surviving Khmer Rouge leaders gets green light from UN after years of debate about funding. 
# 2005 October - Prime minister signs a controversial border agreement with Vietnam. Legal action is taken against some critics of the deal, prompting international concern. 
# 2006 July - Ta Mok, one of the top leaders of the brutal Khmer Rouge regime, dies aged 80. 
# 2007 July - UN-backed court tribunals begin questioning suspects about allegations of genocide by the Khmer Rouge. 
# 2007 September - Most senior surviving member of Khmer Rouge, Nuon Chea - &quot;Brother Number Two&quot; - is arrested and charged with crimes against humanity. 
# 2008 February - Cambodian court sentences 20 member of small Cambodian Freedom Fighters group to prison for attack on prime minister's office in November 2000. 
# 2008 March - Genocide tribunal refuses bail plea of Nuon Chea. 
# 2008 April - US court convicts Cambodian-born Cambodian Freedom Fighters leader Chhun Yasith of masterminding 2000 attack. 
# 2008 July - PM Hun Sen's ruling Cambodian People's Party (CPP) claims victory in parliamentary elections. EU monitors say the vote fell short of international standards. 
# 2008 July - Cambodia and Thailand move troops to disputed land near ancient Preah Vihear temple after decision to list it as UN World Heritage Site fans nationalist emotions on both sides. Officials from both states start talks to resolve standoff.
# (BBC News 2008)

This part would focus on the Khmer Rouge, because it has affected the educational system in Cambodia. On 17 April 1975, after five years of civil war, Cambodia’s capital Phnom Penh fell to the forces of the Communist Party of Kampuchea (CPK), popularly known as the Khmer Rouge (Linton). Pol Pot was the leader of the Khmer Rouge. “The country was cut off from the outside world, its cities emptied, its economy militarized, and its Buddhist and folk cultures destroyed. The leadership’s ‘vision’ of a better Cambodia led to extraordinary horrors on Cambodian people, who had to endure the forced movement of their urban populations, starvation, unlawful killings, forced labor, torture, and cruel, inhumane and degrading treatment and persecution on a massive scale”(Linton). Mao-think that Pol Pot studied in China, the ethnocentric policy, and anti-Vietnam policy became the theoretical support which created the regime of terror by the Khmer Rouge (Miyamoto). Pol Pot sanctioned the use of massive violence because he believed it was the only “practical” way to achieve and protect his utopian vision for a better society in Cambodia (Valentino). By the hands of the Khmer Rouge, mainly educated people (such as teachers and respected local leaders and so on) had been killed, since they were likely to be rebel groups. Furthermore, the Khmer Rouge had denied the value of education at all points. Therefore, it can be said that Cambodia had experienced a systematic destruction of education by the Khmer Rouge, until Heng Samrin robbed the Khmer Rouge of its power in 1979. According to Linton, an estimated 20% of the 1975’s population of 7.7million people lost their lives. After the Khmer Rouge was over, the United Nations peacekeeping force, the United Nations Transitional Authority in Cambodia (UNTAC), began monitoring Cambodia in 1992. “The goal of UNTAC was disarmament of the troops, and national elections for a constituent assembly which took place in 1993. (…) Contrary to the fears of many, the elections were peaceful. Over 90 percent of the registered voters went to the polls in Cambodian freest, fairest, and most secret election since the colonial era. (…) For the first time in Cambodian history, a majority of Cambodians had voted against an armed government such as the Khmer Rouge” (Chandler). Therefore it can be said: Cambodia was able to gradually recover from many years of violence by the Khmer Rouge regime, in the wake of those elections, although not all problems were solved.

=== Educational development strategy and challenge in Cambodia ===
Although all educational systems of Cambodia were abolished under the regime of terror by the Khmer Rouge, according to the World Bank, there have been significant improvements in the past five years in the education sector in Cambodia, particularly in terms of primary net enrollment gains (it’s because the government of Cambodia has put more efforts on the primary education, according to Komai), the introduction of program based-budgeting and the development of a sound pro-poor policy framework. But in Cambodia, several challenges remain. For example, most Cambodian children attend some schooling, but a large share complete only a few grades-with 85 percent of 15 to 19 year olds completing just grade 1, while only 27 percent complete grade 7 (The World Bank). It is partly because parents (especially those in rural areas) are blind to the importance of education. There are also disparities in education participation rates by different regions. Moreover, inefficiency and poor quality in education service delivery at primary, secondary and tertiary levels are other major challenges. For instance, the number and the quality of teachers in Cambodia are insufficient. It is because almost all of experienced old teachers had been killed under the regime of the Khmer Rouge. In fact, one thirds of teachers in the primary school have not finished even lower secondary education (Komai). And the budget for education is extremely restricted, compared to neighboring nations (Komai). Thus, the salary of teachers is too low, which causes teachers’ corrupt practices. And the time for studying in the school is 15 hours per week, which is extremely insufficient, too (Komai). Another challenge is related to child prostitution. “In Cambodia, child prostitution has become a worsening problem since 1990’s. Many Cambodian children have been deceived by words that there are “good” sources of employment in urban areas and have left their home, and have been forced to work in the house of prostitution. In Cambodia, there are more than 8.000～15.000 children who had been damaged by child prostitution” (Social Ecoo). Furthermore, expanding of child prostitution causes spread of HIV/AIDS among Cambodian children. HIV/AIDS is one of educational challenge, since children who are infected with the HIV virus cannot get an education well. 

=== The external actor involved in Cambodian educational development ===
Cambodia has been aided by a number of more developed nations like France, West-Germany, Canada, Australia, the United States, and Japan. The United States has contributed up to $5 million through international relief agencies to children within Cambodia (Sutter). As for Japan, many Japanese organizations such as JICA have supported Cambodian educational development. For example JICA has tried to revive Cambodian educational system, by forming projects including “the project for improvement of science and mathematics education (2000-2004)”, which is aimed to enable teachers engaged in science and mathematics education to acquire skills more than cheap tricks. In addition, The World Bank, UNICEF, UNESCO, Asian Development Bank (ADB), and some NGOs (such as World Vision in the United States, Save the Children in the UK, and so on) have also supported educational development in Cambodia. Among interesting projects formed by those organizations is the World Bank’s “Education Quality Improvement Project (1999 - 2004)”. This project aimed to develop and put into practice a participatory approach to school quality improvement and performance-based resource management. The goal was to have participating schools eventually show improvements in student enrollment, attendance and achievement (The World Bank). The result was that Students in more than 1,000 participating schools in Cambodia achieved higher test scores in literacy, had lower drop-out and higher promotion rates. In addition, this project could show that health and vocational training was the important factor in reducing student drop-out, and in improving promotion rates.

=== Works Cited ===
BBC News. 2008. “Timeline: Cambodia”
＜http://news.bbc.co.uk/1/hi/world/asia-pacific/country_profiles/1244006.stm＞

“Cambodian Information Center”. 2008.
＜http://www.cambodia.org/＞

Chandler, David. 2008. “A History of Cambodia”. Westview Press: Colorado, America. 

“Japan International Cooperation Agency”. 1995.
＜http://www.jica.go.jp/Index-j.html＞

Kennsuke, Miyamoto. 1993. “Tounan Asia no Gennzai”. 〔The Present of Southeast Asia〕. Horupu Publication: Tokyo, Japan. 

Komai, Hiroshi. 2001. “Shinnsei Cambodia”. 〔New Birth of Cambodia〕. Akashi Publication: Tokyo, Japan.

Linton, Suzannah. 2004. “Reconciliation in Cambodia”. Documentation Center of Cambodia: Phnom Penh, Cambodia. 

“Social Ecoo”. 2007.
＜http://www.socialecoo.jp/series/archives/2007/07/03/entry135.html＞

Sutter, Robert G. 1991. “The Cambodian Crisis and U.S. Policy Dilemmas”. Westview Press: Colorado, America.

The World Bank Group. 2008. “The World Bank”.
＜http://www.worldbank.org/＞

Valentino, Benjamin A. 2004. “Final Solutions: Mass Killing and Genocide in the Twentieth Century”. Cornell University Press: New York, America.

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The Kingdom of Cambodia is a country in South East Asia with a population of over 3 million people. Cambodia is the successor state of the once-powerful Hindu and Buddhist Khmer Empire, which ruled most of the Indochinese Peninsula between the eleventh and fourteenth centuries. Today, Cambodia is one of the greatest developing countries. Yet, the country is still having some issues such as its educational system. 

== Land ==
[[Image:Cb-map.png|left]]
Cambodia has an area of 181,035 square kilometers, and it borders [[Thailand]] to its west and northwest, Laos to its northeast, and [[Vietnam]] to its east and southeast. Cambodia gained independence from [[France]] on November 9, 1953. Cambodia's main industries are garments, tourism, and construction. As for its language, more than 90% of its population is of Khmer origin, therefore, speaks the Khmer language, the country's official language. The remainder includes Chinese, Vietnamese, and Indians. As for the religion, most Cambodians are Buddhists of Khmer extraction, but the country also has a substantial number of Muslim Cham, as well as small animist tribes.　In addition, according to the report of The World Bank, “Cambodia's economy is showing resilience in spite of the challenging international economic environment. Economic growth in 2001 was estimated at 6.3 percent, driven by an expanding tourism sector and robust garment exports. In 2001, there was nearly zero inflation. In 2002 inflation continued to be low (World Bank)”. That is, Cambodia’s economic growth has been moderately strong over the past decade. However, this growth has been concentrated in urban areas (The World Bank). Therefore the rural-urban income gap is widening these days. And around 40 percent of the population still remains under the national poverty line. 

== History ==
[[Image:TaPhrom_Cambodia.jpg|right|Ta Phrom by Berglind R]]
# 1863 - Cambodia becomes a protectorate of France. French colonial rule lasts for 90 years. 
# 1941 - Prince Norodom Sihanouk becomes king. Cambodia is occupied by Japan during World War II. 
# 1945 - The Japanese occupation ends. 
# 1946 - France re-imposes its protectorate. Communist guerrillas begin an armed campaign against the French. 
# 1953 - Cambodia wins its independence from France. Under King Sihanouk, it becomes the Kingdom of Cambodia. 
# Early 1970s - Cambodian army faces two enemies: the North Vietnamese and communist Khmer Rouge. Gradually, the army loses territory. 
# 1975 - Lon Nol is overthrown as the Khmer Rouge led by Pol Pot occupied Phnom Penh. Sihanouk briefly becomes head of state, the country is re-named Kampuchea. Basic freedoms are curtailed and religion is banned. Hundreds of thousands of the educated middle-classes are tortured and executed in special centres. Others starve, or die from disease or exhaustion. 
# 1976 - The country is re-named Democratic Kampuchea. Sihanouk resigns, Khieu Samphan becomes head of state, Pol Pot is prime minister. 
# 1977 - Fighting breaks out with Vietnam. 
# 1979 January - The Vietnamese take Phnom Penh. Pol Pot and Khmer Rouge forces flee to the border region with Thailand. The People's Republic of Kampuchea is established. Many elements of life before the Khmer Rouge take-over started to be re-established. 
# 1981 - The pro-Vietnamese Kampuchean People's Revolutionary Party wins the elections to the National Assembly. 
# 1989 - Vietnamese troops withdraw. The country is re-named the State of Cambodia. Buddhism is re-established as the state religion. 
# 1991 - A peace agreement is signed in Paris. A UN transitional authority shares power temporarily with representatives of the various factions in Cambodia. Sihanouk becomes head of state. 
# 1993 - General election sees the royalist Funcinpec party win the most seats followed by Hun Sen's Cambodian People's Party (CPP). A three-party coalition is formed with Funcinpec's Prince Norodom Ranariddh as prime minister and Hun Sen as deputy prime minister. The monarchy is restored, Sihanouk becomes king again. The country is re-named the Kingdom of Cambodia.
# 1994 - Thousands of Khmer Rouge guerrillas surrender in government amnesty. 
# 1997 - Hun Sen stages a coup against the prime ministar, Prince Ranariddh, and replaces him with Ung Huot. The coup attracts international condemnation and Cambodia's membership of the Association of Southeast Asian Nations (ASEAN) is delayed. The Khmer Rouge put Pol Pot on trial and sentence him to life imprisonment.
# 1998 - Pol Pot dies in his jungle hideout. Elections in July are won by Hun Sen's CPP, amid allegations of harassment. A coalition is formed between the CPP and Funcinpec. Hun Sen becomes prime minister, Ranariddh is president of the National Assembly. 
# 2001 - Senate approves a law to create a tribunal to bring genocide charges against Khmer Rouge leaders. 
# 2001 June - International donors, encouraged by Cambodia's reform efforts, pledge $560 million in aid at a donor conference in Tokyo. 
# 2001 June - Five Cambodians, three of them US citizens, are sentenced to life for involvement in armed attack by US-based Cambodian Freedom Fighters (CFF) on government buildings in Phnom Penh in 2000. CFF says it will continue campaign to overthrow Prime Minister Hun Sen. 
# 2001 December - First bridge across the Mekong River opens, linking the east and west of the country. The 1.36km bridge cost $56 million. 
# 2002 February - First multi-party local elections; ruling Cambodian People's Party wins in all but 23 out of 1,620 communes. 
# 2004 August - Parliament ratifies kingdom's entry into World Trade Organization (WTO). 
# 2004 October - King Sihanouk abdicates and is succeeded by his son Norodom Sihamoni. 
# 2005 April - Tribunal to try surviving Khmer Rouge leaders gets green light from UN after years of debate about funding. 
# 2005 October - Prime minister signs a controversial border agreement with Vietnam. Legal action is taken against some critics of the deal, prompting international concern. 
# 2006 July - Ta Mok, one of the top leaders of the brutal Khmer Rouge regime, dies aged 80. 
# 2007 July - UN-backed court tribunals begin questioning suspects about allegations of genocide by the Khmer Rouge. 
# 2007 September - Most senior surviving member of Khmer Rouge, Nuon Chea - &quot;Brother Number Two&quot; - is arrested and charged with crimes against humanity. 
# 2008 February - Cambodian court sentences 20 member of small Cambodian Freedom Fighters group to prison for attack on prime minister's office in November 2000. 
# 2008 March - Genocide tribunal refuses bail plea of Nuon Chea. 
# 2008 April - US court convicts Cambodian-born Cambodian Freedom Fighters leader Chhun Yasith of masterminding 2000 attack. 
# 2008 July - PM Hun Sen's ruling Cambodian People's Party (CPP) claims victory in parliamentary elections. EU monitors say the vote fell short of international standards. 
# 2008 July - Cambodia and Thailand move troops to disputed land near ancient Preah Vihear temple after decision to list it as UN World Heritage Site fans nationalist emotions on both sides. Officials from both states start talks to resolve standoff.
# (BBC News 2008)

This part would focus on the Khmer Rouge, because it has affected the educational system in Cambodia. On 17 April 1975, after five years of civil war, Cambodia’s capital Phnom Penh fell to the forces of the Communist Party of Kampuchea (CPK), popularly known as the Khmer Rouge (Linton). Pol Pot was the leader of the Khmer Rouge. “The country was cut off from the outside world, its cities emptied, its economy militarized, and its Buddhist and folk cultures destroyed. The leadership’s ‘vision’ of a better Cambodia led to extraordinary horrors on Cambodian people, who had to endure the forced movement of their urban populations, starvation, unlawful killings, forced labor, torture, and cruel, inhumane and degrading treatment and persecution on a massive scale”(Linton). Mao-think that Pol Pot studied in China, the ethnocentric policy, and anti-Vietnam policy became the theoretical support which created the regime of terror by the Khmer Rouge (Miyamoto). Pol Pot sanctioned the use of massive violence because he believed it was the only “practical” way to achieve and protect his utopian vision for a better society in Cambodia (Valentino). By the hands of the Khmer Rouge, mainly educated people (such as teachers and respected local leaders and so on) had been killed, since they were likely to be rebel groups. Furthermore, the Khmer Rouge had denied the value of education at all points. Therefore, it can be said that Cambodia had experienced a systematic destruction of education by the Khmer Rouge, until Heng Samrin robbed the Khmer Rouge of its power in 1979. According to Linton, an estimated 20% of the 1975’s population of 7.7million people lost their lives. After the Khmer Rouge was over, the United Nations peacekeeping force, the United Nations Transitional Authority in Cambodia (UNTAC), began monitoring Cambodia in 1992. “The goal of UNTAC was disarmament of the troops, and national elections for a constituent assembly which took place in 1993. (…) Contrary to the fears of many, the elections were peaceful. Over 90 percent of the registered voters went to the polls in Cambodian freest, fairest, and most secret election since the colonial era. (…) For the first time in Cambodian history, a majority of Cambodians had voted against an armed government such as the Khmer Rouge” (Chandler). Therefore it can be said: Cambodia was able to gradually recover from many years of violence by the Khmer Rouge regime, in the wake of those elections, although not all problems were solved.

=== Educational development strategy and challenge in Cambodia ===
Although all educational systems of Cambodia were abolished under the regime of terror by the Khmer Rouge, according to the World Bank, there have been significant improvements in the past five years in the education sector in Cambodia, particularly in terms of primary net enrollment gains (it’s because the government of Cambodia has put more efforts on the primary education, according to Komai), the introduction of program based-budgeting and the development of a sound pro-poor policy framework. But in Cambodia, several challenges remain. For example, most Cambodian children attend some schooling, but a large share complete only a few grades-with 85 percent of 15 to 19 year olds completing just grade 1, while only 27 percent complete grade 7 (The World Bank). It is partly because parents (especially those in rural areas) are blind to the importance of education. There are also disparities in education participation rates by different regions. Moreover, inefficiency and poor quality in education service delivery at primary, secondary and tertiary levels are other major challenges. For instance, the number and the quality of teachers in Cambodia are insufficient. It is because almost all of experienced old teachers had been killed under the regime of the Khmer Rouge. In fact, one thirds of teachers in the primary school have not finished even lower secondary education (Komai). And the budget for education is extremely restricted, compared to neighboring nations (Komai). Thus, the salary of teachers is too low, which causes teachers’ corrupt practices. And the time for studying in the school is 15 hours per week, which is extremely insufficient, too (Komai). Another challenge is related to child prostitution. “In Cambodia, child prostitution has become a worsening problem since 1990’s. Many Cambodian children have been deceived by words that there are “good” sources of employment in urban areas and have left their home, and have been forced to work in the house of prostitution. In Cambodia, there are more than 8.000～15.000 children who had been damaged by child prostitution” (Social Ecoo). Furthermore, expanding of child prostitution causes spread of HIV/AIDS among Cambodian children. HIV/AIDS is one of educational challenge, since children who are infected with the HIV virus cannot get an education well. 

=== The external actor involved in Cambodian educational development ===
Cambodia has been aided by a number of more developed nations like France, West-Germany, Canada, Australia, the United States, and Japan. The United States has contributed up to $5 million through international relief agencies to children within Cambodia (Sutter). As for Japan, many Japanese organizations such as JICA have supported Cambodian educational development. For example JICA has tried to revive Cambodian educational system, by forming projects including “the project for improvement of science and mathematics education (2000-2004)”, which is aimed to enable teachers engaged in science and mathematics education to acquire skills more than cheap tricks. In addition, The World Bank, UNICEF, UNESCO, Asian Development Bank (ADB), and some NGOs (such as World Vision in the United States, Save the Children in the UK, and so on) have also supported educational development in Cambodia. Among interesting projects formed by those organizations is the World Bank’s “Education Quality Improvement Project (1999 - 2004)”. This project aimed to develop and put into practice a participatory approach to school quality improvement and performance-based resource management. The goal was to have participating schools eventually show improvements in student enrollment, attendance and achievement (The World Bank). The result was that Students in more than 1,000 participating schools in Cambodia achieved higher test scores in literacy, had lower drop-out and higher promotion rates. In addition, this project could show that health and vocational training was the important factor in reducing student drop-out, and in improving promotion rates.

=== Works Cited ===
BBC News. 2008. “Timeline: Cambodia”
＜http://news.bbc.co.uk/1/hi/world/asia-pacific/country_profiles/1244006.stm＞

“Cambodian Information Center”. 2008.
＜http://www.cambodia.org/＞

Chandler, David. 2008. “A History of Cambodia”. Westview Press: Colorado, America. 

“Japan International Cooperation Agency”. 1995.
＜http://www.jica.go.jp/Index-j.html＞

Kennsuke, Miyamoto. 1993. “Tounan Asia no Gennzai”. 〔The Present of Southeast Asia〕. Horupu Publication: Tokyo, Japan. 

Komai, Hiroshi. 2001. “Shinnsei Cambodia”. 〔New Birth of Cambodia〕. Akashi Publication: Tokyo, Japan.

Linton, Suzannah. 2004. “Reconciliation in Cambodia”. Documentation Center of Cambodia: Phnom Penh, Cambodia. 

“Social Ecoo”. 2007.
＜http://www.socialecoo.jp/series/archives/2007/07/03/entry135.html＞

Sutter, Robert G. 1991. “The Cambodian Crisis and U.S. Policy Dilemmas”. Westview Press: Colorado, America.

The World Bank Group. 2008. “The World Bank”.
＜http://www.worldbank.org/＞

Valentino, Benjamin A. 2004. “Final Solutions: Mass Killing and Genocide in the Twentieth Century”. Cornell University Press: New York, America.

[[Category:Asia]]</text>
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      <comment>/* Adder */</comment>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:adder.20131113.pdf  |pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20211108.pdf  |pdf]] || 
|| [[Media:adder.rca.20140313.pdf  |pdf]] 
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf |pdf]] 
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.1.A.CLA.20211106.pdf  |pdf]] || 
|| [[Media:adder.cla.20140313.pdf  |pdf]] ||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf  |pdf]] ||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf  |pdf]] ||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20211111.pdf  |pdf]] ||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf  |pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf  |pdf]] ||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf |pdf]],  [[Media:adder.cca.20140313.pdf  |pdf]] 
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf |pdf]] 
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf  |pdf]] ||
|| [[Media:adder.ksa.20140409.pdf  |pdf]] || 
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf  |pdf]] ||
|| || 
|-
|| '''9. Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA.20220803.pdf  |pdf]] ||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf |pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf |pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf |pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf |pdf]])
* Carry-Skip Adder
&lt;/br&gt;

== Barrel Shifter ==
* Barrel Shifter Architecture Exploration ([[Media:bshift.20131105.pdf  |bshfit.vhdl]], [[Media:bshift.makefile.20131109.pdf  |bshfit.makefile]])
&lt;/br&gt;

'''Mux Based Barrel Shifter'''
* Analysis  ([[Media:Arith.BShfiter.20151207.pdf  |pdf]]) 
* Implementation 
&lt;/br&gt;

== Multiplier == 
=== Array Multipliers ===
* Analysis  ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf  |pdf]])
&lt;/br&gt;

=== Tree Mulltipliers ===
* Lattice Multiplication  ([[Media:VLSI.Arith.LatticeMult.20170204.pdf  |pdf]])
* Wallace Tree  ([[Media:VLSI.Arith.WallaceTree.20170204.pdf  |pdf]])
* Dadda Tree  ([[Media:VLSI.Arith.DaddaTree.20170701.pdf  |pdf]])
&lt;/br&gt;

=== Booth Multipliers ===
* [[Media:RNS4.BoothEncode.20161005.pdf  |Booth Encoding Note]]
* Booth Multiplier Note ([[Media:BoothMult.20160929.pdf |H1.pdf]])
&lt;/br&gt;

== Divider == 
* Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf  |pdf]])&lt;/br&gt;

&lt;/br&gt;
&lt;/br&gt; 
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      <text bytes="5113" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Applications ([[Media:Laurent.6.Application.6A.20220803.pdf|A.pdf]], [[Media:Laurent.5.Application.6B.20220723.pdf|B.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

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    <title>The necessities in Random Processes</title>
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      <contributor>
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        <id>21186</id>
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      <comment>/* The Temporal Characteristics of Random Processes */</comment>
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      <text bytes="5805" xml:space="preserve">==''' Random Variables '''==

=== Single Random Variables ===
* Random Variables ([[Media:RV1.RVariable.1.A.20200427.pdf |A.pdf]])
* Distribution Function ([[Media:RV1.Distribution.2.A.201200428.pdf |A.pdf]])
* Density Function ([[Media:RV1.Density.3.A.20200429.pdf |A.pdf]])
* Functions of Random Variables ([[Media:RV1.RVFunction.4.A.20220317.pdf |A.pdf]])
* Gaussian Random Variables ([[Media:RV1.4.Gaussian.20200430.pdf |A.pdf]], [[Media:RV1.4B.Gaussian.20180314.pdf |B.pdf]])
* Other Distribution and Density Functions ([[Media:RV1.5.Other.20200501.pdf |A.pdf]])
* Conditional Distribution and Density Functions ([[Media:RV1.6.Conditional.20200506.pdf |A.pdf]])
&lt;/br&gt;

=== The Characteristics of a Single Random Variable ===
* Expected Value ([[Media:RV2.Expectation.1.A.20200506.pdf |A.pdf]])
* Moments ([[Media:RV2.Moment.2.B.20200507.pdf |A.pdf]], [[Media:RV2.Moment.2.B.20180320.pdf |B.pdf]])
* Moment Generating Functions ([[Media:RV2.MFunctions.3.A.20200508.pdf |A.pdf]])
* Transformations of Random Variables ([[Media:RV2.Transform.4.A.20200514.pdf |A.pdf]])
&lt;/br&gt;

=== Multiple Random Variables ===
* Vector Random Variables ([[Media:3MRV.1A.VectorRV.20200515.pdf |A.pdf]])
* Joint Distribution ([[Media:3MRV.2A.JointDist.20200518.pdf |A.pdf]])
* Joint Density ([[Media:3MRV.3A.JointDensity.20200521.pdf |A.pdf]])
* Conditional Joint Distribution and Density ([[Media:3MRV.4A.CondDistrib.20200527.pdf |A.pdf]])
* Statistical Independence ([[Media:3MRV.5A.StatIndep.20200423.pdf |A.pdf]])
* Sums of Random Variables ([[Media:3MRV.6A.RVSum.20200528.pdf |A.pdf]])
* Central Limit Theorem ([[Media:3MRV.7A.CLimit.20200612.pdf |A.pdf]])
&lt;/br&gt;

=== The Characteristics of Multiple Random Variables ===
* Expected Values ([[Media:4MRV.1A.Expect.20200617.pdf |A.pdf]])
* Joint Characteristic Functions ([[Media:4MRV.2A.JChar.20200618.pdf |A.pdf]])
* Jointly Gaussian Random Variables ([[Media:4MRV.3A.JGauss.20200619.pdf |A.pdf]])
* Transformations of Multiple Random Variables ([[Media:4MRV.4A.Transform.20200620.pdf |A.pdf]])
* Linear Transformation of Gaussian Random Variables ([[Media:4MRV.5A.LinearTrans.20200623.pdf |A.pdf]])
* Simulating Multiple Random Variables ([[Media:4MRV.6A.Simulation.20200624.pdf |A.pdf]])
* Sampling and Some Limit Theorem ([[Media:4MRV.7A.LimitTheorem.20200625.pdf |A.pdf]])
* Complex Random Variables ([[Media:4MRV.8A.ComplexRV.20200626.pdf |A.pdf]])
&lt;/br&gt;

==''' Random Processes '''==

=== The Temporal Characteristics of Random Processes ===
* Random Processes ([[Media:5MRV.1A.RandomProcess.20210216.pdf |A.pdf]])
* Joint Distribution, Independence, Processes ([[Media:5MRV.2A.JointDistribution.20210220.pdf |A.pdf]])
* Stationary Random Processes ([[Media:5MRV.3A.Stationary.20220402.pdf |A.pdf]], [[Media:5MRV.3B.Stationary.20220803.pdf |B.pdf]])
* Covariance &amp; Correlation of Random Variables ([[Media:5MRV.4A.CovCorrRV.20210910.pdf |A.pdf]])
* Covariance &amp; Correlation of Random Processses ([[Media:5MRV.5A.CovCorrRP.20210911.pdf |A.pdf]])
* Example Random Processes ([[Media:5MRV.4A.Example.20210227.pdf |A.pdf]])
* Ergodic Random Processes ([[Media:5MRV.5A.Ergodicity.20211022.pdf |A.pdf]], [[Media:5MRV.7B.Ergodicity.20211215.pdf |B.pdf]])
* Measurement of Correlation Functions ([[Media:5MRV.6A.Measure.20201013.pdf |A.pdf]])
* Complex Random Processes ([[Media:5MRV.7A.Complex.20201022.pdf |A.pdf]])
&lt;/br&gt;

=== The Spectral Characteristics of Random Processes ===
* Power Density Spectrum - Continuous Time ([[Media:6MRV.1A.PSpecCT.20210204.pdf |A.pdf]])
* Auto Correlation Function ([[Media:6MRV.2A.AutoCor.20201218.pdf |A.pdf]])
* Power Density Spectrum - Discrete Time ([[Media:6MRV.3A.PSpecDT.20201203.pdf |A.pdf]])
* Cross Power Density Spectrum ([[Media:6MRV.4A.CPSpec.20191108.pdf |A.pdf]])  
* Cross Correlation Function ([[Media:6MRV.5A.CCorrel.20191114.pdf |A.pdf]])  
* Noise Definitions ([[Media:6MRV.6A.Noise.20191121.pdf |A.pdf]]) 
* Power Spectrum of Complex Random Processes ([[Media:6MRV.7A.ComplexProc.20191125.pdf |A.pdf]])

&lt;/br&gt;

=== Linear System with Random Inputs ===
* Continuous Time LTI System ([[Media:7LTI.1A.CTime.20191203.pdf |A.pdf]])
* Discrete Time LTI System ([[Media:7LTI.2A.DTime.20191211.pdf |A.pdf]])
* System Response ([[Media:7LTI.3A.Response.20191224.pdf |A.pdf]])
* Spectral Characteristics ([[Media:7LTI.4A.Spectral.20200104.pdf |A.pdf]])
* Noise Modeling ([[Media:7LTI.5A.Noise.20200122.pdf |A.pdf]])

&lt;br&gt;

=== Optimum Linear System ===
* Maximum SNR ([[Media:8OPT.1A.MaxSNR.20200128.pdf |A.pdf]])
* Minimum Squared Error ([[Media:8OPT.2A.MinSE.20200207.pdf |A.pdf]])

&lt;br&gt;

=== Noise in Some Application Systems ===
* AM Communication Systems ([[Media:9APP.1A.AM.20200212.pdf |A.pdf]])
* FM Communication Systems ([[Media:9APP.2A.FM.20200221.pdf |A.pdf]])
* Control Systems ([[Media:9APP.3A.Control.20200225.pdf |A.pdf]])
* PLL Systems ([[Media:9APP.4A.PLL.20200305.pdf |A.pdf]])
* Random Waveforms ([[Media:9APP.5A.RandWave.20200311.pdf |A.pdf]])
* Radar Systems ([[Media:9APP.6A.Radar.20200313.pdf |A.pdf]])

&lt;br&gt;

==''' Correlation and Power Spectra '''==

# Correlation Functions of Random Signals ([[Media:RAND.1.A.Correlation.20121106.pdf |pdf]])
# Spectra of Random Signals ([[Media:RAND.2.A.Spectra.20121108.pdf |pdf]])
&lt;/br&gt; 

==''' Ergodicity, Statistics, Estimation '''==
&lt;/br&gt;

==''' Random Processes and Linear Systems '''==
&lt;/br&gt;


# Time Domain Techniques for Noisy Signals ([[Media:RAND.3.A.Time.20130205.pdf |A.pdf]], [[Media:Dirichlet.pdf |B.pdf]])
# Frequency Domain Techniques for Noisy Signals
# Correlation v.s. Convolution for Noisy Signals
&lt;/br&gt;


# System Identification ([[Media:RP.SysId.1.A.pdf |pdf]]) 
# Matched Filter &lt;ref&gt;[[Understanding Digital Communications]], See Baseband Mod/Demod Section&lt;/ref&gt;
 
&lt;/br&gt;
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      <text bytes="5805" xml:space="preserve">==''' Random Variables '''==

=== Single Random Variables ===
* Random Variables ([[Media:RV1.RVariable.1.A.20200427.pdf |A.pdf]])
* Distribution Function ([[Media:RV1.Distribution.2.A.201200428.pdf |A.pdf]])
* Density Function ([[Media:RV1.Density.3.A.20200429.pdf |A.pdf]])
* Functions of Random Variables ([[Media:RV1.RVFunction.4.A.20220317.pdf |A.pdf]])
* Gaussian Random Variables ([[Media:RV1.4.Gaussian.20200430.pdf |A.pdf]], [[Media:RV1.4B.Gaussian.20180314.pdf |B.pdf]])
* Other Distribution and Density Functions ([[Media:RV1.5.Other.20200501.pdf |A.pdf]])
* Conditional Distribution and Density Functions ([[Media:RV1.6.Conditional.20200506.pdf |A.pdf]])
&lt;/br&gt;

=== The Characteristics of a Single Random Variable ===
* Expected Value ([[Media:RV2.Expectation.1.A.20200506.pdf |A.pdf]])
* Moments ([[Media:RV2.Moment.2.B.20200507.pdf |A.pdf]], [[Media:RV2.Moment.2.B.20180320.pdf |B.pdf]])
* Moment Generating Functions ([[Media:RV2.MFunctions.3.A.20200508.pdf |A.pdf]])
* Transformations of Random Variables ([[Media:RV2.Transform.4.A.20200514.pdf |A.pdf]])
&lt;/br&gt;

=== Multiple Random Variables ===
* Vector Random Variables ([[Media:3MRV.1A.VectorRV.20200515.pdf |A.pdf]])
* Joint Distribution ([[Media:3MRV.2A.JointDist.20200518.pdf |A.pdf]])
* Joint Density ([[Media:3MRV.3A.JointDensity.20200521.pdf |A.pdf]])
* Conditional Joint Distribution and Density ([[Media:3MRV.4A.CondDistrib.20200527.pdf |A.pdf]])
* Statistical Independence ([[Media:3MRV.5A.StatIndep.20200423.pdf |A.pdf]])
* Sums of Random Variables ([[Media:3MRV.6A.RVSum.20200528.pdf |A.pdf]])
* Central Limit Theorem ([[Media:3MRV.7A.CLimit.20200612.pdf |A.pdf]])
&lt;/br&gt;

=== The Characteristics of Multiple Random Variables ===
* Expected Values ([[Media:4MRV.1A.Expect.20200617.pdf |A.pdf]])
* Joint Characteristic Functions ([[Media:4MRV.2A.JChar.20200618.pdf |A.pdf]])
* Jointly Gaussian Random Variables ([[Media:4MRV.3A.JGauss.20200619.pdf |A.pdf]])
* Transformations of Multiple Random Variables ([[Media:4MRV.4A.Transform.20200620.pdf |A.pdf]])
* Linear Transformation of Gaussian Random Variables ([[Media:4MRV.5A.LinearTrans.20200623.pdf |A.pdf]])
* Simulating Multiple Random Variables ([[Media:4MRV.6A.Simulation.20200624.pdf |A.pdf]])
* Sampling and Some Limit Theorem ([[Media:4MRV.7A.LimitTheorem.20200625.pdf |A.pdf]])
* Complex Random Variables ([[Media:4MRV.8A.ComplexRV.20200626.pdf |A.pdf]])
&lt;/br&gt;

==''' Random Processes '''==

=== The Temporal Characteristics of Random Processes ===
* Random Processes ([[Media:5MRV.1A.RandomProcess.20210216.pdf |A.pdf]])
* Joint Distribution, Independence, Processes ([[Media:5MRV.2A.JointDistribution.20210220.pdf |A.pdf]])
* Stationary Random Processes ([[Media:5MRV.3A.Stationary.20220402.pdf |A.pdf]], [[Media:5MRV.3B.Stationary.20220804.pdf |B.pdf]])
* Covariance &amp; Correlation of Random Variables ([[Media:5MRV.4A.CovCorrRV.20210910.pdf |A.pdf]])
* Covariance &amp; Correlation of Random Processses ([[Media:5MRV.5A.CovCorrRP.20210911.pdf |A.pdf]])
* Example Random Processes ([[Media:5MRV.4A.Example.20210227.pdf |A.pdf]])
* Ergodic Random Processes ([[Media:5MRV.5A.Ergodicity.20211022.pdf |A.pdf]], [[Media:5MRV.7B.Ergodicity.20211215.pdf |B.pdf]])
* Measurement of Correlation Functions ([[Media:5MRV.6A.Measure.20201013.pdf |A.pdf]])
* Complex Random Processes ([[Media:5MRV.7A.Complex.20201022.pdf |A.pdf]])
&lt;/br&gt;

=== The Spectral Characteristics of Random Processes ===
* Power Density Spectrum - Continuous Time ([[Media:6MRV.1A.PSpecCT.20210204.pdf |A.pdf]])
* Auto Correlation Function ([[Media:6MRV.2A.AutoCor.20201218.pdf |A.pdf]])
* Power Density Spectrum - Discrete Time ([[Media:6MRV.3A.PSpecDT.20201203.pdf |A.pdf]])
* Cross Power Density Spectrum ([[Media:6MRV.4A.CPSpec.20191108.pdf |A.pdf]])  
* Cross Correlation Function ([[Media:6MRV.5A.CCorrel.20191114.pdf |A.pdf]])  
* Noise Definitions ([[Media:6MRV.6A.Noise.20191121.pdf |A.pdf]]) 
* Power Spectrum of Complex Random Processes ([[Media:6MRV.7A.ComplexProc.20191125.pdf |A.pdf]])

&lt;/br&gt;

=== Linear System with Random Inputs ===
* Continuous Time LTI System ([[Media:7LTI.1A.CTime.20191203.pdf |A.pdf]])
* Discrete Time LTI System ([[Media:7LTI.2A.DTime.20191211.pdf |A.pdf]])
* System Response ([[Media:7LTI.3A.Response.20191224.pdf |A.pdf]])
* Spectral Characteristics ([[Media:7LTI.4A.Spectral.20200104.pdf |A.pdf]])
* Noise Modeling ([[Media:7LTI.5A.Noise.20200122.pdf |A.pdf]])

&lt;br&gt;

=== Optimum Linear System ===
* Maximum SNR ([[Media:8OPT.1A.MaxSNR.20200128.pdf |A.pdf]])
* Minimum Squared Error ([[Media:8OPT.2A.MinSE.20200207.pdf |A.pdf]])

&lt;br&gt;

=== Noise in Some Application Systems ===
* AM Communication Systems ([[Media:9APP.1A.AM.20200212.pdf |A.pdf]])
* FM Communication Systems ([[Media:9APP.2A.FM.20200221.pdf |A.pdf]])
* Control Systems ([[Media:9APP.3A.Control.20200225.pdf |A.pdf]])
* PLL Systems ([[Media:9APP.4A.PLL.20200305.pdf |A.pdf]])
* Random Waveforms ([[Media:9APP.5A.RandWave.20200311.pdf |A.pdf]])
* Radar Systems ([[Media:9APP.6A.Radar.20200313.pdf |A.pdf]])

&lt;br&gt;

==''' Correlation and Power Spectra '''==

# Correlation Functions of Random Signals ([[Media:RAND.1.A.Correlation.20121106.pdf |pdf]])
# Spectra of Random Signals ([[Media:RAND.2.A.Spectra.20121108.pdf |pdf]])
&lt;/br&gt; 

==''' Ergodicity, Statistics, Estimation '''==
&lt;/br&gt;

==''' Random Processes and Linear Systems '''==
&lt;/br&gt;


# Time Domain Techniques for Noisy Signals ([[Media:RAND.3.A.Time.20130205.pdf |A.pdf]], [[Media:Dirichlet.pdf |B.pdf]])
# Frequency Domain Techniques for Noisy Signals
# Correlation v.s. Convolution for Noisy Signals
&lt;/br&gt;


# System Identification ([[Media:RP.SysId.1.A.pdf |pdf]]) 
# Matched Filter &lt;ref&gt;[[Understanding Digital Communications]], See Baseband Mod/Demod Section&lt;/ref&gt;
 
&lt;/br&gt;
&lt;/br&gt; 
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  <page>
    <title>Ithkuil/Errors</title>
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      <timestamp>2022-08-05T10:27:39Z</timestamp>
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      <text bytes="22153" xml:space="preserve">=== Morpho-phonology 0.19.0 ===
12 Roots are now only up to five consonants in length

15 &quot;MONDADIC&quot; should be &quot;MONADIC&quot;

15 MSS/DPL/RPV/A results in a value of -tţ-, which is phonotactically impermissible

15 [affiliation]/MDS is ambiguous with [affiliation]/ATV/A/RPV, and [affiliation]/MDC is ambiguous with [affiliation]/ATV/N/RPV.

16 &quot;pm -&gt; vmm&quot; should maybe be &quot;pm -&gt; vvm&quot;

16 Sec. 3.6.3 lists outdated Combination Referential Specification forms

18 Due to the -y- infix in level morphemes, modular adjuncts are always ambiguous when encoding any absolute level. For example, the adjunct &quot;oyeha&quot; could encode either just SPLa level or REG aspect + CRO valence

20 VxCs affix for Mood not found in docs; only Modular Adjunct and Mood Adjuncts available

22 The case accessor should read &quot;the party by / thru which / whom X is subjectively interpreted / considered / described&quot;

34 It is not pragmatic for an affix to scope above an entire formative; forms could instead mark scoping in concatenation chains

36 Modular adjunct slots 3 and 4 use a redundant Cn, which can simply be ň; Vh scoping is impractical (what does it mean to scope above Validation+Illocution+Expectation?) and is de facto given by affixual adjunct scoping

39 Agglomerative form v is ambiguous with pi/DET

39 Agglomerative form -tļ- is ambiguous with 2p/BEN+pi/NEU. i.e. an adjunct like ëstļu can parse as ë-s-tļ-u or as ë-s-t-ļ-u. Easily solved by eliminating -ļ- as a combining form for pi/NEU

43 due to geminate referential forms, a word like kapreomma is ambiguous, and may be parsed as k-a-pr-eomm-a, with a VxCs referential affix -eomm-, or as k-a-preo-mm-a with a -preo- referential CsVx and geminated Ca value. This can be resolved by banning referential affixes in slot V or by taking the Nomic -ç- form to form non-geminate forms, e.g. mm→mç, ll→lç, nn→nç, etc.

43 A second ambiguity concerning referential affixes: they are ambiguous with type three case accessors when the latter bear an adverbial case or a spatio-temporal II case. For example, kalöečwa could bear either a type three CON accessor or bear the referential Mx/N/BEN-CON. One solution would be to prohibit using Nomic in referential affixes. For consistency, we could ban all Ca modifications to referentials in affix form, similarly to how it's done in referential formatives

46 - In the definition of the Satiative bias, pleasure is misspelled

46 ''Section 4.9 - Parsing Adjunct'' says to refer back to ''Section 2.2 Paragraph 5'', when it means ''Section 2.3 Paragraph 5''.

51 ''Section 5.4 - Relative Clause Formation'' contains a superfluous note saying that default values in the linear analysis haven't been included, when this has been the case throughout the document.

54 ''Section 5.6 - Mmiäxainļ'' should be ''Mm'''ie'''xainļ; Nniexainļ'' should be ''Nn'''io'''xainļ; Ňňeoxainļ'' should be ''Ňň'''eö'''xainļ''

55 ''Section 5.7 - hwelu-azčojhaillöelyá'' should be '''''hr'''elu-azčojhaillöelyá; Kšölaelwêi'' should be ''kšöl'''aö'''lwêi''

57 ''Section 6.0 - ksialärsirwa'' should be ''wuksärsëirwa; ksialärsëirwiasta should be wuksärsëirwiasta; SPT/5 affix -irw-'' should be ''SPT/5 affix -'''ë'''irw-''

57 Section 6.1 - ''wurḑaušš'' should be ''wurḑauj''

60 Irtysh should be transliterated &quot;irtiš&quot;, Rio Grande should be transliterated &quot;grande&quot;, and&quot;riyadh&quot; should be &quot;Riyadh&quot; (with a capital R); if so, Yangtze River should be &quot;''caň&quot;'' instead of &quot;''caňčyaň''&quot; which means &quot;Long river&quot;, parallel to hwaň &quot;Yellow river&quot;; 

60 ''phwära hle’e'' should be ''phwära hle’i;'' 

60 Ithkuil phonotactic restraints do not apply as long as the name is pronounceable. According to this principle, ''hönkoň'' should be made ''höňkoň'' for Hong Kong, parallel to ''čuňkwo'' for China; Likewise ''tuňkwan'' for Dongguan 

62 ''Papua New Gunea'' should be ''Papua New Guinea'' 

62 Are Liberia ''laibirya'' and Malaysia ''mëleisya'' English pronunciations with local accents? Why using -a instead of schwa? 

62 ''romênia'' should be ''romênìa'' for Romania''?; tonga'' should be ''toňa''

62 ''saintvinsentangrénadinz'' for ''St. Vincent &amp; the Grenadines'' should be ''seintvinsentangrénadinz'' as ''seintkicannevis'' for ''Saint Kitts &amp; Nevis''

63 ''hlaḑtáu-eḑtëila hi kongo'' has a typo in ''hlaḑtáu-eḑtila hi kongo;'' Also, this seems to make more sense'': T1-S1-“democratic government“-OBJ-ASI—S2-“republican government“-OBJ?'' 

=== Roots 0.5.1 ===
005 the third example for -Č- is missing a closing parenthesis

007 Sec. 1.4 lacks a root for frame category and -LMPY- can include concatenation type

009 CTE of -LŢT- has a typo in pursue

033 S3 of -ẒŘ- should be extrusion

024/042 -TV- is similar to -LDR- and -RTV- is similar to -DRR-

038 -LČW- has a typo in hypothesizing

054 S3 of -THW- should not include &quot;utilizing&quot;

057 S3/OBJ of -JJ- should be the medium in which a depression is made

060/209 -MSY- is shared between &quot;energy output differential or redirection via gear mechanism&quot; and &quot;pleasant/mellifluous sound&quot;

062 -ÇJ- is not permissible as a root

067 S3 of -CY- has a typo in crowd

069 S3 of -RPĻ- can include &quot;e.g. twisting/turning a wheel left/right&quot; and S3 of -LXR- can include &quot;e.g. rolling/raveling up a carpet&quot; as examples

072 S2 of -ŇM- should refer to application of gel-like substance [CPT: foam/froth]

072 S3 of -ŇFY- is defined on page 315; lacquer can be used instead

072 -ŇFY- should be merged into one stem; poultice and organic oil can be added

072 -ŇFL- can be 1. disinfectant/sterilant 2. insecticide/pesticide 3. insect repellant; -ŇFR- can be sunscreen, moisturizer/emollient, bronzer; 

076 S2 of -SXR- can refer to massaging/scratching/rubbing together, with brushing being moved to S1

083 -FÇT- has stray ` marks and -LZDR- has a typo

084 CTE of -SSV- is missing

085 S1 of -KŢG- can be screen/display, because computing device is similar to S1 of -KŢGW-

085 -KŢGY- can instead be 1. program/app/software 2. firmware 3. operating system

086 S2 of -KŢGR- should refer to the transformation/compilation of instructions/text into a different format for a computer to execute

086 S2 of -KŢGL- should read &quot;lambda calculus representation&quot;

087/131 -ŠGW- is shared between &quot;cup/bowl/measuring cup&quot; and &quot;detention/confiscation/usurpation&quot;

088 -ŇSL- should be electrical socket

093 OBJ of -ŽTĻ- should be an individual stair

102 S3 of -Ţ- can refer to the opposing direction or origin from which something emerges, so that S0 can refer to the general vicinity of a position

104 -F- is missing a hyphen

110 OBJ/2 of -KM- should be that to which one is (intended to be) early

111 -LPŠ- has a typo in refreshed

118 the stems of -MSX- do not translate into different languages; -VLL- is sufficient

121 -VŽŘ- has a typo in &quot;metonym&quot; in the root title

129 -NN- refers to a root present on pg. 33

135 S2 of -NXW- has a typo in destination

136 -ŠČ- has an extra hyphen

152 S2 of -BČ- has redundant phrases '', by legal adoption, or long-term informal adoption]''

153 -SR- and -LH- have stray slashes

160 -ŠF- can list &quot;understatement&quot; as a derivation

172 -MBR- is missing a bracket and has a hyphen in achieving

174 -ŽT- has a hyphen in unobtrusive

175 -VSP- is missing brackets around CPT

178 -PŠX- has an extra space in the root expression, and also a missing space in &quot;behavior toward&quot;

179/236 -RXŢ- is already shown through -RTK-

205 -ÇŢ- can include whiff, woosh, and sizzle; -ĻB- can include drip drop

211 the musical note affixes have outdated values

217 S2 of -ÇÇKL- can also be spelled &quot;khene&quot;

219/339 S2 of -ŽG- should be olfactory bulb so that the note on -LMW- can be removed

221 -ŠP- refers to informal designation and the paragraph below the color roots refers to incorporating instead of  concatenating

224/293 -VR- should switch values with -VC- to match its affix value

224 -BŢ- should be patterned as a &quot;degree of&quot; root

225 -XẒ- can include &quot;lint/pollen/ash/soot-like&quot;, -SG- can include &quot;silky&quot;, and -ḐĻ- has odd punctuation

231 Section 4.5.7 has a typo in &quot;The 13 roots below...&quot;

237 -LTW- contains misspelling of &quot;camaraderie&quot;

240 -ZGL- contains an extra &quot;to&quot;

248/252 -TGL- is shared between &quot;spear/pike&quot; and &quot;cue stick/broom (curling)/jump rope&quot;

249 -JTÇ- has an extra slash

253 -ŢSKL- contains possible misspelling in xiangqi

256/257 &quot;parasailing&quot; is shared between stem 2 of -ĻKY- and stem 2 of -ÇÇTL-

256/257 &quot;snowkiting&quot; is shared between stem 2 of -ĻKÇ- and stem 2 of -ÇÇTW-

268 -KSTR- has stems 2 and 3 switched and a typo in accessing

271 -FM- is missing a closing parenthesis

280 S2 of -MTW- is missing a closing parenthesis and -KKY- incorrectly uses the term neutronium

284 OBJ of biome roots should be &quot;the impact of X on its surrounding environment.&quot;

285 -VŇ- makes a difference between canyon and gorge; perhaps gully can be used

285 OBJ of -VŇ- should be &quot;...a particular (i.e, named) ''valley''

291 stems of -FPH- should line up with those of -FŠR-

301/451 -FBL- is shared between &quot;spiked circle&quot; and &quot;Adegphagan beetle 3&quot;

303/451 -FBR- is shared between &quot;3-dimensional spiked spheroid&quot; and &quot;Adegphagan beetle 4&quot;

306 CPT forms of -RẒ- can refer to the oxidation states of zero, oxidized, and reduced ions

311 The following roots most likely follow -RẒB- instead of -RẒ-

311 There are 2 roots for ''baking soda'': -ÇPF- and -DPF-

312 -MSDŘ- has a typo in &quot;carcinoembryonic antigen&quot;

312 &quot;vitamin B3&quot; is shared among -PFKW-, -PFKN- and -PFKH-

312 -PFKÇ- has a typo in ''pyridoxine''

312 -PFKM- has an extra comma

312 -PFGR- has a typo in ''phylloquinone''

312 -ŇTPŢ- has a typo in ''calcitonin''

312 -MSGY- has a typo in ''dehydroepiandrosterone''

312 &quot;hemoglobin&quot; is shared between -MFBL- and -MFBŘ-

313 &quot;gastrin&quot; is shared between -ŇFPF- and -ŇFPĻ-

313 -MSV- has a typo in ''fibrinogen''

320/322 -NDK- is shared between &quot;guava/finger cherry&quot; and &quot;allspice&quot;

322/350 -MŽDY- is shared between &quot;stevia&quot; and &quot;rotator cuff&quot;

323 there is a typo in &quot;vegetables&quot;

324 -CK- is not alphabetized

325 The legumes and nuts section is missing S1 of -NḐKN- macadamia nut

332 S1 of -ŇPY- not yet changed to new meanings

335 S2 of -TÇŇ- can be derived by concatenating positional roots; armpit works instead

335 -TÇŇ- has a typo in knee

337/502 -GZG- is shared between small intestine and Hydrangeaceae

338 S3 of -DDV- can list concatenation with -PFW- for &quot;premolar&quot;

341 S3 of -LPÇN- should be monocyte

342 distal carpal bones are missing; S3 of -LÇTY- should be replaced with 'hamate' (so that the carpal and tarsal bone roots are analogous) and -LÇTW-- should be changed to 1. metacarpal/metatarsal 2. 1st, 2nd, or 3rd cuneiform/trapezium, trapezoid, or capitate 3. cuboid/triquetrum; if this should be split into two roots, -LÇTL- can be overwritten because malleoli are mere bone features

342 -LÇPW- and -LÇPL- both include talus; S1 of -LÇTR- can be replaced with pubic symphysis

343 -LÇFN- should be 1. humerus 2. radius 3. ulna to parallel 1. femur 2. tibia 3. fibula

343 -LÇFR- has a typo in caracoid

344 S3 of -LTÇL- has a 1

345 DYN function can refer to muscle contraction or ligament pulling 

345 S2 of -MZXR- has a typo

347 S2 of -MZKY- has a typo in genus

352 -MŽPŢ- shouldn't refer to the uvula itself; since hard palate is given by -LTÇR-, -LDN- should take uvula

354 the titles of -MÇFW- and -MÇFY- names don't match up

357/358 &quot;radial collateral ligament&quot; is shared between S2 of -KŢTL- and S3 of -KŢTH-; more specification needed

357 &quot;ulnar collateral ligament&quot; is shared between S2 of -KŢTR- and S3 of -KŢTÇ-; more specification needed

358 -PŢTW- is missing a closing parenthesis

362 -ŇNY- and -PSX- both list palpitation

380/386 -MMŠTL- and -ŇŇŠPR- are duplicates of -GZDR-

386 -ŇŇSKH- has a typo

388 -ŇŇZDL- has a typo

401 PZE degrees 3-9 and MZE degrees 2-9 should use 'period' instead of 'epoch'

401 MZE/4 has a typo in Triassic

402/463 OBJ specification between animals and plants differs, making it unclear for other lifeforms

403 -NÇV- erroneously listed &quot;Beira&quot; as the generic name for beira; correct generic name should be &quot;Dorcatragus&quot; instead

404 -ČTF- has a typo in &quot;Mephitis&quot;

405 -VḐ- is missing [stem unassigned]

406 S1 of -BZL- is missing the genera Poliocitellus &amp; Sciurotamias, and S2 has Scirillus instead of Syntheosciurus

407 BZKH- is missing a parenthesis and has an extra comma

407 -BZKÇ- has a misspelling in &quot;Calomyscus&quot;

408 -LGY- has an extra apostrophe

416 -MSKW- thru -MSKL- are the only avian roots built upon an unrelated root (-MSK- necessity)

420 -ŇSGŘ- and -NSKF- are missing [stem unassigned] and in S1 of -NSKY- Stiphromis should be Stiphrornis

424 S2 of -PSTY- has a typo in ginbuna and is missing koi, and -PSTĻ- has an additional stem 3

424 -RPSTL- has a typo in Psilorhynchus

425 -ŘPSNW- has a typo in voladora

428 the second instance of -PŢSP- should be -PŢST-

439 -KFŘ- has an extra closing parenthesis

447 S3 of -ḐPĻ- has a redundant “mantis”

458 -ŘZBY- should have a III

462 S2 of -LZVW- has an extra parenthesis

468 S2 of -SŢW- has an extra parenthesis

470 S3 of -NÇPY- has a typo in ginger

478 -FTL- is missing [stem unassigned]

482 S3 of -RŽTW- is missing tarragon

485 -ŇZBW- is missing [stem unassigned] and -ŇZBL- and -ŇZBR- have an unresolved duplicate

486 S1 of -RẒL- has a typo in Taraxacum

491 -FŠPÇ- has a typo in Lamium and -FŠPS- is missing [stem unassigned]

520 -ŇSXM- is missing descriptions

526 Missing: ‘FABOIDEAE’ XVIII 1. (genus Marina) false prairie-clover 2. (genus Parryella) common dunebroom 3. (genus Psorothamnus) indigo bush, dalea, smoketree / smokethorn

527 -ŘPTHW- is a duplicate of -ŘTP-

531 There are two forms of -LMÇN- ‘EUPHORBIACEAE’ VII and S1 of -RMÇL- and -RMÇR- are the same

542 -SSW- has a typo in Peperomia

543 -RJGY- has a typo in physodermataceous and -RJGL- is missing an open parenthesis

(-RJGL- and -RJGḐ- both include basidiobolaceous fungus, but who knows how the spelling should differ?)

543 -RJTL- redundantly lists phaeohyphomycosis twice

544 S3 of -RJTH- should list Claviceps, making -RJTHW- ‘SORDARIOMYCETES’ III S1. Xylaria 2. Colletotrichum 3. other sordariomycetes fungus

544 -RJXN- has an extra semicolon

546 -PŢPY- includes [stem unassigned] but the fungal family continues into a fourth root

547 -LVMY- is numbered but -BZVY- -BZVW- -NZGY- -NZGL- aren't numbered

547 S1 of -NZGW- uses &quot;genus&quot; and S2 of -BZDV- repeats &quot;fungus&quot;

549 S2 of -LMḐN- should read &quot;(Sagenista, including Labyrinthulomycetes and Eogyrea; several genera)&quot;

550 -NŢKÇ- should read ‘COLPODEA’ II

551 -NŢPR- and -NŢPŘ- share stems 2 and 3, and -NŢPĻ- should be numbered IX

552 S1 of -NŢFR- is capitalized, -RMT- is partially missing stems, and -RMTÇ- and -RMTF- are not numbered

553 -ŇŠKL- is partially missing its stems

556 -LMŠ- is shared between &quot;fusobacteria&quot; and &quot;other bacteria 1&quot;

557 S2 of -RŠTY- includes a number as part of the genus

557 S2 of -LFMW- should have HSV-1 instead of HHV-1

557 S1 of -LFMY- should have papiine betaherpesvirus 3/4 instead of panine betaherpesvirus 3/4

558 -RŠTF- has a typo in Cressdnaviricota

561 S1 of -NXPR- has a typo in spotted orthotospovirus and S3 of -NXFW- has a typo in metapneumovirus

563 -PSFR- has a typo in virus and in

563 -RCXW- should be SPHAEROLIPOVIRIDAE 1. Alphasphaerolipovirus 2. Yingchengvirus 3. Hukuchivirus

564 -FŠTM- has a typo in Virtovirus

=== Affixes 0.7.5 ===
006 DCD hasn't been updated re: the split of Obviative (it also still uses the term &quot;PRA&quot;)

011 TD2/6 should be &quot;for a few months&quot;

012 STT degrees 4 and 6 can be removed to have degrees 1 and 9 be ereyesterday and overmorrow

014 CYC/CYL recurrence affixes are not needed because FQC/FQL can be used without a number to imply 1

019 for P07*, it should be &quot;''position at +X / 0 / 0''&quot; instead of &quot;''position at -X / 0 / 0''&quot; 

020 there are the affixes corresponding to the 3 following roots not included in the affixes doc: -ČḐ- ‘''INTERNAL POINT-LIKE INTERIOR VERTEX/“CORNER” OF AN ENTITY''’, -XḐR- ‘''EXTERNAL LINEAR OUTWARD-FACING “CORNER” EDGE OR “CORNER” SPACE OF AN ENTITY''’, -ČḐR- ‘''INTERNAL LINEAR INTERIOR “CORNER” EDGE OR “CORNER” SPACE OF AN ENTITY''’ 

051 in the NOTE, XXX should be change to EFP 

059 HND values have symmetric duplicates

75/110 the KBP and ENS affixes share the same value

082 IEC has redundant specifications when interpreted as a root

084 TCT doesn't have a Cs value

104 NCD and NCE are not updated with Agglomerative perspective

107 the section name above PG1 has a typo in Differentiation

110 OBN/6 is not pluralized

114 PZE degrees 3-9 and MZE degrees 2-9 should use 'period' instead of 'epoch'

115 MZE/4 has a typo in Triassic

122 KIN is a geminate form, leading to confusion with Ca

140 GPB/6 ''carboxy-'' should be ''carboxylato-''

140 GPD/1 has a typo in ''methylenedioxy''

141 GPG/5 has two typos in ''borinic'' and ''di… borinic''

141 GPG/7 has a typo in ''alkyllithium''

141 GPG/7 has a typo in ''alkylmagnesium''

141 GPH/9 has a typo in ''sulfanylidene-''

142 GPJ/3 has two same values with the above GPH/9: ''carbonothioyl'' and ''sulfanylidene-''

142 GPJ/6 has a typo in ''thioester''

=== Phonotaxis ===

* ç should be an non-sibilant fricative, and it was instead misclassified as a sibilant fricative.
* ç is placed in the wrong position of the table PHONEMIC INVENTORY
* some prohibited conjuncts in sec. 2.9. lack preceding asterisks
* there is a stray ň above section 11
* -ÇÇ-, -ĻĻ-, -ŢŢ- are valid roots
*-ÇẒ- and -ÇJ- (already in use) are impermissible as a root/affix
*-ḐS-, -ḐŠ-, -ḐZ-, -ḐŽ- and -NŇ- are permissible as a root/affix
*kļ is listed as a valid root/affix but is indistinguishable from tļ
*the table for permissible biconsonanantal roots beginning with ç, c and ż should count 24 forms, and beginning with č and j should count 22 forms, so the total should be 684 instead of 679

=== Root Gaps ===
* a note should be made to conjugate pluralia tantum like scissors with DPX instead of noting individual roots
* mortar, pestle, stirrer, pipette, eyedropper, baster
* eyeshadow/eyeliner, mascara, rouge, lipstick/gloss/liner
* Artiopod (trilobite), Megacheira (great appendage arthropod), other extinct Arthropod (e.g. Dinocaridid)
* igneous, sedimentary, metamorphic rock
* plant fiber, thatching/wicker, rope
* stalagtite/stalagmite (ice, brine)
* volcano, geothermal vent, hot spring
* cumuliform/stratiform/cirriform cloud (stems concatenate for distinctions like stratocumulus, cirrocumulus, and cirrostratus; nimbus clouds have a tall/multilevel shape, alto clouds are in the middle of the troposphere)
* pasta/noodle, vinegar, icing/frosting, juice, gelatin, colloid
*caramel, molasses, toffee
* bite (front of mouth), chew (back of mouth)/masticate, swallow
* pad/compress/dressing, bandage, tape/patch/film
* extract/absolute/concrete, wax (MTÇ), essential oil
* blanket/quilt/tarp/mesh/net
*pile carpet/mat/rug
* pillow/cushion/trampoline
* rag/mop/towel
* transistor, diode, LED; 
* spring, crank, roller chain, sprocket, worm gear, Archimedes' screw
* segway/self-balancing scooter/oneway/dicycle
* dolly/barrow/hand truck, pallet truck, garage creeper, snowplow, snowblower
* pion, kaon, rho meson, eta meson, D-meson, B-meson, omega meson, phi meson, psion, upsilon meson, theta meson
* cape, hijab/shayla, balaclava, headband/bandana, scrunchie
* rest (as in, recover from tiredness without sleeping)
*Parts of the head: forehead, brow, temple
*The outer ear: ear, ear canal, earlobe; Auditory ossicles: stapes, incus, malleus; Inner ear: cochlea, vestibule, semicircular canal; Cochlear duct: basilar membrane, vestibular membrane, tectorial membrane; Vestibule: otolith, saccule, utricle; Semicircular canal: osseous ampula, ampullary cupola, hair cell;
*The front eye: cornea, lens, sclera; Uvea: iris, ciliary body, choroid; Back of eye: retina, macula, optic disk; Photoreceptor cell: cone cell, rod cell, intrinsically photosensitive retinal ganglion; Other eye part: eyelid, lacrimal gland, lacrimal sac
*The outer nose: ala (nostril area), philtrum, nasal bridge; Nasal cavity: nasal septum, nasal concha, nasal meatus; Other nose part: sinus, tonsil, eustachian tube
*The upper gastrointestinal tract: pharynx, esophagus, epiglottis; Respiratory system: trachea, bronchus, pulmonary alveolus; Linguistic system: larynx, vocal chord, glottis; Throat: hyoid bone, thyroid cartilage, cricoid cartilage
* Aotidae (night monkeys), Pitheciidae (titis, sakis, uakaris), Atelidae (howler monkeys, spider monkeys, muriquis, woolly monkeys)
* Boselaphus (nilgai), Tetracerus (four-horned antelope)
* Scotocerca (streaked scrub warbler), Erythrocercus (yellow flycatchers)
*Muscles: plantar fascia, semitendinosus, quadriceps femoris, articularis genus, vastus intermedius, quadratus femoris, adductor minimus
*Ligaments: superior ex. / fl. retinaculum, peroneous/fibular retinaculum, lateral retinaculum, medial patellar retinaculum, inguinal ligament, lacunar ligament, pectineal ligament
*Connective tissues: calcaneal tendon, labrum, pubic symphysis, tendinous intersection, linea alba, linea semilunaris

=== Affix Gaps ===

* degree of stretchiness/tightness/looseness/bagginess
* CNM/8 is a redundant form of the PAR case accessor
* associative plurals (&quot;X together with the persons/things associated with it/them; X and co.&quot;), possibly patterned similarly to the OAU suffix; this may be used to derive a pronoun &quot;we&quot; from the first person singular pronoun, for example. (-çẓ and -çj are free Gradient 0 forms which could be suitable.)

=== WRITING SYSTEM FOR NEW ITHKUIL (December 2021) ===
02 &quot;Vr Specification, Function, &amp; Stem, plus Vv Designation &amp; Version&quot; should be &quot;Vr Specification, Function, &amp; Context, plus Vv Version &amp; Stem&quot;

11 &quot;Vlanece&quot; should be &quot;Valence&quot;
[[Category:Ithkuil]]</text>
      <sha1>2xvay66b1ekyvcl0src0tvxexyprlba</sha1>
    </revision>
  </page>
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    <title>Evidence-based assessment/Anorexia nervosa (assessment portfolio)</title>
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        <username>Maddiegray11</username>
        <id>2936309</id>
      </contributor>
      <comment>/* Severity interviews for anorexia nervosa */ deleted extra columns</comment>
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{{medical disclaimer}}
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==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
For background information on what assessment portfolios are, click the link in the heading above.
== [[Evidence based assessment/Preparation phase|'''Preparation phase''']] ==
=== Diagnostic criteria for anorexia nervosa ===
{{blockquotetop}}
'''ICD-11 Criteria'''

Anorexia Nervosa is characterized by significantly low body weight, which is less than minimal normal/expected weight for the individual’s height, sex, age and developmental stage (body mass index (BMI) less than 18.5 kg/m2 in adults and BMI-for-age under fifth percentile in children and adolescents) that is not due to another health condition or to the unavailability of food. Low body weight is accompanied by a persistent pattern of behaviors to prevent restoration of normal weight, which may include behaviors aimed at reducing energy intake (restricted eating), purging behaviors (e.g., self-induced vomiting, misuse of laxatives), and behaviors aimed at increasing energy expenditure (e.g., excessive exercise), typically associated with a fear of weight gain. Low body weight or shape is central to the person's self-evaluation or is inaccurately perceived to be normal or even excessive.

'''Changes in DSM-5'''

* The diagnostic criteria for anorexia nervosa changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here] and [[w:DSM-5|here]].
{{blockquotebottom}}

=== Base rates of anorexia nervosa in different populations and clinical settings ===
This section describes the demographic setting of the population(s) sampled, base rates of diagnosis such as prevalence rates, country/region sampled, and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of adolescent depression that they are likely to see in their clinical practice.

* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']]

{| class=&quot;wikitable&quot;
|-
! Setting (Reference) 
! Base Rate
! Demography
! Diagnostic Method
|-
|Non-clinical: Population-based (NCS-R)&lt;ref&gt;Hudson, James I.; Hiripi, Eva; Pope, Harrison G.; Kessler, Ronald C. (2007-02-01). &quot;The prevalence and correlates of eating disorders in the National Comorbidity Survey Replication&quot;.''Biological Psychiatry'' '''61''' (3): 348–358. doi:10.1016/j.biopsych.2006.03.040. ISSN 0006-3223. PMC 1892232. &lt;nowiki&gt;PMID 16815322&lt;/nowiki&gt;&lt;/ref&gt;
|.9% Female, .3% Male
|Nationally representative US sample of adults
|World Health Organization-Clinical International Diagnostic Interview (WHO-CIDI)
|-
|Non-clinical: Population-based (NCS-A)&lt;ref&gt;Swanson, Sonja A.; Crow, Scott J.; Le Grange, Daniel; Swendsen, Joel; Merikangas, Kathleen R. (2011-07-01). &quot;Prevalence and correlates of eating disorders in adolescents. Results from the national comorbidity survey replication adolescent supplement&quot;. ''Archives of General Psychiatry'' '''68''' (7): 714–723. doi:10.1001/archgenpsychiatry.2011.22. ISSN 1538-3636.&lt;nowiki&gt;PMID 21383252&lt;/nowiki&gt;&lt;/ref&gt;
|.3% Female, .3% Male
|Nationally representative US sample of adolescents
|WHO-CIDI
|-
|Non-clinical: US Population-based prevalence (Adolescent Brain Cognitive Development (ABCD) study&lt;ref&gt;{{Cite journal|last=Rozzell|first=Kaitlin|last2=Moon|first2=Da Yeoun|last3=Klimek|first3=Patrycja|last4=Brown|first4=Tiffany|last5=Blashill|first5=Aaron J.|date=2019-01-01|title=Prevalence of Eating Disorders Among US Children Aged 9 to 10 Years: Data From the Adolescent Brain Cognitive Development (ABCD) Study|url=http://archpedi.jamanetwork.com/article.aspx?doi=10.1001/jamapediatrics.2018.3678|journal=JAMA Pediatrics|language=en|volume=173|issue=1|pages=100|doi=10.1001/jamapediatrics.2018.3678|issn=2168-6203|pmc=PMC6583451|pmid=30476983}}&lt;/ref&gt;
|0.1%, no gender differences
|Nationally representative US sample of 9- and 10-year old children
|DSM-5 using Kiddie Schedule for Affective Disorders and Schizophrenia (KSADS)
|-
|Non-clinical: Latinos in US&lt;ref&gt;Alegria, Margarita; Woo, Meghan; Cao, Zhun; Torres, Maria; Meng, Xiao-li; Striegel-Moore, Ruth (2007-11-01). &quot;Prevalence and correlates of eating disorders in Latinos in the United States&quot;.''The International Journal of Eating Disorders''. 40 Suppl: S15–21. doi:10.1002/eat.20406. ISSN 0276-3478. PMC 2680162. &lt;nowiki&gt;PMID 17584870&lt;/nowiki&gt;&lt;/ref&gt;
|.12% Female, .03% Male
|Latino Households in the US
|WHO-CIDI
|-
|Non-clinical: African Americans and Caribbean Blacks in the US (NSAL)&lt;ref&gt;Taylor, Jacquelyn Y.; Caldwell, Cleopatra Howard; Baser, Raymond E.; Faison, Nakesha; Jackson, James S. (2007-11-01). &quot;Prevalence of eating disorders among Blacks in the National Survey of American Life&quot;. ''The International Journal of Eating Disorders''. 40 Suppl: S10–14. doi:10.1002/eat.20451. ISSN 0276-3478. PMC 2882704. &lt;nowiki&gt;PMID 17879287&lt;/nowiki&gt;&lt;/ref&gt;
|.14% Female, .2% Male
|National probability sample of adult and adolescent African Americans and Caribbean Blacks
|WHO-CIDI
|-
|Non-clinical: National Latino and Asian American Study (NLAAS)
|.12% Female, .05% Male
|Asian American adults in US households
|WHO-CIDI
|-
| Non-clinical; college students&lt;ref&gt;Tyler ID. A true picture of eating disorders among African American women: a review of literature. ABNF J. 2003;14(3):73-4.&lt;/ref&gt; || .0%  || US African American college females || Eating Disorder Diagnostic Questionnaire (EDD-Q)
|-
| Non-clinical; Adolescents&lt;ref&gt;Stice, E., Becker, C. B., &amp; Yokum, S. (2013). Eating disorder prevention: Current evidence-base and future directions. ''Int. J. Eat. Disord. International Journal of Eating Disorders,'' ''46''(5), 478-485.&lt;/ref&gt; || .8% || US Female Adolescents || Eating Disorder Diagnosis Interview (EDDI)
|-
| Non-clinical; student-athletes&lt;ref&gt;Johnson C, Powers PS, Dick R. Athletes and eating disorders: the national collegiate athletic association study. Int J Eat Disord 1999;26:179e88.&lt;/ref&gt; || .0% || US Division-I Varsity Student Athletes || Eating Disorder Inventory-2 (EDI-2)
|-
| Non-clinical; Military&lt;ref&gt;McNulty, PAF. (2001). Prevalence and contributing factors of eating disorder behaviors in active duty service women in the Army, Navy, Air Force and Marines. Military Medicine, 166(1), 53-58. &lt;/ref&gt; || 1.1% || Active duty females in US Army, Navy, 
Airforce, and Marines 
|| EDI-2
|-
|Non-clinical; Commonwealth of Virginia Mid-Atlantic Twin Registry (MATR)&lt;ref&gt;Kendler KS, Walters EE, Neale MC, Kessler R, Heath A, Eaves L. The structure of genetic and environmental risk factors for six major psychiatric disorders in women. Archives of general psychiatry. 1995;52:374–383.&lt;/ref&gt;
|1.62% (narrow), 3.70% (broad)
|US Caucasian female same-sex twins
|Structured Clinical Interview for DSM Disorders (SCID)
|-
|Non-clinical; Health Omnibus Survey (HOS)
|.46% (3 months; combined)
|South Australian older adolescents and adults
|Eating Disorder Examination (EDE)
|-
|Military&lt;ref&gt;{{Cite journal|title=Diagnosed eating disorders in the U.S. Military: a nine year review|url=http://www.ncbi.nlm.nih.gov/pubmed/18821361|journal=Eating Disorders|date=2008-12-01|issn=1532-530X|pmid=18821361|pages=363–377|volume=16|issue=5|doi=10.1080/10640260802370523|first=Amanda J.|last=Antczak|first2=Teresa L.|last2=Brininger}}&lt;/ref&gt;
|.04% (combined)
|US Military
|ICD codes from electronic records
|-
|Non-clinical; Military
|.02% (7 years) Female, 
0.0% (7 years) Males
|US Military Academy cadets
|[https://www.eat-26.com/ Eating Attitudes Test- 26 items] (EAT-26)&lt;ref&gt;{{Cite web|url=https://www.eat-26.com/|title=EAT-26: Eating Attitudes Test &amp; Eating Disorder Testing – Use the EAT-26 to help you determine if you need to speak to a mental health professional to get help for an eating disorder.|last=admin|language=en|access-date=2022-05-30}}&lt;/ref&gt;
|-
|Non-clinical; Military
|1.1% (current &amp; past) Female
|US Navy female nurses
|DSM-III
|-
|Non-clinical; Military
|.04% Female, .005% Males
|US veterans
|ICD-9-CM
|-
|Non-clinical; Military
|2.5% Males
|Active duty males in US Navy
|N/A
|-
|Clinical; Collaborative Study on the Genetics of Alcoholism (COGA)&lt;ref&gt;{{Cite journal|title=Anorexia nervosa and bulimia nervosa in alcohol-dependent men and women and their relatives|url=http://www.ncbi.nlm.nih.gov/pubmed/8540597|journal=The American Journal of Psychiatry|date=1996-01-01|issn=0002-953X|pmid=8540597|pages=74–82|volume=153|issue=1|doi=10.1176/ajp.153.1.74|first=M. A.|last=Schuckit|first2=J. E.|last2=Tipp|first3=R. M.|last3=Anthenelli|first4=K. K.|last4=Bucholz|first5=V. M.|last5=Hesselbrock|first6=J. I.|last6=Nurnberger}}&lt;/ref&gt;
|1.41% (lifetime) Female,
.00% (lifetime) Male
|US alcohol-dependent adults from San Diego, St. Lois, Iowa City, Farmington, New York, &amp; Indianapolis
|SSAGA
|-
|Non-clinical; healthcare members
|.0269% (current)
|Healthcare provider records
|
|-
|Non-clinical; high school students
|. 00% (point), .45% (lifetime) Female
.00% (point), .00% (lifetime) Males
|US high school students in west central Oregon
|DSM-III-R&lt;sub&gt;4&lt;/sub&gt;
|-
|Clinical; substance users
|.4% (lifetime), .3% (current) Female
.4% (lifetime), .3% (current) Males
|Canadian treatment-seeking substance users
|DIS&lt;sub&gt;9&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Europe'''
|-
|Non-clinical; adolescents
|.3% Female
|Adolescent females residing in Navarra, Spain
|EAT-40&lt;sub&gt;3&lt;/sub&gt;
|-
|Non-clinical; adolescents
|.7% (lifetime) Female, .2% Male
|Adolescents in secondary schools in Sør-Trøndelag, County in Norway
|SEDs&lt;sub&gt;10&lt;/sub&gt;
|-
|Non-clinical; adolescents
|.7% (point; age 15), 1.8% (lifetime, age 15), .00% (point, age 18), 2.6% (lifetime, age 18), .9% (3 years) Female
.00% (point &amp; lifetime) Males
|Adolescents in a comprehensive school in Ostrobothnia district in Finland
|RAB-T&lt;sub&gt;11&lt;/sub&gt;&amp;nbsp;&amp; RAB-R&lt;sub&gt;12&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Australia'''
|-
|Non-clinical; adolescents
|.00% (full), 1.8% (partial) Female
|Adolescent females residing in Victoria, Australia
|BET&lt;sub&gt;13&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Central &amp; South America'''
|-
|Non-clinical; college students
|.00% Female
|Mexican first &amp; second year college females
|EAT-40&lt;sub&gt;3&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''East Asia'''
|-
|Clinical; eating disorder patients
|.53% Female
|Adolescent and adult Japanese patients at a university hospital
|DSM-III-R&lt;sub&gt;4&lt;/sub&gt;
|-
|Non-clinical; Korean Epidemiologic Catchment Area (KECA) Study
|.1% (lifetime), .1% (12 months) Female
.2% (lifetime), .00% (12 months)

Males
|Korean adults
|K-CIDI&lt;sub&gt;15&lt;/sub&gt;&amp;nbsp;2.1
|-
|Centers participating in the Collaborative Study on the Genetics of  Alcoholism in San Diego; St. Louis; Iowa City; Farmington, CN; New York; &amp; Indianapolis
|1.41% Females
|Alcohol-dependent adults
|Semi-Structured Assessment for the Genetics of Alcoholism; criteria based on DSM-III-R
|}

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==

=== Psychometric properties of screening instruments for anorexia nervosa ===
The following section contains a list of screening and diagnostic instruments for anorexia nerova. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here.]]

* ''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments '''click here.''']''

{| class=&quot;wikitable&quot;
|-
! Measure 
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Inter-rater reliability
!Test-retest reliability
!Construct validity
!Content validity
!Highly recommended
|-
|Structured Clinical Interview Diagnosis for DSM-IV (SCID-IV) Module H

* [https://www.columbiapsychiatry.org/research/research-areas/services-policy-and-law/structured-clinical-interview-dsm-disorders-scid-1 SCID-IV (for purchase)]
* [[wikipedia:Structured_Clinical_Interview_for_DSM-IV|Wiki Page]]
|Interview
(Patient)
|
|
|G
|L
|A
|E
|X
|-
|Eating Disorder Examination (EDE)

*OSF
*[[wikipedia:Eating_Disorder_Examination_Interview|Wiki Page]]
|Interview
(Patient)
|
|45min-1.25hrs
|E
|A
|A
|E
|X
|-
|[http://www.corc.uk.net/media/1273/ede-q_quesionnaire.pdf EDE-Q (Eating Disorder Examination Questionnaire)]
*OSF
*[[wikipedia:Eating_Disorder_Examination_Interview|Wiki Page]]
|Questionnaire

(Patient)
|
|15-20 min
|NA
|L
|U
|E
|
|-
|[http://www.ori.org/files/Static%20Page%20Files/EDDSDSM-5_10_14.pdf EDDS (Eating Disorder Diagnosis Scale)]
*OSF
*[[wikipedia:Eating_Disorder_Diagnostic_Scale|Wiki Page]]
|Questionnaire 
(Patient)
|
|10-15 min
|NA
|A
|A
|G
|X
|}
&lt;ref&gt;Hunsley, J., &amp; Mash, E. J. (2008). Guide to Assessments that Work. Cary, NC, USA: Oxford University Press, USA. Retrieved from &lt;nowiki&gt;http://www.ebrary.com&lt;/nowiki&gt;&lt;/ref&gt;'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Likelihood ratios and AUCs of screening instruments for anorexia nervosa===
* ''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments '''click here.''']''
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! &lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Mintz|first=L. B.|last2=O'Halloran|first2=M. S.|date=June 2000|title=The Eating Attitudes Test: validation with DSM-IV eating disorder criteria|url=https://www.ncbi.nlm.nih.gov/pubmed/10900574|journal=Journal of Personality Assessment|volume=74|issue=3|pages=489–503|doi=10.1207/S15327752JPA7403_11|issn=0022-3891|pmid=10900574}}&lt;/ref&gt;Screening Measure (Primary Reference)
! AUC
! DiLR+ (Score)
! DiLR- (Score)
!Clinical Generalizability
!Download
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''Biological &amp; Physical Measures'''
|
|-
|Serum leptin level&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Föcker|first=M.|last2=Timmesfeld|first2=N.|last3=Scherag|first3=S.|last4=Bühren|first4=K.|last5=Langkamp|first5=M.|last6=Dempfle|first6=A.|last7=Sheridan|first7=E. M.|last8=Zwaan|first8=M. de|last9=Fleischhaker|first9=C.|date=2011-04-01|title=Screening for anorexia nervosa via measurement of serum leptin levels|url=https://link.springer.com/article/10.1007/s00702-010-0551-z|journal=Journal of Neural Transmission|language=en|volume=118|issue=4|pages=571–578|doi=10.1007/s00702-010-0551-z|issn=0300-9564}}&lt;/ref&gt;
|0.984 (N=139)
|14.72 (&lt;2.31)
|0.10 (2.31+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
|Serum leptin standard deviation score (SDS)&lt;ref name=&quot;:7&quot; /&gt;
|0.939 (N=139)
|5.89 (&lt;-0.45)
|0.09 (-0.45+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
|Body Mass Index (BMI)&lt;ref name=&quot;:7&quot; /&gt;
|0.936 (N=139)
|5.89 (&lt;17.10)
|0.11 (17.10+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''REDS-C&lt;sub&gt;1&lt;/sub&gt;'''
|
|-
|REDS-C&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Boraska|first=V.|last2=Franklin|first2=C. S.|last3=Floyd|first3=J. a. B.|last4=Thornton|first4=L. M.|last5=Huckins|first5=L. M.|last6=Southam|first6=L.|last7=Rayner|first7=N. W.|last8=Tachmazidou|first8=I.|last9=Klump|first9=K. L.|date=October 2014|title=A genome-wide association study of anorexia nervosa|url=https://www.ncbi.nlm.nih.gov/pubmed/24514567|journal=Molecular Psychiatry|volume=19|issue=10|pages=1085–1094|doi=10.1038/mp.2013.187|issn=1476-5578|pmc=PMC4325090|pmid=24514567}}&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt;
|0.658 (N=333), 0.624 (N=236, &gt; 13 years old), 0.772 (N=97, ≤ 13 years old)
| --
| --
|Children and adolescents (8-18 years old) at a clinic being treated for eating and weight concerns. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDI&lt;sub&gt;2&lt;/sub&gt;'''
|
|-
|EDI-2 &lt;ref name=&quot;:3&quot;&gt;Garner, D. M. (1991). ''Manual for Eating Disorder Inventory.'' Odessa, FL: Psychological Assessment Resources, Inc.&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt; 
|0.556 (N=77)
| --
| --
|Children and adolescents (10-18 years old) at a clinic being treated for eating and weight concerns. Eating disorders were not separated.
|
|-
|EDI-2 - Drive for Thinness Sub-scale&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Segura-Garcia|first=Cristina|last2=Ramacciotti|first2=Carla|last3=Rania|first3=Marianna|last4=Aloi|first4=Matteo|last5=Caroleo|first5=Mariarita|last6=Bruni|first6=Antonella|last7=Gazzarrini|first7=Denise|last8=Sinopoli|first8=Flora|last9=De Fazio|first9=Pasquale|date=June 2015|title=The prevalence of orthorexia nervosa among eating disorder patients after treatment|url=https://www.ncbi.nlm.nih.gov/pubmed/25543324|journal=Eating and weight disorders: EWD|volume=20|issue=2|pages=161–166|doi=10.1007/s40519-014-0171-y|issn=1590-1262|pmid=25543324}}&lt;/ref&gt; 
|0.97 (N=92)
|29.39 (14+)
|0.274 (DT≤14)
|Italian individuals with a DSM-IV-TR eating disorder versus individuals at risk for eating disorders. Eating disorders were not separated.
|
|-
|EDI-3 - Drive for Thinness Subscale (Garner, 2004)

Lehmann et al., 2013
|.903 (N=1298)
| --
| --
|&amp;nbsp;
|
|-
|EDI-3 - Eating Disorder Risk Composite &lt;ref name=&quot;:9&quot; /&gt;(EDRC) (Garner, 2004)
|0.942 (N=92)
|52.08 (EDRC 75+, adults; 90+ adolescents)
|0.379 (EDRC≤75, adults; ≤90 adolescents)
|Italian individuals with a DSM-IV-TR eating disorder versus individuals at risk for eating disorders. Eating disorders were not separated.
|
|-
|EDI-3 - Interoceptive Deficits Subscale &lt;ref name=&quot;:10&quot;&gt;{{Cite journal|last=Clausen|first=Loa|last2=Rosenvinge|first2=Jan H.|last3=Friborg|first3=Oddgeir|last4=Rokkedal|first4=Kristian|date=2011-03-01|title=Validating the Eating Disorder Inventory-3 (EDI-3): A Comparison Between 561 Female Eating Disorders Patients and 878 Females from the General Population|url=https://link.springer.com/article/10.1007/s10862-010-9207-4|journal=Journal of Psychopathology and Behavioral Assessment|language=en|volume=33|issue=1|pages=101–110|doi=10.1007/s10862-010-9207-4|issn=0882-2689}}&lt;/ref&gt;(Garner, 2004)
|.911 (N=2561)
|4.0 (9+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interoceptive Deficits Subscale (Garner, 2004)

Lehmann et al., 2013
|.901 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Asceticism Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.886 (N=2561)
|6.5 (9+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Asceticism Subscale (Garner, 2004)

Lehmann et al., 2013
|.902 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Low Self-Esteem Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.884 (N=2561)
|5.8 (10+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited fm an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Low Self-Esteem Subscale (Garner, 2004)

Lehmann et al., 2013
|.906 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Personal Alienation Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.88 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Personal Alienation Subscale (Garner, 2004)

Lehmann et al., 2013
|.899 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Emotional Dysregulation Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.81 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Emotional Dysregulation Subscale (Garner, 2004)

Lehmann et al., 2013
|.779 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Interpersonal Alienation Subscale &lt;ref name=&quot;:10&quot; /&gt;(Garner, 2004)
|.79 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interpersonal Alienation Subscale (Garner, 2004)

Lehmann et al., 2013
|.743 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Perfectionism Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.79 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Perfectionism Subscale (Garner, 2004)

Lehmann et al., 2013
|.768 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Maturity Fears Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.77 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Maturity Fears Subscale (Garner, 2004)

Lehmann et al., 2013
|.678 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Bulimia Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.76 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Bulimia Subscale (Garner, 2004)

Lehmann et al., 2013
|.776 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Interpersonal Insecurity Subscale &lt;ref name=&quot;:10&quot; /&gt;(Garner, 2004)
|.76 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interpersonal Insecurity Subscale (Garner, 2004)

Lehmann et al., 2013
|.697 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Body Dissatisfaction Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.722 (N=2561)
|1.7 (15+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Body Dissatisfaction Subscale (Garner, 2004)

Lehmann et al., 2013
|.849 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EAT&lt;sub&gt;3&lt;/sub&gt;'''
|
|-
|EAT-26&amp;nbsp;&lt;ref name=&quot;:4&quot; /&gt;&lt;ref&gt;{{Cite journal|title = The eating attitudes test: psychometric features and clinical correlates|url = http://www.ncbi.nlm.nih.gov/pubmed/6961471|journal = Psychological Medicine|date = 1982-11-01|issn = 0033-2917|pmid = 6961471|pages = 871–878|volume = 12|issue = 4|first = D. M.|last = Garner|first2 = M. P.|last2 = Olmsted|first3 = Y.|last3 = Bohr|first4 = P. E.|last4 = Garfinkel}}&lt;/ref&gt;
|.90 (N=129)
|12.83 (20+)
|.24 (&lt;20)
|Low-moderate: College women with no eating disorder versus college women with a DSM-IV eating disorder. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDE-Q&lt;sub&gt;4&lt;/sub&gt;'''
|
|-
|EDE-Q&amp;nbsp;&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|last=Aardoom|first=Jiska J.|last2=Dingemans|first2=Alexandra E.|last3=Slof Op't Landt|first3=Margarita C. T.|last4=Van Furth|first4=Eric F.|date=December 2012|title=Norms and discriminative validity of the Eating Disorder Examination Questionnaire (EDE-Q)|url=https://www.ncbi.nlm.nih.gov/pubmed/23121779|journal=Eating Behaviors|volume=13|issue=4|pages=305–309|doi=10.1016/j.eatbeh.2012.09.002|issn=1873-7358|pmid=23121779}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|title = Assessment of eating disorders: interview or self-report questionnaire?|url = http://www.ncbi.nlm.nih.gov/pubmed/7866415|journal = The International Journal of Eating Disorders|date = 1994-12-01|issn = 0276-3478|pmid = 7866415|pages = 363–370|volume = 16|issue = 4|first = C. G.|last = Fairburn|first2 = S. J.|last2 = Beglin}}&lt;/ref&gt;
|.96 (N=1170)
| --
| --
|Moderate: Dutch treatment-seeking females meeting DSM-IV criteria for an eating disorder versus female adult general population sample recruited through advertisements and personal contacts. Eating disorders were not separated.
|
|-
|EDE-Q&amp;nbsp;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Fairburn|first=C. G.|last2=Beglin|first2=S. J.|date=December 1994|title=Assessment of eating disorders: interview or self-report questionnaire?|url=https://www.ncbi.nlm.nih.gov/pubmed/7866415|journal=The International Journal of Eating Disorders|volume=16|issue=4|pages=363–370|issn=0276-3478|pmid=7866415}}&lt;/ref&gt;
| --
|6.57 (2.3+)
|0.09 (&lt;2.3)
|Moderate: “Clinically significant eating disorder” from a community sample versus female adultsindividuals without “clinically significant eating disorder” from same sample. Eating disorders were not separated.
|
|-
|EDE-Q &lt;ref name=&quot;:1&quot; /&gt;&lt;ref&gt;Rø, Ø., Reas, D. L., &amp; Stedal, K. (2015). Eating Disorder Examination Questionnaire (EDE‐Q) in Norwegian Adults: Discrimination between Female Controls and Eating Disorder Patients. ''European Eating Disorders Review'', ''23''(5), 408-412.&lt;/ref&gt;
|0.89 (N=2465)
| --
| --
|Norwegian adult females with an eating disorder recruited from eating disorder specialist centers versus controls determined using the DSM-IV.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDQ-O&lt;sub&gt;5&lt;/sub&gt;'''
|
|-
|EDQ-Q&amp;nbsp;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;
|0.72 (N=134)
|1.00 (all criteria met)
|0.92 (≥1 criteria not met)
|Dutch adults recruited from an eating disorder specialist center determined using the DSM-IV-TR.
|
|}

=== Interpreting anorexia nervosa screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]
* Also see the page on [https://en.wikipedia.org/wiki/Likelihood_ratios_in_diagnostic_testing likelihood ratios in diagnostic testing] for more information.

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==
===Gold standard diagnostic interviews===

* For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

===Recommended diagnostic interviews for anorexia nervosa===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! colspan=&quot;10&quot; |Diagnostic instruments for anorexia nervosa
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| [https://dawba.info/a0.html Development and Well-Being Assessment (DAWBA)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
| Structured; child/adolescent and parent versions
| 11-18
|10-20 minutes for the eating disorder module
|
|
|
|
|
|
*
|-
|[https://www.phenxtoolkit.org/protocols/view/230101 Children's Eating Disorder Examination- Child (ChEDE)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
|Semistructured; child version
|8-16 
|45-75 minutes; mean 1 hour
|E
|
|
|
|
|
|-
|[https://www.credo-oxford.com/7.2.html Eating Disorder Examination (EDE)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt; &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|date=2008|publisher=Oxford University Press|others=John Hunsley, Eric J. Mash|isbn=0-19-804245-0|location=New York|oclc=314222270}}&lt;/ref&gt;
|Semistructured; adolescent version
|16+
|45-75 minutes
|E
|A
|A
|E
|X
|
|-
|[https://mccaed.slam.nhs.uk/wp-content/uploads/2022/02/PARDI-Self-14-V.2.1-Oct-2020.pdf Pica, ARFID, and Rumination Disorder Interview] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
|Semistructured; multi-informant and different versions
|2-22
|Mean of 39 minutes
|G (ARFID diagnosis)
|
|
|
|
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Severity interviews for anorexia nervosa===
{| class=&quot;wikitable&quot; border=&quot;1&quot;
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
!Free and Accessible Measures
|-
| Placeholder example (CDRS-R)
| Structured Interview&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Mayes|first=Taryn L.|last2=Bernstein|first2=Ira H.|last3=Haley|first3=Charlotte L.|last4=Kennard|first4=Betsy D.|last5=Emslie|first5=Graham J.|date=2010-12|title=Psychometric Properties of the Children's Depression Rating Scale–Revised in Adolescents|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3003451/|journal=Journal of Child and Adolescent Psychopharmacology|volume=20|issue=6|pages=513–516|doi=10.1089/cap.2010.0063|issn=1044-5463|pmc=PMC3003451|pmid=21186970}}&lt;/ref&gt;
| 6-12
| 15-20 minutes
|
* Link to purchase [https://www.wpspublish.com/store/p/2703/cdrs-r-children-s-depression-rating-scale-revised#purchase-product CDRS-R]
*[http://www.opapc.com/uploads/documents/CDRS-R.pdf PDF] (excerpt)
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a brief overview of treatment options for anorexia nervosa and a list of process and outcome measures for anorexia nervosa. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information on the differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]].

===Process measures===
Body weight is commonly monitored by clinicians throughout the AN treatment process as helping individuals regain and maintain a healthy weight is a primary treatment goal for AN. Many treatment centers have policies prohibiting patients from seeing their weight.

Motivational Stages of Change may be used to monitor individuals’ readiness to take action against eating disorder behaviors. It has demonstrated predictive validity in a sample of female adolescents attending eating disorder treatment groups.&lt;ref&gt;{{Cite journal|last=Gusella|first=Joanne|last2=Butler|first2=Gordon|last3=Nichols|first3=Laura|last4=Bird|first4=Debbie|date=2003-01-01|title=A brief questionnaire to assess readiness to change in adolescents with eating disorders: its applications to group therapy|url=http://onlinelibrary.wiley.com/doi/10.1002/erv.481/abstract|journal=European Eating Disorders Review|language=en|volume=11|issue=1|pages=58–71|doi=10.1002/erv.481|issn=1099-0968}}&lt;/ref&gt; See Appendix E.

=== Outcome and severity measures ===
This table includes clinically significant benchmarks for anorexia nervosa specific outcome measures
* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
*''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
| colspan=&quot;8&quot; |'''Clinically significant change benchmarks with common instruments for anorexia nervosa'''
|-
| colspan=&quot;8&quot;  span style=&quot;font-size:110%; text-align:center;&quot; | &lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; |&lt;b&gt; Measure&lt;/b&gt;
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; | &lt;b&gt; Subscale&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; | &lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; | &lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-
| rowspan=&quot;5&quot; style=&quot;text-align:center;&quot; |'''EDE-Q &lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Mond|first=J. M.|last2=Hay|first2=P. J.|last3=Rodgers|first3=B.|last4=Owen|first4=C.|last5=Beumont|first5=P. J. V.|date=May 2004|title=Validity of the Eating Disorder Examination Questionnaire (EDE-Q) in screening for eating disorders in community samples|url=https://www.ncbi.nlm.nih.gov/pubmed/15033501|journal=Behaviour Research and Therapy|volume=42|issue=5|pages=551–567|doi=10.1016/S0005-7967(03)00161-X|issn=0005-7967|pmid=15033501}}&lt;/ref&gt;'''
| style=&quot;text-align:right;&quot; |''Global''
|1.4
|3.2
|2.3
|.7
|.6
|.3
|-
| style=&quot;text-align:right;&quot; |''Restraint''
|(-.3)
|3.6
|1.8
|1.5
|1.2
|.8
|-
| style=&quot;text-align:right;&quot; |''Eating Concern''
|.1
|2.0
|1.2
|1.1
|.9
|.6
|-
| style=&quot;text-align:right;&quot; |''Weight Concern''
|1.5
|3.9
|2.6
|1.0
|.9
|.5
|-
| style=&quot;text-align:right;&quot; |''Shape Concern''
|2.1
|4.8
|3.2
|.9
|.7
|.4
|-
| rowspan=&quot;5&quot; style=&quot;text-align:center;&quot; |'''EDE &lt;ref name=&quot;:2&quot; /&gt;'''
| style=&quot;text-align:right;&quot; |''Global''
|1.7
|2.3
|1.9
|1.9
|1.6
|1.0
|-
| style=&quot;text-align:right;&quot; |''Restraint''
|.3
|3.3
|1.9
|1.8
|1.5
|.9
|-
| style=&quot;text-align:right;&quot; |''Eating Concern''
|(-.5)
|.9
|.5
|.8
|.7
|.4
|-
| style=&quot;text-align:right;&quot; |''Weight Concern''
|2.0
|2.8
|2.4
|1.3
|1.1
|.7
|-
| style=&quot;text-align:right;&quot; |''Shape Concern''
|2.0
|3.2
|2.6
|1.2
|1.0
|.6
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; |'''EAT-26 &lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Mintz|first=L. B.|last2=O'Halloran|first2=M. S.|date=2000-06-01|title=The Eating Attitudes Test: validation with DSM-IV eating disorder criteria|url=http://www.ncbi.nlm.nih.gov/pubmed/10900574|journal=Journal of Personality Assessment|volume=74|issue=3|pages=489–503|doi=10.1207/S15327752JPA7403_11|issn=0022-3891|pmid=10900574}}&lt;/ref&gt;'''
| style=&quot;text-align:right;&quot; |''Total''
|6.5
|19.6
|15.0
|7.9
|6.7
|4.0
|}

'''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean.

'''Search terms:''' [Anorexia Nervosa OR eating disorder] AND [validity OR clinical significance] in Google Scholar

=== Treatment ===
{{collapse top| Click here for more information on treatment for AN}}
* Treatment of AN typically consists of restoring the individual to a healthy weight and addressing thoughts and behaviors which are related to the eating disorder. It may involve re-nutrition, psychotherapy, nutritional counseling, and medication. 
* Literature reviews of existing research indicate that evidence supporting AN treatment is lacking. A systematic review of AN treatment efficacy studies by Bulik&lt;ref&gt;{{Cite journal|title = Anorexia nervosa treatment: a systematic review of randomized controlled trials|url = http://www.ncbi.nlm.nih.gov/pubmed/17370290|journal = The International Journal of Eating Disorders|date = 2007-05-01|issn = 0276-3478|pmid = 17370290|pages = 310–320|volume = 40|issue = 4|doi = 10.1002/eat.20367|first = Cynthia M.|last = Bulik|first2 = Nancy D.|last2 = Berkman|first3 = Kimberly A.|last3 = Brownley|first4 = Jan A.|last4 = Sedway|first5 = Kathleen N.|last5 = Lohr}}&lt;/ref&gt; found that evidence supporting medications, medications and behavioral interventions, and behavioral interventions alone in adults is weak. 
* There is moderately strong evidence suggesting that behavioral interventions may be helpful for adolescents. In particular, adolescents may benefit from family therapy. 
* Clinical trials investigating AN treatment suffer from high rates of attrition, as key features of AN (e.g., denial, fear of weight gain) may contribute to low motivation for remaining in treatment.
* More severe cases of AN may be treated in inpatient settings, which are equipped to manage the re-nutrition process and provide medical monitoring. 
* Partial hospitalization and intensive outpatient programs may provide intermediate levels of treatment intensity to assist individuals in the transition from intensive care to outpatient care after weight restoration.
{{collapse bottom}}
* Please refer to the page on [https://en.wikipedia.org/wiki/Anorexia_nervosa anorexia nervosa] for more information on available treatment or go to [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/eating-body-image-problems/ the Effective Child Therapy page for Eating &amp; Body Image Problems] for a curated resource on effective treatments for anorexia nervosa.

== '''External resources''' ==
# [http://apps.who.int/classifications/icd10/browse/2010/en#/F50.0 ICD-10 diagnostic criteria]
# [https://en.wikiversity.org/w/index.php?title=Helping_Give_Away_Psychological_Science/Resources/Annotated_List_of_Where_and_How_to_Find_a_Therapist&amp;wteswitched=1#Other_low-cost_options Find-a-Therapist] 
##This is a curated list of find-a-therapist websites where you can find a provider
# NIMH: [https://www.nimh.nih.gov/health/publications/eating-disorders/index.shtml ''Eating Disorders--About More Than Food'']  and [https://www.nimh.nih.gov/health/topics/eating-disorders/index.shtml ''Eating Disorders'']
##These NIMH website posts provide more information on anorexia nervosa
#[https://www.hopkinsmedicine.org/psychiatry/specialty_areas/eating_disorders/ John's Hopkins Resource] (guide about anorexia nervosa, treatment, and more)
# OMIM (Online Mendelian Inheritance in Man)
##[https://www.omim.org/entry/606788?search=anorexia%20nervosa&amp;highlight=nervosa%20anorexia Anorexia nervosa]
#[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/eating-body-image-problems/ Effective Child Therapy page for anorexia nervosa]
##Effective Child Therapy is website sponsored by Division 53 of the American Psychological Association (APA), or The [https://sccap53.org Society of Clinical Child and Adolescent Psychology] (SCCAP), in collaboration with the Association for Behavioral and Cognitive Therapies (ABCT). Use for information on symptoms and available treatments.

== '''References''' ==
{{collapse top|Click here for references}}
{{Reflist|30em}}

[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]
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==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
For background information on what assessment portfolios are, click the link in the heading above.
Want even 'more' information about this topic? There's an extended version of this page [Evidence-based assessment/Anorexia nervosa (assessment portfolio)/extended version|here]

== [[Evidence based assessment/Preparation phase|'''Preparation phase''']] ==
=== Diagnostic criteria for anorexia nervosa ===
{{blockquotetop}}
'''ICD-11 Criteria'''

Anorexia Nervosa is characterized by significantly low body weight, which is less than minimal normal/expected weight for the individual’s height, sex, age and developmental stage (body mass index (BMI) less than 18.5 kg/m2 in adults and BMI-for-age under fifth percentile in children and adolescents) that is not due to another health condition or to the unavailability of food. Low body weight is accompanied by a persistent pattern of behaviors to prevent restoration of normal weight, which may include behaviors aimed at reducing energy intake (restricted eating), purging behaviors (e.g., self-induced vomiting, misuse of laxatives), and behaviors aimed at increasing energy expenditure (e.g., excessive exercise), typically associated with a fear of weight gain. Low body weight or shape is central to the person's self-evaluation or is inaccurately perceived to be normal or even excessive.

'''Changes in DSM-5'''

* The diagnostic criteria for anorexia nervosa changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here] and [[w:DSM-5|here]].
{{blockquotebottom}}

=== Base rates of anorexia nervosa in different populations and clinical settings ===
This section describes the demographic setting of the population(s) sampled, base rates of diagnosis such as prevalence rates, country/region sampled, and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of adolescent depression that they are likely to see in their clinical practice.

* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']]

{| class=&quot;wikitable&quot;
|-
! Setting (Reference) 
! Base Rate
! Demography
! Diagnostic Method
|-
|Non-clinical: Population-based (NCS-R)&lt;ref&gt;Hudson, James I.; Hiripi, Eva; Pope, Harrison G.; Kessler, Ronald C. (2007-02-01). &quot;The prevalence and correlates of eating disorders in the National Comorbidity Survey Replication&quot;.''Biological Psychiatry'' '''61''' (3): 348–358. doi:10.1016/j.biopsych.2006.03.040. ISSN 0006-3223. PMC 1892232. &lt;nowiki&gt;PMID 16815322&lt;/nowiki&gt;&lt;/ref&gt;
|.9% Female, .3% Male
|Nationally representative US sample of adults
|World Health Organization-Clinical International Diagnostic Interview (WHO-CIDI)
|-
|Non-clinical: Population-based (NCS-A)&lt;ref&gt;Swanson, Sonja A.; Crow, Scott J.; Le Grange, Daniel; Swendsen, Joel; Merikangas, Kathleen R. (2011-07-01). &quot;Prevalence and correlates of eating disorders in adolescents. Results from the national comorbidity survey replication adolescent supplement&quot;. ''Archives of General Psychiatry'' '''68''' (7): 714–723. doi:10.1001/archgenpsychiatry.2011.22. ISSN 1538-3636.&lt;nowiki&gt;PMID 21383252&lt;/nowiki&gt;&lt;/ref&gt;
|.3% Female, .3% Male
|Nationally representative US sample of adolescents
|WHO-CIDI
|-
|Non-clinical: US Population-based prevalence (Adolescent Brain Cognitive Development (ABCD) study&lt;ref&gt;{{Cite journal|last=Rozzell|first=Kaitlin|last2=Moon|first2=Da Yeoun|last3=Klimek|first3=Patrycja|last4=Brown|first4=Tiffany|last5=Blashill|first5=Aaron J.|date=2019-01-01|title=Prevalence of Eating Disorders Among US Children Aged 9 to 10 Years: Data From the Adolescent Brain Cognitive Development (ABCD) Study|url=http://archpedi.jamanetwork.com/article.aspx?doi=10.1001/jamapediatrics.2018.3678|journal=JAMA Pediatrics|language=en|volume=173|issue=1|pages=100|doi=10.1001/jamapediatrics.2018.3678|issn=2168-6203|pmc=PMC6583451|pmid=30476983}}&lt;/ref&gt;
|0.1%, no gender differences
|Nationally representative US sample of 9- and 10-year old children
|DSM-5 using Kiddie Schedule for Affective Disorders and Schizophrenia (KSADS)
|-
|Non-clinical: Latinos in US&lt;ref&gt;Alegria, Margarita; Woo, Meghan; Cao, Zhun; Torres, Maria; Meng, Xiao-li; Striegel-Moore, Ruth (2007-11-01). &quot;Prevalence and correlates of eating disorders in Latinos in the United States&quot;.''The International Journal of Eating Disorders''. 40 Suppl: S15–21. doi:10.1002/eat.20406. ISSN 0276-3478. PMC 2680162. &lt;nowiki&gt;PMID 17584870&lt;/nowiki&gt;&lt;/ref&gt;
|.12% Female, .03% Male
|Latino Households in the US
|WHO-CIDI
|-
|Non-clinical: African Americans and Caribbean Blacks in the US (NSAL)&lt;ref&gt;Taylor, Jacquelyn Y.; Caldwell, Cleopatra Howard; Baser, Raymond E.; Faison, Nakesha; Jackson, James S. (2007-11-01). &quot;Prevalence of eating disorders among Blacks in the National Survey of American Life&quot;. ''The International Journal of Eating Disorders''. 40 Suppl: S10–14. doi:10.1002/eat.20451. ISSN 0276-3478. PMC 2882704. &lt;nowiki&gt;PMID 17879287&lt;/nowiki&gt;&lt;/ref&gt;
|.14% Female, .2% Male
|National probability sample of adult and adolescent African Americans and Caribbean Blacks
|WHO-CIDI
|-
|Non-clinical: National Latino and Asian American Study (NLAAS)
|.12% Female, .05% Male
|Asian American adults in US households
|WHO-CIDI
|-
| Non-clinical; college students&lt;ref&gt;Tyler ID. A true picture of eating disorders among African American women: a review of literature. ABNF J. 2003;14(3):73-4.&lt;/ref&gt; || .0%  || US African American college females || Eating Disorder Diagnostic Questionnaire (EDD-Q)
|-
| Non-clinical; Adolescents&lt;ref&gt;Stice, E., Becker, C. B., &amp; Yokum, S. (2013). Eating disorder prevention: Current evidence-base and future directions. ''Int. J. Eat. Disord. International Journal of Eating Disorders,'' ''46''(5), 478-485.&lt;/ref&gt; || .8% || US Female Adolescents || Eating Disorder Diagnosis Interview (EDDI)
|-
| Non-clinical; student-athletes&lt;ref&gt;Johnson C, Powers PS, Dick R. Athletes and eating disorders: the national collegiate athletic association study. Int J Eat Disord 1999;26:179e88.&lt;/ref&gt; || .0% || US Division-I Varsity Student Athletes || Eating Disorder Inventory-2 (EDI-2)
|-
| Non-clinical; Military&lt;ref&gt;McNulty, PAF. (2001). Prevalence and contributing factors of eating disorder behaviors in active duty service women in the Army, Navy, Air Force and Marines. Military Medicine, 166(1), 53-58. &lt;/ref&gt; || 1.1% || Active duty females in US Army, Navy, 
Airforce, and Marines 
|| EDI-2
|-
|Non-clinical; Commonwealth of Virginia Mid-Atlantic Twin Registry (MATR)&lt;ref&gt;Kendler KS, Walters EE, Neale MC, Kessler R, Heath A, Eaves L. The structure of genetic and environmental risk factors for six major psychiatric disorders in women. Archives of general psychiatry. 1995;52:374–383.&lt;/ref&gt;
|1.62% (narrow), 3.70% (broad)
|US Caucasian female same-sex twins
|Structured Clinical Interview for DSM Disorders (SCID)
|-
|Non-clinical; Health Omnibus Survey (HOS)
|.46% (3 months; combined)
|South Australian older adolescents and adults
|Eating Disorder Examination (EDE)
|-
|Military&lt;ref&gt;{{Cite journal|title=Diagnosed eating disorders in the U.S. Military: a nine year review|url=http://www.ncbi.nlm.nih.gov/pubmed/18821361|journal=Eating Disorders|date=2008-12-01|issn=1532-530X|pmid=18821361|pages=363–377|volume=16|issue=5|doi=10.1080/10640260802370523|first=Amanda J.|last=Antczak|first2=Teresa L.|last2=Brininger}}&lt;/ref&gt;
|.04% (combined)
|US Military
|ICD codes from electronic records
|-
|Non-clinical; Military
|.02% (7 years) Female, 
0.0% (7 years) Males
|US Military Academy cadets
|[https://www.eat-26.com/ Eating Attitudes Test- 26 items] (EAT-26)&lt;ref&gt;{{Cite web|url=https://www.eat-26.com/|title=EAT-26: Eating Attitudes Test &amp; Eating Disorder Testing – Use the EAT-26 to help you determine if you need to speak to a mental health professional to get help for an eating disorder.|last=admin|language=en|access-date=2022-05-30}}&lt;/ref&gt;
|-
|Non-clinical; Military
|1.1% (current &amp; past) Female
|US Navy female nurses
|DSM-III
|-
|Non-clinical; Military
|.04% Female, .005% Males
|US veterans
|ICD-9-CM
|-
|Non-clinical; Military
|2.5% Males
|Active duty males in US Navy
|N/A
|-
|Clinical; Collaborative Study on the Genetics of Alcoholism (COGA)&lt;ref&gt;{{Cite journal|title=Anorexia nervosa and bulimia nervosa in alcohol-dependent men and women and their relatives|url=http://www.ncbi.nlm.nih.gov/pubmed/8540597|journal=The American Journal of Psychiatry|date=1996-01-01|issn=0002-953X|pmid=8540597|pages=74–82|volume=153|issue=1|doi=10.1176/ajp.153.1.74|first=M. A.|last=Schuckit|first2=J. E.|last2=Tipp|first3=R. M.|last3=Anthenelli|first4=K. K.|last4=Bucholz|first5=V. M.|last5=Hesselbrock|first6=J. I.|last6=Nurnberger}}&lt;/ref&gt;
|1.41% (lifetime) Female,
.00% (lifetime) Male
|US alcohol-dependent adults from San Diego, St. Lois, Iowa City, Farmington, New York, &amp; Indianapolis
|SSAGA
|-
|Non-clinical; healthcare members
|.0269% (current)
|Healthcare provider records
|
|-
|Non-clinical; high school students
|. 00% (point), .45% (lifetime) Female
.00% (point), .00% (lifetime) Males
|US high school students in west central Oregon
|DSM-III-R&lt;sub&gt;4&lt;/sub&gt;
|-
|Clinical; substance users
|.4% (lifetime), .3% (current) Female
.4% (lifetime), .3% (current) Males
|Canadian treatment-seeking substance users
|DIS&lt;sub&gt;9&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Europe'''
|-
|Non-clinical; adolescents
|.3% Female
|Adolescent females residing in Navarra, Spain
|EAT-40&lt;sub&gt;3&lt;/sub&gt;
|-
|Non-clinical; adolescents
|.7% (lifetime) Female, .2% Male
|Adolescents in secondary schools in Sør-Trøndelag, County in Norway
|SEDs&lt;sub&gt;10&lt;/sub&gt;
|-
|Non-clinical; adolescents
|.7% (point; age 15), 1.8% (lifetime, age 15), .00% (point, age 18), 2.6% (lifetime, age 18), .9% (3 years) Female
.00% (point &amp; lifetime) Males
|Adolescents in a comprehensive school in Ostrobothnia district in Finland
|RAB-T&lt;sub&gt;11&lt;/sub&gt;&amp;nbsp;&amp; RAB-R&lt;sub&gt;12&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Australia'''
|-
|Non-clinical; adolescents
|.00% (full), 1.8% (partial) Female
|Adolescent females residing in Victoria, Australia
|BET&lt;sub&gt;13&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Central &amp; South America'''
|-
|Non-clinical; college students
|.00% Female
|Mexican first &amp; second year college females
|EAT-40&lt;sub&gt;3&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''East Asia'''
|-
|Clinical; eating disorder patients
|.53% Female
|Adolescent and adult Japanese patients at a university hospital
|DSM-III-R&lt;sub&gt;4&lt;/sub&gt;
|-
|Non-clinical; Korean Epidemiologic Catchment Area (KECA) Study
|.1% (lifetime), .1% (12 months) Female
.2% (lifetime), .00% (12 months)

Males
|Korean adults
|K-CIDI&lt;sub&gt;15&lt;/sub&gt;&amp;nbsp;2.1
|-
|Centers participating in the Collaborative Study on the Genetics of  Alcoholism in San Diego; St. Louis; Iowa City; Farmington, CN; New York; &amp; Indianapolis
|1.41% Females
|Alcohol-dependent adults
|Semi-Structured Assessment for the Genetics of Alcoholism; criteria based on DSM-III-R
|}

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==

=== Psychometric properties of screening instruments for anorexia nervosa ===
The following section contains a list of screening and diagnostic instruments for anorexia nerova. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here.]]

* ''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments '''click here.''']''

{| class=&quot;wikitable&quot;
|-
! Measure 
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Inter-rater reliability
!Test-retest reliability
!Construct validity
!Content validity
!Highly recommended
|-
|Structured Clinical Interview Diagnosis for DSM-IV (SCID-IV) Module H

* [https://www.columbiapsychiatry.org/research/research-areas/services-policy-and-law/structured-clinical-interview-dsm-disorders-scid-1 SCID-IV (for purchase)]
* [[wikipedia:Structured_Clinical_Interview_for_DSM-IV|Wiki Page]]
|Interview
(Patient)
|
|
|G
|L
|A
|E
|X
|-
|Eating Disorder Examination (EDE)

*OSF
*[[wikipedia:Eating_Disorder_Examination_Interview|Wiki Page]]
|Interview
(Patient)
|
|45min-1.25hrs
|E
|A
|A
|E
|X
|-
|[http://www.corc.uk.net/media/1273/ede-q_quesionnaire.pdf EDE-Q (Eating Disorder Examination Questionnaire)]
*OSF
*[[wikipedia:Eating_Disorder_Examination_Interview|Wiki Page]]
|Questionnaire

(Patient)
|
|15-20 min
|NA
|L
|U
|E
|
|-
|[http://www.ori.org/files/Static%20Page%20Files/EDDSDSM-5_10_14.pdf EDDS (Eating Disorder Diagnosis Scale)]
*OSF
*[[wikipedia:Eating_Disorder_Diagnostic_Scale|Wiki Page]]
|Questionnaire 
(Patient)
|
|10-15 min
|NA
|A
|A
|G
|X
|}
&lt;ref&gt;Hunsley, J., &amp; Mash, E. J. (2008). Guide to Assessments that Work. Cary, NC, USA: Oxford University Press, USA. Retrieved from &lt;nowiki&gt;http://www.ebrary.com&lt;/nowiki&gt;&lt;/ref&gt;'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Likelihood ratios and AUCs of screening instruments for anorexia nervosa===
* ''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments '''click here.''']''
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! &lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Mintz|first=L. B.|last2=O'Halloran|first2=M. S.|date=June 2000|title=The Eating Attitudes Test: validation with DSM-IV eating disorder criteria|url=https://www.ncbi.nlm.nih.gov/pubmed/10900574|journal=Journal of Personality Assessment|volume=74|issue=3|pages=489–503|doi=10.1207/S15327752JPA7403_11|issn=0022-3891|pmid=10900574}}&lt;/ref&gt;Screening Measure (Primary Reference)
! AUC
! DiLR+ (Score)
! DiLR- (Score)
!Clinical Generalizability
!Download
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''Biological &amp; Physical Measures'''
|
|-
|Serum leptin level&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Föcker|first=M.|last2=Timmesfeld|first2=N.|last3=Scherag|first3=S.|last4=Bühren|first4=K.|last5=Langkamp|first5=M.|last6=Dempfle|first6=A.|last7=Sheridan|first7=E. M.|last8=Zwaan|first8=M. de|last9=Fleischhaker|first9=C.|date=2011-04-01|title=Screening for anorexia nervosa via measurement of serum leptin levels|url=https://link.springer.com/article/10.1007/s00702-010-0551-z|journal=Journal of Neural Transmission|language=en|volume=118|issue=4|pages=571–578|doi=10.1007/s00702-010-0551-z|issn=0300-9564}}&lt;/ref&gt;
|0.984 (N=139)
|14.72 (&lt;2.31)
|0.10 (2.31+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
|Serum leptin standard deviation score (SDS)&lt;ref name=&quot;:7&quot; /&gt;
|0.939 (N=139)
|5.89 (&lt;-0.45)
|0.09 (-0.45+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
|Body Mass Index (BMI)&lt;ref name=&quot;:7&quot; /&gt;
|0.936 (N=139)
|5.89 (&lt;17.10)
|0.11 (17.10+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''REDS-C&lt;sub&gt;1&lt;/sub&gt;'''
|
|-
|REDS-C&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Boraska|first=V.|last2=Franklin|first2=C. S.|last3=Floyd|first3=J. a. B.|last4=Thornton|first4=L. M.|last5=Huckins|first5=L. M.|last6=Southam|first6=L.|last7=Rayner|first7=N. W.|last8=Tachmazidou|first8=I.|last9=Klump|first9=K. L.|date=October 2014|title=A genome-wide association study of anorexia nervosa|url=https://www.ncbi.nlm.nih.gov/pubmed/24514567|journal=Molecular Psychiatry|volume=19|issue=10|pages=1085–1094|doi=10.1038/mp.2013.187|issn=1476-5578|pmc=PMC4325090|pmid=24514567}}&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt;
|0.658 (N=333), 0.624 (N=236, &gt; 13 years old), 0.772 (N=97, ≤ 13 years old)
| --
| --
|Children and adolescents (8-18 years old) at a clinic being treated for eating and weight concerns. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDI&lt;sub&gt;2&lt;/sub&gt;'''
|
|-
|EDI-2 &lt;ref name=&quot;:3&quot;&gt;Garner, D. M. (1991). ''Manual for Eating Disorder Inventory.'' Odessa, FL: Psychological Assessment Resources, Inc.&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt; 
|0.556 (N=77)
| --
| --
|Children and adolescents (10-18 years old) at a clinic being treated for eating and weight concerns. Eating disorders were not separated.
|
|-
|EDI-2 - Drive for Thinness Sub-scale&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Segura-Garcia|first=Cristina|last2=Ramacciotti|first2=Carla|last3=Rania|first3=Marianna|last4=Aloi|first4=Matteo|last5=Caroleo|first5=Mariarita|last6=Bruni|first6=Antonella|last7=Gazzarrini|first7=Denise|last8=Sinopoli|first8=Flora|last9=De Fazio|first9=Pasquale|date=June 2015|title=The prevalence of orthorexia nervosa among eating disorder patients after treatment|url=https://www.ncbi.nlm.nih.gov/pubmed/25543324|journal=Eating and weight disorders: EWD|volume=20|issue=2|pages=161–166|doi=10.1007/s40519-014-0171-y|issn=1590-1262|pmid=25543324}}&lt;/ref&gt; 
|0.97 (N=92)
|29.39 (14+)
|0.274 (DT≤14)
|Italian individuals with a DSM-IV-TR eating disorder versus individuals at risk for eating disorders. Eating disorders were not separated.
|
|-
|EDI-3 - Drive for Thinness Subscale (Garner, 2004)

Lehmann et al., 2013
|.903 (N=1298)
| --
| --
|&amp;nbsp;
|
|-
|EDI-3 - Eating Disorder Risk Composite &lt;ref name=&quot;:9&quot; /&gt;(EDRC) (Garner, 2004)
|0.942 (N=92)
|52.08 (EDRC 75+, adults; 90+ adolescents)
|0.379 (EDRC≤75, adults; ≤90 adolescents)
|Italian individuals with a DSM-IV-TR eating disorder versus individuals at risk for eating disorders. Eating disorders were not separated.
|
|-
|EDI-3 - Interoceptive Deficits Subscale &lt;ref name=&quot;:10&quot;&gt;{{Cite journal|last=Clausen|first=Loa|last2=Rosenvinge|first2=Jan H.|last3=Friborg|first3=Oddgeir|last4=Rokkedal|first4=Kristian|date=2011-03-01|title=Validating the Eating Disorder Inventory-3 (EDI-3): A Comparison Between 561 Female Eating Disorders Patients and 878 Females from the General Population|url=https://link.springer.com/article/10.1007/s10862-010-9207-4|journal=Journal of Psychopathology and Behavioral Assessment|language=en|volume=33|issue=1|pages=101–110|doi=10.1007/s10862-010-9207-4|issn=0882-2689}}&lt;/ref&gt;(Garner, 2004)
|.911 (N=2561)
|4.0 (9+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interoceptive Deficits Subscale (Garner, 2004)

Lehmann et al., 2013
|.901 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Asceticism Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.886 (N=2561)
|6.5 (9+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Asceticism Subscale (Garner, 2004)

Lehmann et al., 2013
|.902 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Low Self-Esteem Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.884 (N=2561)
|5.8 (10+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited fm an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Low Self-Esteem Subscale (Garner, 2004)

Lehmann et al., 2013
|.906 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Personal Alienation Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.88 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Personal Alienation Subscale (Garner, 2004)

Lehmann et al., 2013
|.899 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Emotional Dysregulation Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.81 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Emotional Dysregulation Subscale (Garner, 2004)

Lehmann et al., 2013
|.779 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Interpersonal Alienation Subscale &lt;ref name=&quot;:10&quot; /&gt;(Garner, 2004)
|.79 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interpersonal Alienation Subscale (Garner, 2004)

Lehmann et al., 2013
|.743 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Perfectionism Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.79 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Perfectionism Subscale (Garner, 2004)

Lehmann et al., 2013
|.768 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Maturity Fears Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.77 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Maturity Fears Subscale (Garner, 2004)

Lehmann et al., 2013
|.678 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Bulimia Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.76 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Bulimia Subscale (Garner, 2004)

Lehmann et al., 2013
|.776 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Interpersonal Insecurity Subscale &lt;ref name=&quot;:10&quot; /&gt;(Garner, 2004)
|.76 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interpersonal Insecurity Subscale (Garner, 2004)

Lehmann et al., 2013
|.697 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Body Dissatisfaction Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.722 (N=2561)
|1.7 (15+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Body Dissatisfaction Subscale (Garner, 2004)

Lehmann et al., 2013
|.849 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EAT&lt;sub&gt;3&lt;/sub&gt;'''
|
|-
|EAT-26&amp;nbsp;&lt;ref name=&quot;:4&quot; /&gt;&lt;ref&gt;{{Cite journal|title = The eating attitudes test: psychometric features and clinical correlates|url = http://www.ncbi.nlm.nih.gov/pubmed/6961471|journal = Psychological Medicine|date = 1982-11-01|issn = 0033-2917|pmid = 6961471|pages = 871–878|volume = 12|issue = 4|first = D. M.|last = Garner|first2 = M. P.|last2 = Olmsted|first3 = Y.|last3 = Bohr|first4 = P. E.|last4 = Garfinkel}}&lt;/ref&gt;
|.90 (N=129)
|12.83 (20+)
|.24 (&lt;20)
|Low-moderate: College women with no eating disorder versus college women with a DSM-IV eating disorder. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDE-Q&lt;sub&gt;4&lt;/sub&gt;'''
|
|-
|EDE-Q&amp;nbsp;&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|last=Aardoom|first=Jiska J.|last2=Dingemans|first2=Alexandra E.|last3=Slof Op't Landt|first3=Margarita C. T.|last4=Van Furth|first4=Eric F.|date=December 2012|title=Norms and discriminative validity of the Eating Disorder Examination Questionnaire (EDE-Q)|url=https://www.ncbi.nlm.nih.gov/pubmed/23121779|journal=Eating Behaviors|volume=13|issue=4|pages=305–309|doi=10.1016/j.eatbeh.2012.09.002|issn=1873-7358|pmid=23121779}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|title = Assessment of eating disorders: interview or self-report questionnaire?|url = http://www.ncbi.nlm.nih.gov/pubmed/7866415|journal = The International Journal of Eating Disorders|date = 1994-12-01|issn = 0276-3478|pmid = 7866415|pages = 363–370|volume = 16|issue = 4|first = C. G.|last = Fairburn|first2 = S. J.|last2 = Beglin}}&lt;/ref&gt;
|.96 (N=1170)
| --
| --
|Moderate: Dutch treatment-seeking females meeting DSM-IV criteria for an eating disorder versus female adult general population sample recruited through advertisements and personal contacts. Eating disorders were not separated.
|
|-
|EDE-Q&amp;nbsp;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Fairburn|first=C. G.|last2=Beglin|first2=S. J.|date=December 1994|title=Assessment of eating disorders: interview or self-report questionnaire?|url=https://www.ncbi.nlm.nih.gov/pubmed/7866415|journal=The International Journal of Eating Disorders|volume=16|issue=4|pages=363–370|issn=0276-3478|pmid=7866415}}&lt;/ref&gt;
| --
|6.57 (2.3+)
|0.09 (&lt;2.3)
|Moderate: “Clinically significant eating disorder” from a community sample versus female adultsindividuals without “clinically significant eating disorder” from same sample. Eating disorders were not separated.
|
|-
|EDE-Q &lt;ref name=&quot;:1&quot; /&gt;&lt;ref&gt;Rø, Ø., Reas, D. L., &amp; Stedal, K. (2015). Eating Disorder Examination Questionnaire (EDE‐Q) in Norwegian Adults: Discrimination between Female Controls and Eating Disorder Patients. ''European Eating Disorders Review'', ''23''(5), 408-412.&lt;/ref&gt;
|0.89 (N=2465)
| --
| --
|Norwegian adult females with an eating disorder recruited from eating disorder specialist centers versus controls determined using the DSM-IV.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDQ-O&lt;sub&gt;5&lt;/sub&gt;'''
|
|-
|EDQ-Q&amp;nbsp;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;
|0.72 (N=134)
|1.00 (all criteria met)
|0.92 (≥1 criteria not met)
|Dutch adults recruited from an eating disorder specialist center determined using the DSM-IV-TR.
|
|}

=== Interpreting anorexia nervosa screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]
* Also see the page on [https://en.wikipedia.org/wiki/Likelihood_ratios_in_diagnostic_testing likelihood ratios in diagnostic testing] for more information.

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==
===Gold standard diagnostic interviews===

* For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

===Recommended diagnostic interviews for anorexia nervosa===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! colspan=&quot;10&quot; |Diagnostic instruments for anorexia nervosa
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| [https://dawba.info/a0.html Development and Well-Being Assessment (DAWBA)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
| Structured; child/adolescent and parent versions
| 11-18
|10-20 minutes for the eating disorder module
|
|
|
|
|
|
*
|-
|[https://www.phenxtoolkit.org/protocols/view/230101 Children's Eating Disorder Examination- Child (ChEDE)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
|Semistructured; child version
|8-16 
|45-75 minutes; mean 1 hour
|E
|
|
|
|
|
|-
|[https://www.credo-oxford.com/7.2.html Eating Disorder Examination (EDE)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt; &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|date=2008|publisher=Oxford University Press|others=John Hunsley, Eric J. Mash|isbn=0-19-804245-0|location=New York|oclc=314222270}}&lt;/ref&gt;
|Semistructured; adolescent version
|16+
|45-75 minutes
|E
|A
|A
|E
|X
|
|-
|[https://mccaed.slam.nhs.uk/wp-content/uploads/2022/02/PARDI-Self-14-V.2.1-Oct-2020.pdf Pica, ARFID, and Rumination Disorder Interview] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
|Semistructured; multi-informant and different versions
|2-22
|Mean of 39 minutes
|G (ARFID diagnosis)
|
|
|
|
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Severity interviews for anorexia nervosa===
{| class=&quot;wikitable&quot; border=&quot;1&quot;
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
!Free and Accessible Measures
|-
| Placeholder example (CDRS-R)
| Structured Interview&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Mayes|first=Taryn L.|last2=Bernstein|first2=Ira H.|last3=Haley|first3=Charlotte L.|last4=Kennard|first4=Betsy D.|last5=Emslie|first5=Graham J.|date=2010-12|title=Psychometric Properties of the Children's Depression Rating Scale–Revised in Adolescents|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3003451/|journal=Journal of Child and Adolescent Psychopharmacology|volume=20|issue=6|pages=513–516|doi=10.1089/cap.2010.0063|issn=1044-5463|pmc=PMC3003451|pmid=21186970}}&lt;/ref&gt;
| 6-12
| 15-20 minutes
|
* Link to purchase [https://www.wpspublish.com/store/p/2703/cdrs-r-children-s-depression-rating-scale-revised#purchase-product CDRS-R]
*[http://www.opapc.com/uploads/documents/CDRS-R.pdf PDF] (excerpt)
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a brief overview of treatment options for anorexia nervosa and a list of process and outcome measures for anorexia nervosa. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information on the differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]].

===Process measures===
Body weight is commonly monitored by clinicians throughout the AN treatment process as helping individuals regain and maintain a healthy weight is a primary treatment goal for AN. Many treatment centers have policies prohibiting patients from seeing their weight.

Motivational Stages of Change may be used to monitor individuals’ readiness to take action against eating disorder behaviors. It has demonstrated predictive validity in a sample of female adolescents attending eating disorder treatment groups.&lt;ref&gt;{{Cite journal|last=Gusella|first=Joanne|last2=Butler|first2=Gordon|last3=Nichols|first3=Laura|last4=Bird|first4=Debbie|date=2003-01-01|title=A brief questionnaire to assess readiness to change in adolescents with eating disorders: its applications to group therapy|url=http://onlinelibrary.wiley.com/doi/10.1002/erv.481/abstract|journal=European Eating Disorders Review|language=en|volume=11|issue=1|pages=58–71|doi=10.1002/erv.481|issn=1099-0968}}&lt;/ref&gt; See Appendix E.

=== Outcome and severity measures ===
This table includes clinically significant benchmarks for anorexia nervosa specific outcome measures
* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
*''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
| colspan=&quot;8&quot; |'''Clinically significant change benchmarks with common instruments for anorexia nervosa'''
|-
| colspan=&quot;8&quot;  span style=&quot;font-size:110%; text-align:center;&quot; | &lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; |&lt;b&gt; Measure&lt;/b&gt;
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; | &lt;b&gt; Subscale&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; | &lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; | &lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-
| rowspan=&quot;5&quot; style=&quot;text-align:center;&quot; |'''EDE-Q &lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Mond|first=J. M.|last2=Hay|first2=P. J.|last3=Rodgers|first3=B.|last4=Owen|first4=C.|last5=Beumont|first5=P. J. V.|date=May 2004|title=Validity of the Eating Disorder Examination Questionnaire (EDE-Q) in screening for eating disorders in community samples|url=https://www.ncbi.nlm.nih.gov/pubmed/15033501|journal=Behaviour Research and Therapy|volume=42|issue=5|pages=551–567|doi=10.1016/S0005-7967(03)00161-X|issn=0005-7967|pmid=15033501}}&lt;/ref&gt;'''
| style=&quot;text-align:right;&quot; |''Global''
|1.4
|3.2
|2.3
|.7
|.6
|.3
|-
| style=&quot;text-align:right;&quot; |''Restraint''
|(-.3)
|3.6
|1.8
|1.5
|1.2
|.8
|-
| style=&quot;text-align:right;&quot; |''Eating Concern''
|.1
|2.0
|1.2
|1.1
|.9
|.6
|-
| style=&quot;text-align:right;&quot; |''Weight Concern''
|1.5
|3.9
|2.6
|1.0
|.9
|.5
|-
| style=&quot;text-align:right;&quot; |''Shape Concern''
|2.1
|4.8
|3.2
|.9
|.7
|.4
|-
| rowspan=&quot;5&quot; style=&quot;text-align:center;&quot; |'''EDE &lt;ref name=&quot;:2&quot; /&gt;'''
| style=&quot;text-align:right;&quot; |''Global''
|1.7
|2.3
|1.9
|1.9
|1.6
|1.0
|-
| style=&quot;text-align:right;&quot; |''Restraint''
|.3
|3.3
|1.9
|1.8
|1.5
|.9
|-
| style=&quot;text-align:right;&quot; |''Eating Concern''
|(-.5)
|.9
|.5
|.8
|.7
|.4
|-
| style=&quot;text-align:right;&quot; |''Weight Concern''
|2.0
|2.8
|2.4
|1.3
|1.1
|.7
|-
| style=&quot;text-align:right;&quot; |''Shape Concern''
|2.0
|3.2
|2.6
|1.2
|1.0
|.6
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; |'''EAT-26 &lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Mintz|first=L. B.|last2=O'Halloran|first2=M. S.|date=2000-06-01|title=The Eating Attitudes Test: validation with DSM-IV eating disorder criteria|url=http://www.ncbi.nlm.nih.gov/pubmed/10900574|journal=Journal of Personality Assessment|volume=74|issue=3|pages=489–503|doi=10.1207/S15327752JPA7403_11|issn=0022-3891|pmid=10900574}}&lt;/ref&gt;'''
| style=&quot;text-align:right;&quot; |''Total''
|6.5
|19.6
|15.0
|7.9
|6.7
|4.0
|}

'''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean.

'''Search terms:''' [Anorexia Nervosa OR eating disorder] AND [validity OR clinical significance] in Google Scholar

=== Treatment ===
{{collapse top| Click here for more information on treatment for AN}}
* Treatment of AN typically consists of restoring the individual to a healthy weight and addressing thoughts and behaviors which are related to the eating disorder. It may involve re-nutrition, psychotherapy, nutritional counseling, and medication. 
* Literature reviews of existing research indicate that evidence supporting AN treatment is lacking. A systematic review of AN treatment efficacy studies by Bulik&lt;ref&gt;{{Cite journal|title = Anorexia nervosa treatment: a systematic review of randomized controlled trials|url = http://www.ncbi.nlm.nih.gov/pubmed/17370290|journal = The International Journal of Eating Disorders|date = 2007-05-01|issn = 0276-3478|pmid = 17370290|pages = 310–320|volume = 40|issue = 4|doi = 10.1002/eat.20367|first = Cynthia M.|last = Bulik|first2 = Nancy D.|last2 = Berkman|first3 = Kimberly A.|last3 = Brownley|first4 = Jan A.|last4 = Sedway|first5 = Kathleen N.|last5 = Lohr}}&lt;/ref&gt; found that evidence supporting medications, medications and behavioral interventions, and behavioral interventions alone in adults is weak. 
* There is moderately strong evidence suggesting that behavioral interventions may be helpful for adolescents. In particular, adolescents may benefit from family therapy. 
* Clinical trials investigating AN treatment suffer from high rates of attrition, as key features of AN (e.g., denial, fear of weight gain) may contribute to low motivation for remaining in treatment.
* More severe cases of AN may be treated in inpatient settings, which are equipped to manage the re-nutrition process and provide medical monitoring. 
* Partial hospitalization and intensive outpatient programs may provide intermediate levels of treatment intensity to assist individuals in the transition from intensive care to outpatient care after weight restoration.
{{collapse bottom}}
* Please refer to the page on [https://en.wikipedia.org/wiki/Anorexia_nervosa anorexia nervosa] for more information on available treatment or go to [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/eating-body-image-problems/ the Effective Child Therapy page for Eating &amp; Body Image Problems] for a curated resource on effective treatments for anorexia nervosa.

== '''External resources''' ==
# [http://apps.who.int/classifications/icd10/browse/2010/en#/F50.0 ICD-10 diagnostic criteria]
# [https://en.wikiversity.org/w/index.php?title=Helping_Give_Away_Psychological_Science/Resources/Annotated_List_of_Where_and_How_to_Find_a_Therapist&amp;wteswitched=1#Other_low-cost_options Find-a-Therapist] 
##This is a curated list of find-a-therapist websites where you can find a provider
# NIMH: [https://www.nimh.nih.gov/health/publications/eating-disorders/index.shtml ''Eating Disorders--About More Than Food'']  and [https://www.nimh.nih.gov/health/topics/eating-disorders/index.shtml ''Eating Disorders'']
##These NIMH website posts provide more information on anorexia nervosa
#[https://www.hopkinsmedicine.org/psychiatry/specialty_areas/eating_disorders/ John's Hopkins Resource] (guide about anorexia nervosa, treatment, and more)
# OMIM (Online Mendelian Inheritance in Man)
##[https://www.omim.org/entry/606788?search=anorexia%20nervosa&amp;highlight=nervosa%20anorexia Anorexia nervosa]
#[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/eating-body-image-problems/ Effective Child Therapy page for anorexia nervosa]
##Effective Child Therapy is website sponsored by Division 53 of the American Psychological Association (APA), or The [https://sccap53.org Society of Clinical Child and Adolescent Psychology] (SCCAP), in collaboration with the Association for Behavioral and Cognitive Therapies (ABCT). Use for information on symptoms and available treatments.

== '''References''' ==
{{collapse top|Click here for references}}
{{Reflist|30em}}

[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]
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      <sha1>1eco38108agk8xb98i9k00nccp9d9mu</sha1>
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      <timestamp>2022-08-04T22:55:43Z</timestamp>
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        <username>Aherman012</username>
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{{medical disclaimer}}
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==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
For background information on what assessment portfolios are, click the link in the heading above.
Want even 'more' information about this topic? There's an extended version of this page [Evidence-based assessment/Anorexia nervosa (assessment portfolio)/extended version|here]].

== [[Evidence based assessment/Preparation phase|'''Preparation phase''']] ==
=== Diagnostic criteria for anorexia nervosa ===
{{blockquotetop}}
'''ICD-11 Criteria'''

Anorexia Nervosa is characterized by significantly low body weight, which is less than minimal normal/expected weight for the individual’s height, sex, age and developmental stage (body mass index (BMI) less than 18.5 kg/m2 in adults and BMI-for-age under fifth percentile in children and adolescents) that is not due to another health condition or to the unavailability of food. Low body weight is accompanied by a persistent pattern of behaviors to prevent restoration of normal weight, which may include behaviors aimed at reducing energy intake (restricted eating), purging behaviors (e.g., self-induced vomiting, misuse of laxatives), and behaviors aimed at increasing energy expenditure (e.g., excessive exercise), typically associated with a fear of weight gain. Low body weight or shape is central to the person's self-evaluation or is inaccurately perceived to be normal or even excessive.

'''Changes in DSM-5'''

* The diagnostic criteria for anorexia nervosa changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here] and [[w:DSM-5|here]].
{{blockquotebottom}}

=== Base rates of anorexia nervosa in different populations and clinical settings ===
This section describes the demographic setting of the population(s) sampled, base rates of diagnosis such as prevalence rates, country/region sampled, and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of adolescent depression that they are likely to see in their clinical practice.

* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']]

{| class=&quot;wikitable&quot;
|-
! Setting (Reference) 
! Base Rate
! Demography
! Diagnostic Method
|-
|Non-clinical: Population-based (NCS-R)&lt;ref&gt;Hudson, James I.; Hiripi, Eva; Pope, Harrison G.; Kessler, Ronald C. (2007-02-01). &quot;The prevalence and correlates of eating disorders in the National Comorbidity Survey Replication&quot;.''Biological Psychiatry'' '''61''' (3): 348–358. doi:10.1016/j.biopsych.2006.03.040. ISSN 0006-3223. PMC 1892232. &lt;nowiki&gt;PMID 16815322&lt;/nowiki&gt;&lt;/ref&gt;
|.9% Female, .3% Male
|Nationally representative US sample of adults
|World Health Organization-Clinical International Diagnostic Interview (WHO-CIDI)
|-
|Non-clinical: Population-based (NCS-A)&lt;ref&gt;Swanson, Sonja A.; Crow, Scott J.; Le Grange, Daniel; Swendsen, Joel; Merikangas, Kathleen R. (2011-07-01). &quot;Prevalence and correlates of eating disorders in adolescents. Results from the national comorbidity survey replication adolescent supplement&quot;. ''Archives of General Psychiatry'' '''68''' (7): 714–723. doi:10.1001/archgenpsychiatry.2011.22. ISSN 1538-3636.&lt;nowiki&gt;PMID 21383252&lt;/nowiki&gt;&lt;/ref&gt;
|.3% Female, .3% Male
|Nationally representative US sample of adolescents
|WHO-CIDI
|-
|Non-clinical: US Population-based prevalence (Adolescent Brain Cognitive Development (ABCD) study&lt;ref&gt;{{Cite journal|last=Rozzell|first=Kaitlin|last2=Moon|first2=Da Yeoun|last3=Klimek|first3=Patrycja|last4=Brown|first4=Tiffany|last5=Blashill|first5=Aaron J.|date=2019-01-01|title=Prevalence of Eating Disorders Among US Children Aged 9 to 10 Years: Data From the Adolescent Brain Cognitive Development (ABCD) Study|url=http://archpedi.jamanetwork.com/article.aspx?doi=10.1001/jamapediatrics.2018.3678|journal=JAMA Pediatrics|language=en|volume=173|issue=1|pages=100|doi=10.1001/jamapediatrics.2018.3678|issn=2168-6203|pmc=PMC6583451|pmid=30476983}}&lt;/ref&gt;
|0.1%, no gender differences
|Nationally representative US sample of 9- and 10-year old children
|DSM-5 using Kiddie Schedule for Affective Disorders and Schizophrenia (KSADS)
|-
|Non-clinical: Latinos in US&lt;ref&gt;Alegria, Margarita; Woo, Meghan; Cao, Zhun; Torres, Maria; Meng, Xiao-li; Striegel-Moore, Ruth (2007-11-01). &quot;Prevalence and correlates of eating disorders in Latinos in the United States&quot;.''The International Journal of Eating Disorders''. 40 Suppl: S15–21. doi:10.1002/eat.20406. ISSN 0276-3478. PMC 2680162. &lt;nowiki&gt;PMID 17584870&lt;/nowiki&gt;&lt;/ref&gt;
|.12% Female, .03% Male
|Latino Households in the US
|WHO-CIDI
|-
|Non-clinical: African Americans and Caribbean Blacks in the US (NSAL)&lt;ref&gt;Taylor, Jacquelyn Y.; Caldwell, Cleopatra Howard; Baser, Raymond E.; Faison, Nakesha; Jackson, James S. (2007-11-01). &quot;Prevalence of eating disorders among Blacks in the National Survey of American Life&quot;. ''The International Journal of Eating Disorders''. 40 Suppl: S10–14. doi:10.1002/eat.20451. ISSN 0276-3478. PMC 2882704. &lt;nowiki&gt;PMID 17879287&lt;/nowiki&gt;&lt;/ref&gt;
|.14% Female, .2% Male
|National probability sample of adult and adolescent African Americans and Caribbean Blacks
|WHO-CIDI
|-
|Non-clinical: National Latino and Asian American Study (NLAAS)
|.12% Female, .05% Male
|Asian American adults in US households
|WHO-CIDI
|-
| Non-clinical; college students&lt;ref&gt;Tyler ID. A true picture of eating disorders among African American women: a review of literature. ABNF J. 2003;14(3):73-4.&lt;/ref&gt; || .0%  || US African American college females || Eating Disorder Diagnostic Questionnaire (EDD-Q)
|-
| Non-clinical; Adolescents&lt;ref&gt;Stice, E., Becker, C. B., &amp; Yokum, S. (2013). Eating disorder prevention: Current evidence-base and future directions. ''Int. J. Eat. Disord. International Journal of Eating Disorders,'' ''46''(5), 478-485.&lt;/ref&gt; || .8% || US Female Adolescents || Eating Disorder Diagnosis Interview (EDDI)
|-
| Non-clinical; student-athletes&lt;ref&gt;Johnson C, Powers PS, Dick R. Athletes and eating disorders: the national collegiate athletic association study. Int J Eat Disord 1999;26:179e88.&lt;/ref&gt; || .0% || US Division-I Varsity Student Athletes || Eating Disorder Inventory-2 (EDI-2)
|-
| Non-clinical; Military&lt;ref&gt;McNulty, PAF. (2001). Prevalence and contributing factors of eating disorder behaviors in active duty service women in the Army, Navy, Air Force and Marines. Military Medicine, 166(1), 53-58. &lt;/ref&gt; || 1.1% || Active duty females in US Army, Navy, 
Airforce, and Marines 
|| EDI-2
|-
|Non-clinical; Commonwealth of Virginia Mid-Atlantic Twin Registry (MATR)&lt;ref&gt;Kendler KS, Walters EE, Neale MC, Kessler R, Heath A, Eaves L. The structure of genetic and environmental risk factors for six major psychiatric disorders in women. Archives of general psychiatry. 1995;52:374–383.&lt;/ref&gt;
|1.62% (narrow), 3.70% (broad)
|US Caucasian female same-sex twins
|Structured Clinical Interview for DSM Disorders (SCID)
|-
|Non-clinical; Health Omnibus Survey (HOS)
|.46% (3 months; combined)
|South Australian older adolescents and adults
|Eating Disorder Examination (EDE)
|-
|Military&lt;ref&gt;{{Cite journal|title=Diagnosed eating disorders in the U.S. Military: a nine year review|url=http://www.ncbi.nlm.nih.gov/pubmed/18821361|journal=Eating Disorders|date=2008-12-01|issn=1532-530X|pmid=18821361|pages=363–377|volume=16|issue=5|doi=10.1080/10640260802370523|first=Amanda J.|last=Antczak|first2=Teresa L.|last2=Brininger}}&lt;/ref&gt;
|.04% (combined)
|US Military
|ICD codes from electronic records
|-
|Non-clinical; Military
|.02% (7 years) Female, 
0.0% (7 years) Males
|US Military Academy cadets
|[https://www.eat-26.com/ Eating Attitudes Test- 26 items] (EAT-26)&lt;ref&gt;{{Cite web|url=https://www.eat-26.com/|title=EAT-26: Eating Attitudes Test &amp; Eating Disorder Testing – Use the EAT-26 to help you determine if you need to speak to a mental health professional to get help for an eating disorder.|last=admin|language=en|access-date=2022-05-30}}&lt;/ref&gt;
|-
|Non-clinical; Military
|1.1% (current &amp; past) Female
|US Navy female nurses
|DSM-III
|-
|Non-clinical; Military
|.04% Female, .005% Males
|US veterans
|ICD-9-CM
|-
|Non-clinical; Military
|2.5% Males
|Active duty males in US Navy
|N/A
|-
|Clinical; Collaborative Study on the Genetics of Alcoholism (COGA)&lt;ref&gt;{{Cite journal|title=Anorexia nervosa and bulimia nervosa in alcohol-dependent men and women and their relatives|url=http://www.ncbi.nlm.nih.gov/pubmed/8540597|journal=The American Journal of Psychiatry|date=1996-01-01|issn=0002-953X|pmid=8540597|pages=74–82|volume=153|issue=1|doi=10.1176/ajp.153.1.74|first=M. A.|last=Schuckit|first2=J. E.|last2=Tipp|first3=R. M.|last3=Anthenelli|first4=K. K.|last4=Bucholz|first5=V. M.|last5=Hesselbrock|first6=J. I.|last6=Nurnberger}}&lt;/ref&gt;
|1.41% (lifetime) Female,
.00% (lifetime) Male
|US alcohol-dependent adults from San Diego, St. Lois, Iowa City, Farmington, New York, &amp; Indianapolis
|SSAGA
|-
|Non-clinical; healthcare members
|.0269% (current)
|Healthcare provider records
|
|-
|Non-clinical; high school students
|. 00% (point), .45% (lifetime) Female
.00% (point), .00% (lifetime) Males
|US high school students in west central Oregon
|DSM-III-R&lt;sub&gt;4&lt;/sub&gt;
|-
|Clinical; substance users
|.4% (lifetime), .3% (current) Female
.4% (lifetime), .3% (current) Males
|Canadian treatment-seeking substance users
|DIS&lt;sub&gt;9&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Europe'''
|-
|Non-clinical; adolescents
|.3% Female
|Adolescent females residing in Navarra, Spain
|EAT-40&lt;sub&gt;3&lt;/sub&gt;
|-
|Non-clinical; adolescents
|.7% (lifetime) Female, .2% Male
|Adolescents in secondary schools in Sør-Trøndelag, County in Norway
|SEDs&lt;sub&gt;10&lt;/sub&gt;
|-
|Non-clinical; adolescents
|.7% (point; age 15), 1.8% (lifetime, age 15), .00% (point, age 18), 2.6% (lifetime, age 18), .9% (3 years) Female
.00% (point &amp; lifetime) Males
|Adolescents in a comprehensive school in Ostrobothnia district in Finland
|RAB-T&lt;sub&gt;11&lt;/sub&gt;&amp;nbsp;&amp; RAB-R&lt;sub&gt;12&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Australia'''
|-
|Non-clinical; adolescents
|.00% (full), 1.8% (partial) Female
|Adolescent females residing in Victoria, Australia
|BET&lt;sub&gt;13&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Central &amp; South America'''
|-
|Non-clinical; college students
|.00% Female
|Mexican first &amp; second year college females
|EAT-40&lt;sub&gt;3&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''East Asia'''
|-
|Clinical; eating disorder patients
|.53% Female
|Adolescent and adult Japanese patients at a university hospital
|DSM-III-R&lt;sub&gt;4&lt;/sub&gt;
|-
|Non-clinical; Korean Epidemiologic Catchment Area (KECA) Study
|.1% (lifetime), .1% (12 months) Female
.2% (lifetime), .00% (12 months)

Males
|Korean adults
|K-CIDI&lt;sub&gt;15&lt;/sub&gt;&amp;nbsp;2.1
|-
|Centers participating in the Collaborative Study on the Genetics of  Alcoholism in San Diego; St. Louis; Iowa City; Farmington, CN; New York; &amp; Indianapolis
|1.41% Females
|Alcohol-dependent adults
|Semi-Structured Assessment for the Genetics of Alcoholism; criteria based on DSM-III-R
|}

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==

=== Psychometric properties of screening instruments for anorexia nervosa ===
The following section contains a list of screening and diagnostic instruments for anorexia nerova. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here.]]

* ''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments '''click here.''']''

{| class=&quot;wikitable&quot;
|-
! Measure 
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Inter-rater reliability
!Test-retest reliability
!Construct validity
!Content validity
!Highly recommended
|-
|Structured Clinical Interview Diagnosis for DSM-IV (SCID-IV) Module H

* [https://www.columbiapsychiatry.org/research/research-areas/services-policy-and-law/structured-clinical-interview-dsm-disorders-scid-1 SCID-IV (for purchase)]
* [[wikipedia:Structured_Clinical_Interview_for_DSM-IV|Wiki Page]]
|Interview
(Patient)
|
|
|G
|L
|A
|E
|X
|-
|Eating Disorder Examination (EDE)

*OSF
*[[wikipedia:Eating_Disorder_Examination_Interview|Wiki Page]]
|Interview
(Patient)
|
|45min-1.25hrs
|E
|A
|A
|E
|X
|-
|[http://www.corc.uk.net/media/1273/ede-q_quesionnaire.pdf EDE-Q (Eating Disorder Examination Questionnaire)]
*OSF
*[[wikipedia:Eating_Disorder_Examination_Interview|Wiki Page]]
|Questionnaire

(Patient)
|
|15-20 min
|NA
|L
|U
|E
|
|-
|[http://www.ori.org/files/Static%20Page%20Files/EDDSDSM-5_10_14.pdf EDDS (Eating Disorder Diagnosis Scale)]
*OSF
*[[wikipedia:Eating_Disorder_Diagnostic_Scale|Wiki Page]]
|Questionnaire 
(Patient)
|
|10-15 min
|NA
|A
|A
|G
|X
|}
&lt;ref&gt;Hunsley, J., &amp; Mash, E. J. (2008). Guide to Assessments that Work. Cary, NC, USA: Oxford University Press, USA. Retrieved from &lt;nowiki&gt;http://www.ebrary.com&lt;/nowiki&gt;&lt;/ref&gt;'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Likelihood ratios and AUCs of screening instruments for anorexia nervosa===
* ''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments '''click here.''']''
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! &lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Mintz|first=L. B.|last2=O'Halloran|first2=M. S.|date=June 2000|title=The Eating Attitudes Test: validation with DSM-IV eating disorder criteria|url=https://www.ncbi.nlm.nih.gov/pubmed/10900574|journal=Journal of Personality Assessment|volume=74|issue=3|pages=489–503|doi=10.1207/S15327752JPA7403_11|issn=0022-3891|pmid=10900574}}&lt;/ref&gt;Screening Measure (Primary Reference)
! AUC
! DiLR+ (Score)
! DiLR- (Score)
!Clinical Generalizability
!Download
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''Biological &amp; Physical Measures'''
|
|-
|Serum leptin level&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Föcker|first=M.|last2=Timmesfeld|first2=N.|last3=Scherag|first3=S.|last4=Bühren|first4=K.|last5=Langkamp|first5=M.|last6=Dempfle|first6=A.|last7=Sheridan|first7=E. M.|last8=Zwaan|first8=M. de|last9=Fleischhaker|first9=C.|date=2011-04-01|title=Screening for anorexia nervosa via measurement of serum leptin levels|url=https://link.springer.com/article/10.1007/s00702-010-0551-z|journal=Journal of Neural Transmission|language=en|volume=118|issue=4|pages=571–578|doi=10.1007/s00702-010-0551-z|issn=0300-9564}}&lt;/ref&gt;
|0.984 (N=139)
|14.72 (&lt;2.31)
|0.10 (2.31+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
|Serum leptin standard deviation score (SDS)&lt;ref name=&quot;:7&quot; /&gt;
|0.939 (N=139)
|5.89 (&lt;-0.45)
|0.09 (-0.45+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
|Body Mass Index (BMI)&lt;ref name=&quot;:7&quot; /&gt;
|0.936 (N=139)
|5.89 (&lt;17.10)
|0.11 (17.10+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''REDS-C&lt;sub&gt;1&lt;/sub&gt;'''
|
|-
|REDS-C&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Boraska|first=V.|last2=Franklin|first2=C. S.|last3=Floyd|first3=J. a. B.|last4=Thornton|first4=L. M.|last5=Huckins|first5=L. M.|last6=Southam|first6=L.|last7=Rayner|first7=N. W.|last8=Tachmazidou|first8=I.|last9=Klump|first9=K. L.|date=October 2014|title=A genome-wide association study of anorexia nervosa|url=https://www.ncbi.nlm.nih.gov/pubmed/24514567|journal=Molecular Psychiatry|volume=19|issue=10|pages=1085–1094|doi=10.1038/mp.2013.187|issn=1476-5578|pmc=PMC4325090|pmid=24514567}}&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt;
|0.658 (N=333), 0.624 (N=236, &gt; 13 years old), 0.772 (N=97, ≤ 13 years old)
| --
| --
|Children and adolescents (8-18 years old) at a clinic being treated for eating and weight concerns. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDI&lt;sub&gt;2&lt;/sub&gt;'''
|
|-
|EDI-2 &lt;ref name=&quot;:3&quot;&gt;Garner, D. M. (1991). ''Manual for Eating Disorder Inventory.'' Odessa, FL: Psychological Assessment Resources, Inc.&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt; 
|0.556 (N=77)
| --
| --
|Children and adolescents (10-18 years old) at a clinic being treated for eating and weight concerns. Eating disorders were not separated.
|
|-
|EDI-2 - Drive for Thinness Sub-scale&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Segura-Garcia|first=Cristina|last2=Ramacciotti|first2=Carla|last3=Rania|first3=Marianna|last4=Aloi|first4=Matteo|last5=Caroleo|first5=Mariarita|last6=Bruni|first6=Antonella|last7=Gazzarrini|first7=Denise|last8=Sinopoli|first8=Flora|last9=De Fazio|first9=Pasquale|date=June 2015|title=The prevalence of orthorexia nervosa among eating disorder patients after treatment|url=https://www.ncbi.nlm.nih.gov/pubmed/25543324|journal=Eating and weight disorders: EWD|volume=20|issue=2|pages=161–166|doi=10.1007/s40519-014-0171-y|issn=1590-1262|pmid=25543324}}&lt;/ref&gt; 
|0.97 (N=92)
|29.39 (14+)
|0.274 (DT≤14)
|Italian individuals with a DSM-IV-TR eating disorder versus individuals at risk for eating disorders. Eating disorders were not separated.
|
|-
|EDI-3 - Drive for Thinness Subscale (Garner, 2004)

Lehmann et al., 2013
|.903 (N=1298)
| --
| --
|&amp;nbsp;
|
|-
|EDI-3 - Eating Disorder Risk Composite &lt;ref name=&quot;:9&quot; /&gt;(EDRC) (Garner, 2004)
|0.942 (N=92)
|52.08 (EDRC 75+, adults; 90+ adolescents)
|0.379 (EDRC≤75, adults; ≤90 adolescents)
|Italian individuals with a DSM-IV-TR eating disorder versus individuals at risk for eating disorders. Eating disorders were not separated.
|
|-
|EDI-3 - Interoceptive Deficits Subscale &lt;ref name=&quot;:10&quot;&gt;{{Cite journal|last=Clausen|first=Loa|last2=Rosenvinge|first2=Jan H.|last3=Friborg|first3=Oddgeir|last4=Rokkedal|first4=Kristian|date=2011-03-01|title=Validating the Eating Disorder Inventory-3 (EDI-3): A Comparison Between 561 Female Eating Disorders Patients and 878 Females from the General Population|url=https://link.springer.com/article/10.1007/s10862-010-9207-4|journal=Journal of Psychopathology and Behavioral Assessment|language=en|volume=33|issue=1|pages=101–110|doi=10.1007/s10862-010-9207-4|issn=0882-2689}}&lt;/ref&gt;(Garner, 2004)
|.911 (N=2561)
|4.0 (9+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interoceptive Deficits Subscale (Garner, 2004)

Lehmann et al., 2013
|.901 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Asceticism Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.886 (N=2561)
|6.5 (9+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Asceticism Subscale (Garner, 2004)

Lehmann et al., 2013
|.902 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Low Self-Esteem Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.884 (N=2561)
|5.8 (10+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited fm an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Low Self-Esteem Subscale (Garner, 2004)

Lehmann et al., 2013
|.906 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Personal Alienation Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.88 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Personal Alienation Subscale (Garner, 2004)

Lehmann et al., 2013
|.899 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Emotional Dysregulation Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.81 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Emotional Dysregulation Subscale (Garner, 2004)

Lehmann et al., 2013
|.779 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Interpersonal Alienation Subscale &lt;ref name=&quot;:10&quot; /&gt;(Garner, 2004)
|.79 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interpersonal Alienation Subscale (Garner, 2004)

Lehmann et al., 2013
|.743 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Perfectionism Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.79 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Perfectionism Subscale (Garner, 2004)

Lehmann et al., 2013
|.768 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Maturity Fears Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.77 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Maturity Fears Subscale (Garner, 2004)

Lehmann et al., 2013
|.678 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Bulimia Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.76 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Bulimia Subscale (Garner, 2004)

Lehmann et al., 2013
|.776 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Interpersonal Insecurity Subscale &lt;ref name=&quot;:10&quot; /&gt;(Garner, 2004)
|.76 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interpersonal Insecurity Subscale (Garner, 2004)

Lehmann et al., 2013
|.697 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Body Dissatisfaction Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.722 (N=2561)
|1.7 (15+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Body Dissatisfaction Subscale (Garner, 2004)

Lehmann et al., 2013
|.849 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EAT&lt;sub&gt;3&lt;/sub&gt;'''
|
|-
|EAT-26&amp;nbsp;&lt;ref name=&quot;:4&quot; /&gt;&lt;ref&gt;{{Cite journal|title = The eating attitudes test: psychometric features and clinical correlates|url = http://www.ncbi.nlm.nih.gov/pubmed/6961471|journal = Psychological Medicine|date = 1982-11-01|issn = 0033-2917|pmid = 6961471|pages = 871–878|volume = 12|issue = 4|first = D. M.|last = Garner|first2 = M. P.|last2 = Olmsted|first3 = Y.|last3 = Bohr|first4 = P. E.|last4 = Garfinkel}}&lt;/ref&gt;
|.90 (N=129)
|12.83 (20+)
|.24 (&lt;20)
|Low-moderate: College women with no eating disorder versus college women with a DSM-IV eating disorder. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDE-Q&lt;sub&gt;4&lt;/sub&gt;'''
|
|-
|EDE-Q&amp;nbsp;&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|last=Aardoom|first=Jiska J.|last2=Dingemans|first2=Alexandra E.|last3=Slof Op't Landt|first3=Margarita C. T.|last4=Van Furth|first4=Eric F.|date=December 2012|title=Norms and discriminative validity of the Eating Disorder Examination Questionnaire (EDE-Q)|url=https://www.ncbi.nlm.nih.gov/pubmed/23121779|journal=Eating Behaviors|volume=13|issue=4|pages=305–309|doi=10.1016/j.eatbeh.2012.09.002|issn=1873-7358|pmid=23121779}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|title = Assessment of eating disorders: interview or self-report questionnaire?|url = http://www.ncbi.nlm.nih.gov/pubmed/7866415|journal = The International Journal of Eating Disorders|date = 1994-12-01|issn = 0276-3478|pmid = 7866415|pages = 363–370|volume = 16|issue = 4|first = C. G.|last = Fairburn|first2 = S. J.|last2 = Beglin}}&lt;/ref&gt;
|.96 (N=1170)
| --
| --
|Moderate: Dutch treatment-seeking females meeting DSM-IV criteria for an eating disorder versus female adult general population sample recruited through advertisements and personal contacts. Eating disorders were not separated.
|
|-
|EDE-Q&amp;nbsp;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Fairburn|first=C. G.|last2=Beglin|first2=S. J.|date=December 1994|title=Assessment of eating disorders: interview or self-report questionnaire?|url=https://www.ncbi.nlm.nih.gov/pubmed/7866415|journal=The International Journal of Eating Disorders|volume=16|issue=4|pages=363–370|issn=0276-3478|pmid=7866415}}&lt;/ref&gt;
| --
|6.57 (2.3+)
|0.09 (&lt;2.3)
|Moderate: “Clinically significant eating disorder” from a community sample versus female adultsindividuals without “clinically significant eating disorder” from same sample. Eating disorders were not separated.
|
|-
|EDE-Q &lt;ref name=&quot;:1&quot; /&gt;&lt;ref&gt;Rø, Ø., Reas, D. L., &amp; Stedal, K. (2015). Eating Disorder Examination Questionnaire (EDE‐Q) in Norwegian Adults: Discrimination between Female Controls and Eating Disorder Patients. ''European Eating Disorders Review'', ''23''(5), 408-412.&lt;/ref&gt;
|0.89 (N=2465)
| --
| --
|Norwegian adult females with an eating disorder recruited from eating disorder specialist centers versus controls determined using the DSM-IV.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDQ-O&lt;sub&gt;5&lt;/sub&gt;'''
|
|-
|EDQ-Q&amp;nbsp;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;
|0.72 (N=134)
|1.00 (all criteria met)
|0.92 (≥1 criteria not met)
|Dutch adults recruited from an eating disorder specialist center determined using the DSM-IV-TR.
|
|}

=== Interpreting anorexia nervosa screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]
* Also see the page on [https://en.wikipedia.org/wiki/Likelihood_ratios_in_diagnostic_testing likelihood ratios in diagnostic testing] for more information.

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==
===Gold standard diagnostic interviews===

* For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

===Recommended diagnostic interviews for anorexia nervosa===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! colspan=&quot;10&quot; |Diagnostic instruments for anorexia nervosa
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| [https://dawba.info/a0.html Development and Well-Being Assessment (DAWBA)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
| Structured; child/adolescent and parent versions
| 11-18
|10-20 minutes for the eating disorder module
|
|
|
|
|
|
*
|-
|[https://www.phenxtoolkit.org/protocols/view/230101 Children's Eating Disorder Examination- Child (ChEDE)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
|Semistructured; child version
|8-16 
|45-75 minutes; mean 1 hour
|E
|
|
|
|
|
|-
|[https://www.credo-oxford.com/7.2.html Eating Disorder Examination (EDE)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt; &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|date=2008|publisher=Oxford University Press|others=John Hunsley, Eric J. Mash|isbn=0-19-804245-0|location=New York|oclc=314222270}}&lt;/ref&gt;
|Semistructured; adolescent version
|16+
|45-75 minutes
|E
|A
|A
|E
|X
|
|-
|[https://mccaed.slam.nhs.uk/wp-content/uploads/2022/02/PARDI-Self-14-V.2.1-Oct-2020.pdf Pica, ARFID, and Rumination Disorder Interview] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
|Semistructured; multi-informant and different versions
|2-22
|Mean of 39 minutes
|G (ARFID diagnosis)
|
|
|
|
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Severity interviews for anorexia nervosa===
{| class=&quot;wikitable&quot; border=&quot;1&quot;
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
!Free and Accessible Measures
|-
| Placeholder example (CDRS-R)
| Structured Interview&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Mayes|first=Taryn L.|last2=Bernstein|first2=Ira H.|last3=Haley|first3=Charlotte L.|last4=Kennard|first4=Betsy D.|last5=Emslie|first5=Graham J.|date=2010-12|title=Psychometric Properties of the Children's Depression Rating Scale–Revised in Adolescents|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3003451/|journal=Journal of Child and Adolescent Psychopharmacology|volume=20|issue=6|pages=513–516|doi=10.1089/cap.2010.0063|issn=1044-5463|pmc=PMC3003451|pmid=21186970}}&lt;/ref&gt;
| 6-12
| 15-20 minutes
|
* Link to purchase [https://www.wpspublish.com/store/p/2703/cdrs-r-children-s-depression-rating-scale-revised#purchase-product CDRS-R]
*[http://www.opapc.com/uploads/documents/CDRS-R.pdf PDF] (excerpt)
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a brief overview of treatment options for anorexia nervosa and a list of process and outcome measures for anorexia nervosa. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information on the differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]].

===Process measures===
Body weight is commonly monitored by clinicians throughout the AN treatment process as helping individuals regain and maintain a healthy weight is a primary treatment goal for AN. Many treatment centers have policies prohibiting patients from seeing their weight.

Motivational Stages of Change may be used to monitor individuals’ readiness to take action against eating disorder behaviors. It has demonstrated predictive validity in a sample of female adolescents attending eating disorder treatment groups.&lt;ref&gt;{{Cite journal|last=Gusella|first=Joanne|last2=Butler|first2=Gordon|last3=Nichols|first3=Laura|last4=Bird|first4=Debbie|date=2003-01-01|title=A brief questionnaire to assess readiness to change in adolescents with eating disorders: its applications to group therapy|url=http://onlinelibrary.wiley.com/doi/10.1002/erv.481/abstract|journal=European Eating Disorders Review|language=en|volume=11|issue=1|pages=58–71|doi=10.1002/erv.481|issn=1099-0968}}&lt;/ref&gt; See Appendix E.

=== Outcome and severity measures ===
This table includes clinically significant benchmarks for anorexia nervosa specific outcome measures
* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
*''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
| colspan=&quot;8&quot; |'''Clinically significant change benchmarks with common instruments for anorexia nervosa'''
|-
| colspan=&quot;8&quot;  span style=&quot;font-size:110%; text-align:center;&quot; | &lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; |&lt;b&gt; Measure&lt;/b&gt;
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; | &lt;b&gt; Subscale&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; | &lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; | &lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-
| rowspan=&quot;5&quot; style=&quot;text-align:center;&quot; |'''EDE-Q &lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Mond|first=J. M.|last2=Hay|first2=P. J.|last3=Rodgers|first3=B.|last4=Owen|first4=C.|last5=Beumont|first5=P. J. V.|date=May 2004|title=Validity of the Eating Disorder Examination Questionnaire (EDE-Q) in screening for eating disorders in community samples|url=https://www.ncbi.nlm.nih.gov/pubmed/15033501|journal=Behaviour Research and Therapy|volume=42|issue=5|pages=551–567|doi=10.1016/S0005-7967(03)00161-X|issn=0005-7967|pmid=15033501}}&lt;/ref&gt;'''
| style=&quot;text-align:right;&quot; |''Global''
|1.4
|3.2
|2.3
|.7
|.6
|.3
|-
| style=&quot;text-align:right;&quot; |''Restraint''
|(-.3)
|3.6
|1.8
|1.5
|1.2
|.8
|-
| style=&quot;text-align:right;&quot; |''Eating Concern''
|.1
|2.0
|1.2
|1.1
|.9
|.6
|-
| style=&quot;text-align:right;&quot; |''Weight Concern''
|1.5
|3.9
|2.6
|1.0
|.9
|.5
|-
| style=&quot;text-align:right;&quot; |''Shape Concern''
|2.1
|4.8
|3.2
|.9
|.7
|.4
|-
| rowspan=&quot;5&quot; style=&quot;text-align:center;&quot; |'''EDE &lt;ref name=&quot;:2&quot; /&gt;'''
| style=&quot;text-align:right;&quot; |''Global''
|1.7
|2.3
|1.9
|1.9
|1.6
|1.0
|-
| style=&quot;text-align:right;&quot; |''Restraint''
|.3
|3.3
|1.9
|1.8
|1.5
|.9
|-
| style=&quot;text-align:right;&quot; |''Eating Concern''
|(-.5)
|.9
|.5
|.8
|.7
|.4
|-
| style=&quot;text-align:right;&quot; |''Weight Concern''
|2.0
|2.8
|2.4
|1.3
|1.1
|.7
|-
| style=&quot;text-align:right;&quot; |''Shape Concern''
|2.0
|3.2
|2.6
|1.2
|1.0
|.6
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; |'''EAT-26 &lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Mintz|first=L. B.|last2=O'Halloran|first2=M. S.|date=2000-06-01|title=The Eating Attitudes Test: validation with DSM-IV eating disorder criteria|url=http://www.ncbi.nlm.nih.gov/pubmed/10900574|journal=Journal of Personality Assessment|volume=74|issue=3|pages=489–503|doi=10.1207/S15327752JPA7403_11|issn=0022-3891|pmid=10900574}}&lt;/ref&gt;'''
| style=&quot;text-align:right;&quot; |''Total''
|6.5
|19.6
|15.0
|7.9
|6.7
|4.0
|}

'''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean.

'''Search terms:''' [Anorexia Nervosa OR eating disorder] AND [validity OR clinical significance] in Google Scholar

=== Treatment ===
{{collapse top| Click here for more information on treatment for AN}}
* Treatment of AN typically consists of restoring the individual to a healthy weight and addressing thoughts and behaviors which are related to the eating disorder. It may involve re-nutrition, psychotherapy, nutritional counseling, and medication. 
* Literature reviews of existing research indicate that evidence supporting AN treatment is lacking. A systematic review of AN treatment efficacy studies by Bulik&lt;ref&gt;{{Cite journal|title = Anorexia nervosa treatment: a systematic review of randomized controlled trials|url = http://www.ncbi.nlm.nih.gov/pubmed/17370290|journal = The International Journal of Eating Disorders|date = 2007-05-01|issn = 0276-3478|pmid = 17370290|pages = 310–320|volume = 40|issue = 4|doi = 10.1002/eat.20367|first = Cynthia M.|last = Bulik|first2 = Nancy D.|last2 = Berkman|first3 = Kimberly A.|last3 = Brownley|first4 = Jan A.|last4 = Sedway|first5 = Kathleen N.|last5 = Lohr}}&lt;/ref&gt; found that evidence supporting medications, medications and behavioral interventions, and behavioral interventions alone in adults is weak. 
* There is moderately strong evidence suggesting that behavioral interventions may be helpful for adolescents. In particular, adolescents may benefit from family therapy. 
* Clinical trials investigating AN treatment suffer from high rates of attrition, as key features of AN (e.g., denial, fear of weight gain) may contribute to low motivation for remaining in treatment.
* More severe cases of AN may be treated in inpatient settings, which are equipped to manage the re-nutrition process and provide medical monitoring. 
* Partial hospitalization and intensive outpatient programs may provide intermediate levels of treatment intensity to assist individuals in the transition from intensive care to outpatient care after weight restoration.
{{collapse bottom}}
* Please refer to the page on [https://en.wikipedia.org/wiki/Anorexia_nervosa anorexia nervosa] for more information on available treatment or go to [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/eating-body-image-problems/ the Effective Child Therapy page for Eating &amp; Body Image Problems] for a curated resource on effective treatments for anorexia nervosa.

== '''External resources''' ==
# [http://apps.who.int/classifications/icd10/browse/2010/en#/F50.0 ICD-10 diagnostic criteria]
# [https://en.wikiversity.org/w/index.php?title=Helping_Give_Away_Psychological_Science/Resources/Annotated_List_of_Where_and_How_to_Find_a_Therapist&amp;wteswitched=1#Other_low-cost_options Find-a-Therapist] 
##This is a curated list of find-a-therapist websites where you can find a provider
# NIMH: [https://www.nimh.nih.gov/health/publications/eating-disorders/index.shtml ''Eating Disorders--About More Than Food'']  and [https://www.nimh.nih.gov/health/topics/eating-disorders/index.shtml ''Eating Disorders'']
##These NIMH website posts provide more information on anorexia nervosa
#[https://www.hopkinsmedicine.org/psychiatry/specialty_areas/eating_disorders/ John's Hopkins Resource] (guide about anorexia nervosa, treatment, and more)
# OMIM (Online Mendelian Inheritance in Man)
##[https://www.omim.org/entry/606788?search=anorexia%20nervosa&amp;highlight=nervosa%20anorexia Anorexia nervosa]
#[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/eating-body-image-problems/ Effective Child Therapy page for anorexia nervosa]
##Effective Child Therapy is website sponsored by Division 53 of the American Psychological Association (APA), or The [https://sccap53.org Society of Clinical Child and Adolescent Psychology] (SCCAP), in collaboration with the Association for Behavioral and Cognitive Therapies (ABCT). Use for information on symptoms and available treatments.

== '''References''' ==
{{collapse top|Click here for references}}
{{Reflist|30em}}

[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]
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==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
For background information on what assessment portfolios are, click the link in the heading above.
Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Anorexia nervosa (assessment portfolio)/extended version|here]].

== [[Evidence based assessment/Preparation phase|'''Preparation phase''']] ==
=== Diagnostic criteria for anorexia nervosa ===
{{blockquotetop}}
'''ICD-11 Criteria'''

Anorexia Nervosa is characterized by significantly low body weight, which is less than minimal normal/expected weight for the individual’s height, sex, age and developmental stage (body mass index (BMI) less than 18.5 kg/m2 in adults and BMI-for-age under fifth percentile in children and adolescents) that is not due to another health condition or to the unavailability of food. Low body weight is accompanied by a persistent pattern of behaviors to prevent restoration of normal weight, which may include behaviors aimed at reducing energy intake (restricted eating), purging behaviors (e.g., self-induced vomiting, misuse of laxatives), and behaviors aimed at increasing energy expenditure (e.g., excessive exercise), typically associated with a fear of weight gain. Low body weight or shape is central to the person's self-evaluation or is inaccurately perceived to be normal or even excessive.

'''Changes in DSM-5'''

* The diagnostic criteria for anorexia nervosa changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here] and [[w:DSM-5|here]].
{{blockquotebottom}}

=== Base rates of anorexia nervosa in different populations and clinical settings ===
This section describes the demographic setting of the population(s) sampled, base rates of diagnosis such as prevalence rates, country/region sampled, and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of adolescent depression that they are likely to see in their clinical practice.

* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']]

{| class=&quot;wikitable&quot;
|-
! Setting (Reference) 
! Base Rate
! Demography
! Diagnostic Method
|-
|Non-clinical: Population-based (NCS-R)&lt;ref&gt;Hudson, James I.; Hiripi, Eva; Pope, Harrison G.; Kessler, Ronald C. (2007-02-01). &quot;The prevalence and correlates of eating disorders in the National Comorbidity Survey Replication&quot;.''Biological Psychiatry'' '''61''' (3): 348–358. doi:10.1016/j.biopsych.2006.03.040. ISSN 0006-3223. PMC 1892232. &lt;nowiki&gt;PMID 16815322&lt;/nowiki&gt;&lt;/ref&gt;
|.9% Female, .3% Male
|Nationally representative US sample of adults
|World Health Organization-Clinical International Diagnostic Interview (WHO-CIDI)
|-
|Non-clinical: Population-based (NCS-A)&lt;ref&gt;Swanson, Sonja A.; Crow, Scott J.; Le Grange, Daniel; Swendsen, Joel; Merikangas, Kathleen R. (2011-07-01). &quot;Prevalence and correlates of eating disorders in adolescents. Results from the national comorbidity survey replication adolescent supplement&quot;. ''Archives of General Psychiatry'' '''68''' (7): 714–723. doi:10.1001/archgenpsychiatry.2011.22. ISSN 1538-3636.&lt;nowiki&gt;PMID 21383252&lt;/nowiki&gt;&lt;/ref&gt;
|.3% Female, .3% Male
|Nationally representative US sample of adolescents
|WHO-CIDI
|-
|Non-clinical: US Population-based prevalence (Adolescent Brain Cognitive Development (ABCD) study&lt;ref&gt;{{Cite journal|last=Rozzell|first=Kaitlin|last2=Moon|first2=Da Yeoun|last3=Klimek|first3=Patrycja|last4=Brown|first4=Tiffany|last5=Blashill|first5=Aaron J.|date=2019-01-01|title=Prevalence of Eating Disorders Among US Children Aged 9 to 10 Years: Data From the Adolescent Brain Cognitive Development (ABCD) Study|url=http://archpedi.jamanetwork.com/article.aspx?doi=10.1001/jamapediatrics.2018.3678|journal=JAMA Pediatrics|language=en|volume=173|issue=1|pages=100|doi=10.1001/jamapediatrics.2018.3678|issn=2168-6203|pmc=PMC6583451|pmid=30476983}}&lt;/ref&gt;
|0.1%, no gender differences
|Nationally representative US sample of 9- and 10-year old children
|DSM-5 using Kiddie Schedule for Affective Disorders and Schizophrenia (KSADS)
|-
|Non-clinical: Latinos in US&lt;ref&gt;Alegria, Margarita; Woo, Meghan; Cao, Zhun; Torres, Maria; Meng, Xiao-li; Striegel-Moore, Ruth (2007-11-01). &quot;Prevalence and correlates of eating disorders in Latinos in the United States&quot;.''The International Journal of Eating Disorders''. 40 Suppl: S15–21. doi:10.1002/eat.20406. ISSN 0276-3478. PMC 2680162. &lt;nowiki&gt;PMID 17584870&lt;/nowiki&gt;&lt;/ref&gt;
|.12% Female, .03% Male
|Latino Households in the US
|WHO-CIDI
|-
|Non-clinical: African Americans and Caribbean Blacks in the US (NSAL)&lt;ref&gt;Taylor, Jacquelyn Y.; Caldwell, Cleopatra Howard; Baser, Raymond E.; Faison, Nakesha; Jackson, James S. (2007-11-01). &quot;Prevalence of eating disorders among Blacks in the National Survey of American Life&quot;. ''The International Journal of Eating Disorders''. 40 Suppl: S10–14. doi:10.1002/eat.20451. ISSN 0276-3478. PMC 2882704. &lt;nowiki&gt;PMID 17879287&lt;/nowiki&gt;&lt;/ref&gt;
|.14% Female, .2% Male
|National probability sample of adult and adolescent African Americans and Caribbean Blacks
|WHO-CIDI
|-
|Non-clinical: National Latino and Asian American Study (NLAAS)
|.12% Female, .05% Male
|Asian American adults in US households
|WHO-CIDI
|-
| Non-clinical; college students&lt;ref&gt;Tyler ID. A true picture of eating disorders among African American women: a review of literature. ABNF J. 2003;14(3):73-4.&lt;/ref&gt; || .0%  || US African American college females || Eating Disorder Diagnostic Questionnaire (EDD-Q)
|-
| Non-clinical; Adolescents&lt;ref&gt;Stice, E., Becker, C. B., &amp; Yokum, S. (2013). Eating disorder prevention: Current evidence-base and future directions. ''Int. J. Eat. Disord. International Journal of Eating Disorders,'' ''46''(5), 478-485.&lt;/ref&gt; || .8% || US Female Adolescents || Eating Disorder Diagnosis Interview (EDDI)
|-
| Non-clinical; student-athletes&lt;ref&gt;Johnson C, Powers PS, Dick R. Athletes and eating disorders: the national collegiate athletic association study. Int J Eat Disord 1999;26:179e88.&lt;/ref&gt; || .0% || US Division-I Varsity Student Athletes || Eating Disorder Inventory-2 (EDI-2)
|-
| Non-clinical; Military&lt;ref&gt;McNulty, PAF. (2001). Prevalence and contributing factors of eating disorder behaviors in active duty service women in the Army, Navy, Air Force and Marines. Military Medicine, 166(1), 53-58. &lt;/ref&gt; || 1.1% || Active duty females in US Army, Navy, 
Airforce, and Marines 
|| EDI-2
|-
|Non-clinical; Commonwealth of Virginia Mid-Atlantic Twin Registry (MATR)&lt;ref&gt;Kendler KS, Walters EE, Neale MC, Kessler R, Heath A, Eaves L. The structure of genetic and environmental risk factors for six major psychiatric disorders in women. Archives of general psychiatry. 1995;52:374–383.&lt;/ref&gt;
|1.62% (narrow), 3.70% (broad)
|US Caucasian female same-sex twins
|Structured Clinical Interview for DSM Disorders (SCID)
|-
|Non-clinical; Health Omnibus Survey (HOS)
|.46% (3 months; combined)
|South Australian older adolescents and adults
|Eating Disorder Examination (EDE)
|-
|Military&lt;ref&gt;{{Cite journal|title=Diagnosed eating disorders in the U.S. Military: a nine year review|url=http://www.ncbi.nlm.nih.gov/pubmed/18821361|journal=Eating Disorders|date=2008-12-01|issn=1532-530X|pmid=18821361|pages=363–377|volume=16|issue=5|doi=10.1080/10640260802370523|first=Amanda J.|last=Antczak|first2=Teresa L.|last2=Brininger}}&lt;/ref&gt;
|.04% (combined)
|US Military
|ICD codes from electronic records
|-
|Non-clinical; Military
|.02% (7 years) Female, 
0.0% (7 years) Males
|US Military Academy cadets
|[https://www.eat-26.com/ Eating Attitudes Test- 26 items] (EAT-26)&lt;ref&gt;{{Cite web|url=https://www.eat-26.com/|title=EAT-26: Eating Attitudes Test &amp; Eating Disorder Testing – Use the EAT-26 to help you determine if you need to speak to a mental health professional to get help for an eating disorder.|last=admin|language=en|access-date=2022-05-30}}&lt;/ref&gt;
|-
|Non-clinical; Military
|1.1% (current &amp; past) Female
|US Navy female nurses
|DSM-III
|-
|Non-clinical; Military
|.04% Female, .005% Males
|US veterans
|ICD-9-CM
|-
|Non-clinical; Military
|2.5% Males
|Active duty males in US Navy
|N/A
|-
|Clinical; Collaborative Study on the Genetics of Alcoholism (COGA)&lt;ref&gt;{{Cite journal|title=Anorexia nervosa and bulimia nervosa in alcohol-dependent men and women and their relatives|url=http://www.ncbi.nlm.nih.gov/pubmed/8540597|journal=The American Journal of Psychiatry|date=1996-01-01|issn=0002-953X|pmid=8540597|pages=74–82|volume=153|issue=1|doi=10.1176/ajp.153.1.74|first=M. A.|last=Schuckit|first2=J. E.|last2=Tipp|first3=R. M.|last3=Anthenelli|first4=K. K.|last4=Bucholz|first5=V. M.|last5=Hesselbrock|first6=J. I.|last6=Nurnberger}}&lt;/ref&gt;
|1.41% (lifetime) Female,
.00% (lifetime) Male
|US alcohol-dependent adults from San Diego, St. Lois, Iowa City, Farmington, New York, &amp; Indianapolis
|SSAGA
|-
|Non-clinical; healthcare members
|.0269% (current)
|Healthcare provider records
|
|-
|Non-clinical; high school students
|. 00% (point), .45% (lifetime) Female
.00% (point), .00% (lifetime) Males
|US high school students in west central Oregon
|DSM-III-R&lt;sub&gt;4&lt;/sub&gt;
|-
|Clinical; substance users
|.4% (lifetime), .3% (current) Female
.4% (lifetime), .3% (current) Males
|Canadian treatment-seeking substance users
|DIS&lt;sub&gt;9&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Europe'''
|-
|Non-clinical; adolescents
|.3% Female
|Adolescent females residing in Navarra, Spain
|EAT-40&lt;sub&gt;3&lt;/sub&gt;
|-
|Non-clinical; adolescents
|.7% (lifetime) Female, .2% Male
|Adolescents in secondary schools in Sør-Trøndelag, County in Norway
|SEDs&lt;sub&gt;10&lt;/sub&gt;
|-
|Non-clinical; adolescents
|.7% (point; age 15), 1.8% (lifetime, age 15), .00% (point, age 18), 2.6% (lifetime, age 18), .9% (3 years) Female
.00% (point &amp; lifetime) Males
|Adolescents in a comprehensive school in Ostrobothnia district in Finland
|RAB-T&lt;sub&gt;11&lt;/sub&gt;&amp;nbsp;&amp; RAB-R&lt;sub&gt;12&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Australia'''
|-
|Non-clinical; adolescents
|.00% (full), 1.8% (partial) Female
|Adolescent females residing in Victoria, Australia
|BET&lt;sub&gt;13&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Central &amp; South America'''
|-
|Non-clinical; college students
|.00% Female
|Mexican first &amp; second year college females
|EAT-40&lt;sub&gt;3&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''East Asia'''
|-
|Clinical; eating disorder patients
|.53% Female
|Adolescent and adult Japanese patients at a university hospital
|DSM-III-R&lt;sub&gt;4&lt;/sub&gt;
|-
|Non-clinical; Korean Epidemiologic Catchment Area (KECA) Study
|.1% (lifetime), .1% (12 months) Female
.2% (lifetime), .00% (12 months)

Males
|Korean adults
|K-CIDI&lt;sub&gt;15&lt;/sub&gt;&amp;nbsp;2.1
|-
|Centers participating in the Collaborative Study on the Genetics of  Alcoholism in San Diego; St. Louis; Iowa City; Farmington, CN; New York; &amp; Indianapolis
|1.41% Females
|Alcohol-dependent adults
|Semi-Structured Assessment for the Genetics of Alcoholism; criteria based on DSM-III-R
|}

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==

=== Psychometric properties of screening instruments for anorexia nervosa ===
The following section contains a list of screening and diagnostic instruments for anorexia nerova. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here.]]

* ''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments '''click here.''']''

{| class=&quot;wikitable&quot;
|-
! Measure 
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Inter-rater reliability
!Test-retest reliability
!Construct validity
!Content validity
!Highly recommended
|-
|Structured Clinical Interview Diagnosis for DSM-IV (SCID-IV) Module H

* [https://www.columbiapsychiatry.org/research/research-areas/services-policy-and-law/structured-clinical-interview-dsm-disorders-scid-1 SCID-IV (for purchase)]
* [[wikipedia:Structured_Clinical_Interview_for_DSM-IV|Wiki Page]]
|Interview
(Patient)
|
|
|G
|L
|A
|E
|X
|-
|Eating Disorder Examination (EDE)

*OSF
*[[wikipedia:Eating_Disorder_Examination_Interview|Wiki Page]]
|Interview
(Patient)
|
|45min-1.25hrs
|E
|A
|A
|E
|X
|-
|[http://www.corc.uk.net/media/1273/ede-q_quesionnaire.pdf EDE-Q (Eating Disorder Examination Questionnaire)]
*OSF
*[[wikipedia:Eating_Disorder_Examination_Interview|Wiki Page]]
|Questionnaire

(Patient)
|
|15-20 min
|NA
|L
|U
|E
|
|-
|[http://www.ori.org/files/Static%20Page%20Files/EDDSDSM-5_10_14.pdf EDDS (Eating Disorder Diagnosis Scale)]
*OSF
*[[wikipedia:Eating_Disorder_Diagnostic_Scale|Wiki Page]]
|Questionnaire 
(Patient)
|
|10-15 min
|NA
|A
|A
|G
|X
|}
&lt;ref&gt;Hunsley, J., &amp; Mash, E. J. (2008). Guide to Assessments that Work. Cary, NC, USA: Oxford University Press, USA. Retrieved from &lt;nowiki&gt;http://www.ebrary.com&lt;/nowiki&gt;&lt;/ref&gt;'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Likelihood ratios and AUCs of screening instruments for anorexia nervosa===
* ''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments '''click here.''']''
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! &lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Mintz|first=L. B.|last2=O'Halloran|first2=M. S.|date=June 2000|title=The Eating Attitudes Test: validation with DSM-IV eating disorder criteria|url=https://www.ncbi.nlm.nih.gov/pubmed/10900574|journal=Journal of Personality Assessment|volume=74|issue=3|pages=489–503|doi=10.1207/S15327752JPA7403_11|issn=0022-3891|pmid=10900574}}&lt;/ref&gt;Screening Measure (Primary Reference)
! AUC
! DiLR+ (Score)
! DiLR- (Score)
!Clinical Generalizability
!Download
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''Biological &amp; Physical Measures'''
|
|-
|Serum leptin level&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Föcker|first=M.|last2=Timmesfeld|first2=N.|last3=Scherag|first3=S.|last4=Bühren|first4=K.|last5=Langkamp|first5=M.|last6=Dempfle|first6=A.|last7=Sheridan|first7=E. M.|last8=Zwaan|first8=M. de|last9=Fleischhaker|first9=C.|date=2011-04-01|title=Screening for anorexia nervosa via measurement of serum leptin levels|url=https://link.springer.com/article/10.1007/s00702-010-0551-z|journal=Journal of Neural Transmission|language=en|volume=118|issue=4|pages=571–578|doi=10.1007/s00702-010-0551-z|issn=0300-9564}}&lt;/ref&gt;
|0.984 (N=139)
|14.72 (&lt;2.31)
|0.10 (2.31+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
|Serum leptin standard deviation score (SDS)&lt;ref name=&quot;:7&quot; /&gt;
|0.939 (N=139)
|5.89 (&lt;-0.45)
|0.09 (-0.45+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
|Body Mass Index (BMI)&lt;ref name=&quot;:7&quot; /&gt;
|0.936 (N=139)
|5.89 (&lt;17.10)
|0.11 (17.10+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''REDS-C&lt;sub&gt;1&lt;/sub&gt;'''
|
|-
|REDS-C&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Boraska|first=V.|last2=Franklin|first2=C. S.|last3=Floyd|first3=J. a. B.|last4=Thornton|first4=L. M.|last5=Huckins|first5=L. M.|last6=Southam|first6=L.|last7=Rayner|first7=N. W.|last8=Tachmazidou|first8=I.|last9=Klump|first9=K. L.|date=October 2014|title=A genome-wide association study of anorexia nervosa|url=https://www.ncbi.nlm.nih.gov/pubmed/24514567|journal=Molecular Psychiatry|volume=19|issue=10|pages=1085–1094|doi=10.1038/mp.2013.187|issn=1476-5578|pmc=PMC4325090|pmid=24514567}}&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt;
|0.658 (N=333), 0.624 (N=236, &gt; 13 years old), 0.772 (N=97, ≤ 13 years old)
| --
| --
|Children and adolescents (8-18 years old) at a clinic being treated for eating and weight concerns. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDI&lt;sub&gt;2&lt;/sub&gt;'''
|
|-
|EDI-2 &lt;ref name=&quot;:3&quot;&gt;Garner, D. M. (1991). ''Manual for Eating Disorder Inventory.'' Odessa, FL: Psychological Assessment Resources, Inc.&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt; 
|0.556 (N=77)
| --
| --
|Children and adolescents (10-18 years old) at a clinic being treated for eating and weight concerns. Eating disorders were not separated.
|
|-
|EDI-2 - Drive for Thinness Sub-scale&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Segura-Garcia|first=Cristina|last2=Ramacciotti|first2=Carla|last3=Rania|first3=Marianna|last4=Aloi|first4=Matteo|last5=Caroleo|first5=Mariarita|last6=Bruni|first6=Antonella|last7=Gazzarrini|first7=Denise|last8=Sinopoli|first8=Flora|last9=De Fazio|first9=Pasquale|date=June 2015|title=The prevalence of orthorexia nervosa among eating disorder patients after treatment|url=https://www.ncbi.nlm.nih.gov/pubmed/25543324|journal=Eating and weight disorders: EWD|volume=20|issue=2|pages=161–166|doi=10.1007/s40519-014-0171-y|issn=1590-1262|pmid=25543324}}&lt;/ref&gt; 
|0.97 (N=92)
|29.39 (14+)
|0.274 (DT≤14)
|Italian individuals with a DSM-IV-TR eating disorder versus individuals at risk for eating disorders. Eating disorders were not separated.
|
|-
|EDI-3 - Drive for Thinness Subscale (Garner, 2004)

Lehmann et al., 2013
|.903 (N=1298)
| --
| --
|&amp;nbsp;
|
|-
|EDI-3 - Eating Disorder Risk Composite &lt;ref name=&quot;:9&quot; /&gt;(EDRC) (Garner, 2004)
|0.942 (N=92)
|52.08 (EDRC 75+, adults; 90+ adolescents)
|0.379 (EDRC≤75, adults; ≤90 adolescents)
|Italian individuals with a DSM-IV-TR eating disorder versus individuals at risk for eating disorders. Eating disorders were not separated.
|
|-
|EDI-3 - Interoceptive Deficits Subscale &lt;ref name=&quot;:10&quot;&gt;{{Cite journal|last=Clausen|first=Loa|last2=Rosenvinge|first2=Jan H.|last3=Friborg|first3=Oddgeir|last4=Rokkedal|first4=Kristian|date=2011-03-01|title=Validating the Eating Disorder Inventory-3 (EDI-3): A Comparison Between 561 Female Eating Disorders Patients and 878 Females from the General Population|url=https://link.springer.com/article/10.1007/s10862-010-9207-4|journal=Journal of Psychopathology and Behavioral Assessment|language=en|volume=33|issue=1|pages=101–110|doi=10.1007/s10862-010-9207-4|issn=0882-2689}}&lt;/ref&gt;(Garner, 2004)
|.911 (N=2561)
|4.0 (9+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interoceptive Deficits Subscale (Garner, 2004)

Lehmann et al., 2013
|.901 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Asceticism Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.886 (N=2561)
|6.5 (9+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Asceticism Subscale (Garner, 2004)

Lehmann et al., 2013
|.902 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Low Self-Esteem Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.884 (N=2561)
|5.8 (10+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited fm an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Low Self-Esteem Subscale (Garner, 2004)

Lehmann et al., 2013
|.906 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Personal Alienation Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.88 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Personal Alienation Subscale (Garner, 2004)

Lehmann et al., 2013
|.899 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Emotional Dysregulation Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.81 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Emotional Dysregulation Subscale (Garner, 2004)

Lehmann et al., 2013
|.779 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Interpersonal Alienation Subscale &lt;ref name=&quot;:10&quot; /&gt;(Garner, 2004)
|.79 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interpersonal Alienation Subscale (Garner, 2004)

Lehmann et al., 2013
|.743 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Perfectionism Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.79 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Perfectionism Subscale (Garner, 2004)

Lehmann et al., 2013
|.768 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Maturity Fears Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.77 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Maturity Fears Subscale (Garner, 2004)

Lehmann et al., 2013
|.678 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Bulimia Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.76 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Bulimia Subscale (Garner, 2004)

Lehmann et al., 2013
|.776 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Interpersonal Insecurity Subscale &lt;ref name=&quot;:10&quot; /&gt;(Garner, 2004)
|.76 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interpersonal Insecurity Subscale (Garner, 2004)

Lehmann et al., 2013
|.697 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Body Dissatisfaction Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.722 (N=2561)
|1.7 (15+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Body Dissatisfaction Subscale (Garner, 2004)

Lehmann et al., 2013
|.849 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EAT&lt;sub&gt;3&lt;/sub&gt;'''
|
|-
|EAT-26&amp;nbsp;&lt;ref name=&quot;:4&quot; /&gt;&lt;ref&gt;{{Cite journal|title = The eating attitudes test: psychometric features and clinical correlates|url = http://www.ncbi.nlm.nih.gov/pubmed/6961471|journal = Psychological Medicine|date = 1982-11-01|issn = 0033-2917|pmid = 6961471|pages = 871–878|volume = 12|issue = 4|first = D. M.|last = Garner|first2 = M. P.|last2 = Olmsted|first3 = Y.|last3 = Bohr|first4 = P. E.|last4 = Garfinkel}}&lt;/ref&gt;
|.90 (N=129)
|12.83 (20+)
|.24 (&lt;20)
|Low-moderate: College women with no eating disorder versus college women with a DSM-IV eating disorder. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDE-Q&lt;sub&gt;4&lt;/sub&gt;'''
|
|-
|EDE-Q&amp;nbsp;&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|last=Aardoom|first=Jiska J.|last2=Dingemans|first2=Alexandra E.|last3=Slof Op't Landt|first3=Margarita C. T.|last4=Van Furth|first4=Eric F.|date=December 2012|title=Norms and discriminative validity of the Eating Disorder Examination Questionnaire (EDE-Q)|url=https://www.ncbi.nlm.nih.gov/pubmed/23121779|journal=Eating Behaviors|volume=13|issue=4|pages=305–309|doi=10.1016/j.eatbeh.2012.09.002|issn=1873-7358|pmid=23121779}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|title = Assessment of eating disorders: interview or self-report questionnaire?|url = http://www.ncbi.nlm.nih.gov/pubmed/7866415|journal = The International Journal of Eating Disorders|date = 1994-12-01|issn = 0276-3478|pmid = 7866415|pages = 363–370|volume = 16|issue = 4|first = C. G.|last = Fairburn|first2 = S. J.|last2 = Beglin}}&lt;/ref&gt;
|.96 (N=1170)
| --
| --
|Moderate: Dutch treatment-seeking females meeting DSM-IV criteria for an eating disorder versus female adult general population sample recruited through advertisements and personal contacts. Eating disorders were not separated.
|
|-
|EDE-Q&amp;nbsp;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Fairburn|first=C. G.|last2=Beglin|first2=S. J.|date=December 1994|title=Assessment of eating disorders: interview or self-report questionnaire?|url=https://www.ncbi.nlm.nih.gov/pubmed/7866415|journal=The International Journal of Eating Disorders|volume=16|issue=4|pages=363–370|issn=0276-3478|pmid=7866415}}&lt;/ref&gt;
| --
|6.57 (2.3+)
|0.09 (&lt;2.3)
|Moderate: “Clinically significant eating disorder” from a community sample versus female adultsindividuals without “clinically significant eating disorder” from same sample. Eating disorders were not separated.
|
|-
|EDE-Q &lt;ref name=&quot;:1&quot; /&gt;&lt;ref&gt;Rø, Ø., Reas, D. L., &amp; Stedal, K. (2015). Eating Disorder Examination Questionnaire (EDE‐Q) in Norwegian Adults: Discrimination between Female Controls and Eating Disorder Patients. ''European Eating Disorders Review'', ''23''(5), 408-412.&lt;/ref&gt;
|0.89 (N=2465)
| --
| --
|Norwegian adult females with an eating disorder recruited from eating disorder specialist centers versus controls determined using the DSM-IV.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDQ-O&lt;sub&gt;5&lt;/sub&gt;'''
|
|-
|EDQ-Q&amp;nbsp;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;
|0.72 (N=134)
|1.00 (all criteria met)
|0.92 (≥1 criteria not met)
|Dutch adults recruited from an eating disorder specialist center determined using the DSM-IV-TR.
|
|}

=== Interpreting anorexia nervosa screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]
* Also see the page on [https://en.wikipedia.org/wiki/Likelihood_ratios_in_diagnostic_testing likelihood ratios in diagnostic testing] for more information.

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==
===Gold standard diagnostic interviews===

* For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

===Recommended diagnostic interviews for anorexia nervosa===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! colspan=&quot;10&quot; |Diagnostic instruments for anorexia nervosa
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| [https://dawba.info/a0.html Development and Well-Being Assessment (DAWBA)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
| Structured; child/adolescent and parent versions
| 11-18
|10-20 minutes for the eating disorder module
|
|
|
|
|
|
*
|-
|[https://www.phenxtoolkit.org/protocols/view/230101 Children's Eating Disorder Examination- Child (ChEDE)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
|Semistructured; child version
|8-16 
|45-75 minutes; mean 1 hour
|E
|
|
|
|
|
|-
|[https://www.credo-oxford.com/7.2.html Eating Disorder Examination (EDE)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt; &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|date=2008|publisher=Oxford University Press|others=John Hunsley, Eric J. Mash|isbn=0-19-804245-0|location=New York|oclc=314222270}}&lt;/ref&gt;
|Semistructured; adolescent version
|16+
|45-75 minutes
|E
|A
|A
|E
|X
|
|-
|[https://mccaed.slam.nhs.uk/wp-content/uploads/2022/02/PARDI-Self-14-V.2.1-Oct-2020.pdf Pica, ARFID, and Rumination Disorder Interview] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
|Semistructured; multi-informant and different versions
|2-22
|Mean of 39 minutes
|G (ARFID diagnosis)
|
|
|
|
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Severity interviews for anorexia nervosa===
{| class=&quot;wikitable&quot; border=&quot;1&quot;
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
!Free and Accessible Measures
|-
| Placeholder example (CDRS-R)
| Structured Interview&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Mayes|first=Taryn L.|last2=Bernstein|first2=Ira H.|last3=Haley|first3=Charlotte L.|last4=Kennard|first4=Betsy D.|last5=Emslie|first5=Graham J.|date=2010-12|title=Psychometric Properties of the Children's Depression Rating Scale–Revised in Adolescents|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3003451/|journal=Journal of Child and Adolescent Psychopharmacology|volume=20|issue=6|pages=513–516|doi=10.1089/cap.2010.0063|issn=1044-5463|pmc=PMC3003451|pmid=21186970}}&lt;/ref&gt;
| 6-12
| 15-20 minutes
|
* Link to purchase [https://www.wpspublish.com/store/p/2703/cdrs-r-children-s-depression-rating-scale-revised#purchase-product CDRS-R]
*[http://www.opapc.com/uploads/documents/CDRS-R.pdf PDF] (excerpt)
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a brief overview of treatment options for anorexia nervosa and a list of process and outcome measures for anorexia nervosa. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information on the differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]].

===Process measures===
Body weight is commonly monitored by clinicians throughout the AN treatment process as helping individuals regain and maintain a healthy weight is a primary treatment goal for AN. Many treatment centers have policies prohibiting patients from seeing their weight.

Motivational Stages of Change may be used to monitor individuals’ readiness to take action against eating disorder behaviors. It has demonstrated predictive validity in a sample of female adolescents attending eating disorder treatment groups.&lt;ref&gt;{{Cite journal|last=Gusella|first=Joanne|last2=Butler|first2=Gordon|last3=Nichols|first3=Laura|last4=Bird|first4=Debbie|date=2003-01-01|title=A brief questionnaire to assess readiness to change in adolescents with eating disorders: its applications to group therapy|url=http://onlinelibrary.wiley.com/doi/10.1002/erv.481/abstract|journal=European Eating Disorders Review|language=en|volume=11|issue=1|pages=58–71|doi=10.1002/erv.481|issn=1099-0968}}&lt;/ref&gt; See Appendix E.

=== Outcome and severity measures ===
This table includes clinically significant benchmarks for anorexia nervosa specific outcome measures
* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
*''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
| colspan=&quot;8&quot; |'''Clinically significant change benchmarks with common instruments for anorexia nervosa'''
|-
| colspan=&quot;8&quot;  span style=&quot;font-size:110%; text-align:center;&quot; | &lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; |&lt;b&gt; Measure&lt;/b&gt;
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; | &lt;b&gt; Subscale&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; | &lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; | &lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-
| rowspan=&quot;5&quot; style=&quot;text-align:center;&quot; |'''EDE-Q &lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Mond|first=J. M.|last2=Hay|first2=P. J.|last3=Rodgers|first3=B.|last4=Owen|first4=C.|last5=Beumont|first5=P. J. V.|date=May 2004|title=Validity of the Eating Disorder Examination Questionnaire (EDE-Q) in screening for eating disorders in community samples|url=https://www.ncbi.nlm.nih.gov/pubmed/15033501|journal=Behaviour Research and Therapy|volume=42|issue=5|pages=551–567|doi=10.1016/S0005-7967(03)00161-X|issn=0005-7967|pmid=15033501}}&lt;/ref&gt;'''
| style=&quot;text-align:right;&quot; |''Global''
|1.4
|3.2
|2.3
|.7
|.6
|.3
|-
| style=&quot;text-align:right;&quot; |''Restraint''
|(-.3)
|3.6
|1.8
|1.5
|1.2
|.8
|-
| style=&quot;text-align:right;&quot; |''Eating Concern''
|.1
|2.0
|1.2
|1.1
|.9
|.6
|-
| style=&quot;text-align:right;&quot; |''Weight Concern''
|1.5
|3.9
|2.6
|1.0
|.9
|.5
|-
| style=&quot;text-align:right;&quot; |''Shape Concern''
|2.1
|4.8
|3.2
|.9
|.7
|.4
|-
| rowspan=&quot;5&quot; style=&quot;text-align:center;&quot; |'''EDE &lt;ref name=&quot;:2&quot; /&gt;'''
| style=&quot;text-align:right;&quot; |''Global''
|1.7
|2.3
|1.9
|1.9
|1.6
|1.0
|-
| style=&quot;text-align:right;&quot; |''Restraint''
|.3
|3.3
|1.9
|1.8
|1.5
|.9
|-
| style=&quot;text-align:right;&quot; |''Eating Concern''
|(-.5)
|.9
|.5
|.8
|.7
|.4
|-
| style=&quot;text-align:right;&quot; |''Weight Concern''
|2.0
|2.8
|2.4
|1.3
|1.1
|.7
|-
| style=&quot;text-align:right;&quot; |''Shape Concern''
|2.0
|3.2
|2.6
|1.2
|1.0
|.6
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; |'''EAT-26 &lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Mintz|first=L. B.|last2=O'Halloran|first2=M. S.|date=2000-06-01|title=The Eating Attitudes Test: validation with DSM-IV eating disorder criteria|url=http://www.ncbi.nlm.nih.gov/pubmed/10900574|journal=Journal of Personality Assessment|volume=74|issue=3|pages=489–503|doi=10.1207/S15327752JPA7403_11|issn=0022-3891|pmid=10900574}}&lt;/ref&gt;'''
| style=&quot;text-align:right;&quot; |''Total''
|6.5
|19.6
|15.0
|7.9
|6.7
|4.0
|}

'''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean.

'''Search terms:''' [Anorexia Nervosa OR eating disorder] AND [validity OR clinical significance] in Google Scholar

=== Treatment ===
{{collapse top| Click here for more information on treatment for AN}}
* Treatment of AN typically consists of restoring the individual to a healthy weight and addressing thoughts and behaviors which are related to the eating disorder. It may involve re-nutrition, psychotherapy, nutritional counseling, and medication. 
* Literature reviews of existing research indicate that evidence supporting AN treatment is lacking. A systematic review of AN treatment efficacy studies by Bulik&lt;ref&gt;{{Cite journal|title = Anorexia nervosa treatment: a systematic review of randomized controlled trials|url = http://www.ncbi.nlm.nih.gov/pubmed/17370290|journal = The International Journal of Eating Disorders|date = 2007-05-01|issn = 0276-3478|pmid = 17370290|pages = 310–320|volume = 40|issue = 4|doi = 10.1002/eat.20367|first = Cynthia M.|last = Bulik|first2 = Nancy D.|last2 = Berkman|first3 = Kimberly A.|last3 = Brownley|first4 = Jan A.|last4 = Sedway|first5 = Kathleen N.|last5 = Lohr}}&lt;/ref&gt; found that evidence supporting medications, medications and behavioral interventions, and behavioral interventions alone in adults is weak. 
* There is moderately strong evidence suggesting that behavioral interventions may be helpful for adolescents. In particular, adolescents may benefit from family therapy. 
* Clinical trials investigating AN treatment suffer from high rates of attrition, as key features of AN (e.g., denial, fear of weight gain) may contribute to low motivation for remaining in treatment.
* More severe cases of AN may be treated in inpatient settings, which are equipped to manage the re-nutrition process and provide medical monitoring. 
* Partial hospitalization and intensive outpatient programs may provide intermediate levels of treatment intensity to assist individuals in the transition from intensive care to outpatient care after weight restoration.
{{collapse bottom}}
* Please refer to the page on [https://en.wikipedia.org/wiki/Anorexia_nervosa anorexia nervosa] for more information on available treatment or go to [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/eating-body-image-problems/ the Effective Child Therapy page for Eating &amp; Body Image Problems] for a curated resource on effective treatments for anorexia nervosa.

== '''External resources''' ==
# [http://apps.who.int/classifications/icd10/browse/2010/en#/F50.0 ICD-10 diagnostic criteria]
# [https://en.wikiversity.org/w/index.php?title=Helping_Give_Away_Psychological_Science/Resources/Annotated_List_of_Where_and_How_to_Find_a_Therapist&amp;wteswitched=1#Other_low-cost_options Find-a-Therapist] 
##This is a curated list of find-a-therapist websites where you can find a provider
# NIMH: [https://www.nimh.nih.gov/health/publications/eating-disorders/index.shtml ''Eating Disorders--About More Than Food'']  and [https://www.nimh.nih.gov/health/topics/eating-disorders/index.shtml ''Eating Disorders'']
##These NIMH website posts provide more information on anorexia nervosa
#[https://www.hopkinsmedicine.org/psychiatry/specialty_areas/eating_disorders/ John's Hopkins Resource] (guide about anorexia nervosa, treatment, and more)
# OMIM (Online Mendelian Inheritance in Man)
##[https://www.omim.org/entry/606788?search=anorexia%20nervosa&amp;highlight=nervosa%20anorexia Anorexia nervosa]
#[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/eating-body-image-problems/ Effective Child Therapy page for anorexia nervosa]
##Effective Child Therapy is website sponsored by Division 53 of the American Psychological Association (APA), or The [https://sccap53.org Society of Clinical Child and Adolescent Psychology] (SCCAP), in collaboration with the Association for Behavioral and Cognitive Therapies (ABCT). Use for information on symptoms and available treatments.

== '''References''' ==
{{collapse top|Click here for references}}
{{Reflist|30em}}

[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]
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{{medical disclaimer}}
{{:{{BASEPAGENAME}}/Sidebar}}

==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
For background information on what assessment portfolios are, click the link in the heading above.
Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Anorexia nervosa (assessment portfolio)/extended version|here]].

== [[Evidence based assessment/Preparation phase|'''Preparation phase''']] ==
=== Diagnostic criteria for anorexia nervosa ===
{{blockquotetop}}
'''ICD-11 Criteria'''

Anorexia Nervosa is characterized by significantly low body weight, which is less than minimal normal/expected weight for the individual’s height, sex, age and developmental stage (body mass index (BMI) less than 18.5 kg/m2 in adults and BMI-for-age under fifth percentile in children and adolescents) that is not due to another health condition or to the unavailability of food. Low body weight is accompanied by a persistent pattern of behaviors to prevent restoration of normal weight, which may include behaviors aimed at reducing energy intake (restricted eating), purging behaviors (e.g., self-induced vomiting, misuse of laxatives), and behaviors aimed at increasing energy expenditure (e.g., excessive exercise), typically associated with a fear of weight gain. Low body weight or shape is central to the person's self-evaluation or is inaccurately perceived to be normal or even excessive.

'''Changes in DSM-5'''

* The diagnostic criteria for anorexia nervosa changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here] and [[w:DSM-5|here]].
{{blockquotebottom}}

=== Base rates of anorexia nervosa in different populations and clinical settings ===
This section describes the demographic setting of the population(s) sampled, base rates of diagnosis such as prevalence rates, country/region sampled, and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of adolescent depression that they are likely to see in their clinical practice.

* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']]

{| class=&quot;wikitable&quot;
|-
! Setting (Reference) 
! Base Rate
! Demography
! Diagnostic Method
|-
|Non-clinical: Population-based (NCS-R)&lt;ref&gt;Hudson, James I.; Hiripi, Eva; Pope, Harrison G.; Kessler, Ronald C. (2007-02-01). &quot;The prevalence and correlates of eating disorders in the National Comorbidity Survey Replication&quot;.''Biological Psychiatry'' '''61''' (3): 348–358. doi:10.1016/j.biopsych.2006.03.040. ISSN 0006-3223. PMC 1892232. &lt;nowiki&gt;PMID 16815322&lt;/nowiki&gt;&lt;/ref&gt;
|.9% Female, .3% Male
|Nationally representative US sample of adults
|World Health Organization-Clinical International Diagnostic Interview (WHO-CIDI)
|-
|Non-clinical: Population-based (NCS-A)&lt;ref&gt;Swanson, Sonja A.; Crow, Scott J.; Le Grange, Daniel; Swendsen, Joel; Merikangas, Kathleen R. (2011-07-01). &quot;Prevalence and correlates of eating disorders in adolescents. Results from the national comorbidity survey replication adolescent supplement&quot;. ''Archives of General Psychiatry'' '''68''' (7): 714–723. doi:10.1001/archgenpsychiatry.2011.22. ISSN 1538-3636.&lt;nowiki&gt;PMID 21383252&lt;/nowiki&gt;&lt;/ref&gt;
|.3% Female, .3% Male
|Nationally representative US sample of adolescents
|WHO-CIDI
|-
|Non-clinical: US Population-based prevalence (Adolescent Brain Cognitive Development (ABCD) study&lt;ref&gt;{{Cite journal|last=Rozzell|first=Kaitlin|last2=Moon|first2=Da Yeoun|last3=Klimek|first3=Patrycja|last4=Brown|first4=Tiffany|last5=Blashill|first5=Aaron J.|date=2019-01-01|title=Prevalence of Eating Disorders Among US Children Aged 9 to 10 Years: Data From the Adolescent Brain Cognitive Development (ABCD) Study|url=http://archpedi.jamanetwork.com/article.aspx?doi=10.1001/jamapediatrics.2018.3678|journal=JAMA Pediatrics|language=en|volume=173|issue=1|pages=100|doi=10.1001/jamapediatrics.2018.3678|issn=2168-6203|pmc=PMC6583451|pmid=30476983}}&lt;/ref&gt;
|0.1%, no gender differences
|Nationally representative US sample of 9- and 10-year old children
|DSM-5 using Kiddie Schedule for Affective Disorders and Schizophrenia (KSADS)
|-
|Non-clinical: Latinos in US&lt;ref&gt;Alegria, Margarita; Woo, Meghan; Cao, Zhun; Torres, Maria; Meng, Xiao-li; Striegel-Moore, Ruth (2007-11-01). &quot;Prevalence and correlates of eating disorders in Latinos in the United States&quot;.''The International Journal of Eating Disorders''. 40 Suppl: S15–21. doi:10.1002/eat.20406. ISSN 0276-3478. PMC 2680162. &lt;nowiki&gt;PMID 17584870&lt;/nowiki&gt;&lt;/ref&gt;
|.12% Female, .03% Male
|Latino Households in the US
|WHO-CIDI
|-
|Non-clinical: African Americans and Caribbean Blacks in the US (NSAL)&lt;ref&gt;Taylor, Jacquelyn Y.; Caldwell, Cleopatra Howard; Baser, Raymond E.; Faison, Nakesha; Jackson, James S. (2007-11-01). &quot;Prevalence of eating disorders among Blacks in the National Survey of American Life&quot;. ''The International Journal of Eating Disorders''. 40 Suppl: S10–14. doi:10.1002/eat.20451. ISSN 0276-3478. PMC 2882704. &lt;nowiki&gt;PMID 17879287&lt;/nowiki&gt;&lt;/ref&gt;
|.14% Female, .2% Male
|National probability sample of adult and adolescent African Americans and Caribbean Blacks
|WHO-CIDI
|-
|Non-clinical: National Latino and Asian American Study (NLAAS)
|.12% Female, .05% Male
|Asian American adults in US households
|WHO-CIDI
|-
| Non-clinical; college students&lt;ref&gt;Tyler ID. A true picture of eating disorders among African American women: a review of literature. ABNF J. 2003;14(3):73-4.&lt;/ref&gt; || .0%  || US African American college females || Eating Disorder Diagnostic Questionnaire (EDD-Q)
|-
| Non-clinical; Adolescents&lt;ref&gt;Stice, E., Becker, C. B., &amp; Yokum, S. (2013). Eating disorder prevention: Current evidence-base and future directions. ''Int. J. Eat. Disord. International Journal of Eating Disorders,'' ''46''(5), 478-485.&lt;/ref&gt; || .8% || US Female Adolescents || Eating Disorder Diagnosis Interview (EDDI)
|-
| Non-clinical; student-athletes&lt;ref&gt;Johnson C, Powers PS, Dick R. Athletes and eating disorders: the national collegiate athletic association study. Int J Eat Disord 1999;26:179e88.&lt;/ref&gt; || .0% || US Division-I Varsity Student Athletes || Eating Disorder Inventory-2 (EDI-2)
|-
| Non-clinical; Military&lt;ref&gt;McNulty, PAF. (2001). Prevalence and contributing factors of eating disorder behaviors in active duty service women in the Army, Navy, Air Force and Marines. Military Medicine, 166(1), 53-58. &lt;/ref&gt; || 1.1% || Active duty females in US Army, Navy, 
Airforce, and Marines 
|| EDI-2
|-
|Non-clinical; Commonwealth of Virginia Mid-Atlantic Twin Registry (MATR)&lt;ref&gt;Kendler KS, Walters EE, Neale MC, Kessler R, Heath A, Eaves L. The structure of genetic and environmental risk factors for six major psychiatric disorders in women. Archives of general psychiatry. 1995;52:374–383.&lt;/ref&gt;
|1.62% (narrow), 3.70% (broad)
|US Caucasian female same-sex twins
|Structured Clinical Interview for DSM Disorders (SCID)
|-
|Non-clinical; Health Omnibus Survey (HOS)
|.46% (3 months; combined)
|South Australian older adolescents and adults
|Eating Disorder Examination (EDE)
|-
|Military&lt;ref&gt;{{Cite journal|title=Diagnosed eating disorders in the U.S. Military: a nine year review|url=http://www.ncbi.nlm.nih.gov/pubmed/18821361|journal=Eating Disorders|date=2008-12-01|issn=1532-530X|pmid=18821361|pages=363–377|volume=16|issue=5|doi=10.1080/10640260802370523|first=Amanda J.|last=Antczak|first2=Teresa L.|last2=Brininger}}&lt;/ref&gt;
|.04% (combined)
|US Military
|ICD codes from electronic records
|-
|Non-clinical; Military
|.02% (7 years) Female, 
0.0% (7 years) Males
|US Military Academy cadets
|[https://www.eat-26.com/ Eating Attitudes Test- 26 items] (EAT-26)&lt;ref&gt;{{Cite web|url=https://www.eat-26.com/|title=EAT-26: Eating Attitudes Test &amp; Eating Disorder Testing – Use the EAT-26 to help you determine if you need to speak to a mental health professional to get help for an eating disorder.|last=admin|language=en|access-date=2022-05-30}}&lt;/ref&gt;
|-
|Non-clinical; Military
|1.1% (current &amp; past) Female
|US Navy female nurses
|DSM-III
|-
|Non-clinical; Military
|.04% Female, .005% Males
|US veterans
|ICD-9-CM
|-
|Non-clinical; Military
|2.5% Males
|Active duty males in US Navy
|N/A
|-
|Clinical; Collaborative Study on the Genetics of Alcoholism (COGA)&lt;ref&gt;{{Cite journal|title=Anorexia nervosa and bulimia nervosa in alcohol-dependent men and women and their relatives|url=http://www.ncbi.nlm.nih.gov/pubmed/8540597|journal=The American Journal of Psychiatry|date=1996-01-01|issn=0002-953X|pmid=8540597|pages=74–82|volume=153|issue=1|doi=10.1176/ajp.153.1.74|first=M. A.|last=Schuckit|first2=J. E.|last2=Tipp|first3=R. M.|last3=Anthenelli|first4=K. K.|last4=Bucholz|first5=V. M.|last5=Hesselbrock|first6=J. I.|last6=Nurnberger}}&lt;/ref&gt;
|1.41% (lifetime) Female,
.00% (lifetime) Male
|US alcohol-dependent adults from San Diego, St. Lois, Iowa City, Farmington, New York, &amp; Indianapolis
|SSAGA
|-
|Non-clinical; healthcare members
|.0269% (current)
|Healthcare provider records
|
|-
|Non-clinical; high school students
|. 00% (point), .45% (lifetime) Female
.00% (point), .00% (lifetime) Males
|US high school students in west central Oregon
|DSM-III-R&lt;sub&gt;4&lt;/sub&gt;
|-
|Clinical; substance users
|.4% (lifetime), .3% (current) Female
.4% (lifetime), .3% (current) Males
|Canadian treatment-seeking substance users
|DIS&lt;sub&gt;9&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Europe'''
|-
|Non-clinical; adolescents
|.3% Female
|Adolescent females residing in Navarra, Spain
|EAT-40&lt;sub&gt;3&lt;/sub&gt;
|-
|Non-clinical; adolescents
|.7% (lifetime) Female, .2% Male
|Adolescents in secondary schools in Sør-Trøndelag, County in Norway
|SEDs&lt;sub&gt;10&lt;/sub&gt;
|-
|Non-clinical; adolescents
|.7% (point; age 15), 1.8% (lifetime, age 15), .00% (point, age 18), 2.6% (lifetime, age 18), .9% (3 years) Female
.00% (point &amp; lifetime) Males
|Adolescents in a comprehensive school in Ostrobothnia district in Finland
|RAB-T&lt;sub&gt;11&lt;/sub&gt;&amp;nbsp;&amp; RAB-R&lt;sub&gt;12&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Australia'''
|-
|Non-clinical; adolescents
|.00% (full), 1.8% (partial) Female
|Adolescent females residing in Victoria, Australia
|BET&lt;sub&gt;13&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Central &amp; South America'''
|-
|Non-clinical; college students
|.00% Female
|Mexican first &amp; second year college females
|EAT-40&lt;sub&gt;3&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''East Asia'''
|-
|Clinical; eating disorder patients
|.53% Female
|Adolescent and adult Japanese patients at a university hospital
|DSM-III-R&lt;sub&gt;4&lt;/sub&gt;
|-
|Non-clinical; Korean Epidemiologic Catchment Area (KECA) Study
|.1% (lifetime), .1% (12 months) Female
.2% (lifetime), .00% (12 months)

Males
|Korean adults
|K-CIDI&lt;sub&gt;15&lt;/sub&gt;&amp;nbsp;2.1
|-
|Centers participating in the Collaborative Study on the Genetics of  Alcoholism in San Diego; St. Louis; Iowa City; Farmington, CN; New York; &amp; Indianapolis
|1.41% Females
|Alcohol-dependent adults
|Semi-Structured Assessment for the Genetics of Alcoholism; criteria based on DSM-III-R
|}

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==

=== Psychometric properties of screening instruments for anorexia nervosa ===
The following section contains a list of screening and diagnostic instruments for anorexia nerova. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here.]]

* ''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments '''click here.''']''

{| class=&quot;wikitable&quot;
|-
! Measure 
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Inter-rater reliability
!Test-retest reliability
!Construct validity
!Content validity
!Highly recommended
|-
|Structured Clinical Interview Diagnosis for DSM-IV (SCID-IV) Module H

* [https://www.columbiapsychiatry.org/research/research-areas/services-policy-and-law/structured-clinical-interview-dsm-disorders-scid-1 SCID-IV (for purchase)]
* [[wikipedia:Structured_Clinical_Interview_for_DSM-IV|Wiki Page]]
|Interview
(Patient)
|
|
|G
|L
|A
|E
|X
|-
|Eating Disorder Examination (EDE)

*OSF
*[[wikipedia:Eating_Disorder_Examination_Interview|Wiki Page]]
|Interview
(Patient)
|
|45min-1.25hrs
|E
|A
|A
|E
|X
|-
|[http://www.corc.uk.net/media/1273/ede-q_quesionnaire.pdf EDE-Q (Eating Disorder Examination Questionnaire)]
*OSF
*[[wikipedia:Eating_Disorder_Examination_Interview|Wiki Page]]
|Questionnaire

(Patient)
|
|15-20 min
|NA
|L
|U
|E
|
|-
|[http://www.ori.org/files/Static%20Page%20Files/EDDSDSM-5_10_14.pdf EDDS (Eating Disorder Diagnosis Scale)]
*OSF
*[[wikipedia:Eating_Disorder_Diagnostic_Scale|Wiki Page]]
|Questionnaire 
(Patient)
|
|10-15 min
|NA
|A
|A
|G
|X
|}
&lt;ref&gt;Hunsley, J., &amp; Mash, E. J. (2008). Guide to Assessments that Work. Cary, NC, USA: Oxford University Press, USA. Retrieved from &lt;nowiki&gt;http://www.ebrary.com&lt;/nowiki&gt;&lt;/ref&gt;'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Likelihood ratios and AUCs of screening instruments for anorexia nervosa===
* ''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments '''click here.''']''
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! &lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Mintz|first=L. B.|last2=O'Halloran|first2=M. S.|date=June 2000|title=The Eating Attitudes Test: validation with DSM-IV eating disorder criteria|url=https://www.ncbi.nlm.nih.gov/pubmed/10900574|journal=Journal of Personality Assessment|volume=74|issue=3|pages=489–503|doi=10.1207/S15327752JPA7403_11|issn=0022-3891|pmid=10900574}}&lt;/ref&gt;Screening Measure (Primary Reference)
! AUC
! DiLR+ (Score)
! DiLR- (Score)
!Clinical Generalizability
!Download
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''Biological &amp; Physical Measures'''
|
|-
|Serum leptin level&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Föcker|first=M.|last2=Timmesfeld|first2=N.|last3=Scherag|first3=S.|last4=Bühren|first4=K.|last5=Langkamp|first5=M.|last6=Dempfle|first6=A.|last7=Sheridan|first7=E. M.|last8=Zwaan|first8=M. de|last9=Fleischhaker|first9=C.|date=2011-04-01|title=Screening for anorexia nervosa via measurement of serum leptin levels|url=https://link.springer.com/article/10.1007/s00702-010-0551-z|journal=Journal of Neural Transmission|language=en|volume=118|issue=4|pages=571–578|doi=10.1007/s00702-010-0551-z|issn=0300-9564}}&lt;/ref&gt;
|0.984 (N=139)
|14.72 (&lt;2.31)
|0.10 (2.31+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
|Serum leptin standard deviation score (SDS)&lt;ref name=&quot;:7&quot; /&gt;
|0.939 (N=139)
|5.89 (&lt;-0.45)
|0.09 (-0.45+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
|Body Mass Index (BMI)&lt;ref name=&quot;:7&quot; /&gt;
|0.936 (N=139)
|5.89 (&lt;17.10)
|0.11 (17.10+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''REDS-C&lt;sub&gt;1&lt;/sub&gt;'''
|
|-
|REDS-C&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Boraska|first=V.|last2=Franklin|first2=C. S.|last3=Floyd|first3=J. a. B.|last4=Thornton|first4=L. M.|last5=Huckins|first5=L. M.|last6=Southam|first6=L.|last7=Rayner|first7=N. W.|last8=Tachmazidou|first8=I.|last9=Klump|first9=K. L.|date=October 2014|title=A genome-wide association study of anorexia nervosa|url=https://www.ncbi.nlm.nih.gov/pubmed/24514567|journal=Molecular Psychiatry|volume=19|issue=10|pages=1085–1094|doi=10.1038/mp.2013.187|issn=1476-5578|pmc=PMC4325090|pmid=24514567}}&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt;
|0.658 (N=333), 0.624 (N=236, &gt; 13 years old), 0.772 (N=97, ≤ 13 years old)
| --
| --
|Children and adolescents (8-18 years old) at a clinic being treated for eating and weight concerns. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDI&lt;sub&gt;2&lt;/sub&gt;'''
|
|-
|EDI-2 &lt;ref name=&quot;:3&quot;&gt;Garner, D. M. (1991). ''Manual for Eating Disorder Inventory.'' Odessa, FL: Psychological Assessment Resources, Inc.&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt; 
|0.556 (N=77)
| --
| --
|Children and adolescents (10-18 years old) at a clinic being treated for eating and weight concerns. Eating disorders were not separated.
|
|-
|EDI-2 - Drive for Thinness Sub-scale&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Segura-Garcia|first=Cristina|last2=Ramacciotti|first2=Carla|last3=Rania|first3=Marianna|last4=Aloi|first4=Matteo|last5=Caroleo|first5=Mariarita|last6=Bruni|first6=Antonella|last7=Gazzarrini|first7=Denise|last8=Sinopoli|first8=Flora|last9=De Fazio|first9=Pasquale|date=June 2015|title=The prevalence of orthorexia nervosa among eating disorder patients after treatment|url=https://www.ncbi.nlm.nih.gov/pubmed/25543324|journal=Eating and weight disorders: EWD|volume=20|issue=2|pages=161–166|doi=10.1007/s40519-014-0171-y|issn=1590-1262|pmid=25543324}}&lt;/ref&gt; 
|0.97 (N=92)
|29.39 (14+)
|0.274 (DT≤14)
|Italian individuals with a DSM-IV-TR eating disorder versus individuals at risk for eating disorders. Eating disorders were not separated.
|
|-
|EDI-3 - Drive for Thinness Subscale (Garner, 2004)

Lehmann et al., 2013
|.903 (N=1298)
| --
| --
|&amp;nbsp;
|
|-
|EDI-3 - Eating Disorder Risk Composite &lt;ref name=&quot;:9&quot; /&gt;(EDRC) (Garner, 2004)
|0.942 (N=92)
|52.08 (EDRC 75+, adults; 90+ adolescents)
|0.379 (EDRC≤75, adults; ≤90 adolescents)
|Italian individuals with a DSM-IV-TR eating disorder versus individuals at risk for eating disorders. Eating disorders were not separated.
|
|-
|EDI-3 - Interoceptive Deficits Subscale &lt;ref name=&quot;:10&quot;&gt;{{Cite journal|last=Clausen|first=Loa|last2=Rosenvinge|first2=Jan H.|last3=Friborg|first3=Oddgeir|last4=Rokkedal|first4=Kristian|date=2011-03-01|title=Validating the Eating Disorder Inventory-3 (EDI-3): A Comparison Between 561 Female Eating Disorders Patients and 878 Females from the General Population|url=https://link.springer.com/article/10.1007/s10862-010-9207-4|journal=Journal of Psychopathology and Behavioral Assessment|language=en|volume=33|issue=1|pages=101–110|doi=10.1007/s10862-010-9207-4|issn=0882-2689}}&lt;/ref&gt;(Garner, 2004)
|.911 (N=2561)
|4.0 (9+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interoceptive Deficits Subscale (Garner, 2004)

Lehmann et al., 2013
|.901 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Asceticism Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.886 (N=2561)
|6.5 (9+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Asceticism Subscale (Garner, 2004)

Lehmann et al., 2013
|.902 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Low Self-Esteem Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.884 (N=2561)
|5.8 (10+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited fm an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Low Self-Esteem Subscale (Garner, 2004)

Lehmann et al., 2013
|.906 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Personal Alienation Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.88 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Personal Alienation Subscale (Garner, 2004)

Lehmann et al., 2013
|.899 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Emotional Dysregulation Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.81 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Emotional Dysregulation Subscale (Garner, 2004)

Lehmann et al., 2013
|.779 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Interpersonal Alienation Subscale &lt;ref name=&quot;:10&quot; /&gt;(Garner, 2004)
|.79 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interpersonal Alienation Subscale (Garner, 2004)

Lehmann et al., 2013
|.743 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Perfectionism Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.79 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Perfectionism Subscale (Garner, 2004)

Lehmann et al., 2013
|.768 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Maturity Fears Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.77 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Maturity Fears Subscale (Garner, 2004)

Lehmann et al., 2013
|.678 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Bulimia Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.76 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Bulimia Subscale (Garner, 2004)

Lehmann et al., 2013
|.776 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Interpersonal Insecurity Subscale &lt;ref name=&quot;:10&quot; /&gt;(Garner, 2004)
|.76 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interpersonal Insecurity Subscale (Garner, 2004)

Lehmann et al., 2013
|.697 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Body Dissatisfaction Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.722 (N=2561)
|1.7 (15+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Body Dissatisfaction Subscale (Garner, 2004)

Lehmann et al., 2013
|.849 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EAT&lt;sub&gt;3&lt;/sub&gt;'''
|
|-
|EAT-26&amp;nbsp;&lt;ref name=&quot;:4&quot; /&gt;&lt;ref&gt;{{Cite journal|title = The eating attitudes test: psychometric features and clinical correlates|url = http://www.ncbi.nlm.nih.gov/pubmed/6961471|journal = Psychological Medicine|date = 1982-11-01|issn = 0033-2917|pmid = 6961471|pages = 871–878|volume = 12|issue = 4|first = D. M.|last = Garner|first2 = M. P.|last2 = Olmsted|first3 = Y.|last3 = Bohr|first4 = P. E.|last4 = Garfinkel}}&lt;/ref&gt;
|.90 (N=129)
|12.83 (20+)
|.24 (&lt;20)
|Low-moderate: College women with no eating disorder versus college women with a DSM-IV eating disorder. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDE-Q&lt;sub&gt;4&lt;/sub&gt;'''
|
|-
|EDE-Q&amp;nbsp;&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|last=Aardoom|first=Jiska J.|last2=Dingemans|first2=Alexandra E.|last3=Slof Op't Landt|first3=Margarita C. T.|last4=Van Furth|first4=Eric F.|date=December 2012|title=Norms and discriminative validity of the Eating Disorder Examination Questionnaire (EDE-Q)|url=https://www.ncbi.nlm.nih.gov/pubmed/23121779|journal=Eating Behaviors|volume=13|issue=4|pages=305–309|doi=10.1016/j.eatbeh.2012.09.002|issn=1873-7358|pmid=23121779}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|title = Assessment of eating disorders: interview or self-report questionnaire?|url = http://www.ncbi.nlm.nih.gov/pubmed/7866415|journal = The International Journal of Eating Disorders|date = 1994-12-01|issn = 0276-3478|pmid = 7866415|pages = 363–370|volume = 16|issue = 4|first = C. G.|last = Fairburn|first2 = S. J.|last2 = Beglin}}&lt;/ref&gt;
|.96 (N=1170)
| --
| --
|Moderate: Dutch treatment-seeking females meeting DSM-IV criteria for an eating disorder versus female adult general population sample recruited through advertisements and personal contacts. Eating disorders were not separated.
|
|-
|EDE-Q&amp;nbsp;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Fairburn|first=C. G.|last2=Beglin|first2=S. J.|date=December 1994|title=Assessment of eating disorders: interview or self-report questionnaire?|url=https://www.ncbi.nlm.nih.gov/pubmed/7866415|journal=The International Journal of Eating Disorders|volume=16|issue=4|pages=363–370|issn=0276-3478|pmid=7866415}}&lt;/ref&gt;
| --
|6.57 (2.3+)
|0.09 (&lt;2.3)
|Moderate: “Clinically significant eating disorder” from a community sample versus female adultsindividuals without “clinically significant eating disorder” from same sample. Eating disorders were not separated.
|
|-
|EDE-Q &lt;ref name=&quot;:1&quot; /&gt;&lt;ref&gt;Rø, Ø., Reas, D. L., &amp; Stedal, K. (2015). Eating Disorder Examination Questionnaire (EDE‐Q) in Norwegian Adults: Discrimination between Female Controls and Eating Disorder Patients. ''European Eating Disorders Review'', ''23''(5), 408-412.&lt;/ref&gt;
|0.89 (N=2465)
| --
| --
|Norwegian adult females with an eating disorder recruited from eating disorder specialist centers versus controls determined using the DSM-IV.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDQ-O&lt;sub&gt;5&lt;/sub&gt;'''
|
|-
|EDQ-Q&amp;nbsp;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;
|0.72 (N=134)
|1.00 (all criteria met)
|0.92 (≥1 criteria not met)
|Dutch adults recruited from an eating disorder specialist center determined using the DSM-IV-TR.
|
|}

=== Interpreting anorexia nervosa screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]
* Also see the page on [https://en.wikipedia.org/wiki/Likelihood_ratios_in_diagnostic_testing likelihood ratios in diagnostic testing] for more information.

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==
===Gold standard diagnostic interviews===

* For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

===Recommended diagnostic interviews for anorexia nervosa===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! colspan=&quot;10&quot; |Diagnostic instruments for anorexia nervosa
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| [https://dawba.info/a0.html Development and Well-Being Assessment (DAWBA)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
| Structured; child/adolescent and parent versions
| 11-18
|10-20 minutes for the eating disorder module
|
|
|
|
|
|
*
|-
|[https://www.phenxtoolkit.org/protocols/view/230101 Children's Eating Disorder Examination- Child (ChEDE)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
|Semistructured; child version
|8-16 
|45-75 minutes; mean 1 hour
|E
|
|
|
|
|
|-
|[https://www.credo-oxford.com/7.2.html Eating Disorder Examination (EDE)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt; &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|date=2008|publisher=Oxford University Press|others=John Hunsley, Eric J. Mash|isbn=0-19-804245-0|location=New York|oclc=314222270}}&lt;/ref&gt;
|Semistructured; adolescent version
|16+
|45-75 minutes
|E
|A
|A
|E
|X
|
|-
|[https://mccaed.slam.nhs.uk/wp-content/uploads/2022/02/PARDI-Self-14-V.2.1-Oct-2020.pdf Pica, ARFID, and Rumination Disorder Interview] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
|Semistructured; multi-informant and different versions
|2-22
|Mean of 39 minutes
|G (ARFID diagnosis)
|
|
|
|
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a brief overview of treatment options for anorexia nervosa and a list of process and outcome measures for anorexia nervosa. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information on the differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]].

===Process measures===
Body weight is commonly monitored by clinicians throughout the AN treatment process as helping individuals regain and maintain a healthy weight is a primary treatment goal for AN. Many treatment centers have policies prohibiting patients from seeing their weight.

Motivational Stages of Change may be used to monitor individuals’ readiness to take action against eating disorder behaviors. It has demonstrated predictive validity in a sample of female adolescents attending eating disorder treatment groups.&lt;ref&gt;{{Cite journal|last=Gusella|first=Joanne|last2=Butler|first2=Gordon|last3=Nichols|first3=Laura|last4=Bird|first4=Debbie|date=2003-01-01|title=A brief questionnaire to assess readiness to change in adolescents with eating disorders: its applications to group therapy|url=http://onlinelibrary.wiley.com/doi/10.1002/erv.481/abstract|journal=European Eating Disorders Review|language=en|volume=11|issue=1|pages=58–71|doi=10.1002/erv.481|issn=1099-0968}}&lt;/ref&gt; See Appendix E.

=== Outcome and severity measures ===
This table includes clinically significant benchmarks for anorexia nervosa specific outcome measures
* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
*''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
| colspan=&quot;8&quot; |'''Clinically significant change benchmarks with common instruments for anorexia nervosa'''
|-
| colspan=&quot;8&quot;  span style=&quot;font-size:110%; text-align:center;&quot; | &lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; |&lt;b&gt; Measure&lt;/b&gt;
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; | &lt;b&gt; Subscale&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; | &lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; | &lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-
| rowspan=&quot;5&quot; style=&quot;text-align:center;&quot; |'''EDE-Q &lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Mond|first=J. M.|last2=Hay|first2=P. J.|last3=Rodgers|first3=B.|last4=Owen|first4=C.|last5=Beumont|first5=P. J. V.|date=May 2004|title=Validity of the Eating Disorder Examination Questionnaire (EDE-Q) in screening for eating disorders in community samples|url=https://www.ncbi.nlm.nih.gov/pubmed/15033501|journal=Behaviour Research and Therapy|volume=42|issue=5|pages=551–567|doi=10.1016/S0005-7967(03)00161-X|issn=0005-7967|pmid=15033501}}&lt;/ref&gt;'''
| style=&quot;text-align:right;&quot; |''Global''
|1.4
|3.2
|2.3
|.7
|.6
|.3
|-
| style=&quot;text-align:right;&quot; |''Restraint''
|(-.3)
|3.6
|1.8
|1.5
|1.2
|.8
|-
| style=&quot;text-align:right;&quot; |''Eating Concern''
|.1
|2.0
|1.2
|1.1
|.9
|.6
|-
| style=&quot;text-align:right;&quot; |''Weight Concern''
|1.5
|3.9
|2.6
|1.0
|.9
|.5
|-
| style=&quot;text-align:right;&quot; |''Shape Concern''
|2.1
|4.8
|3.2
|.9
|.7
|.4
|-
| rowspan=&quot;5&quot; style=&quot;text-align:center;&quot; |'''EDE &lt;ref name=&quot;:2&quot; /&gt;'''
| style=&quot;text-align:right;&quot; |''Global''
|1.7
|2.3
|1.9
|1.9
|1.6
|1.0
|-
| style=&quot;text-align:right;&quot; |''Restraint''
|.3
|3.3
|1.9
|1.8
|1.5
|.9
|-
| style=&quot;text-align:right;&quot; |''Eating Concern''
|(-.5)
|.9
|.5
|.8
|.7
|.4
|-
| style=&quot;text-align:right;&quot; |''Weight Concern''
|2.0
|2.8
|2.4
|1.3
|1.1
|.7
|-
| style=&quot;text-align:right;&quot; |''Shape Concern''
|2.0
|3.2
|2.6
|1.2
|1.0
|.6
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; |'''EAT-26 &lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Mintz|first=L. B.|last2=O'Halloran|first2=M. S.|date=2000-06-01|title=The Eating Attitudes Test: validation with DSM-IV eating disorder criteria|url=http://www.ncbi.nlm.nih.gov/pubmed/10900574|journal=Journal of Personality Assessment|volume=74|issue=3|pages=489–503|doi=10.1207/S15327752JPA7403_11|issn=0022-3891|pmid=10900574}}&lt;/ref&gt;'''
| style=&quot;text-align:right;&quot; |''Total''
|6.5
|19.6
|15.0
|7.9
|6.7
|4.0
|}

'''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean.

'''Search terms:''' [Anorexia Nervosa OR eating disorder] AND [validity OR clinical significance] in Google Scholar

=== Treatment ===
{{collapse top| Click here for more information on treatment for AN}}
* Treatment of AN typically consists of restoring the individual to a healthy weight and addressing thoughts and behaviors which are related to the eating disorder. It may involve re-nutrition, psychotherapy, nutritional counseling, and medication. 
* Literature reviews of existing research indicate that evidence supporting AN treatment is lacking. A systematic review of AN treatment efficacy studies by Bulik&lt;ref&gt;{{Cite journal|title = Anorexia nervosa treatment: a systematic review of randomized controlled trials|url = http://www.ncbi.nlm.nih.gov/pubmed/17370290|journal = The International Journal of Eating Disorders|date = 2007-05-01|issn = 0276-3478|pmid = 17370290|pages = 310–320|volume = 40|issue = 4|doi = 10.1002/eat.20367|first = Cynthia M.|last = Bulik|first2 = Nancy D.|last2 = Berkman|first3 = Kimberly A.|last3 = Brownley|first4 = Jan A.|last4 = Sedway|first5 = Kathleen N.|last5 = Lohr}}&lt;/ref&gt; found that evidence supporting medications, medications and behavioral interventions, and behavioral interventions alone in adults is weak. 
* There is moderately strong evidence suggesting that behavioral interventions may be helpful for adolescents. In particular, adolescents may benefit from family therapy. 
* Clinical trials investigating AN treatment suffer from high rates of attrition, as key features of AN (e.g., denial, fear of weight gain) may contribute to low motivation for remaining in treatment.
* More severe cases of AN may be treated in inpatient settings, which are equipped to manage the re-nutrition process and provide medical monitoring. 
* Partial hospitalization and intensive outpatient programs may provide intermediate levels of treatment intensity to assist individuals in the transition from intensive care to outpatient care after weight restoration.
{{collapse bottom}}
* Please refer to the page on [https://en.wikipedia.org/wiki/Anorexia_nervosa anorexia nervosa] for more information on available treatment or go to [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/eating-body-image-problems/ the Effective Child Therapy page for Eating &amp; Body Image Problems] for a curated resource on effective treatments for anorexia nervosa.

== '''External resources''' ==
# [http://apps.who.int/classifications/icd10/browse/2010/en#/F50.0 ICD-10 diagnostic criteria]
# [https://en.wikiversity.org/w/index.php?title=Helping_Give_Away_Psychological_Science/Resources/Annotated_List_of_Where_and_How_to_Find_a_Therapist&amp;wteswitched=1#Other_low-cost_options Find-a-Therapist] 
##This is a curated list of find-a-therapist websites where you can find a provider
# NIMH: [https://www.nimh.nih.gov/health/publications/eating-disorders/index.shtml ''Eating Disorders--About More Than Food'']  and [https://www.nimh.nih.gov/health/topics/eating-disorders/index.shtml ''Eating Disorders'']
##These NIMH website posts provide more information on anorexia nervosa
#[https://www.hopkinsmedicine.org/psychiatry/specialty_areas/eating_disorders/ John's Hopkins Resource] (guide about anorexia nervosa, treatment, and more)
# OMIM (Online Mendelian Inheritance in Man)
##[https://www.omim.org/entry/606788?search=anorexia%20nervosa&amp;highlight=nervosa%20anorexia Anorexia nervosa]
#[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/eating-body-image-problems/ Effective Child Therapy page for anorexia nervosa]
##Effective Child Therapy is website sponsored by Division 53 of the American Psychological Association (APA), or The [https://sccap53.org Society of Clinical Child and Adolescent Psychology] (SCCAP), in collaboration with the Association for Behavioral and Cognitive Therapies (ABCT). Use for information on symptoms and available treatments.

== '''References''' ==
{{collapse top|Click here for references}}
{{Reflist|30em}}

[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]
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{{:{{BASEPAGENAME}}/Sidebar}}

==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
For background information on what assessment portfolios are, click the link in the heading above.
Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Anorexia nervosa (assessment portfolio)/extended version|here]].

== [[Evidence based assessment/Preparation phase|'''Preparation phase''']] ==
=== Diagnostic criteria for anorexia nervosa ===
{{blockquotetop}}
'''ICD-11 Criteria'''

Anorexia Nervosa is characterized by significantly low body weight, which is less than minimal normal/expected weight for the individual’s height, sex, age and developmental stage (body mass index (BMI) less than 18.5 kg/m2 in adults and BMI-for-age under fifth percentile in children and adolescents) that is not due to another health condition or to the unavailability of food. Low body weight is accompanied by a persistent pattern of behaviors to prevent restoration of normal weight, which may include behaviors aimed at reducing energy intake (restricted eating), purging behaviors (e.g., self-induced vomiting, misuse of laxatives), and behaviors aimed at increasing energy expenditure (e.g., excessive exercise), typically associated with a fear of weight gain. Low body weight or shape is central to the person's self-evaluation or is inaccurately perceived to be normal or even excessive.

'''Changes in DSM-5'''

* The diagnostic criteria for anorexia nervosa changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here] and [[w:DSM-5|here]].
{{blockquotebottom}}

=== Base rates of anorexia nervosa in different populations and clinical settings ===
This section describes the demographic setting of the population(s) sampled, base rates of diagnosis such as prevalence rates, country/region sampled, and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of adolescent depression that they are likely to see in their clinical practice.

* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']]

{| class=&quot;wikitable&quot;
|-
! Setting (Reference) 
! Base Rate
! Demography
! Diagnostic Method
|-
|Non-clinical: Population-based (NCS-R)&lt;ref&gt;Hudson, James I.; Hiripi, Eva; Pope, Harrison G.; Kessler, Ronald C. (2007-02-01). &quot;The prevalence and correlates of eating disorders in the National Comorbidity Survey Replication&quot;.''Biological Psychiatry'' '''61''' (3): 348–358. doi:10.1016/j.biopsych.2006.03.040. ISSN 0006-3223. PMC 1892232. &lt;nowiki&gt;PMID 16815322&lt;/nowiki&gt;&lt;/ref&gt;
|.9% Female, .3% Male
|Nationally representative US sample of adults
|World Health Organization-Clinical International Diagnostic Interview (WHO-CIDI)
|-
|Non-clinical: Population-based (NCS-A)&lt;ref&gt;Swanson, Sonja A.; Crow, Scott J.; Le Grange, Daniel; Swendsen, Joel; Merikangas, Kathleen R. (2011-07-01). &quot;Prevalence and correlates of eating disorders in adolescents. Results from the national comorbidity survey replication adolescent supplement&quot;. ''Archives of General Psychiatry'' '''68''' (7): 714–723. doi:10.1001/archgenpsychiatry.2011.22. ISSN 1538-3636.&lt;nowiki&gt;PMID 21383252&lt;/nowiki&gt;&lt;/ref&gt;
|.3% Female, .3% Male
|Nationally representative US sample of adolescents
|WHO-CIDI
|-
|Non-clinical: US Population-based prevalence (Adolescent Brain Cognitive Development (ABCD) study&lt;ref&gt;{{Cite journal|last=Rozzell|first=Kaitlin|last2=Moon|first2=Da Yeoun|last3=Klimek|first3=Patrycja|last4=Brown|first4=Tiffany|last5=Blashill|first5=Aaron J.|date=2019-01-01|title=Prevalence of Eating Disorders Among US Children Aged 9 to 10 Years: Data From the Adolescent Brain Cognitive Development (ABCD) Study|url=http://archpedi.jamanetwork.com/article.aspx?doi=10.1001/jamapediatrics.2018.3678|journal=JAMA Pediatrics|language=en|volume=173|issue=1|pages=100|doi=10.1001/jamapediatrics.2018.3678|issn=2168-6203|pmc=PMC6583451|pmid=30476983}}&lt;/ref&gt;
|0.1%, no gender differences
|Nationally representative US sample of 9- and 10-year old children
|DSM-5 using Kiddie Schedule for Affective Disorders and Schizophrenia (KSADS)
|-
|Non-clinical: Latinos in US&lt;ref&gt;Alegria, Margarita; Woo, Meghan; Cao, Zhun; Torres, Maria; Meng, Xiao-li; Striegel-Moore, Ruth (2007-11-01). &quot;Prevalence and correlates of eating disorders in Latinos in the United States&quot;.''The International Journal of Eating Disorders''. 40 Suppl: S15–21. doi:10.1002/eat.20406. ISSN 0276-3478. PMC 2680162. &lt;nowiki&gt;PMID 17584870&lt;/nowiki&gt;&lt;/ref&gt;
|.12% Female, .03% Male
|Latino Households in the US
|WHO-CIDI
|-
|Non-clinical: African Americans and Caribbean Blacks in the US (NSAL)&lt;ref&gt;Taylor, Jacquelyn Y.; Caldwell, Cleopatra Howard; Baser, Raymond E.; Faison, Nakesha; Jackson, James S. (2007-11-01). &quot;Prevalence of eating disorders among Blacks in the National Survey of American Life&quot;. ''The International Journal of Eating Disorders''. 40 Suppl: S10–14. doi:10.1002/eat.20451. ISSN 0276-3478. PMC 2882704. &lt;nowiki&gt;PMID 17879287&lt;/nowiki&gt;&lt;/ref&gt;
|.14% Female, .2% Male
|National probability sample of adult and adolescent African Americans and Caribbean Blacks
|WHO-CIDI
|-
|Non-clinical: National Latino and Asian American Study (NLAAS)
|.12% Female, .05% Male
|Asian American adults in US households
|WHO-CIDI
|-
| Non-clinical; college students&lt;ref&gt;Tyler ID. A true picture of eating disorders among African American women: a review of literature. ABNF J. 2003;14(3):73-4.&lt;/ref&gt; || .0%  || US African American college females || Eating Disorder Diagnostic Questionnaire (EDD-Q)
|-
| Non-clinical; Adolescents&lt;ref&gt;Stice, E., Becker, C. B., &amp; Yokum, S. (2013). Eating disorder prevention: Current evidence-base and future directions. ''Int. J. Eat. Disord. International Journal of Eating Disorders,'' ''46''(5), 478-485.&lt;/ref&gt; || .8% || US Female Adolescents || Eating Disorder Diagnosis Interview (EDDI)
|-
| Non-clinical; student-athletes&lt;ref&gt;Johnson C, Powers PS, Dick R. Athletes and eating disorders: the national collegiate athletic association study. Int J Eat Disord 1999;26:179e88.&lt;/ref&gt; || .0% || US Division-I Varsity Student Athletes || Eating Disorder Inventory-2 (EDI-2)
|-
| Non-clinical; Military&lt;ref&gt;McNulty, PAF. (2001). Prevalence and contributing factors of eating disorder behaviors in active duty service women in the Army, Navy, Air Force and Marines. Military Medicine, 166(1), 53-58. &lt;/ref&gt; || 1.1% || Active duty females in US Army, Navy, 
Airforce, and Marines 
|| EDI-2
|-
|Non-clinical; Commonwealth of Virginia Mid-Atlantic Twin Registry (MATR)&lt;ref&gt;Kendler KS, Walters EE, Neale MC, Kessler R, Heath A, Eaves L. The structure of genetic and environmental risk factors for six major psychiatric disorders in women. Archives of general psychiatry. 1995;52:374–383.&lt;/ref&gt;
|1.62% (narrow), 3.70% (broad)
|US Caucasian female same-sex twins
|Structured Clinical Interview for DSM Disorders (SCID)
|-
|Non-clinical; Health Omnibus Survey (HOS)
|.46% (3 months; combined)
|South Australian older adolescents and adults
|Eating Disorder Examination (EDE)
|-
|Military&lt;ref&gt;{{Cite journal|title=Diagnosed eating disorders in the U.S. Military: a nine year review|url=http://www.ncbi.nlm.nih.gov/pubmed/18821361|journal=Eating Disorders|date=2008-12-01|issn=1532-530X|pmid=18821361|pages=363–377|volume=16|issue=5|doi=10.1080/10640260802370523|first=Amanda J.|last=Antczak|first2=Teresa L.|last2=Brininger}}&lt;/ref&gt;
|.04% (combined)
|US Military
|ICD codes from electronic records
|-
|Non-clinical; Military
|.02% (7 years) Female, 
0.0% (7 years) Males
|US Military Academy cadets
|[https://www.eat-26.com/ Eating Attitudes Test- 26 items] (EAT-26)&lt;ref&gt;{{Cite web|url=https://www.eat-26.com/|title=EAT-26: Eating Attitudes Test &amp; Eating Disorder Testing – Use the EAT-26 to help you determine if you need to speak to a mental health professional to get help for an eating disorder.|last=admin|language=en|access-date=2022-05-30}}&lt;/ref&gt;
|-
|Non-clinical; Military
|1.1% (current &amp; past) Female
|US Navy female nurses
|DSM-III
|-
|Non-clinical; Military
|.04% Female, .005% Males
|US veterans
|ICD-9-CM
|-
|Non-clinical; Military
|2.5% Males
|Active duty males in US Navy
|N/A
|-
|Clinical; Collaborative Study on the Genetics of Alcoholism (COGA)&lt;ref&gt;{{Cite journal|title=Anorexia nervosa and bulimia nervosa in alcohol-dependent men and women and their relatives|url=http://www.ncbi.nlm.nih.gov/pubmed/8540597|journal=The American Journal of Psychiatry|date=1996-01-01|issn=0002-953X|pmid=8540597|pages=74–82|volume=153|issue=1|doi=10.1176/ajp.153.1.74|first=M. A.|last=Schuckit|first2=J. E.|last2=Tipp|first3=R. M.|last3=Anthenelli|first4=K. K.|last4=Bucholz|first5=V. M.|last5=Hesselbrock|first6=J. I.|last6=Nurnberger}}&lt;/ref&gt;
|1.41% (lifetime) Female,
.00% (lifetime) Male
|US alcohol-dependent adults from San Diego, St. Lois, Iowa City, Farmington, New York, &amp; Indianapolis
|SSAGA
|-
|Non-clinical; healthcare members
|.0269% (current)
|Healthcare provider records
|
|-
|Non-clinical; high school students
|. 00% (point), .45% (lifetime) Female
.00% (point), .00% (lifetime) Males
|US high school students in west central Oregon
|DSM-III-R&lt;sub&gt;4&lt;/sub&gt;
|-
|Clinical; substance users
|.4% (lifetime), .3% (current) Female
.4% (lifetime), .3% (current) Males
|Canadian treatment-seeking substance users
|DIS&lt;sub&gt;9&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Europe'''
|-
|Non-clinical; adolescents
|.3% Female
|Adolescent females residing in Navarra, Spain
|EAT-40&lt;sub&gt;3&lt;/sub&gt;
|-
|Non-clinical; adolescents
|.7% (lifetime) Female, .2% Male
|Adolescents in secondary schools in Sør-Trøndelag, County in Norway
|SEDs&lt;sub&gt;10&lt;/sub&gt;
|-
|Non-clinical; adolescents
|.7% (point; age 15), 1.8% (lifetime, age 15), .00% (point, age 18), 2.6% (lifetime, age 18), .9% (3 years) Female
.00% (point &amp; lifetime) Males
|Adolescents in a comprehensive school in Ostrobothnia district in Finland
|RAB-T&lt;sub&gt;11&lt;/sub&gt;&amp;nbsp;&amp; RAB-R&lt;sub&gt;12&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Australia'''
|-
|Non-clinical; adolescents
|.00% (full), 1.8% (partial) Female
|Adolescent females residing in Victoria, Australia
|BET&lt;sub&gt;13&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Central &amp; South America'''
|-
|Non-clinical; college students
|.00% Female
|Mexican first &amp; second year college females
|EAT-40&lt;sub&gt;3&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''East Asia'''
|-
|Clinical; eating disorder patients
|.53% Female
|Adolescent and adult Japanese patients at a university hospital
|DSM-III-R&lt;sub&gt;4&lt;/sub&gt;
|-
|Non-clinical; Korean Epidemiologic Catchment Area (KECA) Study
|.1% (lifetime), .1% (12 months) Female
.2% (lifetime), .00% (12 months)

Males
|Korean adults
|K-CIDI&lt;sub&gt;15&lt;/sub&gt;&amp;nbsp;2.1
|-
|Centers participating in the Collaborative Study on the Genetics of  Alcoholism in San Diego; St. Louis; Iowa City; Farmington, CN; New York; &amp; Indianapolis
|1.41% Females
|Alcohol-dependent adults
|Semi-Structured Assessment for the Genetics of Alcoholism; criteria based on DSM-III-R
|}

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==

=== Psychometric properties of screening instruments for anorexia nervosa ===
The following section contains a list of screening and diagnostic instruments for anorexia nerova. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here.]]

* ''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments '''click here.''']''

{| class=&quot;wikitable&quot;
|-
! Measure 
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Inter-rater reliability
!Test-retest reliability
!Construct validity
!Content validity
!Highly recommended
|-
|Structured Clinical Interview Diagnosis for DSM-IV (SCID-IV) Module H

* [https://www.columbiapsychiatry.org/research/research-areas/services-policy-and-law/structured-clinical-interview-dsm-disorders-scid-1 SCID-IV (for purchase)]
* [[wikipedia:Structured_Clinical_Interview_for_DSM-IV|Wiki Page]]
|Interview
(Patient)
|
|
|G
|L
|A
|E
|X
|-
|Eating Disorder Examination (EDE)

*OSF
*[[wikipedia:Eating_Disorder_Examination_Interview|Wiki Page]]
|Interview
(Patient)
|
|45min-1.25hrs
|E
|A
|A
|E
|X
|-
|[http://www.corc.uk.net/media/1273/ede-q_quesionnaire.pdf EDE-Q (Eating Disorder Examination Questionnaire)]
*OSF
*[[wikipedia:Eating_Disorder_Examination_Interview|Wiki Page]]
|Questionnaire

(Patient)
|
|15-20 min
|NA
|L
|U
|E
|
|-
|[http://www.ori.org/files/Static%20Page%20Files/EDDSDSM-5_10_14.pdf EDDS (Eating Disorder Diagnosis Scale)]
*OSF
*[[wikipedia:Eating_Disorder_Diagnostic_Scale|Wiki Page]]
|Questionnaire 
(Patient)
|
|10-15 min
|NA
|A
|A
|G
|X
|}
&lt;ref&gt;Hunsley, J., &amp; Mash, E. J. (2008). Guide to Assessments that Work. Cary, NC, USA: Oxford University Press, USA. Retrieved from &lt;nowiki&gt;http://www.ebrary.com&lt;/nowiki&gt;&lt;/ref&gt;'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Likelihood ratios and AUCs of screening instruments for anorexia nervosa===
* ''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments '''click here.''']''
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! &lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Mintz|first=L. B.|last2=O'Halloran|first2=M. S.|date=June 2000|title=The Eating Attitudes Test: validation with DSM-IV eating disorder criteria|url=https://www.ncbi.nlm.nih.gov/pubmed/10900574|journal=Journal of Personality Assessment|volume=74|issue=3|pages=489–503|doi=10.1207/S15327752JPA7403_11|issn=0022-3891|pmid=10900574}}&lt;/ref&gt;Screening Measure (Primary Reference)
! AUC
! DiLR+ (Score)
! DiLR- (Score)
!Clinical Generalizability
!Download
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''Biological &amp; Physical Measures'''
|
|-
|Serum leptin level&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Föcker|first=M.|last2=Timmesfeld|first2=N.|last3=Scherag|first3=S.|last4=Bühren|first4=K.|last5=Langkamp|first5=M.|last6=Dempfle|first6=A.|last7=Sheridan|first7=E. M.|last8=Zwaan|first8=M. de|last9=Fleischhaker|first9=C.|date=2011-04-01|title=Screening for anorexia nervosa via measurement of serum leptin levels|url=https://link.springer.com/article/10.1007/s00702-010-0551-z|journal=Journal of Neural Transmission|language=en|volume=118|issue=4|pages=571–578|doi=10.1007/s00702-010-0551-z|issn=0300-9564}}&lt;/ref&gt;
|0.984 (N=139)
|14.72 (&lt;2.31)
|0.10 (2.31+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
|Serum leptin standard deviation score (SDS)&lt;ref name=&quot;:7&quot; /&gt;
|0.939 (N=139)
|5.89 (&lt;-0.45)
|0.09 (-0.45+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
|Body Mass Index (BMI)&lt;ref name=&quot;:7&quot; /&gt;
|0.936 (N=139)
|5.89 (&lt;17.10)
|0.11 (17.10+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''REDS-C&lt;sub&gt;1&lt;/sub&gt;'''
|
|-
|REDS-C&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Boraska|first=V.|last2=Franklin|first2=C. S.|last3=Floyd|first3=J. a. B.|last4=Thornton|first4=L. M.|last5=Huckins|first5=L. M.|last6=Southam|first6=L.|last7=Rayner|first7=N. W.|last8=Tachmazidou|first8=I.|last9=Klump|first9=K. L.|date=October 2014|title=A genome-wide association study of anorexia nervosa|url=https://www.ncbi.nlm.nih.gov/pubmed/24514567|journal=Molecular Psychiatry|volume=19|issue=10|pages=1085–1094|doi=10.1038/mp.2013.187|issn=1476-5578|pmc=PMC4325090|pmid=24514567}}&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt;
|0.658 (N=333), 0.624 (N=236, &gt; 13 years old), 0.772 (N=97, ≤ 13 years old)
| --
| --
|Children and adolescents (8-18 years old) at a clinic being treated for eating and weight concerns. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDI&lt;sub&gt;2&lt;/sub&gt;'''
|
|-
|EDI-2 &lt;ref name=&quot;:3&quot;&gt;Garner, D. M. (1991). ''Manual for Eating Disorder Inventory.'' Odessa, FL: Psychological Assessment Resources, Inc.&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt; 
|0.556 (N=77)
| --
| --
|Children and adolescents (10-18 years old) at a clinic being treated for eating and weight concerns. Eating disorders were not separated.
|
|-
|EDI-2 - Drive for Thinness Sub-scale&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Segura-Garcia|first=Cristina|last2=Ramacciotti|first2=Carla|last3=Rania|first3=Marianna|last4=Aloi|first4=Matteo|last5=Caroleo|first5=Mariarita|last6=Bruni|first6=Antonella|last7=Gazzarrini|first7=Denise|last8=Sinopoli|first8=Flora|last9=De Fazio|first9=Pasquale|date=June 2015|title=The prevalence of orthorexia nervosa among eating disorder patients after treatment|url=https://www.ncbi.nlm.nih.gov/pubmed/25543324|journal=Eating and weight disorders: EWD|volume=20|issue=2|pages=161–166|doi=10.1007/s40519-014-0171-y|issn=1590-1262|pmid=25543324}}&lt;/ref&gt; 
|0.97 (N=92)
|29.39 (14+)
|0.274 (DT≤14)
|Italian individuals with a DSM-IV-TR eating disorder versus individuals at risk for eating disorders. Eating disorders were not separated.
|
|-
|EDI-3 - Drive for Thinness Subscale (Garner, 2004)

Lehmann et al., 2013
|.903 (N=1298)
| --
| --
|&amp;nbsp;
|
|-
|EDI-3 - Eating Disorder Risk Composite &lt;ref name=&quot;:9&quot; /&gt;(EDRC) (Garner, 2004)
|0.942 (N=92)
|52.08 (EDRC 75+, adults; 90+ adolescents)
|0.379 (EDRC≤75, adults; ≤90 adolescents)
|Italian individuals with a DSM-IV-TR eating disorder versus individuals at risk for eating disorders. Eating disorders were not separated.
|
|-
|EDI-3 - Interoceptive Deficits Subscale &lt;ref name=&quot;:10&quot;&gt;{{Cite journal|last=Clausen|first=Loa|last2=Rosenvinge|first2=Jan H.|last3=Friborg|first3=Oddgeir|last4=Rokkedal|first4=Kristian|date=2011-03-01|title=Validating the Eating Disorder Inventory-3 (EDI-3): A Comparison Between 561 Female Eating Disorders Patients and 878 Females from the General Population|url=https://link.springer.com/article/10.1007/s10862-010-9207-4|journal=Journal of Psychopathology and Behavioral Assessment|language=en|volume=33|issue=1|pages=101–110|doi=10.1007/s10862-010-9207-4|issn=0882-2689}}&lt;/ref&gt;(Garner, 2004)
|.911 (N=2561)
|4.0 (9+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interoceptive Deficits Subscale (Garner, 2004)

Lehmann et al., 2013
|.901 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Asceticism Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.886 (N=2561)
|6.5 (9+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Asceticism Subscale (Garner, 2004)

Lehmann et al., 2013
|.902 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Low Self-Esteem Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.884 (N=2561)
|5.8 (10+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited fm an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Low Self-Esteem Subscale (Garner, 2004)

Lehmann et al., 2013
|.906 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Personal Alienation Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.88 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Personal Alienation Subscale (Garner, 2004)

Lehmann et al., 2013
|.899 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Emotional Dysregulation Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.81 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Emotional Dysregulation Subscale (Garner, 2004)

Lehmann et al., 2013
|.779 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Interpersonal Alienation Subscale &lt;ref name=&quot;:10&quot; /&gt;(Garner, 2004)
|.79 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interpersonal Alienation Subscale (Garner, 2004)

Lehmann et al., 2013
|.743 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Perfectionism Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.79 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Perfectionism Subscale (Garner, 2004)

Lehmann et al., 2013
|.768 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Maturity Fears Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.77 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Maturity Fears Subscale (Garner, 2004)

Lehmann et al., 2013
|.678 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Bulimia Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.76 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Bulimia Subscale (Garner, 2004)

Lehmann et al., 2013
|.776 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Interpersonal Insecurity Subscale &lt;ref name=&quot;:10&quot; /&gt;(Garner, 2004)
|.76 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interpersonal Insecurity Subscale (Garner, 2004)

Lehmann et al., 2013
|.697 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Body Dissatisfaction Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.722 (N=2561)
|1.7 (15+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Body Dissatisfaction Subscale (Garner, 2004)

Lehmann et al., 2013
|.849 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EAT&lt;sub&gt;3&lt;/sub&gt;'''
|
|-
|EAT-26&amp;nbsp;&lt;ref name=&quot;:4&quot; /&gt;&lt;ref&gt;{{Cite journal|title = The eating attitudes test: psychometric features and clinical correlates|url = http://www.ncbi.nlm.nih.gov/pubmed/6961471|journal = Psychological Medicine|date = 1982-11-01|issn = 0033-2917|pmid = 6961471|pages = 871–878|volume = 12|issue = 4|first = D. M.|last = Garner|first2 = M. P.|last2 = Olmsted|first3 = Y.|last3 = Bohr|first4 = P. E.|last4 = Garfinkel}}&lt;/ref&gt;
|.90 (N=129)
|12.83 (20+)
|.24 (&lt;20)
|Low-moderate: College women with no eating disorder versus college women with a DSM-IV eating disorder. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDE-Q&lt;sub&gt;4&lt;/sub&gt;'''
|
|-
|EDE-Q&amp;nbsp;&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|last=Aardoom|first=Jiska J.|last2=Dingemans|first2=Alexandra E.|last3=Slof Op't Landt|first3=Margarita C. T.|last4=Van Furth|first4=Eric F.|date=December 2012|title=Norms and discriminative validity of the Eating Disorder Examination Questionnaire (EDE-Q)|url=https://www.ncbi.nlm.nih.gov/pubmed/23121779|journal=Eating Behaviors|volume=13|issue=4|pages=305–309|doi=10.1016/j.eatbeh.2012.09.002|issn=1873-7358|pmid=23121779}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|title = Assessment of eating disorders: interview or self-report questionnaire?|url = http://www.ncbi.nlm.nih.gov/pubmed/7866415|journal = The International Journal of Eating Disorders|date = 1994-12-01|issn = 0276-3478|pmid = 7866415|pages = 363–370|volume = 16|issue = 4|first = C. G.|last = Fairburn|first2 = S. J.|last2 = Beglin}}&lt;/ref&gt;
|.96 (N=1170)
| --
| --
|Moderate: Dutch treatment-seeking females meeting DSM-IV criteria for an eating disorder versus female adult general population sample recruited through advertisements and personal contacts. Eating disorders were not separated.
|
|-
|EDE-Q&amp;nbsp;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Fairburn|first=C. G.|last2=Beglin|first2=S. J.|date=December 1994|title=Assessment of eating disorders: interview or self-report questionnaire?|url=https://www.ncbi.nlm.nih.gov/pubmed/7866415|journal=The International Journal of Eating Disorders|volume=16|issue=4|pages=363–370|issn=0276-3478|pmid=7866415}}&lt;/ref&gt;
| --
|6.57 (2.3+)
|0.09 (&lt;2.3)
|Moderate: “Clinically significant eating disorder” from a community sample versus female adultsindividuals without “clinically significant eating disorder” from same sample. Eating disorders were not separated.
|
|-
|EDE-Q &lt;ref name=&quot;:1&quot; /&gt;&lt;ref&gt;Rø, Ø., Reas, D. L., &amp; Stedal, K. (2015). Eating Disorder Examination Questionnaire (EDE‐Q) in Norwegian Adults: Discrimination between Female Controls and Eating Disorder Patients. ''European Eating Disorders Review'', ''23''(5), 408-412.&lt;/ref&gt;
|0.89 (N=2465)
| --
| --
|Norwegian adult females with an eating disorder recruited from eating disorder specialist centers versus controls determined using the DSM-IV.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDQ-O&lt;sub&gt;5&lt;/sub&gt;'''
|
|-
|EDQ-Q&amp;nbsp;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;
|0.72 (N=134)
|1.00 (all criteria met)
|0.92 (≥1 criteria not met)
|Dutch adults recruited from an eating disorder specialist center determined using the DSM-IV-TR.
|
|}

=== Interpreting anorexia nervosa screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]
* Also see the page on [https://en.wikipedia.org/wiki/Likelihood_ratios_in_diagnostic_testing likelihood ratios in diagnostic testing] for more information.

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==
===Gold standard diagnostic interviews===

* For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

===Recommended diagnostic interviews for anorexia nervosa===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! colspan=&quot;5&quot; |Diagnostic instruments for anorexia nervosa
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
!Where to access
|-
|[https://www.credo-oxford.com/7.2.html Eating Disorder Examination (EDE)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt; &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|date=2008|publisher=Oxford University Press|others=John Hunsley, Eric J. Mash|isbn=0-19-804245-0|location=New York|oclc=314222270}}&lt;/ref&gt;
|Semistructured (child and adult)
|8-16 (child version)
16+ (adult version)
|45-75 minutes
|[https://www.phenxtoolkit.org/protocols/view/230101 Child version PDF][https://www.credo-oxford.com/7.2.html Adult version PDF]
|-
|Structured Clinical Interview for DSM-5-Clinician Version (SCID-5-CV)&lt;ref&gt;{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}&lt;/ref&gt;
|Structured Interview (Adult )
|16+
|Varies
|[https://www.columbiapsychiatry.org/research/research-labs/diagnostic-and-assessment-lab/structured-clinical-interview-dsm-disorders-11 Website to purchase]
|-
| [https://dawba.info/a0.html Development and Well-Being Assessment (DAWBA)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
| Structured (child/adolescent and parent)
| 11-18
|10-20 minutes for the eating disorder module
|English PDF
[https://dawba.info/py/dawbainfo/b1.py Additional languages]
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a brief overview of treatment options for anorexia nervosa and a list of process and outcome measures for anorexia nervosa. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information on the differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]].

===Process measures===
Body weight is commonly monitored by clinicians throughout the AN treatment process as helping individuals regain and maintain a healthy weight is a primary treatment goal for AN. Many treatment centers have policies prohibiting patients from seeing their weight.

Motivational Stages of Change may be used to monitor individuals’ readiness to take action against eating disorder behaviors. It has demonstrated predictive validity in a sample of female adolescents attending eating disorder treatment groups.&lt;ref&gt;{{Cite journal|last=Gusella|first=Joanne|last2=Butler|first2=Gordon|last3=Nichols|first3=Laura|last4=Bird|first4=Debbie|date=2003-01-01|title=A brief questionnaire to assess readiness to change in adolescents with eating disorders: its applications to group therapy|url=http://onlinelibrary.wiley.com/doi/10.1002/erv.481/abstract|journal=European Eating Disorders Review|language=en|volume=11|issue=1|pages=58–71|doi=10.1002/erv.481|issn=1099-0968}}&lt;/ref&gt; See Appendix E.

=== Outcome and severity measures ===
This table includes clinically significant benchmarks for anorexia nervosa specific outcome measures
* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
*''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
| colspan=&quot;8&quot; |'''Clinically significant change benchmarks with common instruments for anorexia nervosa'''
|-
| colspan=&quot;8&quot;  span style=&quot;font-size:110%; text-align:center;&quot; | &lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; |&lt;b&gt; Measure&lt;/b&gt;
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; | &lt;b&gt; Subscale&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; | &lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; | &lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-
| rowspan=&quot;5&quot; style=&quot;text-align:center;&quot; |'''EDE-Q &lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Mond|first=J. M.|last2=Hay|first2=P. J.|last3=Rodgers|first3=B.|last4=Owen|first4=C.|last5=Beumont|first5=P. J. V.|date=May 2004|title=Validity of the Eating Disorder Examination Questionnaire (EDE-Q) in screening for eating disorders in community samples|url=https://www.ncbi.nlm.nih.gov/pubmed/15033501|journal=Behaviour Research and Therapy|volume=42|issue=5|pages=551–567|doi=10.1016/S0005-7967(03)00161-X|issn=0005-7967|pmid=15033501}}&lt;/ref&gt;'''
| style=&quot;text-align:right;&quot; |''Global''
|1.4
|3.2
|2.3
|.7
|.6
|.3
|-
| style=&quot;text-align:right;&quot; |''Restraint''
|(-.3)
|3.6
|1.8
|1.5
|1.2
|.8
|-
| style=&quot;text-align:right;&quot; |''Eating Concern''
|.1
|2.0
|1.2
|1.1
|.9
|.6
|-
| style=&quot;text-align:right;&quot; |''Weight Concern''
|1.5
|3.9
|2.6
|1.0
|.9
|.5
|-
| style=&quot;text-align:right;&quot; |''Shape Concern''
|2.1
|4.8
|3.2
|.9
|.7
|.4
|-
| rowspan=&quot;5&quot; style=&quot;text-align:center;&quot; |'''EDE &lt;ref name=&quot;:2&quot; /&gt;'''
| style=&quot;text-align:right;&quot; |''Global''
|1.7
|2.3
|1.9
|1.9
|1.6
|1.0
|-
| style=&quot;text-align:right;&quot; |''Restraint''
|.3
|3.3
|1.9
|1.8
|1.5
|.9
|-
| style=&quot;text-align:right;&quot; |''Eating Concern''
|(-.5)
|.9
|.5
|.8
|.7
|.4
|-
| style=&quot;text-align:right;&quot; |''Weight Concern''
|2.0
|2.8
|2.4
|1.3
|1.1
|.7
|-
| style=&quot;text-align:right;&quot; |''Shape Concern''
|2.0
|3.2
|2.6
|1.2
|1.0
|.6
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; |'''EAT-26 &lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Mintz|first=L. B.|last2=O'Halloran|first2=M. S.|date=2000-06-01|title=The Eating Attitudes Test: validation with DSM-IV eating disorder criteria|url=http://www.ncbi.nlm.nih.gov/pubmed/10900574|journal=Journal of Personality Assessment|volume=74|issue=3|pages=489–503|doi=10.1207/S15327752JPA7403_11|issn=0022-3891|pmid=10900574}}&lt;/ref&gt;'''
| style=&quot;text-align:right;&quot; |''Total''
|6.5
|19.6
|15.0
|7.9
|6.7
|4.0
|}

'''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean.

'''Search terms:''' [Anorexia Nervosa OR eating disorder] AND [validity OR clinical significance] in Google Scholar

=== Treatment ===
{{collapse top| Click here for more information on treatment for AN}}
* Treatment of AN typically consists of restoring the individual to a healthy weight and addressing thoughts and behaviors which are related to the eating disorder. It may involve re-nutrition, psychotherapy, nutritional counseling, and medication. 
* Literature reviews of existing research indicate that evidence supporting AN treatment is lacking. A systematic review of AN treatment efficacy studies by Bulik&lt;ref&gt;{{Cite journal|title = Anorexia nervosa treatment: a systematic review of randomized controlled trials|url = http://www.ncbi.nlm.nih.gov/pubmed/17370290|journal = The International Journal of Eating Disorders|date = 2007-05-01|issn = 0276-3478|pmid = 17370290|pages = 310–320|volume = 40|issue = 4|doi = 10.1002/eat.20367|first = Cynthia M.|last = Bulik|first2 = Nancy D.|last2 = Berkman|first3 = Kimberly A.|last3 = Brownley|first4 = Jan A.|last4 = Sedway|first5 = Kathleen N.|last5 = Lohr}}&lt;/ref&gt; found that evidence supporting medications, medications and behavioral interventions, and behavioral interventions alone in adults is weak. 
* There is moderately strong evidence suggesting that behavioral interventions may be helpful for adolescents. In particular, adolescents may benefit from family therapy. 
* Clinical trials investigating AN treatment suffer from high rates of attrition, as key features of AN (e.g., denial, fear of weight gain) may contribute to low motivation for remaining in treatment.
* More severe cases of AN may be treated in inpatient settings, which are equipped to manage the re-nutrition process and provide medical monitoring. 
* Partial hospitalization and intensive outpatient programs may provide intermediate levels of treatment intensity to assist individuals in the transition from intensive care to outpatient care after weight restoration.
{{collapse bottom}}
* Please refer to the page on [https://en.wikipedia.org/wiki/Anorexia_nervosa anorexia nervosa] for more information on available treatment or go to [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/eating-body-image-problems/ the Effective Child Therapy page for Eating &amp; Body Image Problems] for a curated resource on effective treatments for anorexia nervosa.

== '''External resources''' ==
# [http://apps.who.int/classifications/icd10/browse/2010/en#/F50.0 ICD-10 diagnostic criteria]
# [https://en.wikiversity.org/w/index.php?title=Helping_Give_Away_Psychological_Science/Resources/Annotated_List_of_Where_and_How_to_Find_a_Therapist&amp;wteswitched=1#Other_low-cost_options Find-a-Therapist] 
##This is a curated list of find-a-therapist websites where you can find a provider
# NIMH: [https://www.nimh.nih.gov/health/publications/eating-disorders/index.shtml ''Eating Disorders--About More Than Food'']  and [https://www.nimh.nih.gov/health/topics/eating-disorders/index.shtml ''Eating Disorders'']
##These NIMH website posts provide more information on anorexia nervosa
#[https://www.hopkinsmedicine.org/psychiatry/specialty_areas/eating_disorders/ John's Hopkins Resource] (guide about anorexia nervosa, treatment, and more)
# OMIM (Online Mendelian Inheritance in Man)
##[https://www.omim.org/entry/606788?search=anorexia%20nervosa&amp;highlight=nervosa%20anorexia Anorexia nervosa]
#[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/eating-body-image-problems/ Effective Child Therapy page for anorexia nervosa]
##Effective Child Therapy is website sponsored by Division 53 of the American Psychological Association (APA), or The [https://sccap53.org Society of Clinical Child and Adolescent Psychology] (SCCAP), in collaboration with the Association for Behavioral and Cognitive Therapies (ABCT). Use for information on symptoms and available treatments.

== '''References''' ==
{{collapse top|Click here for references}}
{{Reflist|30em}}

[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]
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{{medical disclaimer}}
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==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
* For background information on what assessment portfolios are, click the link in the heading above.

Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Generalized anxiety disorder (assessment portfolio)/extended version|here]].

==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]==
=== Diagnostic criteria for generalized anxiety disorder ===
{{blockquotetop}}

&lt;big&gt;'''ICD-11 Diagnostic Criteria&lt;ref&gt;https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1712535455&lt;/ref&gt;'''&lt;/big&gt;
*Generalised anxiety disorder is characterized by marked symptoms of anxiety that persist for at least several months, for more days than not, manifested by either general apprehension (i.e. ‘free-floating anxiety’) or excessive worry focused on multiple everyday events, most often concerning family, health, finances, and school or work, together with additional symptoms such as muscular tension or motor restlessness, sympathetic autonomic over-activity, subjective experience of nervousness, difficulty maintaining concentration, irritability, or sleep disturbance. The symptoms result in significant distress or significant impairment in personal, family, social, educational, occupational, or other important areas of functioning. The symptoms are not a manifestation of another health condition and are not due to the effects of a substance or medication on the central nervous system.

'''Changes in DSM-5'''

* The diagnostic criteria for generalized anxiety disorder changed slightly from DSM-IV-TR to DSM-5. Summaries are available [https://www.ncbi.nlm.nih.gov/books/NBK519712/table/ch3.t9/?report=objectonly here].

{{blockquotebottom}}


=== Base rates of GAD in different clinical settings ===
This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of GAD that they are likely to see in their clinical practice.

* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']] 

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Demography
! Setting
! Base Rate
! Diagnostic Method
|-
| Adults and adolescences in all of U.S.A.
| US National Comorbidity Survey Replication (NCS-R; age &gt; = 13) 
[https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4005415/pdf/nihms-571992.pdf (2012)]&lt;ref name=&quot;KesslerEtAl2012&quot; /&gt;
|0.9% (age 13-17)

2.9% (age 18-64)

1.2% (age &amp;gt;= 65)

2.0% (age &amp;gt;=13)
| Fully-structured Composite International Diagnostic Interview (CIDI Version 3.0)
|-
| Psychiatric outpatients
| Individuals seeking treatment in a Psychiatric Outpatient Clinic (age range not reported) 
([https://ajp.psychiatryonline.org/doi/pdf/10.1176/appi.ajp.162.10.1911 2014])&lt;ref name=&quot;ZimmermanEtAl2005&quot; /&gt;
|21%
| Structured Clinical Interview for DSM-IV (SCID)
|-
| Caucasian youth
| Children seeking treatment in a Child &amp;amp; Adolescent Anxiety Diagnostic Clinic (age 7 – 18 years old) 
([http://journals.sagepub.com/doi/pdf/10.1177/1073191110375792 2011])&lt;ref name=&quot;BrownJacobsenEtAl2011&quot; /&gt;
|0.39% (parent report)

0.38% (child report)
| Anxiety Disorders Interview Schedule for Children for DSM-IV
Spence Children's Anxiety Scale (SCAS)
|-
| Caucasian, African American, Asian American, and Hispanic population
| Collaborative Psychiatric Epidemiology Studies (CPES; age &amp;gt;= 18, data merged from three representative national database) 
([https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3135672/pdf/nihms281333.pdf 2011])&lt;ref&gt;{{Cite journal|last=McLean|first=Carmen P.|last2=Asnaani|first2=Anu|last3=Litz|first3=Brett T.|last4=Hofmann|first4=Stefan G.|date=2011-08-01|title=Gender differences in anxiety disorders: Prevalence, course of illness, comorbidity and burden of illness|url=https://www.sciencedirect.com/science/article/pii/S0022395611000458|journal=Journal of Psychiatric Research|language=en|volume=45|issue=8|pages=1027–1035|doi=10.1016/j.jpsychires.2011.03.006|issn=0022-3956|pmc=PMC3135672|pmid=21439576}}&lt;/ref&gt;
|4.1% (female)

2.1% (male)
| World Mental Health Survey Initiative Version of the World Health Organization Composite International Interview (WMH-CIDI)
|-
| Pennsylvania
| Metropolitan Community Sample, all individuals with eating disorders (ages 13 – 65) 
([https://ajp.psychiatryonline.org/doi/pdf/10.1176/appi.ajp.161.12.2215 2014])&lt;ref name=&quot;KayeEtAl2004&quot; /&gt;
|10%
| Structured Clinical Interview for DSM-IV (SCID)
|-
| Adolescents in all of U.S.A.
| National Comorbidity Survey Replication Adolescent Supplement (NCS-A; ages 3–18 in the continental U.S) 
([https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2946114/pdf/nihms214371.pdf 2011])&lt;ref name=&quot;MerikangasEtAl2010&quot; /&gt;
|2.2%
| World Health Organization Composite International Diagnostic Interview (WHO-CIDI)
|-
|Adolescents in all of U.S.A
|National Comorbidity Survey Replication Adolescent Supplement (NCS-A; ages 3–18 in the continental U.S)&lt;ref name=&quot;:3&quot;&gt;Kessler, R. C., Avenevoli, S., Costello, E. J., Georgiades, K., Green, J. G., Gruber, M. J., . . . Merikangas, K. R. (2012). Prevalence, persistence, and sociodemographic correlates of DSM-IV disorders in the National Comorbidity Survey Replication Adolescent Supplement. Archives of General Psychiatry, 69(4), 372-380. doi:10.1001/archgenpsychiatry.2011.160&lt;/ref&gt;
|5.4%
|Composite International Diagnostic Interview (CIDI)
|-
| North Carolina
| Rural community sample African American and White youth (ages 13-16) 
[https://www.ncbi.nlm.nih.gov/pubmed/12365876 (2002)]&lt;ref&gt;{{Cite journal|last=Angold|first=Adrian|last2=Erkanli|first2=Alaattin|last3=Farmer|first3=Elizabeth M. Z.|last4=Fairbank|first4=John A.|last5=Burns|first5=Barbara J.|last6=Keeler|first6=Gordon|last7=Costello|first7=E. Jane|date=October 2002|title=Psychiatric disorder, impairment, and service use in rural African American and white youth|url=https://www.ncbi.nlm.nih.gov/pubmed/12365876|journal=Archives of General Psychiatry|volume=59|issue=10|pages=893–901|issn=0003-990X|pmid=12365876}}&lt;/ref&gt;
|1.4%
| The Child and Adolescent Psychiatric Assessment (CAPA)
|-
| Texas
| Metropolitan Community Sample (ages 11-17) 
([https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2736593/pdf/nihms30019.pdf 2007])&lt;ref name=&quot;RobertsEtAl2007&quot; /&gt;
|0.4%
| Diagnostic Interview Schedule for Children, Version IV (DISC-IV)
|-
| Midwestern Urban
| Incarcerated adolescents (ages 10-18)&lt;ref&gt;{{Cite journal|last=ABRAM|first=KAREN M.|last2=CHOE|first2=JEANNE Y.|last3=WASHBURN|first3=JASON J.|last4=TEPLIN|first4=LINDA A.|last5=KING|first5=DEVON C.|last6=DULCAN|first6=MINA K.|title=Suicidal Ideation and Behaviors Among Youths in Juvenile Detention|url=http://linkinghub.elsevier.com/retrieve/pii/S0890856709623121|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=47|issue=3|pages=291–300|doi=10.1097/chi.0b013e318160b3ce}}&lt;/ref&gt; 
[http://vb3lk7eb4t.search.serialssolutions.com/?ctx_ver=Z39.88-2004&amp;ctx_enc=info%3Aofi%2Fenc%3AUTF-8&amp;rfr_id=info%3Asid%2Fsummon.serialssolutions.com&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=Suicidal+Ideation+and+Behaviors+Among+Youths+in+Juvenile+Detention&amp;rft.jtitle=Journal+of+the+American+Academy+of+Child+%26+Adolescent+Psychiatry&amp;rft.au=ABRAM%2C+KAREN+M.%2C+Ph.D&amp;rft.au=CHOE%2C+JEANNE+Y.%2C+B.A&amp;rft.au=WASHBURN%2C+JASON+J.%2C+Ph.D.%2C+A.B.P.P&amp;rft.au=TEPLIN%2C+LINDA+A.%2C+Ph.D&amp;rft.date=2008&amp;rft.issn=0890-8567&amp;rft.eissn=1527-5418&amp;rft.volume=47&amp;rft.issue=3&amp;rft.spage=291&amp;rft.epage=300&amp;rft_id=info:doi/10.1097%2FCHI.0b013e318160b3ce&amp;rft.externalDocID=1_s2_0_S0890856709623121 (2002)]
|1%
| Diagnostic Interview Schedule for Children, Version IV (DISC-IV)
|-
|Non-institutionalized general US population
|LGBTQ sample (ages 20-65)&lt;ref&gt;{{Cite journal|last=Bostwick|first=Wendy B.|last2=Boyd|first2=Carol J.|last3=Hughes|first3=Tonda L.|last4=McCabe|first4=Sean Esteban|date=2010-3|title=Dimensions of Sexual Orientation and the Prevalence of Mood and Anxiety Disorders in the United States|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2820045/|journal=American Journal of Public Health|volume=100|issue=3|pages=468–475|doi=10.2105/AJPH.2008.152942|issn=0090-0036|pmc=PMC2820045|pmid=19696380}}&lt;/ref&gt; [http://ajph.aphapublications.org/doi/10.2105/AJPH.2008.152942 (2013)]
|Women:
14.8% same-sex

22.5% bisexual

Men:

16.9% same-sex

11.5% bisexual
|The Alcohol Use Disorder and Associated Disabilities Interview Schedule-IV (AUDADIS-IV)
|-
|Non-institutionalized general US population
|Cross-ethnic American population (ages 18+)&lt;ref&gt;{{Cite journal|last=Asnaani|first=Anu|last2=Richey|first2=J. Anthony|last3=Dimaite|first3=Ruta|last4=Hinton|first4=Devon E.|last5=Hofmann|first5=Stefan G.|date=2010-8|title=A Cross-Ethnic Comparison of Lifetime Prevalence Rates of Anxiety Disorders|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2931265/|journal=The Journal of nervous and mental disease|volume=198|issue=8|pages=551–555|doi=10.1097/NMD.0b013e3181ea169f|issn=0022-3018|pmc=PMC2931265|pmid=20699719}}&lt;/ref&gt; [https://journals.lww.com/jonmd/Abstract/2010/08000/A_Cross_Ethnic_Comparison_of_Lifetime_Prevalence.4.aspx (2018)]
|White 8.6%

African Americans 4.9%

Hispanic Americans 5.8%

Asian Americans 2.4%
|World Mental Health Survey Initiative Version of the World Health Organization Composite International Interview (WMH-CIDI)
|-
|Outpatient clinics worldwide
|Samples across multiple studies worldwide (all ages)&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Rettew|first=David C.|last2=Lynch|first2=Alicia Doyle|last3=Achenbach|first3=Thomas M.|last4=Dumenci|first4=Levent|last5=Ivanova|first5=Masha Y.|date=2009-09|title=Meta-analyses of agreement between diagnoses made from clinical evaluations and standardized diagnostic interviews|url=http://dx.doi.org/10.1002/mpr.289|journal=International Journal of Methods in Psychiatric Research|language=en|volume=18|issue=3|pages=169–184|doi=10.1002/mpr.289|issn=1049-8931}}&lt;/ref&gt;
|5%
|Clinical evaluations
|-
|Outpatient clinic worldwide
|Samples across multiple studies worldwide (all ages)&lt;ref name=&quot;:12&quot; /&gt;
|10%
|Standardized Diagnostic Interviews (SDIs)
|}

'''Search terms:''' [General Anxiety Disorder] AND [youth OR adolescents OR pediatric] AND [prevalence OR incidence] in GoogleScholar and PsycINFO

== [[Evidence based assessment/Prediction phase|'''Prediction phase''']] ==

=== Psychometric properties of screening instruments for GAD ===
The following section contains a list of screening and diagnostic instruments for generalized anxiety disorder. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. 
* '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]'''''

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! colspan=&quot;5&quot; |Screening measures for GAD
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! style=&quot;width:12em&quot; | Where to Access 
|-
| Penn State Worry Questionnaire (PSWQ)&lt;ref name=&quot;:0&quot;&gt;{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|date=2008|publisher=Oxford University Press|author=Hunsley, John |author2=Mash, Eric J.|isbn=9780195310641|location=New York|oclc=314222270}}&lt;/ref&gt;
| Questionnaire (Adult Version, Child Version)
| 18+ (Adult Version), 6-18 (Child Version)
| 4 minutes
|[http://www.midss.org/content/penn-state-worry-questionnaire-pswq PSWQ homepage]

[https://mfr.osf.io/render?url=https://osf.io/s7p38/?action=download%26mode=render PSWQ Adult Version]

[https://mfr.osf.io/render?url=https://osf.io/6q8y9/?action=download%26mode=render PSWQ Child Version]

[https://mfr.osf.io/render?url=https://osf.io/gx5sr/?action=download%26mode=render PSWC-C Korean]

[https://mfr.osf.io/render?url=https://osf.io/hc6n2/?action=download%26mode=render PSWQ-C Danish]

[https://mfr.osf.io/render?url=https://osf.io/fwbes/?action=download%26mode=render Scoring the PSWQ-C]
|-
|[[wikipedia:Screen_for_child_anxiety_related_disorders|Screen for Child Anxiety Related Emotional Disorder (SCARED)]]&lt;ref name=&quot;:0&quot; /&gt;
| Questionnaire (Child, Parent)
| 8-19
| 9 or 16 minutes
|[http://www.midss.org/content/screen-child-anxiety-related-disorders-scared SCARED] homepage

[[wikipedia:Screen_for_child_anxiety_related_disorders#PDFs_and_automated_scoring_for_SCARED|SCARED English + Translations &amp; Automatic Scoring]]
|-
|Child Behavior Checklist (CBCL)&lt;ref name=&quot;:0&quot; /&gt;
|Questionnaire (Parent report)
|6-18
|10 minutes
|[https://aseba.org/ ASEBA homepage][https://store.aseba.org/ Purchase]
|}

=== Likelihood ratios and AUCs of screening instruments for GAD ===
* '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' 
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! Screening Measure (Primary Reference)
! Area Under Curve (AUC)
! LR+ (Score)
! LR- (Score)
! Clinical Generalizability
!Where to Access
|-
| Penn State Worry Questionnaire (PSWQ)&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Fresco|first=David M.|last2=Mennin|first2=Douglas S.|last3=Heimberg|first3=Richard G.|last4=Turk|first4=Cynthia L.|title=Using the Penn State Worry Questionnaire to identify individuals with generalized anxiety disorder: a receiver operating characteristic analysis|url=http://linkinghub.elsevier.com/retrieve/pii/S0005791603000569|journal=Journal of Behavior Therapy and Experimental Psychiatry|volume=34|issue=3-4|pages=283–291|doi=10.1016/j.jbtep.2003.09.001}}&lt;/ref&gt;
| 0.74
(N=164)
| 1.8 (65+)
| 0.5 (&lt; 65)
| Generalized Anxiety Disorder vs. social anxiety disorder, adults presenting to specialty anxiety clinic
|[https://mfr.osf.io/render?url=https://osf.io/s7p38/?action=download%26mode=render PSWQ Adult Version]

[https://mfr.osf.io/render?url=https://osf.io/6q8y9/?action=download%26mode=render PSWQ Child Version]
|-
|[[wikipedia:Screen_for_child_anxiety_related_disorders#PDFs_and_automated_scoring_for_SCARED|Screen for Child Anxiety Related Disorders (SCARED)]]&lt;ref name=&quot;BirmaherEtAl1997&quot; /&gt;
| .70 
(N=243)
| 5.0 (+32)
| .04
| High: Pure anxiety disorder versus non-anxiety psychiatric disorder, excluding children with disruptive disorder and depression
|[http://www.midss.org/content/penn-state-worry-questionnaire-pswq SCARED English + Translations &amp; Automatic Scoring]
|-
|CBCL Anxious/Depressed Scale T-score&lt;ref&gt;{{Cite journal|last=Eimecke|first=Sylvia D.|last2=Remschmidt|first2=Helmut|last3=Mattejat|first3=Fritz|date=2011-03|title=Utility of the Child Behavior Checklist in screening depressive disorders within clinical samples|url=https://linkinghub.elsevier.com/retrieve/pii/S0165032710005458|journal=Journal of Affective Disorders|language=en|volume=129|issue=1-3|pages=191–197|doi=10.1016/j.jad.2010.08.011}}&lt;/ref&gt;
|.75 (N = 1445)
|1.49 (9+)
|.67(9-)
|Inpatient and outpatient children and adolescents
|[https://store.aseba.org/ Purchase]
|}

'''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation&lt;ref&gt;Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., &amp; Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone.&lt;/ref&gt;.

'''Search terms:''' [General Anxiety Disorder] AND [children OR adolescents OR pediatric] AND [sensitivity OR specificity] in GoogleScholar and PsycINFO

=== Interpreting depression screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==
===Gold standard diagnostic interviews===
* For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

=== Recommended diagnostic instruments for GAD ===
{| class=&quot;wikitable sortable&quot;
! colspan=&quot;5&quot; |Diagnostic instruments for GAD
|-
!Measure
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Where to Access
|-
|Anxiety Disorders Interview Schedule for Children/Parent&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|date=2001-08-01|title=Test-Retest Reliability of Anxiety Symptoms and Diagnoses With the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Versions|url=https://www.sciencedirect.com/science/article/pii/S0890856709603427|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|language=en|volume=40|issue=8|pages=937–944|doi=10.1097/00004583-200108000-00016|issn=0890-8567}}&lt;/ref&gt;
|Structured Interview
(Child (ADIS-C), Parent (ADIS-P))
|6-16&lt;ref&gt;{{Cite journal|last=LYNEHAM|first=HEIDI J.|last2=ABBOTT|first2=MAREE J.|last3=RAPEE|first3=RONALD M.|date=2007-06|title=Interrater Reliability of the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Version|url=https://doi.org/10.1097/chi.0b013e3180465a09|journal=Journal of the American Academy of Child &amp;amp; Adolescent Psychiatry|volume=46|issue=6|pages=731–736|doi=10.1097/chi.0b013e3180465a09|issn=0890-8567}}&lt;/ref&gt;
|Varies
|[https://books.google.com/books/about/Anxiety_Disorders_Interview_Schedule_for.html?id=xpR6V3rboxwC Purchase]
|-
|Anxiety and Related Disorders Interview Schedule for DSM-5 (ADIS-5)&lt;ref name=&quot;:0&quot; /&gt;
|Structured Interview (Adult) 
|16+
|Varies
|[https://global.oup.com/academic/product/anxiety-and-related-disorders-interview-schedule-for-dsm-5-adis-5---adult-version-9780199325160?cc=us&amp;lang=en&amp; Purchase]
|-
|Structured Clinical Interview for DSM-5-Clinician Version (SCID-5-CV)&lt;ref&gt;{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}&lt;/ref&gt;
|Structured Interview (Adult )
|16+
|Varies
|[https://www.columbiapsychiatry.org/research/research-labs/diagnostic-and-assessment-lab/structured-clinical-interview-dsm-disorders-11 Website and purchase]
|-
|Generalized Anxiety Disorder Screener (GAD-7)&lt;ref name=&quot;:0&quot; /&gt;
|Questionnaire (Self-report)
|18+
|5 minutes
|[https://www.pfizerpcoa.com/general-anxiety-disorder-7-gad-7-screener GAD-7 homepage]

[https://osf.io/szmpu GAD-7 PDF]
|}

==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a list of process and outcome measures for generalized anxiety disorder. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. 
=== Process measures ===

=== Outcome and severity measures ===
* This table includes clinically significant benchmarks for '''(insert portfolio name here)''' specific outcome measures
* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
*''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
| colspan=&quot;7&quot; |'''Clinically significant change benchmarks with common instruments for GAD'''
|-
| rowspan=1&quot; style=&quot;text-align:center;font-size:130%;&quot; | &lt;b&gt; Measure&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; | &lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; | &lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| colspan=&quot;7&quot;  span style=&quot;font-size:110%; text-align:center;&quot; | &lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| colspan=&quot;1&quot; |
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | &lt;b&gt; GAD-7&lt;/b&gt;
| style=“text-align:center;”| -1
| style=“text-align:center;”| 1.3
| style=“text-align:center;”| 0.5
| style=“text-align:center;”| 0.6
| style=“text-align:center;”| 0.5
| style=“text-align:center;”| 0.3
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | &lt;b&gt; PSWQ&lt;/b&gt;
| style=“text-align:center;”|  51
| style=“text-align:center;”|  73
| style=“text-align:center;”|  59
| style=“text-align:center;”|  9
| style=“text-align:center;”|  8
| style=“text-align:center;”|  4.8
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | &lt;b&gt; SCARED &lt;/b&gt;
| style=“text-align:center;”|  9.9
| style=“text-align:center;”|  18.1
| style=“text-align:center;”|  15.3
| style=“text-align:center;”|  8.9
| style=“text-align:center;”|  7.5
| style=“text-align:center;”|  4.5
|}

'''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean.

'''Search terms:''' [General Anxiety Disorder] AND [children OR adolescents OR pediatric] AND [clinical significance OR outcomes] in GoogleScholar and PsycINFO

=== Treatment ===
{{collapse top| Click here for treatment information}}
Individuals suffering from GAD tend to be high users of outpatient medical care. When treating GAD, physicians should first determine whether pharmacotherapy, psychotherapy, or a combination of the two treatments would be most beneficial to the patient. Literature suggests that treatment of GAD frequently consists of a combination of psychotherapy and pharmacotherapy. Although these therapies have the potential to be effective individually, previous work demonstrates that when combined the degree of clinically significant change increases significantly. Recent studies (e.g., Gorman, 2003&lt;ref name=&quot;Gorman2003&quot; /&gt;; Walkup et al., 2008&lt;ref name=&quot;WalkupEtAl2008&quot; /&gt;) have provided evidence to support this claim with the most efficacious medication and behavioral interventions listed below.

# '''Medication Interventions'''
## ''Sertraline (Zoloft)'' has been shown to reduce experiences and effects of GAD above and beyond that of placebo conditions.
## ''Pregabalin.'' The mean baseline-to-endpoint decreases in total Hamilton anxiety scale score in the patients given 150 mg/day of pregabalin (–9.2) was significantly greater than the decrease in those given placebo (–6.8)&lt;ref name=&quot;PandeEtAl2003&quot; /&gt;.
## ''Paroxetine.'' Remission was achieved by 30% of patients in the 20-mg paroxetine groups compared with 20% given placebo. For all three domains of the Sheehan Disability Scale, significantly greater improvement was seen with paroxetine than placebo&lt;ref name=&quot;RickelsEtAl2003&quot; /&gt;.
# '''Behavioral interventions'''
## ''Cognitive behavioral therapy.'' Fourteen 60-minute sessions, which include CBT in anxiety-management skills, followed by behavioral exposure to anxiety-provoking situations have been shown to be effective in treating GAD. A review of studies by Fisher and Durham (1999)&lt;ref name=&quot;FisherEtAl1999&quot; /&gt; revealed significant recovery rates at a 6 month follow up after CBT. 
## ''Exposure therapy and modeling therapy.'' One meta-analysis found that virtual reality exposure therapy for anxiety disorders had a large effect size (Cohen's d=1.11) compared to controls.&lt;ref&gt;{{Cite journal|last=Powers|first=Mark B.|last2=Emmelkamp|first2=Paul M.G.|title=Virtual reality exposure therapy for anxiety disorders: A meta-analysis|url=https://doi.org/10.1016/j.janxdis.2007.04.006|journal=Journal of Anxiety Disorders|volume=22|issue=3|pages=561–569|doi=10.1016/j.janxdis.2007.04.006}}&lt;/ref&gt; 
## ''Mindfulness meditation.'' New treatment options such as mindfulness meditation-based stress reduction interventions have also shown to reduce symptoms over the long-term.&lt;ref&gt;{{Cite journal|last=Miller|first=J. J.|last2=Fletcher|first2=K.|last3=Kabat-Zinn|first3=J.|date=May 1995|title=Three-year follow-up and clinical implications of a mindfulness meditation-based stress reduction intervention in the treatment of anxiety disorders|url=https://www.ncbi.nlm.nih.gov/pubmed/7649463|journal=General Hospital Psychiatry|volume=17|issue=3|pages=192–200|issn=0163-8343|pmid=7649463}}&lt;/ref&gt;
# '''Combination treatment'''
## Previous research suggests that combination therapy that includes components of psychotherapy and pharmacotherapy are the most efficacious in treating GAD. In a study comparing the efficacies GAD treatments, Walkup and colleagues demonstrated a 21-25% improvement of combination therapy over cognitive behavioral therapy or sertraline alone during short-term treatment. These findings suggest that among effective treatments, combination therapy has the potential to provide the best chance for a positive outcome. See Gorman, 2003&lt;ref name=&quot;Gorman2003&quot; /&gt;; Walkup et al., 2008&lt;ref name=&quot;WalkupEtAl2008&quot; /&gt;.
{{collapse bottom}}
* Please refer to the page on [[wikipedia:Generalized_anxiety_disorder|generalized anxiety disorder]] for more information on available treatment or go to [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/fear-worry-and-anxiety/ Effective Child Therapy] for a curated resource on effective treatments for GAD.
*For information on conducting Exposure Therapy for anxiety disordered youth, see [https://www.bravepracticeforkids.com/ www.BravePracticeForKids.com]

=='''External Resources'''==

# [http://apps.who.int/classifications/icd10/browse/2010/en#/F41.1 ICD-10 diagnostic criteria]
# [https://en.wikiversity.org/w/index.php?title=Helping_Give_Away_Psychological_Science/Resources/Annotated_List_of_Where_and_How_to_Find_a_Therapist&amp;wteswitched=1#Other_low-cost_options Find-a-Therapist] 
#*This is a curated list of find-a-therapist websites where you can find a provider
# [https://www.nimh.nih.gov/health/topics/anxiety-disorders/index.shtml NIMH] entry about anxiety disorders
# OMIM (Online Mendelian Inheritance in Man)
#*[https://www.omim.org/entry/607834 607834]
# [https://emedicine.medscape.com/article/286227-overview#a2 eMedicine entry about anxiety disorders]
#[https://sccap53.org Society of Clinical Child and Adolescent Psychology]
#[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/fear-worry-and-anxiety/ Effective Child Therapy information on Fear, Worry, &amp; Anxiety]
#*Effective Child Therapy is website sponsored by Division 53 of the American Psychological Association (APA), or The [https://sccap53.org Society of Clinical Child and Adolescent Psychology] (SCCAP), in collaboration with the Association for Behavioral and Cognitive Therapies (ABCT). Use for information on symptoms and available treatments.
#[http://pediatricbipolar.pitt.edu/resources/instruments Links to SCARED Child, Parent, and Adult + Translations]

=='''References'''==
{{collapse top|Click here for references}}
{{Reflist|3|refs=

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&lt;ref name=&quot;ChorpitaEtAl2005&quot;&gt;{{cite journal|last1=Chorpita|first1=BF|last2=Moffitt|first2=CE|last3=Gray|first3=J|title=Psychometric properties of the Revised Child Anxiety and Depression Scale in a clinical sample.|journal=Behaviour research and therapy|date=March 2005|volume=43|issue=3|pages=309-22|pmid=15680928}}&lt;/ref&gt;

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&lt;ref name=&quot;Gorman2003&quot;&gt;{{cite journal|last1=Gorman|first1=JM|title=Treating generalized anxiety disorder.|journal=The Journal of clinical psychiatry|date=2003|volume=64 Suppl 2|pages=24-9|pmid=12625796}}&lt;/ref&gt;

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&lt;ref name=&quot;KesslerEtAl2012&quot;&gt;{{cite journal|last1=Kessler|first1=RC|last2=Petukhova|first2=M|last3=Sampson|first3=NA|last4=Zaslavsky|first4=AM|last5=Wittchen H|first5=-U|title=Twelve-month and lifetime prevalence and lifetime morbid risk of anxiety and mood disorders in the United States.|journal=International journal of methods in psychiatric research|date=September 2012|volume=21|issue=3|pages=169-84|pmid=22865617}}&lt;/ref&gt;

&lt;ref name=&quot;LynehamEtAl2007&quot;&gt;{{cite journal|last1=Lyneham|first1=HJ|last2=Abbott|first2=MJ|last3=Rapee|first3=RM|title=Interrater reliability of the Anxiety Disorders Interview Schedule for DSM-IV: child and parent version.|journal=Journal of the American Academy of Child and Adolescent Psychiatry|date=June 2007|volume=46|issue=6|pages=731-6|pmid=17513985}}&lt;/ref&gt;

&lt;ref name=&quot;MarchEtAl1997&quot;&gt;{{cite journal|last1=March|first1=JS|last2=Parker|first2=JD|last3=Sullivan|first3=K|last4=Stallings|first4=P|last5=Conners|first5=CK|title=The Multidimensional Anxiety Scale for Children (MASC): factor structure, reliability, and validity.|journal=Journal of the American Academy of Child and Adolescent Psychiatry|date=April 1997|volume=36|issue=4|pages=554-65|pmid=9100431}}&lt;/ref&gt;

&lt;ref name=&quot;McLeanEtAl2011&quot;&gt;{{cite journal|last1=McLean|first1=CP|last2=Asnaani|first2=A|last3=Litz|first3=BT|last4=Hofmann|first4=SG|title=Gender differences in anxiety disorders: prevalence, course of illness, comorbidity and burden of illness.|journal=Journal of psychiatric research|date=August 2011|volume=45|issue=8|pages=1027-35|pmid=21439576}}&lt;/ref&gt;

&lt;ref name=&quot;MerikangasEtAl2010&quot;&gt;{{cite journal|last1=Merikangas|first1=KR|last2=He|first2=JP|last3=Burstein|first3=M|last4=Swanson|first4=SA|last5=Avenevoli|first5=S|last6=Cui|first6=L|last7=Benjet|first7=C|last8=Georgiades|first8=K|last9=Swendsen|first9=J|title=Lifetime prevalence of mental disorders in U.S. adolescents: results from the National Comorbidity Survey Replication--Adolescent Supplement (NCS-A).|journal=Journal of the American Academy of Child and Adolescent Psychiatry|date=October 2010|volume=49|issue=10|pages=980-9|pmid=20855043}}&lt;/ref&gt;

&lt;ref name=&quot;PandeEtAl2003&quot;&gt;{{cite journal|last1=Pande|first1=AC|last2=Crockatt|first2=JG|last3=Feltner|first3=DE|last4=Janney|first4=CA|last5=Smith|first5=WT|last6=Weisler|first6=R|last7=Londborg|first7=PD|last8=Bielski|first8=RJ|last9=Zimbroff|first9=DL|last10=Davidson|first10=JR|last11=Liu-Dumaw|first11=M|title=Pregabalin in generalized anxiety disorder: a placebo-controlled trial.|journal=The American journal of psychiatry|date=March 2003|volume=160|issue=3|pages=533-40|pmid=12611835}}&lt;/ref&gt;

&lt;ref name=&quot;RickelsEtAl2003&quot;&gt;{{cite journal|last1=Rickels|first1=K|last2=Zaninelli|first2=R|last3=McCafferty|first3=J|last4=Bellew|first4=K|last5=Iyengar|first5=M|last6=Sheehan|first6=D|title=Paroxetine treatment of generalized anxiety disorder: a double-blind, placebo-controlled study.|journal=The American journal of psychiatry|date=April 2003|volume=160|issue=4|pages=749-56|pmid=12668365}}&lt;/ref&gt;

&lt;ref name=&quot;RobertsEtAl2007&quot;&gt;{{cite journal|last1=Roberts|first1=RE|last2=Roberts|first2=CR|last3=Xing|first3=Y|title=Rates of DSM-IV psychiatric disorders among adolescents in a large metropolitan area.|journal=Journal of psychiatric research|date=December 2007|volume=41|issue=11|pages=959-67|pmid=17107689}}&lt;/ref&gt;

&lt;ref name=&quot;SeligmanEtAl2004&quot;&gt;{{cite journal|last1=Seligman|first1=LD|last2=Ollendick|first2=TH|last3=Langley|first3=AK|last4=Baldacci|first4=HB|title=The utility of measures of child and adolescent anxiety: a meta-analytic review of the Revised Children's Manifest Anxiety Scale, the State-Trait Anxiety Inventory for Children, and the Child Behavior Checklist.|journal=Journal of clinical child and adolescent psychology : the official journal for the Society of Clinical Child and Adolescent Psychology, American Psychological Association, Division 53|date=September 2004|volume=33|issue=3|pages=557-65|pmid=15271613}}&lt;/ref&gt;

&lt;ref name=&quot;WalkupEtAl2008&quot;&gt;{{cite journal|last1=Walkup|first1=JT|last2=Albano|first2=AM|last3=Piacentini|first3=J|last4=Birmaher|first4=B|last5=Compton|first5=SN|last6=Sherrill|first6=JT|last7=Ginsburg|first7=GS|last8=Rynn|first8=MA|last9=McCracken|first9=J|last10=Waslick|first10=B|last11=Iyengar|first11=S|last12=March|first12=JS|last13=Kendall|first13=PC|title=Cognitive behavioral therapy, sertraline, or a combination in childhood anxiety.|journal=The New England journal of medicine|date=25 December 2008|volume=359|issue=26|pages=2753-66|pmid=18974308}}&lt;/ref&gt;

&lt;ref name=&quot;WhitakerEtAl1990&quot;&gt;{{cite journal|last1=Whitaker|first1=A|last2=Johnson|first2=J|last3=Shaffer|first3=D|last4=Rapoport|first4=JL|last5=Kalikow|first5=K|last6=Walsh|first6=BT|last7=Davies|first7=M|last8=Braiman|first8=S|last9=Dolinsky|first9=A|title=Uncommon troubles in young people: prevalence estimates of selected psychiatric disorders in a nonreferred adolescent population.|journal=Archives of general psychiatry|date=May 1990|volume=47|issue=5|pages=487-96|pmid=2331210}}&lt;/ref&gt;

&lt;ref name=&quot;SpitzerEtAl2006&quot;&gt;{{cite journal|last1=Spitzer|first1=RL|last2=Kroenke|first2=K|last3=Williams|first3=JB|last4=Löwe|first4=B|title=A brief measure for assessing generalized anxiety disorder: the GAD-7.|journal=Archives of internal medicine|date=22 May 2006|volume=166|issue=10|pages=1092-7|pmid=16717171}}&lt;/ref&gt;

&lt;ref name=&quot;vanGastelEtAl2008&quot;&gt;{{cite journal|last1=van Gastel|first1=W|last2=Ferdinand|first2=RF|title=Screening capacity of the Multidimensional Anxiety Scale for Children (MASC) for DSM-IV anxiety disorders.|journal=Depression and anxiety|date=2008|volume=25|issue=12|pages=1046-52|pmid=18833579}}&lt;/ref&gt;

&lt;ref name=&quot;WoodEtAl2002&quot;&gt;{{cite journal|last1=Wood|first1=JJ|last2=Piacentini|first2=JC|last3=Bergman|first3=RL|last4=McCracken|first4=J|last5=Barrios|first5=V|title=Concurrent validity of the anxiety disorders section of the Anxiety Disorders Interview Schedule for DSM-IV: child and parent versions.|journal=Journal of clinical child and adolescent psychology : the official journal for the Society of Clinical Child and Adolescent Psychology, American Psychological Association, Division 53|date=September 2002|volume=31|issue=3|pages=335-42|pmid=12149971}}&lt;/ref&gt;

&lt;ref name=&quot;ZimmermanEtAl2005&quot;&gt;{{cite journal|last1=Zimmerman|first1=M|last2=Rothschild|first2=L|last3=Chelminski|first3=I|title=The prevalence of DSM-IV personality disorders in psychiatric outpatients.|journal=The American journal of psychiatry|date=October 2005|volume=162|issue=10|pages=1911-8|pmid=16199838}}&lt;/ref&gt;

}}
{{collapse bottom|Click here for references}}
[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]</text>
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    <title>Evidence-based assessment/Oppositional defiant disorder (disorder portfolio)</title>
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      <comment>/* Recommended diagnostic instruments for ODD */ Added in citations</comment>
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{{medical disclaimer}}
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==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
* For background information on what assessment portfolios are, click the link in the heading above.
* Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Oppositional defiant disorder (assessment portfolio)/extended version|here]].

==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]==
=== Diagnostic criteria for oppositional defiant disorder===

{{blockquotetop}}
&lt;big&gt;ICD-11 Diagnostic Criteria&lt;/big&gt;&lt;br&gt;
&lt;br&gt;
'''General Description:'''
Oppositional defiant disorder is a persistent pattern (e.g., 6 months or more) of markedly defiant, disobedient, provocative or spiteful behaviour that occurs more frequently than is typically observed in individuals of comparable age and developmental level and that is not restricted to interaction with siblings. Oppositional defiant disorder may be manifest in prevailing, persistent angry or irritable mood, often accompanied by severe temper outbursts or in headstrong, argumentative and defiant behaviour. The behavior pattern is of sufficient severity to result in significant impairment in personal, family, social, educational, occupational or other important areas of functioning
&lt;br&gt;
&lt;br&gt;
'''Oppositional Defiant Disorder With Chronic Irritability-Anger:''' All definitional requirements for oppositional defiant disorder are met. This form of oppositional defiant disorder is characterized by prevailing, persistent angry or irritable mood that may be present independent of any apparent provocation. The negative mood is often accompanied by regularly occurring severe temper outbursts that are grossly out of proportion in intensity or duration to the provocation. Chronic irritability and anger are characteristic of the individual’s functioning nearly every day, are observable across multiple settings or domains of functioning (e.g., home, school, social relationships), and are not restricted to the individual’s relationship with his/her parents or guardians. The pattern of chronic irritability and anger is not limited to occasional episodes (e.g., developmentally typical irritability) or discrete periods (e.g., irritable mood in the context of manic or depressive episodes).
*Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder with chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions, and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f818792113 here]. 
&lt;br&gt;
'''Oppositional Defiant Disorder Without Chronic Irritability-Anger:''' 
Meets all definitional requirements for oppositional defiant disorder. This form of oppositional defiant disorder is not characterized by prevailing, persistent, angry or irritable mood, but does feature headstrong, argumentative, and defiant behavior.
*Note: The ICD-11 lists 3 additional subcategories of oppositional defiant disorder without chronic irritability-anger (i.e., with limited prosocial emotions, with typical prosocial emotions,  and unspecified). They can be found [https://icd.who.int/browse11/l-m/en#/http%3a%2f%2fid.who.int%2ficd%2fentity%2f736540987 here].
&lt;br&gt;
&lt;big&gt;Changes in DSM-5&lt;/big&gt;
&lt;br&gt;
The diagnostic criteria for ADHD changed slightly from DSM-IV to DSM-5. See the changes [https://www.ncbi.nlm.nih.gov/books/NBK519712/table/ch3.t14/ here]. 

{{blockquotebottom}}&lt;ref&gt;{{Cite web|url=https://icd.who.int/browse11/l-m/en#/http://id.who.int/icd/entity/1545599508|title=ICD-11 for Mortality and Morbidity Statistics|website=icd.who.int|access-date=2022-07-11}}&lt;/ref&gt;

===Base rates of ODD in different clinical settings===
This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of ODD that they are likely to see in their clinical practice.

* '''''To see prevalence rates across multiple disorders, [[Evidence-based assessment/Preparation phase|click here.]]''''' 

{|class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Demography 
! Setting !! Base Rate !! Diagnostic Method
|-
|| Various locations across USA
| Meta-analysis of 38 studies&lt;ref&gt;Canino G, Polanczyk G, Bauermeister JJ, et al. (2010) Does the prevalence of CD and ODD vary across cultures? Social Psychiatry Epidemiology, 45(7):695–704.&lt;/ref&gt; 
|| 3.3% 
|| Varied
|-
|| All of the U.S.  
| Nationally representative large-scale study (N = 3,119) &lt;ref&gt;Nock, M. K., Kazdin, A. E., Hiripi, E., &amp; Kessler, R. C. (2007). Lifetime prevalence, correlates, and persistence of oppositional defiant disorder: results from the National Comorbidity Survey Replication. ''Journal of Child Psychology and Psychiatry, 48''(7), 703-713.&lt;/ref&gt; 
|| 10.2% (overall)
11.2% (males) 

9.2% (females) 
|| World Health Organization (WHO) Composite International Diagnostic Interview (CIDI)&lt;sup&gt;r&lt;/sup&gt;
|-
|Suburban and urban Colorado
|Project to Learn about Youth-Mental health, school-based study for  children from kindergarten to high-school (n = 236)&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last=Danielson|first=Melissa L.|last2=Bitsko|first2=Rebecca H.|last3=Holbrook|first3=Joseph R.|last4=Charania|first4=Sana N.|last5=Claussen|first5=Angelika H.|last6=McKeown|first6=Robert E.|last7=Cuffe|first7=Steven P.|last8=Owens|first8=Julie Sarno|last9=Evans|first9=Steven W.|date=2021-06-01|title=Community-Based Prevalence of Externalizing and Internalizing Disorders among School-Aged Children and Adolescents in Four Geographically Dispersed School Districts in the United States|url=https://doi.org/10.1007/s10578-020-01027-z|journal=Child Psychiatry &amp; Human Development|language=en|volume=52|issue=3|pages=500–514|doi=10.1007/s10578-020-01027-z|issn=1573-3327|pmc=PMC8016018|pmid=32734339}}&lt;/ref&gt;
|6.8%
|Diagnostic Interview Schedule for Children (DISC)
|-
|Urban and suburban Florida
|Project to Learn about Youth-Mental health, school-based study for  children from kindergarten to high-school (n = 289)&lt;ref name=&quot;:1&quot; /&gt;
|6.9%
|Diagnostic Interview Schedule for Children (DISC)
|-
|Rural and suburban Ohio
|Project to Learn about Youth-Mental health, school-based study for  children from elementary to high-school (n = 152)&lt;ref name=&quot;:1&quot; /&gt;
|17.3%
|Diagnostic Interview Schedule for Children (DISC)
|-
|Suburban and rural South Carolina
|Project to Learn about Youth-Mental health, school-based study for  children from elementary to high-school (n = 270)&lt;ref name=&quot;:1&quot; /&gt;
|5.7%
|Diagnostic Interview Schedule for Children (DISC)
|-
|| Semi-rural North Carolina 
| Preschool-aged children from pediatric practices (N = 306; age 2 - 5 years old)&lt;ref&gt;Egger, H.L., &amp; Angold, A. (2006). Common emotional and behavioral disorders in preschool children: Presentation, nosology, and epidemiology. ''Journal of Child Psychiatry and Psychology, 47'', 313–337.&lt;/ref&gt; 
|| 6.6% 
|| Preschool Age Psychiatric Assessment (PAPA)&lt;sup&gt;p&lt;/sup&gt;
|-
|| Western North Carolina  
| The Great Smoky Mountains Study - longitudinal, population-based study of community sample&lt;ref&gt;Costello, E. J., Mustillo, S., Erkanli, A., Keeler, G., &amp; Angold, A. (2003) Prevalence and development of psychiatric disorders in adolescence. ''Arch Gen Psychiatry, 60'', 837-844.&lt;/ref&gt; 
|| 2.33% (overall) 
3.16% (males) 2.75% (females) 
|| Child and Adolescent Psychiatric Assessment (CAPA)&lt;sup&gt;p, y&lt;/sup&gt; 
|-
|| Chicago 
| Preschool-aged children from inner city schools and pediatric practices (N = 796; age 2 - 5 years old)&lt;ref&gt;Lavigne, J. V., LeBailly, S. A., Hopkins, J., Gouze, K. R., &amp; Binns, H. J. (2009). The prevalence of ADHD, ODD, depression, and anxiety in a community sample of 4-year-olds. ''Journal Of Clinical Child And Adolescent Psychology, 38''(3), 315-328. doi:10.1080/15374410902851382&lt;/ref&gt; 
|| 8.3% 
|| Diagnostic Interview Schedule for Children–Parent Scale–Young Child Version (DISC-YC) &lt;sup&gt;p&lt;/sup&gt;
|-
|Germany (Saarbrücken County)
|All school-aged children were examined during a routine school-entry medical examination (N = 1676, mean age = 5.7)&lt;ref&gt;{{Cite journal|last=Niemczyk|first=Justine|last2=Equit|first2=Monika|last3=Braun-Bither|first3=Katrin|last4=Klein|first4=Anna-Maria|last5=von Gontard|first5=Alexander|date=2015-07-01|title=Prevalence of incontinence, attention deficit/hyperactivity disorder and oppositional defiant disorder in preschool children|url=https://doi.org/10.1007/s00787-014-0628-6|journal=European Child &amp; Adolescent Psychiatry|language=en|volume=24|issue=7|pages=837–843|doi=10.1007/s00787-014-0628-6|issn=1435-165X}}&lt;/ref&gt;
|7.3% (males)
5.1%(females)
|DISYPS-II
|-
|South Korea (Seoul)
|Children were randomly surveyed across 6 school districts in Seoul (N = 1645, age 6 - 12 years old)
|5.8% (males)
4.1%(females)
|Diagnostic Interview Schedule for Children–Parent Scale IV (DISC-IV)
|}

&lt;sup&gt;p&lt;/sup&gt; Parent interviewed as part of diagnostic assessment; &lt;sup&gt;y&lt;/sup&gt; youth interviewed as part of diagnostic assessment,  &lt;sup&gt;r&lt;/sup&gt; adult interviewed for retrospective report as part of diagnostic assessment

'''Note:''' Mash and Barkley note that prevalence rates of ODD must be qualified, because the definition of ODD has changed at a fast rate, the rates adolescents meeting criteria in any cross-sectional evaluation may be misleading because of the developmental progressions with and between ODD and Conduct Disorder, and categorical definitions of aggressive patterns may reflect arbitrary numbers of constituent estimates. These factors may lead to misleading prevalence rates. In addition, few studies have investigated the prevalence of ODD in preschool-aged children, and early onset of these behaviors is associated with more severe and stable impairment.&lt;ref&gt;Mash, E., &amp; Barkley, R. (Eds.). (2003). ''Child Psychopathology''. 2nd Edition. New York: Guilford Press.&lt;/ref&gt;

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==
The following section contains a list of screening and diagnostic instruments for ODD. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. 
* '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]'''''

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! colspan=&quot;5&quot; |Screening measures for ODD
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
!Where to access
|-
|Achenbach System of Empirically Based Assessments (ASEBA): Child Behavior Checklist (CBCL), Youth Self-Report (YSR)
|Parent report (CBCL)
Youth self-report (YSR)
|6-18 (CBCL)
11-18 (YSR) 
&lt;ref name=&quot;:9&quot;&gt;{{Cite book|url=https://www.worldcat.org/oclc/1130319849|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4363-2|edition=Fifth edition|location=New York, NY|oclc=1130319849}}&lt;/ref&gt;
|10 - 15 minutes&lt;ref name=&quot;:9&quot; /&gt;
|[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase]  
|-
|Eyberg Child Behavior Inventory (ECBI)
|Parent report
|2-16
|5 minutes
|[https://www.parinc.com/products/pkey/97 Purchase]
|-
|Strengths and Difficulties Questionnaire (SDQ)
|Parent report
|3 - 16 
|3 - 5 minutes
|[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs]
|}

'''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable
=== Likelihood ratios and AUCs of screening instruments for ODD ===
* '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' 
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! Screening Measure (Primary Reference)
! Area Under Curve (AUC)
! LR+ (Score)
! LR- (Score)
! Clinical Generalizability
!Where to Access
|-
|CBCL DSM-Oriented Scales&lt;ref name=&quot;:7&quot;&gt;Achenbach, T. M. (1991a). ''Manual for the Child Behavior Checklist/4–18 and 1991 Profile''. Burlington , VT : University of Vermont Department of Psychiatry.&lt;/ref&gt;
|.71 (N=475)&lt;ref name=&quot;:3&quot;&gt;Ebesutani, C., Bernstein, A., Nakamura, B. J., Chorpita, B. F., Higa-McMillan, C. K., &amp; Weisz, J. R. (2010). Concurrent validity of the child behavior checklist DSM-oriented scales: Correspondence with DSM diagnoses and comparison to syndrome scales. ''Journal of Psychopathology and Behavioral Assessment, 32''(3), 373-384.&lt;/ref&gt;
| 2.80 (60+ to 70+) &lt;ref name=&quot;:4&quot;&gt;Warnick, E. M., Bracken, M. B., &amp; Kasl, S. (2008). Screening efficiency of the Child Behavior Checklist and Strengths and Difficulties Questionnaire: a systematic review. ''Child and Adolescent Mental Health, 13''(3), 140-147.&lt;/ref&gt;
| .52 (60- to 70-)*&lt;ref name=&quot;:4&quot; /&gt;
|Youth aged 5 - 18 seeking out patient treatment across a variety of settings&lt;ref name=&quot;:4&quot; /&gt;
|[https://store.aseba.org/School-Age-6-18-Materials/departments/11/ Purchase]
|-
|SDQ&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|last=Shabani|first=Amir|last2=Masoumian|first2=Samira|last3=Zamirinejad|first3=Somayeh|last4=Hejri|first4=Maryam|last5=Pirmorad|first5=Tahereh|last6=Yaghmaeezadeh|first6=Hooman|date=2021-05|title=Psychometric properties of Structured Clinical Interview for DSM‐5 Disorders‐Clinician Version (SCID‐5‐CV)|url=https://onlinelibrary.wiley.com/doi/10.1002/brb3.1894|journal=Brain and Behavior|language=en|volume=11|issue=5|doi=10.1002/brb3.1894|issn=2162-3279|pmc=PMC8119811|pmid=33729681}}&lt;/ref&gt;
| .81 -.88 (N = 18,416)&lt;ref&gt;{{Cite journal|last=Algorta|first=Guillermo Perez|last2=Dodd|first2=Alyson Lamont|last3=Stringaris|first3=Argyris|last4=Youngstrom|first4=Eric A.|date=2016-09|title=Diagnostic efficiency of the SDQ for parents to identify ADHD in the UK: a ROC analysis|url=http://link.springer.com/10.1007/s00787-015-0815-0|journal=European Child &amp; Adolescent Psychiatry|language=en|volume=25|issue=9|pages=949–957|doi=10.1007/s00787-015-0815-0|issn=1018-8827|pmc=PMC4990620|pmid=26762184}}&lt;/ref&gt;
|7.55 (3+)&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Goodman|first=Robert|date=2001-11-01|title=Psychometric Properties of the Strengths and Difficulties Questionnaire|url=https://www.jaacap.org/article/S0890-8567(09)60543-8/abstract|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|language=English|volume=40|issue=11|pages=1337–1345|doi=10.1097/00004583-200111000-00015|issn=0890-8567|pmid=11699809}}&lt;/ref&gt;
|.35 (3-)&lt;ref name=&quot;:2&quot; /&gt;
|Surveyed youth aged 5 - 15 in the UK &lt;ref name=&quot;:2&quot; /&gt;
|[https://www.sdqinfo.org/a0.html SDQ Homepage][https://www.sdqinfo.org/py/sdqinfo/b0.py PDFs]
|-
|ECBI- Intensity Scale&lt;ref name=&quot;:8&quot;&gt;Eyberg, S. M., &amp; Robinson, E. A. (1983). Conduct problem behavior: Standardization of a behavioral rating scale with adolescents. Journal of Clinical Child Psychology, 12 (3), 347-354.&lt;/ref&gt;
|
|6.92 (131+)&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=LYNEHAM|first=HEIDI J.|last2=ABBOTT|first2=MAREE J.|last3=RAPEE|first3=RONALD M.|date=2007-06|title=Interrater Reliability of the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Version|url=https://doi.org/10.1097/chi.0b013e3180465a09|journal=Journal of the American Academy of Child &amp;amp; Adolescent Psychiatry|volume=46|issue=6|pages=731–736|doi=10.1097/chi.0b013e3180465a09|issn=0890-8567}}&lt;/ref&gt;
|.11 (131-)&lt;ref name=&quot;:5&quot; /&gt;
|Youth aged 7-16 had responses compared to diagnosis&lt;ref name=&quot;:5&quot; /&gt;
|[https://www.parinc.com/products/pkey/97 Purchase]
|}

'''Note:''' “LR+” refers to the change in likelihood ratio associated with a positive test score, and “LR-” is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all. LRs larger than 10 or smaller than .10 are frequently clinically decisive; 5 or .20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinically meaningful changes of formulation&lt;ref&gt;Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., &amp; Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone.&lt;/ref&gt;.

'''Search terms:''' [Oppositional Defiant Disorder] AND [sensitivity OR specificity] in GoogleScholar and PsychINFO; 

=== Interpreting ODD screening measure scores ===

* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]

== [[Evidence based assessment/Prescription phase|'''Prescription phase''']] ==

=== Gold standard diagnostic interviews ===

* For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

=== Recommended diagnostic instruments for ODD ===
{| class=&quot;wikitable sortable&quot;
! colspan=&quot;5&quot; |Diagnostic instruments for ODD
|-
!Measure
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Where to Access
|-
|Kiddie Schedule for Affective Disorders and Schizophrenia Present and Lifetime Version (KSADS-PL)&lt;ref&gt;{{Cite web|url=http://dx.doi.org/10.1037/t03988-000|title=Schedule for Affective Disorders and Schizophrenia for School-Age Children-Present and Lifetime Version|last=Kaufman|first=John|last2=Birmaher|first2=Boris|date=1997|website=PsycTESTS Dataset|access-date=2022-08-04|last3=Brent|first3=David|last4=Rao|first4=Uma|last5=Flynn|first5=Cynthia|last6=Moreci|first6=Paula|last7=Williamson|first7=Douglas|last8=Ryan|first8=Neal}}&lt;/ref&gt;
|Structured interview
|6-28
|45-75 minutes
|[https://www.pediatricbipolar.pitt.edu/resources/instruments Website to access]
|-
|Development and Well-Being Assessment (DAWBA) &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
|Structured (child/adolescent and parent)
|11-18
|10-20 minutes for the eating disorder module
|English PDF
[https://dawba.info/py/dawbainfo/b1.py Additional languages]
|-
|Diagnostic Interview Schedule for Children (DISC-5)&lt;ref&gt;{{Cite web|url=http://dx.doi.org/10.1037/t04405-000|title=National Institute of Mental Health Diagnostic Interview Schedule for Children-IV|last=Shaffer|first=David|last2=Fisher|first2=Prudence|date=2000|website=PsycTESTS Dataset|access-date=2022-08-04|last3=Lucas|first3=Christopher P.|last4=Dulcan|first4=Mina K.|last5=Schwab-Stone|first5=Mary E.}}&lt;/ref&gt;
|Structured Interview (Self report and parent)
|6-17
|About 20 minutes
|[https://telesage.com/netdisc-5/# Coming soon]
|}
'''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable

== [[Evidence based assessment/Process phase|'''Process phase''']] ==
The following section contains a list of process and outcome measures for oppositional defiant disorder. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]].

=== Outcome and severity measures ===

* This table includes clinically significant benchmarks for generalized anxiety disorder specific outcome measures
* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
* ''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
| rowspan=&quot;1&amp;quot;&quot; style=&quot;text-align:center;font-size:130%;&quot; | &lt;b&gt; Measure&lt;/b&gt;
| style=&quot;text-align:center;font-size:130%;&quot; | &lt;b&gt; Subscale&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; | &lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; | &lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| colspan=&quot;8&quot; style=&quot;font-size:110%; text-align:center;&quot; span | &lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| colspan=&quot;2&quot; |
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | &lt;b&gt; CBCL T-scores &lt;br&gt; (2001 Norms)&lt;/b&gt;
| style=&quot;text-align:right;&quot; | &lt;i&gt; Externalizing&lt;/i&gt;
| style=&quot;text-align:center;&quot; | 49
| style=&quot;text-align:center;&quot; | 70
| style=&quot;text-align:center;&quot; | 58
| style=&quot;text-align:center;&quot; | 7
| style=&quot;text-align:center;&quot; | 6
| style=&quot;text-align:center;&quot; | 3.4
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | &lt;b&gt; ECBI Scaled Scores &lt;br&gt; (1983 Norms)&lt;/b&gt;
| style=&quot;text-align:right;&quot; | &lt;i&gt; Intensity&lt;/i&gt;
| style=&quot;text-align:center;&quot; | 80.1
| style=&quot;text-align:center;&quot; | 169.5
| style=&quot;text-align:center;&quot; | 112.9
| style=&quot;text-align:center;&quot; | 9.5
| style=&quot;text-align:center;&quot; | 8
| style=&quot;text-align:center;&quot; | 4.8
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | &lt;b&gt; ECBI Scaled Scores &lt;br&gt; (1983 Norms)&lt;/b&gt;
| style=&quot;text-align:right;&quot; | &lt;i&gt; Problem&lt;/i&gt;
| style=&quot;text-align:center;&quot; | 3.9
| style=&quot;text-align:center;&quot; | 17.7
| style=&quot;text-align:center;&quot; | 11.5
| style=&quot;text-align:center;&quot; | 2.1
| style=&quot;text-align:center;&quot; | 1.8
| style=&quot;text-align:center;&quot; | 1.1
|}

'''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean. 

=== Process measures===
See Section 1.1 for overview of evidence-based measures to use depending on etiology and symptomatology of Oppositional Defiant Disorder.

==Treatment==
===Behavioral parent training===

Behavioral Parent Training is considered the most effective treatment for childhood disruptive behavior disorders (e.g., Oppositional Defiant Disorder), especially for younger children (i.e., 3-8 year-olds). See http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/ a website sponsored by The Society for Child and Adolescent Psychology (APA, Division 53) and the Association for Behavioral and Cognitive Therapies (ABCT), for current summary of evidence-based treatments. 

===Overview of recommendations for assessment and treatment=== 
See the [https://pathways.nice.org.uk/pathways/antisocial-behaviour-and-conduct-disorders-in-children-and-young-people#content=view-info-category%3Aview-about-menu National Institute for Health and Care Excellence (NICE) Practice Guidelines for Childhood Conduct Disorders], for an overview of recommendations for both assessment and treatment of Oppositional Defiant Disorder.

== External Links ==
*[https://sccap53.org Society of Clinical Child and Adolescent Psychology]
*http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/rule-breaking-defiance-and-acting-out/

== References ==
{{collapse top|Click here for references}}
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    <title>Evidence-based assessment/Obsessive-compulsive disorder (assessment portfolio)</title>
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{{medical disclaimer}}
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==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
For background information on what assessment portfolios are, click the link in the heading above.

==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]==

=== Diagnostic criteria for ______ disorder ===
{{blockquotetop}}'''ICD-11 Criteria''' &lt;ref&gt;{{Cite web|url=http://apps.who.int/classifications/icd10/browse/2016/en#/F40-F48|title=ICD-10 Version:2016|website=apps.who.int|language=en|accessdate=2018-03-01}}&lt;/ref&gt;
Obsessive-Compulsive Disorder is characterized by the presence of persistent obsessions or compulsions, or most commonly both. Obsessions are repetitive and persistent thoughts, images, or impulses/urges that are intrusive, unwanted, and are commonly associated with anxiety. The individual attempts to ignore or suppress obsessions or to neutralize them by performing compulsions. Compulsions are repetitive behaviors including repetitive mental acts that the individual feels driven to perform in response to an obsession, according to rigid rules, or to achieve a sense of ‘completeness’. In order for obsessive-compulsive disorder to be diagnosed, obsessions and compulsions must be time consuming (e.g., taking more than an hour per day), and result in significant distress or significant impairment in personal, family, social, educational, occupational or other important areas of functioning.

Inclusions
*anankastic neurosis
*obsessive-compulsive neurosis

Exclusions
*obsessive compulsive behaviour (MB23.4)

{{blockquotebottom}}

=== Demographics ===
This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of OCD that they are likely to see in their clinical practice.
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Demography
! Setting
! Base Rate
! Diagnostic Method
|-
|National (U.S.) adult 
sample (n=2073) 
|National Comorbidity Survey Replication&lt;ref name=&quot;RuscioEtAl2010&quot;/&gt;
|2.3%
|World Health Organization Composite 
International Diagnostic Interview (CIDI 3.0)
|-
|U.S. household sample 
(n=18572) 
|Epidemiological Catchment Area (ECA) Program&lt;ref name=&quot;KarnoEtAl1988&quot;/&gt;
|1.9-3.3%
|Diagnostic Interview Schedule (DIS)
|-
|Iranian adults
(n=25180) 
|Iranian population-based study&lt;ref name=&quot;MohammadiEtAl2004&quot;/&gt;
|1.8%
|DIS
|-
|NSAL adult study
(n=5191) 
|African-American and Caribbean Households (U.S.)&lt;ref name=&quot;HimleEtAl2008&quot;/&gt;
|1.6%
|CIDI Short Form
|-
|Epidemiological sample 
(n=6616) 
|Singapore Mental Health Study&lt;ref name=&quot;SubramaniamEtAl2012&quot;/&gt;
|3.0%
|CIDI 3.0
|}

'''Search terms:'''

[obsessive compulsive disorder OR ocd] AND [prevalence OR incidence] in PsycInfo and PubMed

[obsessive compulsive disorder OR ocd] AND [epidemiology] in PsycInfo and PubMed

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==

=== Diagnosis ===

{{blockquotetop}}'''ICD-11 Criteria''' &lt;ref&gt;{{Cite web|url=http://apps.who.int/classifications/icd10/browse/2016/en#/F40-F48|title=ICD-10 Version:2016|website=apps.who.int|language=en|accessdate=2018-03-01}}&lt;/ref&gt;
Obsessive-Compulsive Disorder is characterized by the presence of persistent obsessions or compulsions, or most commonly both. Obsessions are repetitive and persistent thoughts, images, or impulses/urges that are intrusive, unwanted, and are commonly associated with anxiety. The individual attempts to ignore or suppress obsessions or to neutralize them by performing compulsions. Compulsions are repetitive behaviors including repetitive mental acts that the individual feels driven to perform in response to an obsession, according to rigid rules, or to achieve a sense of ‘completeness’. In order for obsessive-compulsive disorder to be diagnosed, obsessions and compulsions must be time consuming (e.g., taking more than an hour per day), and result in significant distress or significant impairment in personal, family, social, educational, occupational or other important areas of functioning.

Inclusions
*anankastic neurosis
*obsessive-compulsive neurosis

Exclusions
*obsessive compulsive behaviour (MB23.4)

{{blockquotebottom}}

=== Screening and diagnostic instruments for OCD ===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Downloads
|- 
| colspan=&quot;9&quot;  span style=&quot;font-size:110%; text-align:center;&quot; | &lt;b&gt; Diagnostic Measures&lt;/b&gt;
|
|-
| Anxiety Disorders Interview Schedule&lt;ref&gt;Brown, T.A., Di Nardo, P.A., Barlow, D.H., 1994. Anxiety Disorders Interview Schedule for DSM-IV (ADIS-IV-L). Psychological Corporation, San Antonio, TX.&lt;/ref&gt; 
| 
| 
| 
| E
| G
| E
| E
| X
|
|-
| Structured Clinical Interview for DSM-IV (SCID)&lt;ref&gt;First, M. B., &amp; Gibbon, M. (2004). The Structured Clinical Interview for DSM-IV Axis I Disorders (SCID-I) and the Structured Clinical Interview for DSM-IV Axis II Disorders (SCID-II). In M. J. Hilsenroth, D. L. Segal, M. J. Hilsenroth, D. L. Segal (Eds.) , ''Comprehensive handbook of psychological assessment, Vol. 2: Personality assessment'' (pp. 134-143). Hoboken, NJ, US: John Wiley &amp; Sons Inc.&lt;/ref&gt;
| 
|
| 
| A
| A
| E
| E
| 
|
|-
|[https://mfr.osf.io/render?url=https://osf.io/mnzy2/?action=download%26mode=render Yale-Brown Obsessive Compulsive Scale Symptom Checklist]&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Steketee|first=G|title=The Yale-Brown Obsessive Compulsive Scale: Interview versus self-report|url=http://linkinghub.elsevier.com/retrieve/pii/0005796796000368|journal=Behaviour Research and Therapy|volume=34|issue=8|pages=675–684|doi=10.1016/0005-7967(96)00036-8}}&lt;/ref&gt;
| 
| 
| 
| NA
| NA
| E
| G
| X
|
|-
| [https://search.proquest.com/docview/220481418/abstract/4AF689339CB14A0APQ/1?accountid=14244 Brown Assessment of Beliefs Scale]&lt;ref name=&quot;:1&quot;&gt;Eisen, J. L., Phillips, K. A., Baer, L., Beer, D. A., &amp; al, e. (1998). The brown assessment of beliefs scale: Reliability and validity. ''The American Journal of Psychiatry, 155''(1), 102-8. Retrieved from &lt;nowiki&gt;http://libproxy.lib.unc.edu/login?url=https://search.proquest.com/docview/220481418?accountid=14244&lt;/nowiki&gt;&lt;/ref&gt;
| 
| 
| 
| G
| A
| G
| G
| X
|
|-
|Revised Children’s Anxiety and Depression Scale (RCADS)
|Questionnaire (Child)
|6-18
|12 minutes
|G&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Chorpita|first=Bruce F.|last2=Moffitt|first2=Catherine E.|last3=Gray|first3=Jennifer|date=2005-03|title=Psychometric properties of the Revised Child Anxiety and Depression Scale in a clinical sample|url=http://dx.doi.org/10.1016/j.brat.2004.02.004|journal=Behaviour Research and Therapy|volume=43|issue=3|pages=309–322|doi=10.1016/j.brat.2004.02.004|issn=0005-7967}}&lt;/ref&gt;
|G&lt;ref&gt;{{Cite journal|last=Chorpita|first=Bruce F|last2=Yim|first2=Letitia|last3=Moffitt|first3=Catherine|last4=Umemoto|first4=Lori A|last5=Francis|first5=Sarah E|date=2000-08|title=Assessment of symptoms of DSM-IV anxiety and depression in children: a revised child anxiety and depression scale|url=http://dx.doi.org/10.1016/s0005-7967(99)00130-8|journal=Behaviour Research and Therapy|volume=38|issue=8|pages=835–855|doi=10.1016/s0005-7967(99)00130-8|issn=0005-7967}}&lt;/ref&gt;
|G&lt;ref name=&quot;:2&quot; /&gt;
|
|
|
*[http://www.corc.uk.net/outcome-experience-measures/revised-childrens-anxiety-and-depression-scale-and-subscales/ RCADS homepage]

'''PDFs for RCADS'''

*[https://mfr.osf.io/render?url=https://osf.io/s3fu2/?action=download%26mode=render RCADS Child Self-reported (8-18 years)]
*[https://mfr.osf.io/render?url=https://osf.io/fp9mk/?action=download%26mode=render RCADS Parent-reported]
*[https://mfr.osf.io/render?url=https://osf.io/vy7ta/?action=download%26mode=render Child Scoring Aid]
*[https://mfr.osf.io/render?url=https://osf.io/t4bz6/?action=download%26mode=render Parent Scoring Aid]

'''Subscales'''

*[https://mfr.osf.io/render?url=https://osf.io/ca5hk/?action=download%26mode=render OCD Self-reported]

* [https://mfr.osf.io/render?url=https://osf.io/7xkbn/?action=download%26mode=render OCD Parent-reported]

'''Translations'''

'''[https://mfr.osf.io/render?url=https://osf.io/qsjh9/?action=download%26mode=render User Guide]'''

*
|}

'''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable

*[https://mfr.osf.io/render?url=https://osf.io/tn2vg/?action=download%26mode=render Yale-Brown Obsessive Compulsive Scale (Y-BOC)S]&lt;ref&gt;{{Cite journal|last=Goodman|first=Wayne K.|date=1989-11-01|title=The Yale-Brown Obsessive Compulsive Scale|url=http://archpsyc.jamanetwork.com/article.aspx?doi=10.1001/archpsyc.1989.01810110048007|journal=Archives of General Psychiatry|language=en|volume=46|issue=11|doi=10.1001/archpsyc.1989.01810110048007|issn=0003-990X}}&lt;/ref&gt;
**With Symptom Checklist (Y-BOCS-SC) or self-report (Y-BOCS-SR)&lt;ref name=&quot;:0&quot; /&gt;

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==

=== Recommended self-report questionnaires ===
*[http://psycnet.apa.org/record/2010-04450-019 Dimensional Obsessive Compulsive Scale (DOCS)]&lt;ref&gt;{{Cite journal|last=Abramowitz|first=Jonathan S.|last2=Deacon|first2=Brett J.|last3=Olatunji|first3=Bunmi O.|last4=Wheaton|first4=Michael G.|last5=Berman|first5=Noah C.|last6=Losardo|first6=Diane|last7=Timpano|first7=Kiara R.|last8=McGrath|first8=Patrick B.|last9=Riemann|first9=Bradley C.|title=Assessment of obsessive-compulsive symptom dimensions: Development and evaluation of the Dimensional Obsessive-Compulsive Scale.|url=http://dx.doi.org/10.1037/a0018260|journal=Psychological Assessment|language=en|volume=22|issue=1|pages=180–198|doi=10.1037/a0018260}}&lt;/ref&gt;
*[https://www.researchgate.net/profile/Robert_Langner/publication/10973110_The_Obsessive-Compulsive_Inventory_Development_and_validation_of_a_short_version/links/54be1e9a0cf218d4a16a4dc5/The-Obsessive-Compulsive-Inventory-Development-and-validation-of-a-short-version.pdf Obsessive Compulsive Inventory – Revised]&lt;ref&gt;Foa, E. B., Huppert, J. D., Leiberg, S., Langner, R., Kichic, R., Hajcak, G., &amp; Salkovskis, P. M. (2002). The Obsessive-Compulsive Inventory: development and validation of a short version. ''Psychological assessment'', ''14''(4), 485.&lt;/ref&gt;
*[https://ac.els-cdn.com/S0005796700000851/1-s2.0-S0005796700000851-main.pdf?_tid=f904c386-c354-424f-b339-6a53a87ec31a&amp;acdnat=1521761575_8fe72e0787f227a8ee9cdf3592136d64 Interpretation of Intrusions Inventory]&lt;ref&gt;{{Cite journal|title=Development and initial validation of the obsessive beliefs questionnaire and the interpretation of intrusions inventory|url=https://doi.org/10.1016/S0005-7967(00)00085-1|journal=Behaviour Research and Therapy|volume=39|issue=8|pages=987–1006|doi=10.1016/s0005-7967(00)00085-1}}&lt;/ref&gt; 

=== Interpreting obsessive compulsive disorder screening measure scores ===

'''Overview'''

The purpose of this subsection is to use Bayesian probability theory in order to accurately predict the diagnosis of obsessive compulsive disorder, given base diagnosis rate in the region and likelihood ratios in diagnostic likelihood ratios.

'''Area under curve (AUC)'''

The area under the curve (AUC, or AUROC) is equal to the probability that a classifier will rank a randomly chosen positive diagnosis of obsessive compulsive disorder higher than a randomly chosen negative diagnosis of obsessive compulsive disorder.

'''Likelihood ratios'''

Likelihood ratios (also known as likelihood ratios in diagnostic testing) are the proportion of cases with the diagnosis scoring in a given range divided by the proportion of the cases without the diagnosis scoring in the same range&lt;ref name=&quot;EAYfuturedirections&quot;/&gt; &lt;ref name=&quot;strauss2011&quot;/&gt;. The table below shows area under the curve (AUCs) and likelihood ratios in diagnostic testing for potential screening measures for obsessive compulsive disorder. It should be noted that all studies used some version of a K-SADS interview by a trained rater, combined with review by a clinician to establish consensus.

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Likelihood Ratio
! Comments
|-
| Larger than 10, smaller than 0.10
| Frequently clinically decisive
|-
| Ranging from 5 to 10, 0.20
| Helpful in clinical diagnosis
|-
|Between 2.0 and 0.5
| Rarely result in clinically meaningful changes of formulation
|-
| Around 1.0
| Test result did not change clinical impressions at all
|}

'''&quot;LR+&quot;''' refers to the change in likelihood ratio associated with a positive test score, and '''&quot;LR-&quot;''' is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all&lt;ref name=&quot;EAYfuturedirections&quot; /&gt;. On the other hand, likelihood ratios larger than 10 or smaller than 0.10 are frequently clinically decisive, 5 or 0.20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinical significance|clinically meaningful changes of formulation. &lt;ref name= &quot;sackett&quot;/&gt;

=== Psychometric properties of screening instruments for OCD ===

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Screening Measure (Primary Reference)
! Area Under curve (AUC) and Sample Size
! LR+ (Score)
! LR-
! Clinical generalizability
!Download Link
|-
| Y-BOCS-SR&lt;ref name=&quot;SteketeeEtAl1996&quot;/&gt;
| 0.75
(N=162)
| 5.50
(7)
| 0.50
| Moderate: OCD among pregnant and postpartum women
|[http://www.seinstitute.com/wp-content/uploads/2015/08/YBOC-Symptom-Checklist.pdf Y-BOCS-SR]
|-
| OCI-R Total&lt;ref name=&quot;FoaEtAl2002&quot;/&gt;
| 0.81
(N=322)
| 3.66
(14)
| 0.44
| High: OCD (n=167) versus other anxiety disorders (n=155) at outpatient anxiety clinic
|[https://serene.me.uk/tests/oci.pdf OCI-R Total]
|-
| OCI-R Total&lt;ref name=&quot;FoaEtAl2002&quot;/&gt;
| 0.82
(N=458)
| 2.98
(18)
| 0.36
| High: OCD (n=215) versus other anxiety disorders (n=243) at outpatient anxiety clinic
|[https://serene.me.uk/tests/oci.pdf OCI-R Total]
|-
| Dimensional Obsessive-Compulsive Scale&lt;ref name=&quot;AbramowitzEtAl2010&quot;/&gt;
| 0.77
(N=513)
| 2.33
(21)
| 0.43
| High: OCD (n=315) versus other anxiety disorders (n=198) at outpatient clinics across the U.S.
|[https://www.unc.edu/~jonabram/DOCS.pdf DOCS]
|-
|Brown Assessment of Beliefs Scale&lt;ref name=&quot;:1&quot; /&gt;
|(N=50)
|
|
|
|[http://www.veale.co.uk/wp-content/uploads/2010/11/BABS_revised_501.pdf BABS]
|-
|}

==[[Evidence based assessment/Process phase|'''Process phase''']]==

=== Treatments ===

==== Cognitive behavioral therapy (CBT) and exposure and response prevention (ERP) ====

*Behavior therapy, specifically ERP, has been established as the treatment of choice for OCD &lt;ref&gt;Whittal M.L., McLean P.D., Söchting I., Koch W.J., Taylor S., Anderson K., Paterson R.'''OCD treatment outcome using behavioral and cognitive approaches'''

Paper presented at the meeting of the Association for Advancement of Behavior Therapy, Miami Beach, FL (1997)&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Foa|first=Edna B.|last2=Kozak|first2=Michael J.|title=Beyond the efficacy ceiling? Cognitive behavior therapy in search of theory|url=https://doi.org/10.1016/S0005-7894(97)80019-6|journal=Behavior Therapy|volume=28|issue=4|pages=601–611|doi=10.1016/s0005-7894(97)80019-6}}&lt;/ref&gt;.
*Therapy incorporates ERP and emphasizes cognitive change.
**Therapist will help individual identify anxiety-provoking thoughts and situations.
**Therapist will develop a treatment plan and idiographic “fear hierarchy.”
**Individuals will learn to encounter situations that invoke anxiety without engaging in rituals used to dispel anxiety (ERP).
**Exposures will be done gradually at a pace that is comfortable for the client.
**Therapy will include homework assignments and is designed to offer lifelong skills.
*Therapy includes verbal techniques such as psychoeducation and cognitive restructuring.
*Manuals for reference:
**The therapist guide: Mastery of Obsessive-Compulsive Disorder: A Cognitive Behavioral Approach&lt;ref&gt;E. Foa, M. Kozak Mastery of obsessive–compulsive disorder: A cognitive-behavioral approach Graywind Publications (1997)&lt;/ref&gt; 
**Cognitive Therapy of Obsessive-Compulsive Disorder: A Guide for Professionals (Wilhelm &amp; Steketee)
**Obsessive Compulsive Disorder: Advances in Psychotherapy &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/70659789|title=Obsessive compulsive disorder|last=S.|first=Abramowitz, Jonathan|date=2006|publisher=Hogrefe &amp; Huber Publishers|isbn=9780889373167|location=Cambridge, MA|oclc=70659789}}&lt;/ref&gt;
*Treatment alliance is a predictor of subsequent change in OCD symptoms&lt;ref&gt;Keeley, M. L., Geffken, G. R., Ricketts, E., McNamara, J. P., &amp; Storch, E. A. (2011). The therapeutic alliance in the cognitive behavioral treatment of pediatric obsessive–compulsive disorder. ''Journal of Anxiety Disorders'', ''25''(7), 855-863.&lt;/ref&gt;. The therapist should provide a “validating and
:encouraging” environment so that clients can tolerate the emotional arousal associated with exposures.

==== Medication ====

*Selective serotonin reuptake inhibitors (SSRIs) are commonly used to treat OCD.
*These antidepressants include:
**[[wikipedia:Fluvoxamine|fluvoxamine]]  
**[[wikipedia:Fluoxetine|fluoxetine]]
**[[wikipedia:Sertraline|sertraline]]  
**[[wikipedia:Paroxetine|paroxetine]] 
**[[wikipedia:Citalopram|citalopram]]  
**[[wikipedia:Clomipramine|clomipramine]] 
**[[wikipedia:Escitalopram|escitalopram]]  
**[[wikipedia:Venlafaxine|venlafaxine]] 
*High doses (relative to doses prescribed for depression) are needed for individuals with OCD.

=== Process and outcome measures ===

==== Clinically significant change benchmarks with common instruments and mood rating scales ====

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
| rowspan=1&quot; style=&quot;text-align:center;font-size:130%;&quot; | &lt;b&gt; Measure&lt;/b&gt;
| style=&quot;text-align:center;font-size:130%;&quot; | &lt;b&gt; Subscale&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; | &lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; | &lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| colspan=&quot;8&quot;  span style=&quot;font-size:110%; text-align:center;&quot; | &lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| colspan=&quot;2&quot; |
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-

| rowspan=&quot;3&quot; style=&quot;text-align:center;&quot; | &lt;b&gt; Yale-Brown Obsessive Compulsive Scale (Y-BOCS-SR)&lt;/b&gt;
| style=&quot;text-align:right;&quot; | &lt;i&gt; Total&lt;/i&gt;
| style=&quot;text-align:center;&quot;| 10.6
| style=&quot;text-align:center;&quot;| 14.4
| style=&quot;text-align:center;&quot;| 12.6
| style=&quot;text-align:center;&quot;| 4.7
| style=&quot;text-align:center;&quot;| 3.9
| style=&quot;text-align:center;&quot;| 2.4
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Obsessions&lt;/i&gt;
| style=&quot;text-align:center;&quot;| 6.6
| style=&quot;text-align:center;&quot;| 7.6
| style=&quot;text-align:center;&quot;| 7.0
| style=&quot;text-align:center;&quot;| 2.5
| style=&quot;text-align:center;&quot;| 2.1
| style=&quot;text-align:center;&quot;| 1.3
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Compulsions&lt;/i&gt;
| style=&quot;text-align:center;&quot;| 3.5
| style=&quot;text-align:center;&quot;| 8.2
| style=&quot;text-align:center;&quot;| 6.1
| style=&quot;text-align:center;&quot;| 3.6
| style=&quot;text-align:center;&quot;| 3.0
| style=&quot;text-align:center;&quot;| 1.8
|-

| rowspan=&quot;7&quot; style=&quot;text-align:center;&quot; | &lt;b&gt; Obsessive-Compulsive Inventory – Revised (OCI-R)&lt;/b&gt;
| style=&quot;text-align:right;&quot; | &lt;i&gt; Total&lt;/i&gt;
| style=&quot;text-align:center;&quot;| 1.0
| style=&quot;text-align:center;&quot;| 41.0
| style=&quot;text-align:center;&quot;| 23.0
| style=&quot;text-align:center;&quot;| 14.8
| style=&quot;text-align:center;&quot;| 12.5
| style=&quot;text-align:center;&quot;| 7.6
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Washing&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 7.4
| style=&quot;text-align:center;&quot;| 3.1
| style=&quot;text-align:center;&quot;| 3.4
| style=&quot;text-align:center;&quot;| 2.9
| style=&quot;text-align:center;&quot;| 1.7
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Checking&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 8.0
| style=&quot;text-align:center;&quot;| 3.7
| style=&quot;text-align:center;&quot;| 3.0
| style=&quot;text-align:center;&quot;| 2.5
| style=&quot;text-align:center;&quot;| 1.5
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Ordering&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 10.5
| style=&quot;text-align:center;&quot;| 4.6
| style=&quot;text-align:center;&quot;| 3.1
| style=&quot;text-align:center;&quot;| 2.6
| style=&quot;text-align:center;&quot;| 1.6
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Obsessing&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 8.3
| style=&quot;text-align:center;&quot;| 4.7
| style=&quot;text-align:center;&quot;| 3.8
| style=&quot;text-align:center;&quot;| 3.2
| style=&quot;text-align:center;&quot;| 1.9
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Hoarding&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 9.8
| style=&quot;text-align:center;&quot;| 4.1
| style=&quot;text-align:center;&quot;| 2.8
| style=&quot;text-align:center;&quot;| 2.4
| style=&quot;text-align:center;&quot;| 1.4
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Neutralizing&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 6.2
| style=&quot;text-align:center;&quot;| 2.3
| style=&quot;text-align:center;&quot;| 3.0
| style=&quot;text-align:center;&quot;| 2.5
| style=&quot;text-align:center;&quot;| 1.5
|-

| rowspan=&quot;7&quot; style=&quot;text-align:center;&quot; | &lt;b&gt; Dimensional Obsessive Compulsive Scale (DOCS)&lt;/b&gt;
| style=&quot;text-align:right;&quot; | &lt;i&gt; Total&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 31.7
| style=&quot;text-align:center;&quot;| 19.0
| style=&quot;text-align:center;&quot;| 10.3
| style=&quot;text-align:center;&quot;| 8.7
| style=&quot;text-align:center;&quot;| 5.3
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Contamination&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 7.8
| style=&quot;text-align:center;&quot;| 3.4
| style=&quot;text-align:center;&quot;| 2.4
| style=&quot;text-align:center;&quot;| 2.0
| style=&quot;text-align:center;&quot;| 1.2
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Responsibility for Harm&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 8.7
| style=&quot;text-align:center;&quot;| 4.4
| style=&quot;text-align:center;&quot;| 2.4
| style=&quot;text-align:center;&quot;| 2.0
| style=&quot;text-align:center;&quot;| 1.2
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Unacceptable Thoughts&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 9.6
| style=&quot;text-align:center;&quot;| 5.4
| style=&quot;text-align:center;&quot;| 2.5
| style=&quot;text-align:center;&quot;| 2.1
| style=&quot;text-align:center;&quot;| 1.3
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Symmetry&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 7.9
| style=&quot;text-align:center;&quot;| 3.6
| style=&quot;text-align:center;&quot;| 2.2
| style=&quot;text-align:center;&quot;| 1.8
| style=&quot;text-align:center;&quot;| 1.1
|-
|}

'''Note:''' “A” = Away from the clinical range – moving at least 2 standard deviations away from clinical mean; “B” = Back into the nonclinical range – moving within 2 standard deviations of the nonclinical mean; “C” = Closer to the nonclinical than clinical mean – crossing the weighted average of the two groups.

=== Process measures ===
*Quality of Life
**Sheehan Disability Scale&lt;ref&gt;Sheehan DV, Harnett-Sheehan K, Raj BA. 1996. The measurement of disability. Int Clin Psychopharmacol 11(Suppl 3): 89–95.&lt;/ref&gt;
**[http://www.jstor.org/stable/pdf/3765819.pdf?refreqid=excelsior%3A9dcc715f829676edec9bc2c7be1478fa Medical Outcomes Study (MOS) 36-Item Short Form (SF-36) Health Survey]&lt;ref&gt;McHorney, C., Ware, J., &amp; Raczek, A. (1993). The MOS 36-Item Short-Form Health Survey (SF-36): II. Psychometric and Clinical Tests of Validity in Measuring Physical and Mental Health Constructs. ''Medical Care,'' ''31''(3), 247-263. Retrieved from &lt;nowiki&gt;http://www.jstor.org/stable/3765819&lt;/nowiki&gt;&lt;/ref&gt;
*Compulsions scale of YBOCS
*SUDS Ratings

== External Links == 
*[https://sccap53.org Society of Clinical Child and Adolescent Psychology]
*[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/self-injurious-thoughts-and-behaviors/ EffectiveChildTherapy.Org information on rule-breaking, defiance, and acting out]
*For information on conducting Exposure Therapy for anxiety disordered youth, see [https://www.bravepracticeforkids.com/ www.BravePracticeForKids.com]

== References ==
{{collapse top|Click here for references}}
{{Reflist|2|refs=

&lt;ref name=&quot;EAYfuturedirections&quot;&gt; Youngstrom, E. A. (2013). Future directions in psychological assessment: Combining evidence-based medicine innovations with psychology's historical strengths to enhance utility. Journal of Clinical Child and Adolescent Psychology, 42(1), 139-159. &lt;/ref&gt;

&lt;ref name=&quot;strauss2011&quot;&gt; Strauss, S. E., Glasziou, P., Richardson, W. S., &amp; Haynes, R. B. (2011). Evidence-based medicine: How to practice and teach EBM (4th ed.). New York, NY: Churchill Livingstone. &lt;/ref&gt;

&lt;ref name=&quot;sackett&quot;&gt; Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., &amp; Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone. &lt;/ref&gt;

&lt;ref name=&quot;RuscioEtAl2010&quot;&gt;{{cite journal|last1=Ruscio|first1=AM|last2=Stein|first2=DJ|last3=Chiu|first3=WT|last4=Kessler|first4=RC|title=The epidemiology of obsessive-compulsive disorder in the National Comorbidity Survey Replication.|journal=Molecular psychiatry|date=January 2010|volume=15|issue=1|pages=53-63|pmid=18725912}}&lt;/ref&gt;

&lt;ref name=&quot;KarnoEtAl1988&quot;&gt;{{cite journal|last1=Karno|first1=M|last2=Golding|first2=JM|last3=Sorenson|first3=SB|last4=Burnam|first4=MA|title=The epidemiology of obsessive-compulsive disorder in five US communities.|journal=Archives of general psychiatry|date=December 1988|volume=45|issue=12|pages=1094-9|pmid=3264144}}&lt;/ref&gt;

&lt;ref name=&quot;MohammadiEtAl2004&quot;&gt;{{cite journal|last1=Mohammadi|first1=MR|last2=Ghanizadeh|first2=A|last3=Rahgozar|first3=M|last4=Noorbala|first4=AA|last5=Davidian|first5=H|last6=Afzali|first6=HM|last7=Naghavi|first7=HR|last8=Yazdi|first8=SA|last9=Saberi|first9=SM|last10=Mesgarpour|first10=B|last11=Akhondzadeh|first11=S|last12=Alaghebandrad|first12=J|last13=Tehranidoost|first13=M|title=Prevalence of obsessive-compulsive disorder in Iran.|journal=BMC psychiatry|date=14 February 2004|volume=4|pages=2|pmid=15018627}}&lt;/ref&gt;

&lt;ref name=&quot;HimleEtAl2008&quot;&gt;{{cite journal|last1=Himle|first1=JA|last2=Muroff|first2=JR|last3=Taylor|first3=RJ|last4=Baser|first4=RE|last5=Abelson|first5=JM|last6=Hanna|first6=GL|last7=Abelson|first7=JL|last8=Jackson|first8=JS|title=Obsessive-compulsive disorder among African Americans and blacks of Caribbean descent: results from the National Survey of American Life.|journal=Depression and anxiety|date=2008|volume=25|issue=12|pages=993-1005|pmid=18833577}}&lt;/ref&gt;

&lt;ref name=&quot;SubramaniamEtAl2012&quot;&gt;{{cite journal|last1=Subramaniam|first1=M|last2=Abdin|first2=E|last3=Vaingankar|first3=JA|last4=Chong|first4=SA|title=Obsessive--compulsive disorder: prevalence, correlates, help-seeking and quality of life in a multiracial Asian population.|journal=Social psychiatry and psychiatric epidemiology|date=December 2012|volume=47|issue=12|pages=2035-43|pmid=22526825}}&lt;/ref&gt;

&lt;ref name=&quot;LordEtAl2011&quot;&gt;{{cite journal|last1=Lord|first1=C|last2=Hall|first2=G|last3=Soares|first3=CN|last4=Steiner|first4=M|title=Physiological stress response in postpartum women with obsessive-compulsive disorder: A pilot study.|journal=Psychoneuroendocrinology|date=January 2011|volume=36|issue=1|pages=133-8|pmid=20537805}}&lt;/ref&gt;

&lt;ref name=&quot;FoaEtAl2002&quot;&gt;{{cite journal|last1=Foa|first1=EB|last2=Huppert|first2=JD|last3=Leiberg|first3=S|last4=Langner|first4=R|last5=Kichic|first5=R|last6=Hajcak|first6=G|last7=Salkovskis|first7=PM|title=The Obsessive-Compulsive Inventory: development and validation of a short version.|journal=Psychological assessment|date=December 2002|volume=14|issue=4|pages=485-96|pmid=12501574}}&lt;/ref&gt;

&lt;ref name=&quot;SteketeeEtAl1996&quot;&gt;{{cite journal|last1=Steketee|first1=G|last2=Frost|first2=R|last3=Bogart|first3=K|title=The Yale-Brown Obsessive Compulsive Scale: interview versus self-report.|journal=Behaviour research and therapy|date=August 1996|volume=34|issue=8|pages=675-84|pmid=8870295}}&lt;/ref&gt;

&lt;ref name=&quot;AbramowitzDeacon2005&quot;&gt;{{cite journal|last1=Abramowitz|first1=Jonathan S.|last2=Deacon|first2=Brett J.|title=Psychometric properties and construct validity of the Obsessive–Compulsive Inventory—Revised: Replication and extension with a clinical sample|journal=Journal of Anxiety Disorders|date=January 2006|volume=20|issue=8|pages=1016–1035|doi=10.1016/j.janxdis.2006.03.001}}&lt;/ref&gt;

&lt;ref name=&quot;AbramowitzEtAl2010&quot;&gt;{{cite journal|last1=Abramowitz|first1=JS|last2=Deacon|first2=BJ|last3=Olatunji|first3=BO|last4=Wheaton|first4=MG|last5=Berman|first5=NC|last6=Losardo|first6=D|last7=Timpano|first7=KR|last8=McGrath|first8=PB|last9=Riemann|first9=BC|last10=Adams|first10=T|last11=Björgvinsson|first11=T|last12=Storch|first12=EA|last13=Hale|first13=LR|title=Assessment of obsessive-compulsive symptom dimensions: development and evaluation of the Dimensional Obsessive-Compulsive Scale.|journal=Psychological assessment|date=March 2010|volume=22|issue=1|pages=180-98|pmid=20230164}}&lt;/ref&gt;

}}
{{collapse bottom|Click here for references}}
[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]</text>
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      <parentid>2411972</parentid>
      <timestamp>2022-08-04T23:02:08Z</timestamp>
      <contributor>
        <username>Maddiegray11</username>
        <id>2936309</id>
      </contributor>
      <comment>/* Screening and diagnostic instruments for OCD */ removed scid and diagnostic instrument label</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="28775" xml:space="preserve">&lt;noinclude&gt;{{Helping Give Away Psychological Science Banner}}&lt;/noinclude&gt;

{{medical disclaimer}}
{{:{{BASEPAGENAME}}/Sidebar}}

==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
For background information on what assessment portfolios are, click the link in the heading above.

==[[Evidence based assessment/Preparation phase|'''Preparation phase''']]==

=== Diagnostic criteria for ______ disorder ===
{{blockquotetop}}'''ICD-11 Criteria''' &lt;ref&gt;{{Cite web|url=http://apps.who.int/classifications/icd10/browse/2016/en#/F40-F48|title=ICD-10 Version:2016|website=apps.who.int|language=en|accessdate=2018-03-01}}&lt;/ref&gt;
Obsessive-Compulsive Disorder is characterized by the presence of persistent obsessions or compulsions, or most commonly both. Obsessions are repetitive and persistent thoughts, images, or impulses/urges that are intrusive, unwanted, and are commonly associated with anxiety. The individual attempts to ignore or suppress obsessions or to neutralize them by performing compulsions. Compulsions are repetitive behaviors including repetitive mental acts that the individual feels driven to perform in response to an obsession, according to rigid rules, or to achieve a sense of ‘completeness’. In order for obsessive-compulsive disorder to be diagnosed, obsessions and compulsions must be time consuming (e.g., taking more than an hour per day), and result in significant distress or significant impairment in personal, family, social, educational, occupational or other important areas of functioning.

Inclusions
*anankastic neurosis
*obsessive-compulsive neurosis

Exclusions
*obsessive compulsive behaviour (MB23.4)

{{blockquotebottom}}

=== Demographics ===
This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of OCD that they are likely to see in their clinical practice.
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Demography
! Setting
! Base Rate
! Diagnostic Method
|-
|National (U.S.) adult 
sample (n=2073) 
|National Comorbidity Survey Replication&lt;ref name=&quot;RuscioEtAl2010&quot;/&gt;
|2.3%
|World Health Organization Composite 
International Diagnostic Interview (CIDI 3.0)
|-
|U.S. household sample 
(n=18572) 
|Epidemiological Catchment Area (ECA) Program&lt;ref name=&quot;KarnoEtAl1988&quot;/&gt;
|1.9-3.3%
|Diagnostic Interview Schedule (DIS)
|-
|Iranian adults
(n=25180) 
|Iranian population-based study&lt;ref name=&quot;MohammadiEtAl2004&quot;/&gt;
|1.8%
|DIS
|-
|NSAL adult study
(n=5191) 
|African-American and Caribbean Households (U.S.)&lt;ref name=&quot;HimleEtAl2008&quot;/&gt;
|1.6%
|CIDI Short Form
|-
|Epidemiological sample 
(n=6616) 
|Singapore Mental Health Study&lt;ref name=&quot;SubramaniamEtAl2012&quot;/&gt;
|3.0%
|CIDI 3.0
|}

'''Search terms:'''

[obsessive compulsive disorder OR ocd] AND [prevalence OR incidence] in PsycInfo and PubMed

[obsessive compulsive disorder OR ocd] AND [epidemiology] in PsycInfo and PubMed

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==

=== Diagnosis ===

{{blockquotetop}}'''ICD-11 Criteria''' &lt;ref&gt;{{Cite web|url=http://apps.who.int/classifications/icd10/browse/2016/en#/F40-F48|title=ICD-10 Version:2016|website=apps.who.int|language=en|accessdate=2018-03-01}}&lt;/ref&gt;
Obsessive-Compulsive Disorder is characterized by the presence of persistent obsessions or compulsions, or most commonly both. Obsessions are repetitive and persistent thoughts, images, or impulses/urges that are intrusive, unwanted, and are commonly associated with anxiety. The individual attempts to ignore or suppress obsessions or to neutralize them by performing compulsions. Compulsions are repetitive behaviors including repetitive mental acts that the individual feels driven to perform in response to an obsession, according to rigid rules, or to achieve a sense of ‘completeness’. In order for obsessive-compulsive disorder to be diagnosed, obsessions and compulsions must be time consuming (e.g., taking more than an hour per day), and result in significant distress or significant impairment in personal, family, social, educational, occupational or other important areas of functioning.

Inclusions
*anankastic neurosis
*obsessive-compulsive neurosis

Exclusions
*obsessive compulsive behaviour (MB23.4)

{{blockquotebottom}}

=== Screening instruments for OCD ===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Downloads
|-
| Anxiety Disorders Interview Schedule&lt;ref&gt;Brown, T.A., Di Nardo, P.A., Barlow, D.H., 1994. Anxiety Disorders Interview Schedule for DSM-IV (ADIS-IV-L). Psychological Corporation, San Antonio, TX.&lt;/ref&gt; 
| 
| 
| 
| E
| G
| E
| E
| X
|
|-
|[https://mfr.osf.io/render?url=https://osf.io/mnzy2/?action=download%26mode=render Yale-Brown Obsessive Compulsive Scale Symptom Checklist]&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Steketee|first=G|title=The Yale-Brown Obsessive Compulsive Scale: Interview versus self-report|url=http://linkinghub.elsevier.com/retrieve/pii/0005796796000368|journal=Behaviour Research and Therapy|volume=34|issue=8|pages=675–684|doi=10.1016/0005-7967(96)00036-8}}&lt;/ref&gt;
| 
| 
| 
| NA
| NA
| E
| G
| X
|
|-
| [https://search.proquest.com/docview/220481418/abstract/4AF689339CB14A0APQ/1?accountid=14244 Brown Assessment of Beliefs Scale]&lt;ref name=&quot;:1&quot;&gt;Eisen, J. L., Phillips, K. A., Baer, L., Beer, D. A., &amp; al, e. (1998). The brown assessment of beliefs scale: Reliability and validity. ''The American Journal of Psychiatry, 155''(1), 102-8. Retrieved from &lt;nowiki&gt;http://libproxy.lib.unc.edu/login?url=https://search.proquest.com/docview/220481418?accountid=14244&lt;/nowiki&gt;&lt;/ref&gt;
| 
| 
| 
| G
| A
| G
| G
| X
|
|-
|Revised Children’s Anxiety and Depression Scale (RCADS)
|Questionnaire (Child)
|6-18
|12 minutes
|G&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Chorpita|first=Bruce F.|last2=Moffitt|first2=Catherine E.|last3=Gray|first3=Jennifer|date=2005-03|title=Psychometric properties of the Revised Child Anxiety and Depression Scale in a clinical sample|url=http://dx.doi.org/10.1016/j.brat.2004.02.004|journal=Behaviour Research and Therapy|volume=43|issue=3|pages=309–322|doi=10.1016/j.brat.2004.02.004|issn=0005-7967}}&lt;/ref&gt;
|G&lt;ref&gt;{{Cite journal|last=Chorpita|first=Bruce F|last2=Yim|first2=Letitia|last3=Moffitt|first3=Catherine|last4=Umemoto|first4=Lori A|last5=Francis|first5=Sarah E|date=2000-08|title=Assessment of symptoms of DSM-IV anxiety and depression in children: a revised child anxiety and depression scale|url=http://dx.doi.org/10.1016/s0005-7967(99)00130-8|journal=Behaviour Research and Therapy|volume=38|issue=8|pages=835–855|doi=10.1016/s0005-7967(99)00130-8|issn=0005-7967}}&lt;/ref&gt;
|G&lt;ref name=&quot;:2&quot; /&gt;
|
|
|
*[http://www.corc.uk.net/outcome-experience-measures/revised-childrens-anxiety-and-depression-scale-and-subscales/ RCADS homepage]

'''PDFs for RCADS'''

*[https://mfr.osf.io/render?url=https://osf.io/s3fu2/?action=download%26mode=render RCADS Child Self-reported (8-18 years)]
*[https://mfr.osf.io/render?url=https://osf.io/fp9mk/?action=download%26mode=render RCADS Parent-reported]
*[https://mfr.osf.io/render?url=https://osf.io/vy7ta/?action=download%26mode=render Child Scoring Aid]
*[https://mfr.osf.io/render?url=https://osf.io/t4bz6/?action=download%26mode=render Parent Scoring Aid]

'''Subscales'''

*[https://mfr.osf.io/render?url=https://osf.io/ca5hk/?action=download%26mode=render OCD Self-reported]

* [https://mfr.osf.io/render?url=https://osf.io/7xkbn/?action=download%26mode=render OCD Parent-reported]

'''Translations'''

'''[https://mfr.osf.io/render?url=https://osf.io/qsjh9/?action=download%26mode=render User Guide]'''

*
|}

'''Note:''' L = Less than adequate; A = Adequate; G = Good; E = Excellent; U = Unavailable; NA = Not applicable

*[https://mfr.osf.io/render?url=https://osf.io/tn2vg/?action=download%26mode=render Yale-Brown Obsessive Compulsive Scale (Y-BOC)S]&lt;ref&gt;{{Cite journal|last=Goodman|first=Wayne K.|date=1989-11-01|title=The Yale-Brown Obsessive Compulsive Scale|url=http://archpsyc.jamanetwork.com/article.aspx?doi=10.1001/archpsyc.1989.01810110048007|journal=Archives of General Psychiatry|language=en|volume=46|issue=11|doi=10.1001/archpsyc.1989.01810110048007|issn=0003-990X}}&lt;/ref&gt;
**With Symptom Checklist (Y-BOCS-SC) or self-report (Y-BOCS-SR)&lt;ref name=&quot;:0&quot; /&gt;

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==

=== Recommended self-report questionnaires ===
*[http://psycnet.apa.org/record/2010-04450-019 Dimensional Obsessive Compulsive Scale (DOCS)]&lt;ref&gt;{{Cite journal|last=Abramowitz|first=Jonathan S.|last2=Deacon|first2=Brett J.|last3=Olatunji|first3=Bunmi O.|last4=Wheaton|first4=Michael G.|last5=Berman|first5=Noah C.|last6=Losardo|first6=Diane|last7=Timpano|first7=Kiara R.|last8=McGrath|first8=Patrick B.|last9=Riemann|first9=Bradley C.|title=Assessment of obsessive-compulsive symptom dimensions: Development and evaluation of the Dimensional Obsessive-Compulsive Scale.|url=http://dx.doi.org/10.1037/a0018260|journal=Psychological Assessment|language=en|volume=22|issue=1|pages=180–198|doi=10.1037/a0018260}}&lt;/ref&gt;
*[https://www.researchgate.net/profile/Robert_Langner/publication/10973110_The_Obsessive-Compulsive_Inventory_Development_and_validation_of_a_short_version/links/54be1e9a0cf218d4a16a4dc5/The-Obsessive-Compulsive-Inventory-Development-and-validation-of-a-short-version.pdf Obsessive Compulsive Inventory – Revised]&lt;ref&gt;Foa, E. B., Huppert, J. D., Leiberg, S., Langner, R., Kichic, R., Hajcak, G., &amp; Salkovskis, P. M. (2002). The Obsessive-Compulsive Inventory: development and validation of a short version. ''Psychological assessment'', ''14''(4), 485.&lt;/ref&gt;
*[https://ac.els-cdn.com/S0005796700000851/1-s2.0-S0005796700000851-main.pdf?_tid=f904c386-c354-424f-b339-6a53a87ec31a&amp;acdnat=1521761575_8fe72e0787f227a8ee9cdf3592136d64 Interpretation of Intrusions Inventory]&lt;ref&gt;{{Cite journal|title=Development and initial validation of the obsessive beliefs questionnaire and the interpretation of intrusions inventory|url=https://doi.org/10.1016/S0005-7967(00)00085-1|journal=Behaviour Research and Therapy|volume=39|issue=8|pages=987–1006|doi=10.1016/s0005-7967(00)00085-1}}&lt;/ref&gt; 

=== Interpreting obsessive compulsive disorder screening measure scores ===

'''Overview'''

The purpose of this subsection is to use Bayesian probability theory in order to accurately predict the diagnosis of obsessive compulsive disorder, given base diagnosis rate in the region and likelihood ratios in diagnostic likelihood ratios.

'''Area under curve (AUC)'''

The area under the curve (AUC, or AUROC) is equal to the probability that a classifier will rank a randomly chosen positive diagnosis of obsessive compulsive disorder higher than a randomly chosen negative diagnosis of obsessive compulsive disorder.

'''Likelihood ratios'''

Likelihood ratios (also known as likelihood ratios in diagnostic testing) are the proportion of cases with the diagnosis scoring in a given range divided by the proportion of the cases without the diagnosis scoring in the same range&lt;ref name=&quot;EAYfuturedirections&quot;/&gt; &lt;ref name=&quot;strauss2011&quot;/&gt;. The table below shows area under the curve (AUCs) and likelihood ratios in diagnostic testing for potential screening measures for obsessive compulsive disorder. It should be noted that all studies used some version of a K-SADS interview by a trained rater, combined with review by a clinician to establish consensus.

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Likelihood Ratio
! Comments
|-
| Larger than 10, smaller than 0.10
| Frequently clinically decisive
|-
| Ranging from 5 to 10, 0.20
| Helpful in clinical diagnosis
|-
|Between 2.0 and 0.5
| Rarely result in clinically meaningful changes of formulation
|-
| Around 1.0
| Test result did not change clinical impressions at all
|}

'''&quot;LR+&quot;''' refers to the change in likelihood ratio associated with a positive test score, and '''&quot;LR-&quot;''' is the likelihood ratio for a low score. Likelihood ratios of 1 indicate that the test result did not change impressions at all&lt;ref name=&quot;EAYfuturedirections&quot; /&gt;. On the other hand, likelihood ratios larger than 10 or smaller than 0.10 are frequently clinically decisive, 5 or 0.20 are helpful, and between 2.0 and .5 are small enough that they rarely result in clinical significance|clinically meaningful changes of formulation. &lt;ref name= &quot;sackett&quot;/&gt;

=== Psychometric properties of screening instruments for OCD ===

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Screening Measure (Primary Reference)
! Area Under curve (AUC) and Sample Size
! LR+ (Score)
! LR-
! Clinical generalizability
!Download Link
|-
| Y-BOCS-SR&lt;ref name=&quot;SteketeeEtAl1996&quot;/&gt;
| 0.75
(N=162)
| 5.50
(7)
| 0.50
| Moderate: OCD among pregnant and postpartum women
|[http://www.seinstitute.com/wp-content/uploads/2015/08/YBOC-Symptom-Checklist.pdf Y-BOCS-SR]
|-
| OCI-R Total&lt;ref name=&quot;FoaEtAl2002&quot;/&gt;
| 0.81
(N=322)
| 3.66
(14)
| 0.44
| High: OCD (n=167) versus other anxiety disorders (n=155) at outpatient anxiety clinic
|[https://serene.me.uk/tests/oci.pdf OCI-R Total]
|-
| OCI-R Total&lt;ref name=&quot;FoaEtAl2002&quot;/&gt;
| 0.82
(N=458)
| 2.98
(18)
| 0.36
| High: OCD (n=215) versus other anxiety disorders (n=243) at outpatient anxiety clinic
|[https://serene.me.uk/tests/oci.pdf OCI-R Total]
|-
| Dimensional Obsessive-Compulsive Scale&lt;ref name=&quot;AbramowitzEtAl2010&quot;/&gt;
| 0.77
(N=513)
| 2.33
(21)
| 0.43
| High: OCD (n=315) versus other anxiety disorders (n=198) at outpatient clinics across the U.S.
|[https://www.unc.edu/~jonabram/DOCS.pdf DOCS]
|-
|Brown Assessment of Beliefs Scale&lt;ref name=&quot;:1&quot; /&gt;
|(N=50)
|
|
|
|[http://www.veale.co.uk/wp-content/uploads/2010/11/BABS_revised_501.pdf BABS]
|-
|}

==[[Evidence based assessment/Process phase|'''Process phase''']]==

=== Treatments ===

==== Cognitive behavioral therapy (CBT) and exposure and response prevention (ERP) ====

*Behavior therapy, specifically ERP, has been established as the treatment of choice for OCD &lt;ref&gt;Whittal M.L., McLean P.D., Söchting I., Koch W.J., Taylor S., Anderson K., Paterson R.'''OCD treatment outcome using behavioral and cognitive approaches'''

Paper presented at the meeting of the Association for Advancement of Behavior Therapy, Miami Beach, FL (1997)&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Foa|first=Edna B.|last2=Kozak|first2=Michael J.|title=Beyond the efficacy ceiling? Cognitive behavior therapy in search of theory|url=https://doi.org/10.1016/S0005-7894(97)80019-6|journal=Behavior Therapy|volume=28|issue=4|pages=601–611|doi=10.1016/s0005-7894(97)80019-6}}&lt;/ref&gt;.
*Therapy incorporates ERP and emphasizes cognitive change.
**Therapist will help individual identify anxiety-provoking thoughts and situations.
**Therapist will develop a treatment plan and idiographic “fear hierarchy.”
**Individuals will learn to encounter situations that invoke anxiety without engaging in rituals used to dispel anxiety (ERP).
**Exposures will be done gradually at a pace that is comfortable for the client.
**Therapy will include homework assignments and is designed to offer lifelong skills.
*Therapy includes verbal techniques such as psychoeducation and cognitive restructuring.
*Manuals for reference:
**The therapist guide: Mastery of Obsessive-Compulsive Disorder: A Cognitive Behavioral Approach&lt;ref&gt;E. Foa, M. Kozak Mastery of obsessive–compulsive disorder: A cognitive-behavioral approach Graywind Publications (1997)&lt;/ref&gt; 
**Cognitive Therapy of Obsessive-Compulsive Disorder: A Guide for Professionals (Wilhelm &amp; Steketee)
**Obsessive Compulsive Disorder: Advances in Psychotherapy &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/70659789|title=Obsessive compulsive disorder|last=S.|first=Abramowitz, Jonathan|date=2006|publisher=Hogrefe &amp; Huber Publishers|isbn=9780889373167|location=Cambridge, MA|oclc=70659789}}&lt;/ref&gt;
*Treatment alliance is a predictor of subsequent change in OCD symptoms&lt;ref&gt;Keeley, M. L., Geffken, G. R., Ricketts, E., McNamara, J. P., &amp; Storch, E. A. (2011). The therapeutic alliance in the cognitive behavioral treatment of pediatric obsessive–compulsive disorder. ''Journal of Anxiety Disorders'', ''25''(7), 855-863.&lt;/ref&gt;. The therapist should provide a “validating and
:encouraging” environment so that clients can tolerate the emotional arousal associated with exposures.

==== Medication ====

*Selective serotonin reuptake inhibitors (SSRIs) are commonly used to treat OCD.
*These antidepressants include:
**[[wikipedia:Fluvoxamine|fluvoxamine]]  
**[[wikipedia:Fluoxetine|fluoxetine]]
**[[wikipedia:Sertraline|sertraline]]  
**[[wikipedia:Paroxetine|paroxetine]] 
**[[wikipedia:Citalopram|citalopram]]  
**[[wikipedia:Clomipramine|clomipramine]] 
**[[wikipedia:Escitalopram|escitalopram]]  
**[[wikipedia:Venlafaxine|venlafaxine]] 
*High doses (relative to doses prescribed for depression) are needed for individuals with OCD.

=== Process and outcome measures ===

==== Clinically significant change benchmarks with common instruments and mood rating scales ====

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
| rowspan=1&quot; style=&quot;text-align:center;font-size:130%;&quot; | &lt;b&gt; Measure&lt;/b&gt;
| style=&quot;text-align:center;font-size:130%;&quot; | &lt;b&gt; Subscale&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; | &lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; | &lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| colspan=&quot;8&quot;  span style=&quot;font-size:110%; text-align:center;&quot; | &lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| colspan=&quot;2&quot; |
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; | &lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-

| rowspan=&quot;3&quot; style=&quot;text-align:center;&quot; | &lt;b&gt; Yale-Brown Obsessive Compulsive Scale (Y-BOCS-SR)&lt;/b&gt;
| style=&quot;text-align:right;&quot; | &lt;i&gt; Total&lt;/i&gt;
| style=&quot;text-align:center;&quot;| 10.6
| style=&quot;text-align:center;&quot;| 14.4
| style=&quot;text-align:center;&quot;| 12.6
| style=&quot;text-align:center;&quot;| 4.7
| style=&quot;text-align:center;&quot;| 3.9
| style=&quot;text-align:center;&quot;| 2.4
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Obsessions&lt;/i&gt;
| style=&quot;text-align:center;&quot;| 6.6
| style=&quot;text-align:center;&quot;| 7.6
| style=&quot;text-align:center;&quot;| 7.0
| style=&quot;text-align:center;&quot;| 2.5
| style=&quot;text-align:center;&quot;| 2.1
| style=&quot;text-align:center;&quot;| 1.3
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Compulsions&lt;/i&gt;
| style=&quot;text-align:center;&quot;| 3.5
| style=&quot;text-align:center;&quot;| 8.2
| style=&quot;text-align:center;&quot;| 6.1
| style=&quot;text-align:center;&quot;| 3.6
| style=&quot;text-align:center;&quot;| 3.0
| style=&quot;text-align:center;&quot;| 1.8
|-

| rowspan=&quot;7&quot; style=&quot;text-align:center;&quot; | &lt;b&gt; Obsessive-Compulsive Inventory – Revised (OCI-R)&lt;/b&gt;
| style=&quot;text-align:right;&quot; | &lt;i&gt; Total&lt;/i&gt;
| style=&quot;text-align:center;&quot;| 1.0
| style=&quot;text-align:center;&quot;| 41.0
| style=&quot;text-align:center;&quot;| 23.0
| style=&quot;text-align:center;&quot;| 14.8
| style=&quot;text-align:center;&quot;| 12.5
| style=&quot;text-align:center;&quot;| 7.6
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Washing&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 7.4
| style=&quot;text-align:center;&quot;| 3.1
| style=&quot;text-align:center;&quot;| 3.4
| style=&quot;text-align:center;&quot;| 2.9
| style=&quot;text-align:center;&quot;| 1.7
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Checking&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 8.0
| style=&quot;text-align:center;&quot;| 3.7
| style=&quot;text-align:center;&quot;| 3.0
| style=&quot;text-align:center;&quot;| 2.5
| style=&quot;text-align:center;&quot;| 1.5
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Ordering&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 10.5
| style=&quot;text-align:center;&quot;| 4.6
| style=&quot;text-align:center;&quot;| 3.1
| style=&quot;text-align:center;&quot;| 2.6
| style=&quot;text-align:center;&quot;| 1.6
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Obsessing&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 8.3
| style=&quot;text-align:center;&quot;| 4.7
| style=&quot;text-align:center;&quot;| 3.8
| style=&quot;text-align:center;&quot;| 3.2
| style=&quot;text-align:center;&quot;| 1.9
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Hoarding&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 9.8
| style=&quot;text-align:center;&quot;| 4.1
| style=&quot;text-align:center;&quot;| 2.8
| style=&quot;text-align:center;&quot;| 2.4
| style=&quot;text-align:center;&quot;| 1.4
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Neutralizing&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 6.2
| style=&quot;text-align:center;&quot;| 2.3
| style=&quot;text-align:center;&quot;| 3.0
| style=&quot;text-align:center;&quot;| 2.5
| style=&quot;text-align:center;&quot;| 1.5
|-

| rowspan=&quot;7&quot; style=&quot;text-align:center;&quot; | &lt;b&gt; Dimensional Obsessive Compulsive Scale (DOCS)&lt;/b&gt;
| style=&quot;text-align:right;&quot; | &lt;i&gt; Total&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 31.7
| style=&quot;text-align:center;&quot;| 19.0
| style=&quot;text-align:center;&quot;| 10.3
| style=&quot;text-align:center;&quot;| 8.7
| style=&quot;text-align:center;&quot;| 5.3
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Contamination&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 7.8
| style=&quot;text-align:center;&quot;| 3.4
| style=&quot;text-align:center;&quot;| 2.4
| style=&quot;text-align:center;&quot;| 2.0
| style=&quot;text-align:center;&quot;| 1.2
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Responsibility for Harm&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 8.7
| style=&quot;text-align:center;&quot;| 4.4
| style=&quot;text-align:center;&quot;| 2.4
| style=&quot;text-align:center;&quot;| 2.0
| style=&quot;text-align:center;&quot;| 1.2
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Unacceptable Thoughts&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 9.6
| style=&quot;text-align:center;&quot;| 5.4
| style=&quot;text-align:center;&quot;| 2.5
| style=&quot;text-align:center;&quot;| 2.1
| style=&quot;text-align:center;&quot;| 1.3
|-
| style=&quot;text-align:right;&quot; | &lt;i&gt; Symmetry&lt;/i&gt;
| style=&quot;text-align:center;&quot;| n/a
| style=&quot;text-align:center;&quot;| 7.9
| style=&quot;text-align:center;&quot;| 3.6
| style=&quot;text-align:center;&quot;| 2.2
| style=&quot;text-align:center;&quot;| 1.8
| style=&quot;text-align:center;&quot;| 1.1
|-
|}

'''Note:''' “A” = Away from the clinical range – moving at least 2 standard deviations away from clinical mean; “B” = Back into the nonclinical range – moving within 2 standard deviations of the nonclinical mean; “C” = Closer to the nonclinical than clinical mean – crossing the weighted average of the two groups.

=== Process measures ===
*Quality of Life
**Sheehan Disability Scale&lt;ref&gt;Sheehan DV, Harnett-Sheehan K, Raj BA. 1996. The measurement of disability. Int Clin Psychopharmacol 11(Suppl 3): 89–95.&lt;/ref&gt;
**[http://www.jstor.org/stable/pdf/3765819.pdf?refreqid=excelsior%3A9dcc715f829676edec9bc2c7be1478fa Medical Outcomes Study (MOS) 36-Item Short Form (SF-36) Health Survey]&lt;ref&gt;McHorney, C., Ware, J., &amp; Raczek, A. (1993). The MOS 36-Item Short-Form Health Survey (SF-36): II. Psychometric and Clinical Tests of Validity in Measuring Physical and Mental Health Constructs. ''Medical Care,'' ''31''(3), 247-263. Retrieved from &lt;nowiki&gt;http://www.jstor.org/stable/3765819&lt;/nowiki&gt;&lt;/ref&gt;
*Compulsions scale of YBOCS
*SUDS Ratings

== External Links == 
*[https://sccap53.org Society of Clinical Child and Adolescent Psychology]
*[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/self-injurious-thoughts-and-behaviors/ EffectiveChildTherapy.Org information on rule-breaking, defiance, and acting out]
*For information on conducting Exposure Therapy for anxiety disordered youth, see [https://www.bravepracticeforkids.com/ www.BravePracticeForKids.com]

== References ==
{{collapse top|Click here for references}}
{{Reflist|2|refs=

&lt;ref name=&quot;EAYfuturedirections&quot;&gt; Youngstrom, E. A. (2013). Future directions in psychological assessment: Combining evidence-based medicine innovations with psychology's historical strengths to enhance utility. Journal of Clinical Child and Adolescent Psychology, 42(1), 139-159. &lt;/ref&gt;

&lt;ref name=&quot;strauss2011&quot;&gt; Strauss, S. E., Glasziou, P., Richardson, W. S., &amp; Haynes, R. B. (2011). Evidence-based medicine: How to practice and teach EBM (4th ed.). New York, NY: Churchill Livingstone. &lt;/ref&gt;

&lt;ref name=&quot;sackett&quot;&gt; Sackett, D. L., Straus, S. E., Richardson, W. S., Rosenberg, W., &amp; Haynes, R. B. (2000). Evidence-based medicine: How to practice and teach EBM. Edinburgh: Churchill Livingstone. &lt;/ref&gt;

&lt;ref name=&quot;RuscioEtAl2010&quot;&gt;{{cite journal|last1=Ruscio|first1=AM|last2=Stein|first2=DJ|last3=Chiu|first3=WT|last4=Kessler|first4=RC|title=The epidemiology of obsessive-compulsive disorder in the National Comorbidity Survey Replication.|journal=Molecular psychiatry|date=January 2010|volume=15|issue=1|pages=53-63|pmid=18725912}}&lt;/ref&gt;

&lt;ref name=&quot;KarnoEtAl1988&quot;&gt;{{cite journal|last1=Karno|first1=M|last2=Golding|first2=JM|last3=Sorenson|first3=SB|last4=Burnam|first4=MA|title=The epidemiology of obsessive-compulsive disorder in five US communities.|journal=Archives of general psychiatry|date=December 1988|volume=45|issue=12|pages=1094-9|pmid=3264144}}&lt;/ref&gt;

&lt;ref name=&quot;MohammadiEtAl2004&quot;&gt;{{cite journal|last1=Mohammadi|first1=MR|last2=Ghanizadeh|first2=A|last3=Rahgozar|first3=M|last4=Noorbala|first4=AA|last5=Davidian|first5=H|last6=Afzali|first6=HM|last7=Naghavi|first7=HR|last8=Yazdi|first8=SA|last9=Saberi|first9=SM|last10=Mesgarpour|first10=B|last11=Akhondzadeh|first11=S|last12=Alaghebandrad|first12=J|last13=Tehranidoost|first13=M|title=Prevalence of obsessive-compulsive disorder in Iran.|journal=BMC psychiatry|date=14 February 2004|volume=4|pages=2|pmid=15018627}}&lt;/ref&gt;

&lt;ref name=&quot;HimleEtAl2008&quot;&gt;{{cite journal|last1=Himle|first1=JA|last2=Muroff|first2=JR|last3=Taylor|first3=RJ|last4=Baser|first4=RE|last5=Abelson|first5=JM|last6=Hanna|first6=GL|last7=Abelson|first7=JL|last8=Jackson|first8=JS|title=Obsessive-compulsive disorder among African Americans and blacks of Caribbean descent: results from the National Survey of American Life.|journal=Depression and anxiety|date=2008|volume=25|issue=12|pages=993-1005|pmid=18833577}}&lt;/ref&gt;

&lt;ref name=&quot;SubramaniamEtAl2012&quot;&gt;{{cite journal|last1=Subramaniam|first1=M|last2=Abdin|first2=E|last3=Vaingankar|first3=JA|last4=Chong|first4=SA|title=Obsessive--compulsive disorder: prevalence, correlates, help-seeking and quality of life in a multiracial Asian population.|journal=Social psychiatry and psychiatric epidemiology|date=December 2012|volume=47|issue=12|pages=2035-43|pmid=22526825}}&lt;/ref&gt;

&lt;ref name=&quot;LordEtAl2011&quot;&gt;{{cite journal|last1=Lord|first1=C|last2=Hall|first2=G|last3=Soares|first3=CN|last4=Steiner|first4=M|title=Physiological stress response in postpartum women with obsessive-compulsive disorder: A pilot study.|journal=Psychoneuroendocrinology|date=January 2011|volume=36|issue=1|pages=133-8|pmid=20537805}}&lt;/ref&gt;

&lt;ref name=&quot;FoaEtAl2002&quot;&gt;{{cite journal|last1=Foa|first1=EB|last2=Huppert|first2=JD|last3=Leiberg|first3=S|last4=Langner|first4=R|last5=Kichic|first5=R|last6=Hajcak|first6=G|last7=Salkovskis|first7=PM|title=The Obsessive-Compulsive Inventory: development and validation of a short version.|journal=Psychological assessment|date=December 2002|volume=14|issue=4|pages=485-96|pmid=12501574}}&lt;/ref&gt;

&lt;ref name=&quot;SteketeeEtAl1996&quot;&gt;{{cite journal|last1=Steketee|first1=G|last2=Frost|first2=R|last3=Bogart|first3=K|title=The Yale-Brown Obsessive Compulsive Scale: interview versus self-report.|journal=Behaviour research and therapy|date=August 1996|volume=34|issue=8|pages=675-84|pmid=8870295}}&lt;/ref&gt;

&lt;ref name=&quot;AbramowitzDeacon2005&quot;&gt;{{cite journal|last1=Abramowitz|first1=Jonathan S.|last2=Deacon|first2=Brett J.|title=Psychometric properties and construct validity of the Obsessive–Compulsive Inventory—Revised: Replication and extension with a clinical sample|journal=Journal of Anxiety Disorders|date=January 2006|volume=20|issue=8|pages=1016–1035|doi=10.1016/j.janxdis.2006.03.001}}&lt;/ref&gt;

&lt;ref name=&quot;AbramowitzEtAl2010&quot;&gt;{{cite journal|last1=Abramowitz|first1=JS|last2=Deacon|first2=BJ|last3=Olatunji|first3=BO|last4=Wheaton|first4=MG|last5=Berman|first5=NC|last6=Losardo|first6=D|last7=Timpano|first7=KR|last8=McGrath|first8=PB|last9=Riemann|first9=BC|last10=Adams|first10=T|last11=Björgvinsson|first11=T|last12=Storch|first12=EA|last13=Hale|first13=LR|title=Assessment of obsessive-compulsive symptom dimensions: development and evaluation of the Dimensional Obsessive-Compulsive Scale.|journal=Psychological assessment|date=March 2010|volume=22|issue=1|pages=180-98|pmid=20230164}}&lt;/ref&gt;

}}
{{collapse bottom|Click here for references}}
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==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
* For background information on what assessment portfolios are, click the link in the heading above.
* Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Posttraumatic stress disorder (disorder portfolio)/extended version|here]].

== [[Evidence based assessment/Preparation phase|'''Preparation phase''']] ==
=== Diagnostic criteria for post traumatic stress disorder ===

===='''ICD-11 Diagnostic Criteria'''====
{{blockquotetop}}
===='''ICD-11 Criteria'''====
*Post-traumatic stress disorder (PTSD) is a syndrome that develops following exposure to an extremely threatening or horrific event or series of events that is characterized by all of the following: 
**1) re-experiencing the traumatic event or events in the present in the form of vivid intrusive memories, flashbacks, or nightmares, which are typically accompanied by strong and overwhelming emotions such as fear or horror and strong physical sensations, or feelings of being overwhelmed or immersed in the same intense emotions that were experienced during the traumatic event; 
**2) avoidance of thoughts and memories of the event or events, or avoidance of activities, situations, or people reminiscent of the event or events
**3) persistent perceptions of heightened current threat, for example as indicated by hypervigilance or an enhanced startle reaction to stimuli such as unexpected noises. 
*The symptoms must persist for at least several weeks and cause significant impairment in personal, family, social, educational, occupational or other important areas of functioning. 

Additionally, ICD 11 includes a category called &quot;Complex post-traumatic stress disorder,&quot; which is described as:
*A disorder that may develop following exposure to an event or series of events of an extremely threatening or horrific nature, most commonly prolonged or repetitive events from which escape is difficult or impossible (e.g., torture, slavery, genocide campaigns, prolonged domestic violence, repeated childhood sexual or physical abuse). 
*The disorder is characterized by the core symptoms of PTSD; that is, all diagnostic requirements for PTSD have been met at some point during the course of the disorder. 
*In addition, Complex PTSD is characterized by:
**1) severe and pervasive problems in affect regulation; 
**2) persistent beliefs about oneself as diminished, defeated or worthless, accompanied by deep and pervasive feelings of shame, guilt or failure related to the traumatic event
**3) persistent difficulties in sustaining relationships and in feeling close to others. The disturbance causes significant impairment in personal, family, social, educational, occupational or other important areas of functioning.
{{blockquotebottom}}

===='''Changes in DSM-5'''====
* The diagnostic criteria for post-traumatic stress disorder changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here] and [[w:DSM-5|here]].


=== Developmental sensitivities ===
{{collapse top| Click for more information}}
*Diagnostic thresholds have been lowered for children and adolescents to account for development.
*Separate and additional criteria have been added for children age 6 or younger.
*Child sexual abuse has been found to have a substantial effect on the development of PTSD.&lt;ref&gt;{{cite journal|last1=Paolucci|first1=Elizabeth ODDONE|last2=Genuis|first2=Mark L.|last3=Violato|first3=Claudio|title=A Meta-Analysis of the Published Research on the Effects of Child Sexual Abuse|journal=The Journal of Psychology|date=January 2001|volume=135|issue=1|pages=17–36|doi=10.1080/00223980109603677}}&lt;/ref&gt;
*Children with higher exposure to trauma, less social support, and other major life events are more likely to have continued PTSD symptoms 7 months after a trauma.&lt;ref name=&quot;:2&quot;&gt;{{cite journal |last1=La Greca|first1=A|last2=Silverman|first2=WK|last3=Vernberg|first3=EM|last4=Prinstein|first4=MJ|title=Symptoms of posttraumatic stress in children after Hurricane Andrew: a prospective study.|journal=Journal of consulting and clinical psychology|date=August 1996|volume=64|issue=4|pages=712-23|pmid=8803361}}&lt;/ref&gt; 10 months after a trauma, however, only experience of a major life event remained predictive of continuing PTSD symptoms.&lt;ref name=&quot;:2&quot; /&gt; 
*Lack of social support, specifically lack of support by a teacher, was predictive of higher PTSD symptoms among children who had gone through an environmental trauma.&lt;ref name=&quot;:2&quot; /&gt;  
*Children who reported using blame and anger as strategies for coping had higher levels of PTSD symptoms 10 months after a trauma.&lt;ref name=&quot;:2&quot; /&gt;  
*Negative affect in children before a traumatic event was predictive of development of PTSD symptoms after a traumatic event.&lt;ref&gt;{{cite journal|last1=Weems|first1=CF|last2=Pina|first2=AA|last3=Costa|first3=NM|last4=Watts|first4=SE|last5=Taylor|first5=LK|last6=Cannon|first6=MF|title=Predisaster trait anxiety and negative affect predict posttraumatic stress in youths after hurricane Katrina.|journal=Journal of consulting and clinical psychology|date=February 2007|volume=75|issue=1|pages=154-9|doi=10.1037/0022-006X.75.1.154|pmid=17295574}}&lt;/ref&gt;  

[[wikipedia:Posttraumatic_stress_disorder|Posttraumatic stress disorder]] is now more sensitive to development in that diagnostic thresholds have been lowered for children and adolescents. Furthermore, separate and additional criteria have been added for children age 6 years of age or younger.
{{collapse bottom}}

=== Base rates of PTSD in different clinical settings and populations ===
This section describes the demographic settings of the populations sampled, base rates of PTSD diagnoses, country/region sampled, and the diagnostic methods that were used. Using this information, clinicians will be able to anchor the most appropriate rate of PTSD that they are likely to see in their clinical practice.
* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']] 

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Setting
! Base Rate
! Demography
! Diagnostic Method
|-

| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Kessler|first=Ronald C.|last2=Berglund|first2=Patricia|last3=Demler|first3=Olga|last4=Jin|first4=Robert|last5=Merikangas|first5=Kathleen R.|last6=Walters|first6=Ellen E.|date=2005-06-01|title=Lifetime Prevalence and Age-of-Onset Distributions of DSM-IV Disorders in the National Comorbidity Survey Replication|url=http://archpsyc.jamanetwork.com/article.aspx?doi=10.1001/archpsyc.62.6.593|journal=Archives of General Psychiatry|language=en|volume=62|issue=6|doi=10.1001/archpsyc.62.6.593|issn=0003-990X}}&lt;/ref&gt;
| 6.8%
| United States, nationally representative, age 18 and older
| National Comorbidity Survey - Replication
|-

| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=de Vries|first=Giel-Jan|last2=Olff|first2=Miranda|date=2009-08-01|title=The lifetime prevalence of traumatic events and posttraumatic stress disorder in the Netherlands|url=http://onlinelibrary.wiley.com/doi/10.1002/jts.20429/abstract|journal=Journal of Traumatic Stress|language=en|volume=22|issue=4|pages=259–267|doi=10.1002/jts.20429|issn=1573-6598}}&lt;/ref&gt;
| 7.4%
| Netherlands, nationally representative, age 18-80
| Composite International Diagnostic Interview (CIDI)
|-

| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Pietrzak|first=Robert H.|last2=Goldstein|first2=Risë B.|last3=Southwick|first3=Steven M.|last4=Grant|first4=Bridget F.|title=Prevalence and Axis I comorbidity of full and partial posttraumatic stress disorder in the United States: Results from Wave 2 of the National Epidemiologic Survey on Alcohol and Related Conditions|url=http://linkinghub.elsevier.com/retrieve/pii/S0887618510002288|journal=Journal of Anxiety Disorders|volume=25|issue=3|pages=456–465|doi=10.1016/j.janxdis.2010.11.010}}&lt;/ref&gt;
| 6.4%
| United States, nationally representative, age 18 and older
| Wave 2 National Epidemiologic Survey on Alcohol and Related Conditions
|-

| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Bunting|first=Brendan P.|last2=Ferry|first2=Finola R.|last3=Murphy|first3=Samuel D.|last4=O'Neill|first4=Siobhan M.|last5=Bolton|first5=David|date=2013-02-01|title=Trauma Associated With Civil Conflict and Posttraumatic Stress Disorder: Evidence From the Northern Ireland Study of Health and Stress|url=http://onlinelibrary.wiley.com/doi/10.1002/jts.21766/abstract|journal=Journal of Traumatic Stress|language=en|volume=26|issue=1|pages=134–141|doi=10.1002/jts.21766|issn=1573-6598}}&lt;/ref&gt;
| 8.8%
| Northern Ireland, representative sample, age 18 and older
| Northern Ireland Study of Health and Stress
|-

| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Atwoli|first=Lukoye|last2=Stein|first2=Dan J.|last3=Williams|first3=David R.|last4=Mclaughlin|first4=Katie A.|last5=Petukhova|first5=Maria|last6=Kessler|first6=Ronald C.|last7=Koenen|first7=Karestan C.|date=2013-07-03|title=Trauma and posttraumatic stress disorder in South Africa: analysis from the South African Stress and Health Study|url=https://doi.org/10.1186/1471-244X-13-182|journal=BMC Psychiatry|volume=13|pages=182|doi=10.1186/1471-244x-13-182|issn=1471-244X}}&lt;/ref&gt;
| 2.3%
| South Africa, nationally representative sample, age 18 and older
| South African Stress and Health Study, using the Composite International Diagnostic Interview (CIDI)
|-

| U.S. Service Members&lt;ref&gt;{{Cite journal|last=Hoge|first=Charles W.|last2=Castro|first2=Carl A.|last3=Messer|first3=Stephen C.|last4=McGurk|first4=Dennis|last5=Cotting|first5=Dave I.|last6=Koffman|first6=Robert L.|date=2004-07-01|title=Combat Duty in Iraq and Afghanistan, Mental Health Problems, and Barriers to Care|url=http://dx.doi.org/10.1056/NEJMoa040603|journal=New England Journal of Medicine|volume=351|issue=1|pages=13–22|doi=10.1056/nejmoa040603|issn=0028-4793|pmid=15229303}}&lt;/ref&gt;
|11.5% - 19.5% ♦
| U.S. Army and Marine Soldiers Deployed to Iraq and Afghanistan
| PTSD Checklist
|-
|Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Merikangas|first=Kathleen Ries|last2=He|first2=Jian-ping|last3=Burstein|first3=Marcy|last4=Swanson|first4=Sonja A.|last5=Avenevoli|first5=Shelli|last6=Cui|first6=Lihong|last7=Benjet|first7=Corina|last8=Georgiades|first8=Katholiki|last9=Swendsen|first9=Joel|title=Lifetime Prevalence of Mental Disorders in U.S. Adolescents: Results from the National Comorbidity Survey Replication–Adolescent Supplement (NCS-A)|url=https://doi.org/10.1016/j.jaac.2010.05.017|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=49|issue=10|pages=980–989|doi=10.1016/j.jaac.2010.05.017}}&lt;/ref&gt;
|5.0%
|United States, nationally representative, ages 13-18
|National Comorbidity Survey Replication—Adolescent Supplement (NCS-A)
|}
♦ '''Note''': These rates were using broad PTSD Checklist scoring criteria of being scored positive if subjects reported at least one intrusion symptom, three avoidance symptoms, and two hyperarousal symptom that were categorized as at the moderate level. The 11.5% is for soldiers returning from deployment in Iraq, 19.5% is for soldiers returning from Afghanistan. Another common practice is to use a strict cutoff of 50 on the PCL, above which someone screens positive for PTSD. With this cutoff, rates are 6.2% and 12.9% for Service Members returned from Afghanistan and Iraqi, respectively.

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==

=== Psychometric properties of screening for PTSD ===
The following section contains a list of screening and diagnostic instruments for PTSD. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here.]]

* '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]'''''
{| class=&quot;wikitable sortable&quot;
|-
! Measure 
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Highly recommended
|-
|[https://www.ptsd.va.gov/professional/assessment/adult-sr/ptsd-checklist.asp PCL (PTSD Checklist for DSM-5)]
|Self-Report 
|adult
|5-10 minutes
|X
|-
|[https://www.ptsd.va.gov/professional/assessment/adult-int/caps.asp CAPS (Clinician Administered PTSD Scale)]
|Clinician Administered Interview 
|adult and child versions available 
|40-60 minutes 
|X
|-
|[https://www.appi.org/products/structured-clinical-interview-for-dsm-5-scid-5 SCID-IV (Structured Clinical Interview for DSM-IV)]
|Clinician Administered Interview
|adult
|1-2 hours 
|X
|-
|[https://www.reactionindex.com/index.php/ UCLA PTSD Reaction Index for DSM-5]
|Clinician Administered
|child, school age, adolescents
|
|Yes
|-
|[https://www.aacap.org/App_Themes/AACAP/docs/resource_centers/resources/misc/child_ptsd_symptom_scale.pdf CPSS (Children's PTSD Symptom Scale)]
|Self-Report
|child
|10-20 minutes
|XX (new rec)
|-
|[https://adaa.org/sites/default/files/Yusko%20_210.pdf PDS (Post-traumatic Diagnosis Scale)]
|Self-Report
|adult
|10-20 minutes
|X
|-
|[https://medicine.tulane.edu/infant-institute Young Child PTSD Checklist (YCPC)]
|Parent Report
|child
|13 items
|Yes
|}

'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

=== Likelihood ratios and AUCs of screening measures for PTSD ===

* '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' 

{| class=&quot;wikitable sortable&quot;
|-
! Screening Measure (Primary Reference) !! AUC (sample size) !! DLR+ (score) !! DLR- (score) !! Clinical Generalizability
!Download
|-            
| ||  ||  ||  || 
|
|-
| ||  ||  ||  || 
|
|- 
| ||  ||  ||  || 
|
|-
| ||  ||  ||  || 
|
|-
| ||  ||  ||  || 
|
|-
|}

=== Interpreting PTSD screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]
* Also see the page on [https://en.wikipedia.org/wiki/Likelihood_ratios_in_diagnostic_testing likelihood ratios in diagnostic testing] for more information

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==
===Gold standard diagnostic interviews===
* For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

===Recommended diagnostic interviews for PTSD===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! colspan=&quot;10&quot; |Diagnostic instruments for PTSD
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| Children’s PTSD Inventory (CPTSD-I)
| Diagnostic Interview
| 6-18 years
| 50 items
| G
| 
| G
| 
| 
|
|-
| 
| 
| 
| 
| 
| 
| 
| 
| 
|

|-
| 
|
|
|
| 
| 
| 
| 
| 
|
|-
| 
|
| 
| 
| 
| 
| 
| 
| 
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Severity interviews for PTSD===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| [https://www.kennedykrieger.org/sites/default/files/library/documents/faculty/ksads-dsm-5-screener.pdf Schedule for Affective Disorders and Schizophrenia for School Age Children (K-SADS)]
| Diagnostic Interview
| 6-18 years
| 18 PTSD items
| G
| 
| G
| 
| 
|
|-
| Anxiety Disorders Interview Schedule, Child Version (ADIS-C)
| Diagnostic Interview
| 7-17 years
| 26 PTSD items
| G
| 
| G
| 
| 
|

|-
| Children’s Interview for Psychiatric Symptoms (ChIPS)
|Diagnostic Interview
|6 – 18 years
|31 PTSD items
| G
| 
| G
| 
| 
|
|-
| Diagnostic Infant and Preschool Assessment (DIPA)
|Diagnostic Interview
| Age 6 and younger
| 46 PTSD items
| G
| 
| G
| 
| 
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable
==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a brief overview of treatment options for PTSD and list of process and outcome measures for PTSD. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]].

=== Process measures ===

* Information coming soon

=== Outcome and severity measures ===
This table includes clinically significant benchmarks for PTSD specific outcome measures

* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
* ''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
| colspan=&quot;7&quot; |'''Clinically significant change benchmarks with common instruments for PTSD'''
|-
| colspan=&quot;7&quot; style=&quot;font-size:110%; text-align:center;&quot; span |&lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; |&lt;b&gt; Measure&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; |&lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; |&lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | Primary Care PTSD Screen
| style=&quot;text-align:center;&quot; | 1.0
| style=&quot;text-align:center;&quot; | 3.1
| style=&quot;text-align:center;&quot; | 2.0
| style=&quot;text-align:center;&quot; | 1.0
| style=&quot;text-align:center;&quot; | .8
| style=&quot;text-align:center;&quot; | .5
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | PTSD Checklist Scores 
| style=&quot;text-align:center;&quot; | 28.8
| style=&quot;text-align:center;&quot; | 40.8
| style=&quot;text-align:center;&quot; | 34.9
| style=&quot;text-align:center;&quot; | 4.6
| style=&quot;text-align:center;&quot; | 3.8
| style=&quot;text-align:center;&quot; | 2.3
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | Clinician Administered PTSD Scale
| style=&quot;text-align:center;&quot; | 28.8
| style=&quot;text-align:center;&quot; | 40.8
| style=&quot;text-align:center;&quot; | 34.9
| style=&quot;text-align:center;&quot; | 8.3
| style=&quot;text-align:center;&quot; | 7.0
| style=&quot;text-align:center;&quot; | 4.2
|}
=== Treatment ===
* Please refer to the Wikipedia page on [[wikipedia:Posttraumatic_stress_disorder|PTSD]] for more information on available treatment for PTSD or go to the [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/post-traumatic-stress-disorder/ Effective Child Therapy] page for for a curated resource on effective treatments for PTSD.

{{collapse top| Click here for more information}}
'''&lt;big&gt;Behavioral interventions&lt;/big&gt;'''

Recommended (have ''significant benefit'') &lt;ref name=&quot;:0&quot;&gt;Department of Veterans Affairs and Department of Defense. . (2010). VA/DoD clinical practice guidelines: management of post-traumatic stress. Washington, D.C.: Veterans Health Administration, Department of Defense.&lt;/ref&gt;:

# Brief Cognitive Behavioral Therapy (CBT; 4-5 sessions)
#* This includes stress inoculation training, trauma-focused therapy including components of cognitive restructuring, Cognitive Processing Therapy (CPT), imaginal, virtual, and in-vivo exposure as in Prolonged Exposure psychotherapy (PE)&lt;sup&gt;2&lt;/sup&gt;
#* Patient education is recommended as part of psychotherapy for patients and family members
# EMDR may help both acute and chronic PTSD, especially individuals who have trouble with prolonged exposure or have trouble verbalizing their trauma.&lt;ref name=&quot;:1&quot;&gt;Ursano, R. J., Bell, C., Eth, S., Friedman, M., Norwood, A., Pfefferbaum, B., . . . McIntyre, J. S. (2004). ''Practice guideline for the treatment of patients with acute stress disorder and posttraumatic stress disorder'': American Psychiatric Publ
&lt;/ref&gt; 
## Long term gains require further study.

Treatments with weaker evidence (have ''some benefit'') &lt;ref name=&quot;:0&quot; /&gt;:

# Patient education,
# Imagery rehearsal therapy,
# Psychodynamic therapy,
# Hypnosis,
# Relaxation techniques,  
# and Group therapy.

Treatment with ''unknown benefit'' &lt;ref name=&quot;:0&quot; /&gt;: 

# Web-based CBT,
# Acceptance and commitment therapy,
# and Dialectical Behavioral Therapy. 

&lt;big&gt;'''Medication'''&lt;/big&gt;

*SSRIs are more effective than placebo in treating PTSD.&lt;ref name=&quot;:1&quot; /&gt;
*There is no evidence to support a medication to prevent the development of PTSD.&lt;ref name=&quot;:0&quot; /&gt;
**Imipramine, propranolol, prazosin, other antidepressants, anticonvulsants, and atypical antipsychotics have ''unknown benefit.''
**Strongly recommend against the use of benzodiazepines&lt;sup&gt;2&lt;/sup&gt; and typical antipsychotics since they have ''no benefit and potential harm.''

{{collapse bottom}}

== '''External resources''' ==

=== For professionals ===
* [http://www.som.uq.edu.au/ptsd Post Traumatic Stress Disorder Information Resource] from [http://www.uq.edu.au/ The University of Queensland School of Medicine]
* [http://www.apa.org/ptsd-guideline/ APA practice parameters for assessment and treatment for PTSD (Updated 2017)]
* [http://www.ptsd.va.gov/professional/index.asp Resources for professionals] from [http://www.ptsd.va.gov/index.asp VA National PTSD Center]
* [https://www.integration.samhsa.gov/clinical-practice/trauma-informed SAMHSA's Trauma-Informed Care Resources and Training]
* [https://www.integration.samhsa.gov/clinical-practice/trauma-informed SAMHSA's Resources on Intimate Partner Violence (IPV) for Clinicians, Patients, Families, and Organizations]

=== For caregivers ===
* [http://www.nctsn.org/sites/default/files/assets/pdfs/tips_for_parents_media_final.pdf Tips for parents on shooting media coverage]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/parents_guidelines_for_helping_teens_after_the_recent_attacks.pdf Parent guidelines to helping youths after a shooting]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/helping_teens_with_traumatic_grief_caregivers_final.pdf Caregiver tips for helping teens with traumatic grief]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/10things_schoolage_ctg.pdf Caregiver tips for helping school-age children with traumatic grief]
*[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/post-traumatic-stress-disorder/ EffectiveChildTherapy.Org information on PTSD]
*[https://sccap53.org Society of Clinical Child and Adolescent Psychology]

=== For educators ===
* [http://www.nctsn.org/sites/default/files/assets/pdfs/pfa_for_schools_appendix_a_teachers.pdf Teacher tips for providing psychological first aid]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/helping_youth_after_community_violence_educators.pdf Educator tips for helping youths after a community trauma]

=== For public ===
* [https://complicatedgrief.columbia.edu/for-the-public/resources/ Resources for grief and loss]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/psychological_information_sheet_two_pager.pdf Psychological impacts of recent shootings]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/youth_journalists.pdf Tips to talking to youths about a shooting]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/talking_to_children_about_the_shooting.pdf Tips to talking to children about a shooting]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/helping_young_children_heal_crisis.pdf Helping young children heal after a crisis]
* [http://www.nctsn.org/content/psychological-first-aid-schoolspfa Psychological first aid for schools]
* [http://www.nctsn.org/content/psychological-first-aid Resources and manuals on psychological first aid]
* [http://www.ptsd.va.gov/public/index.asp Resources for the public] from [http://www.ptsd.va.gov/index.asp VA National PTSD Center]
* [https://curlie.org/Health/Mental_Health/Disorders/Anxiety/Post-traumatic_Stress Posttraumatic stress disorder] at Curlie (based on DMOZ)

=='''References'''==
{{collapse top|Click here for references}}
{{Reflist|30em}}

[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]
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[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]</text>
      <sha1>k1tefrgatqpvvyv1jjrc2u057lb45wa</sha1>
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    <revision>
      <id>2411964</id>
      <parentid>2411963</parentid>
      <timestamp>2022-08-04T22:36:33Z</timestamp>
      <contributor>
        <username>Ardenguo</username>
        <id>2944162</id>
      </contributor>
      <comment>/* Psychometric properties of screening for PTSD */ added administration time for UCLA PTSD Reaction Index for DSM-5 measure</comment>
      <model>wikitext</model>
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{{medical disclaimer}}
{{:{{BASEPAGENAME}}/Sidebar}}

==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
* For background information on what assessment portfolios are, click the link in the heading above.
* Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Posttraumatic stress disorder (disorder portfolio)/extended version|here]].

== [[Evidence based assessment/Preparation phase|'''Preparation phase''']] ==
=== Diagnostic criteria for post traumatic stress disorder ===

===='''ICD-11 Diagnostic Criteria'''====
{{blockquotetop}}
===='''ICD-11 Criteria'''====
*Post-traumatic stress disorder (PTSD) is a syndrome that develops following exposure to an extremely threatening or horrific event or series of events that is characterized by all of the following: 
**1) re-experiencing the traumatic event or events in the present in the form of vivid intrusive memories, flashbacks, or nightmares, which are typically accompanied by strong and overwhelming emotions such as fear or horror and strong physical sensations, or feelings of being overwhelmed or immersed in the same intense emotions that were experienced during the traumatic event; 
**2) avoidance of thoughts and memories of the event or events, or avoidance of activities, situations, or people reminiscent of the event or events
**3) persistent perceptions of heightened current threat, for example as indicated by hypervigilance or an enhanced startle reaction to stimuli such as unexpected noises. 
*The symptoms must persist for at least several weeks and cause significant impairment in personal, family, social, educational, occupational or other important areas of functioning. 

Additionally, ICD 11 includes a category called &quot;Complex post-traumatic stress disorder,&quot; which is described as:
*A disorder that may develop following exposure to an event or series of events of an extremely threatening or horrific nature, most commonly prolonged or repetitive events from which escape is difficult or impossible (e.g., torture, slavery, genocide campaigns, prolonged domestic violence, repeated childhood sexual or physical abuse). 
*The disorder is characterized by the core symptoms of PTSD; that is, all diagnostic requirements for PTSD have been met at some point during the course of the disorder. 
*In addition, Complex PTSD is characterized by:
**1) severe and pervasive problems in affect regulation; 
**2) persistent beliefs about oneself as diminished, defeated or worthless, accompanied by deep and pervasive feelings of shame, guilt or failure related to the traumatic event
**3) persistent difficulties in sustaining relationships and in feeling close to others. The disturbance causes significant impairment in personal, family, social, educational, occupational or other important areas of functioning.
{{blockquotebottom}}

===='''Changes in DSM-5'''====
* The diagnostic criteria for post-traumatic stress disorder changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here] and [[w:DSM-5|here]].


=== Developmental sensitivities ===
{{collapse top| Click for more information}}
*Diagnostic thresholds have been lowered for children and adolescents to account for development.
*Separate and additional criteria have been added for children age 6 or younger.
*Child sexual abuse has been found to have a substantial effect on the development of PTSD.&lt;ref&gt;{{cite journal|last1=Paolucci|first1=Elizabeth ODDONE|last2=Genuis|first2=Mark L.|last3=Violato|first3=Claudio|title=A Meta-Analysis of the Published Research on the Effects of Child Sexual Abuse|journal=The Journal of Psychology|date=January 2001|volume=135|issue=1|pages=17–36|doi=10.1080/00223980109603677}}&lt;/ref&gt;
*Children with higher exposure to trauma, less social support, and other major life events are more likely to have continued PTSD symptoms 7 months after a trauma.&lt;ref name=&quot;:2&quot;&gt;{{cite journal |last1=La Greca|first1=A|last2=Silverman|first2=WK|last3=Vernberg|first3=EM|last4=Prinstein|first4=MJ|title=Symptoms of posttraumatic stress in children after Hurricane Andrew: a prospective study.|journal=Journal of consulting and clinical psychology|date=August 1996|volume=64|issue=4|pages=712-23|pmid=8803361}}&lt;/ref&gt; 10 months after a trauma, however, only experience of a major life event remained predictive of continuing PTSD symptoms.&lt;ref name=&quot;:2&quot; /&gt; 
*Lack of social support, specifically lack of support by a teacher, was predictive of higher PTSD symptoms among children who had gone through an environmental trauma.&lt;ref name=&quot;:2&quot; /&gt;  
*Children who reported using blame and anger as strategies for coping had higher levels of PTSD symptoms 10 months after a trauma.&lt;ref name=&quot;:2&quot; /&gt;  
*Negative affect in children before a traumatic event was predictive of development of PTSD symptoms after a traumatic event.&lt;ref&gt;{{cite journal|last1=Weems|first1=CF|last2=Pina|first2=AA|last3=Costa|first3=NM|last4=Watts|first4=SE|last5=Taylor|first5=LK|last6=Cannon|first6=MF|title=Predisaster trait anxiety and negative affect predict posttraumatic stress in youths after hurricane Katrina.|journal=Journal of consulting and clinical psychology|date=February 2007|volume=75|issue=1|pages=154-9|doi=10.1037/0022-006X.75.1.154|pmid=17295574}}&lt;/ref&gt;  

[[wikipedia:Posttraumatic_stress_disorder|Posttraumatic stress disorder]] is now more sensitive to development in that diagnostic thresholds have been lowered for children and adolescents. Furthermore, separate and additional criteria have been added for children age 6 years of age or younger.
{{collapse bottom}}

=== Base rates of PTSD in different clinical settings and populations ===
This section describes the demographic settings of the populations sampled, base rates of PTSD diagnoses, country/region sampled, and the diagnostic methods that were used. Using this information, clinicians will be able to anchor the most appropriate rate of PTSD that they are likely to see in their clinical practice.
* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']] 

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Setting
! Base Rate
! Demography
! Diagnostic Method
|-

| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Kessler|first=Ronald C.|last2=Berglund|first2=Patricia|last3=Demler|first3=Olga|last4=Jin|first4=Robert|last5=Merikangas|first5=Kathleen R.|last6=Walters|first6=Ellen E.|date=2005-06-01|title=Lifetime Prevalence and Age-of-Onset Distributions of DSM-IV Disorders in the National Comorbidity Survey Replication|url=http://archpsyc.jamanetwork.com/article.aspx?doi=10.1001/archpsyc.62.6.593|journal=Archives of General Psychiatry|language=en|volume=62|issue=6|doi=10.1001/archpsyc.62.6.593|issn=0003-990X}}&lt;/ref&gt;
| 6.8%
| United States, nationally representative, age 18 and older
| National Comorbidity Survey - Replication
|-

| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=de Vries|first=Giel-Jan|last2=Olff|first2=Miranda|date=2009-08-01|title=The lifetime prevalence of traumatic events and posttraumatic stress disorder in the Netherlands|url=http://onlinelibrary.wiley.com/doi/10.1002/jts.20429/abstract|journal=Journal of Traumatic Stress|language=en|volume=22|issue=4|pages=259–267|doi=10.1002/jts.20429|issn=1573-6598}}&lt;/ref&gt;
| 7.4%
| Netherlands, nationally representative, age 18-80
| Composite International Diagnostic Interview (CIDI)
|-

| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Pietrzak|first=Robert H.|last2=Goldstein|first2=Risë B.|last3=Southwick|first3=Steven M.|last4=Grant|first4=Bridget F.|title=Prevalence and Axis I comorbidity of full and partial posttraumatic stress disorder in the United States: Results from Wave 2 of the National Epidemiologic Survey on Alcohol and Related Conditions|url=http://linkinghub.elsevier.com/retrieve/pii/S0887618510002288|journal=Journal of Anxiety Disorders|volume=25|issue=3|pages=456–465|doi=10.1016/j.janxdis.2010.11.010}}&lt;/ref&gt;
| 6.4%
| United States, nationally representative, age 18 and older
| Wave 2 National Epidemiologic Survey on Alcohol and Related Conditions
|-

| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Bunting|first=Brendan P.|last2=Ferry|first2=Finola R.|last3=Murphy|first3=Samuel D.|last4=O'Neill|first4=Siobhan M.|last5=Bolton|first5=David|date=2013-02-01|title=Trauma Associated With Civil Conflict and Posttraumatic Stress Disorder: Evidence From the Northern Ireland Study of Health and Stress|url=http://onlinelibrary.wiley.com/doi/10.1002/jts.21766/abstract|journal=Journal of Traumatic Stress|language=en|volume=26|issue=1|pages=134–141|doi=10.1002/jts.21766|issn=1573-6598}}&lt;/ref&gt;
| 8.8%
| Northern Ireland, representative sample, age 18 and older
| Northern Ireland Study of Health and Stress
|-

| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Atwoli|first=Lukoye|last2=Stein|first2=Dan J.|last3=Williams|first3=David R.|last4=Mclaughlin|first4=Katie A.|last5=Petukhova|first5=Maria|last6=Kessler|first6=Ronald C.|last7=Koenen|first7=Karestan C.|date=2013-07-03|title=Trauma and posttraumatic stress disorder in South Africa: analysis from the South African Stress and Health Study|url=https://doi.org/10.1186/1471-244X-13-182|journal=BMC Psychiatry|volume=13|pages=182|doi=10.1186/1471-244x-13-182|issn=1471-244X}}&lt;/ref&gt;
| 2.3%
| South Africa, nationally representative sample, age 18 and older
| South African Stress and Health Study, using the Composite International Diagnostic Interview (CIDI)
|-

| U.S. Service Members&lt;ref&gt;{{Cite journal|last=Hoge|first=Charles W.|last2=Castro|first2=Carl A.|last3=Messer|first3=Stephen C.|last4=McGurk|first4=Dennis|last5=Cotting|first5=Dave I.|last6=Koffman|first6=Robert L.|date=2004-07-01|title=Combat Duty in Iraq and Afghanistan, Mental Health Problems, and Barriers to Care|url=http://dx.doi.org/10.1056/NEJMoa040603|journal=New England Journal of Medicine|volume=351|issue=1|pages=13–22|doi=10.1056/nejmoa040603|issn=0028-4793|pmid=15229303}}&lt;/ref&gt;
|11.5% - 19.5% ♦
| U.S. Army and Marine Soldiers Deployed to Iraq and Afghanistan
| PTSD Checklist
|-
|Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Merikangas|first=Kathleen Ries|last2=He|first2=Jian-ping|last3=Burstein|first3=Marcy|last4=Swanson|first4=Sonja A.|last5=Avenevoli|first5=Shelli|last6=Cui|first6=Lihong|last7=Benjet|first7=Corina|last8=Georgiades|first8=Katholiki|last9=Swendsen|first9=Joel|title=Lifetime Prevalence of Mental Disorders in U.S. Adolescents: Results from the National Comorbidity Survey Replication–Adolescent Supplement (NCS-A)|url=https://doi.org/10.1016/j.jaac.2010.05.017|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=49|issue=10|pages=980–989|doi=10.1016/j.jaac.2010.05.017}}&lt;/ref&gt;
|5.0%
|United States, nationally representative, ages 13-18
|National Comorbidity Survey Replication—Adolescent Supplement (NCS-A)
|}
♦ '''Note''': These rates were using broad PTSD Checklist scoring criteria of being scored positive if subjects reported at least one intrusion symptom, three avoidance symptoms, and two hyperarousal symptom that were categorized as at the moderate level. The 11.5% is for soldiers returning from deployment in Iraq, 19.5% is for soldiers returning from Afghanistan. Another common practice is to use a strict cutoff of 50 on the PCL, above which someone screens positive for PTSD. With this cutoff, rates are 6.2% and 12.9% for Service Members returned from Afghanistan and Iraqi, respectively.

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==

=== Psychometric properties of screening for PTSD ===
The following section contains a list of screening and diagnostic instruments for PTSD. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here.]]

* '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]'''''
{| class=&quot;wikitable sortable&quot;
|-
! Measure 
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Highly recommended
|-
|[https://www.ptsd.va.gov/professional/assessment/adult-sr/ptsd-checklist.asp PCL (PTSD Checklist for DSM-5)]
|Self-Report 
|adult
|5-10 minutes
|X
|-
|[https://www.ptsd.va.gov/professional/assessment/adult-int/caps.asp CAPS (Clinician Administered PTSD Scale)]
|Clinician Administered Interview 
|adult and child versions available 
|40-60 minutes 
|X
|-
|[https://www.appi.org/products/structured-clinical-interview-for-dsm-5-scid-5 SCID-IV (Structured Clinical Interview for DSM-IV)]
|Clinician Administered Interview
|adult
|1-2 hours 
|X
|-
|[https://www.reactionindex.com/index.php/ UCLA PTSD Reaction Index for DSM-5]
|Clinician Administered
|child, school age, adolescents
|20-30 minutes
|Yes
|-
|[https://www.aacap.org/App_Themes/AACAP/docs/resource_centers/resources/misc/child_ptsd_symptom_scale.pdf CPSS (Children's PTSD Symptom Scale)]
|Self-Report
|child
|10-20 minutes
|XX (new rec)
|-
|[https://adaa.org/sites/default/files/Yusko%20_210.pdf PDS (Post-traumatic Diagnosis Scale)]
|Self-Report
|adult
|10-20 minutes
|X
|-
|[https://medicine.tulane.edu/infant-institute Young Child PTSD Checklist (YCPC)]
|Parent Report
|child
|13 items
|Yes
|}

'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

=== Likelihood ratios and AUCs of screening measures for PTSD ===

* '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' 

{| class=&quot;wikitable sortable&quot;
|-
! Screening Measure (Primary Reference) !! AUC (sample size) !! DLR+ (score) !! DLR- (score) !! Clinical Generalizability
!Download
|-            
| ||  ||  ||  || 
|
|-
| ||  ||  ||  || 
|
|- 
| ||  ||  ||  || 
|
|-
| ||  ||  ||  || 
|
|-
| ||  ||  ||  || 
|
|-
|}

=== Interpreting PTSD screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]
* Also see the page on [https://en.wikipedia.org/wiki/Likelihood_ratios_in_diagnostic_testing likelihood ratios in diagnostic testing] for more information

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==
===Gold standard diagnostic interviews===
* For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

===Recommended diagnostic interviews for PTSD===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! colspan=&quot;10&quot; |Diagnostic instruments for PTSD
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| Children’s PTSD Inventory (CPTSD-I)
| Diagnostic Interview
| 6-18 years
| 50 items
| G
| 
| G
| 
| 
|
|-
| 
| 
| 
| 
| 
| 
| 
| 
| 
|

|-
| 
|
|
|
| 
| 
| 
| 
| 
|
|-
| 
|
| 
| 
| 
| 
| 
| 
| 
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Severity interviews for PTSD===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| [https://www.kennedykrieger.org/sites/default/files/library/documents/faculty/ksads-dsm-5-screener.pdf Schedule for Affective Disorders and Schizophrenia for School Age Children (K-SADS)]
| Diagnostic Interview
| 6-18 years
| 18 PTSD items
| G
| 
| G
| 
| 
|
|-
| Anxiety Disorders Interview Schedule, Child Version (ADIS-C)
| Diagnostic Interview
| 7-17 years
| 26 PTSD items
| G
| 
| G
| 
| 
|

|-
| Children’s Interview for Psychiatric Symptoms (ChIPS)
|Diagnostic Interview
|6 – 18 years
|31 PTSD items
| G
| 
| G
| 
| 
|
|-
| Diagnostic Infant and Preschool Assessment (DIPA)
|Diagnostic Interview
| Age 6 and younger
| 46 PTSD items
| G
| 
| G
| 
| 
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable
==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a brief overview of treatment options for PTSD and list of process and outcome measures for PTSD. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]].

=== Process measures ===

* Information coming soon

=== Outcome and severity measures ===
This table includes clinically significant benchmarks for PTSD specific outcome measures

* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
* ''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
| colspan=&quot;7&quot; |'''Clinically significant change benchmarks with common instruments for PTSD'''
|-
| colspan=&quot;7&quot; style=&quot;font-size:110%; text-align:center;&quot; span |&lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; |&lt;b&gt; Measure&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; |&lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; |&lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | Primary Care PTSD Screen
| style=&quot;text-align:center;&quot; | 1.0
| style=&quot;text-align:center;&quot; | 3.1
| style=&quot;text-align:center;&quot; | 2.0
| style=&quot;text-align:center;&quot; | 1.0
| style=&quot;text-align:center;&quot; | .8
| style=&quot;text-align:center;&quot; | .5
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | PTSD Checklist Scores 
| style=&quot;text-align:center;&quot; | 28.8
| style=&quot;text-align:center;&quot; | 40.8
| style=&quot;text-align:center;&quot; | 34.9
| style=&quot;text-align:center;&quot; | 4.6
| style=&quot;text-align:center;&quot; | 3.8
| style=&quot;text-align:center;&quot; | 2.3
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | Clinician Administered PTSD Scale
| style=&quot;text-align:center;&quot; | 28.8
| style=&quot;text-align:center;&quot; | 40.8
| style=&quot;text-align:center;&quot; | 34.9
| style=&quot;text-align:center;&quot; | 8.3
| style=&quot;text-align:center;&quot; | 7.0
| style=&quot;text-align:center;&quot; | 4.2
|}
=== Treatment ===
* Please refer to the Wikipedia page on [[wikipedia:Posttraumatic_stress_disorder|PTSD]] for more information on available treatment for PTSD or go to the [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/post-traumatic-stress-disorder/ Effective Child Therapy] page for for a curated resource on effective treatments for PTSD.

{{collapse top| Click here for more information}}
'''&lt;big&gt;Behavioral interventions&lt;/big&gt;'''

Recommended (have ''significant benefit'') &lt;ref name=&quot;:0&quot;&gt;Department of Veterans Affairs and Department of Defense. . (2010). VA/DoD clinical practice guidelines: management of post-traumatic stress. Washington, D.C.: Veterans Health Administration, Department of Defense.&lt;/ref&gt;:

# Brief Cognitive Behavioral Therapy (CBT; 4-5 sessions)
#* This includes stress inoculation training, trauma-focused therapy including components of cognitive restructuring, Cognitive Processing Therapy (CPT), imaginal, virtual, and in-vivo exposure as in Prolonged Exposure psychotherapy (PE)&lt;sup&gt;2&lt;/sup&gt;
#* Patient education is recommended as part of psychotherapy for patients and family members
# EMDR may help both acute and chronic PTSD, especially individuals who have trouble with prolonged exposure or have trouble verbalizing their trauma.&lt;ref name=&quot;:1&quot;&gt;Ursano, R. J., Bell, C., Eth, S., Friedman, M., Norwood, A., Pfefferbaum, B., . . . McIntyre, J. S. (2004). ''Practice guideline for the treatment of patients with acute stress disorder and posttraumatic stress disorder'': American Psychiatric Publ
&lt;/ref&gt; 
## Long term gains require further study.

Treatments with weaker evidence (have ''some benefit'') &lt;ref name=&quot;:0&quot; /&gt;:

# Patient education,
# Imagery rehearsal therapy,
# Psychodynamic therapy,
# Hypnosis,
# Relaxation techniques,  
# and Group therapy.

Treatment with ''unknown benefit'' &lt;ref name=&quot;:0&quot; /&gt;: 

# Web-based CBT,
# Acceptance and commitment therapy,
# and Dialectical Behavioral Therapy. 

&lt;big&gt;'''Medication'''&lt;/big&gt;

*SSRIs are more effective than placebo in treating PTSD.&lt;ref name=&quot;:1&quot; /&gt;
*There is no evidence to support a medication to prevent the development of PTSD.&lt;ref name=&quot;:0&quot; /&gt;
**Imipramine, propranolol, prazosin, other antidepressants, anticonvulsants, and atypical antipsychotics have ''unknown benefit.''
**Strongly recommend against the use of benzodiazepines&lt;sup&gt;2&lt;/sup&gt; and typical antipsychotics since they have ''no benefit and potential harm.''

{{collapse bottom}}

== '''External resources''' ==

=== For professionals ===
* [http://www.som.uq.edu.au/ptsd Post Traumatic Stress Disorder Information Resource] from [http://www.uq.edu.au/ The University of Queensland School of Medicine]
* [http://www.apa.org/ptsd-guideline/ APA practice parameters for assessment and treatment for PTSD (Updated 2017)]
* [http://www.ptsd.va.gov/professional/index.asp Resources for professionals] from [http://www.ptsd.va.gov/index.asp VA National PTSD Center]
* [https://www.integration.samhsa.gov/clinical-practice/trauma-informed SAMHSA's Trauma-Informed Care Resources and Training]
* [https://www.integration.samhsa.gov/clinical-practice/trauma-informed SAMHSA's Resources on Intimate Partner Violence (IPV) for Clinicians, Patients, Families, and Organizations]

=== For caregivers ===
* [http://www.nctsn.org/sites/default/files/assets/pdfs/tips_for_parents_media_final.pdf Tips for parents on shooting media coverage]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/parents_guidelines_for_helping_teens_after_the_recent_attacks.pdf Parent guidelines to helping youths after a shooting]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/helping_teens_with_traumatic_grief_caregivers_final.pdf Caregiver tips for helping teens with traumatic grief]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/10things_schoolage_ctg.pdf Caregiver tips for helping school-age children with traumatic grief]
*[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/post-traumatic-stress-disorder/ EffectiveChildTherapy.Org information on PTSD]
*[https://sccap53.org Society of Clinical Child and Adolescent Psychology]

=== For educators ===
* [http://www.nctsn.org/sites/default/files/assets/pdfs/pfa_for_schools_appendix_a_teachers.pdf Teacher tips for providing psychological first aid]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/helping_youth_after_community_violence_educators.pdf Educator tips for helping youths after a community trauma]

=== For public ===
* [https://complicatedgrief.columbia.edu/for-the-public/resources/ Resources for grief and loss]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/psychological_information_sheet_two_pager.pdf Psychological impacts of recent shootings]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/youth_journalists.pdf Tips to talking to youths about a shooting]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/talking_to_children_about_the_shooting.pdf Tips to talking to children about a shooting]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/helping_young_children_heal_crisis.pdf Helping young children heal after a crisis]
* [http://www.nctsn.org/content/psychological-first-aid-schoolspfa Psychological first aid for schools]
* [http://www.nctsn.org/content/psychological-first-aid Resources and manuals on psychological first aid]
* [http://www.ptsd.va.gov/public/index.asp Resources for the public] from [http://www.ptsd.va.gov/index.asp VA National PTSD Center]
* [https://curlie.org/Health/Mental_Health/Disorders/Anxiety/Post-traumatic_Stress Posttraumatic stress disorder] at Curlie (based on DMOZ)

=='''References'''==
{{collapse top|Click here for references}}
{{Reflist|30em}}

[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]
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==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
* For background information on what assessment portfolios are, click the link in the heading above.
* Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Posttraumatic stress disorder (disorder portfolio)/extended version|here]].

== [[Evidence based assessment/Preparation phase|'''Preparation phase''']] ==
=== Diagnostic criteria for post traumatic stress disorder ===

===='''ICD-11 Diagnostic Criteria'''====
{{blockquotetop}}
===='''ICD-11 Criteria'''====
*Post-traumatic stress disorder (PTSD) is a syndrome that develops following exposure to an extremely threatening or horrific event or series of events that is characterized by all of the following: 
**1) re-experiencing the traumatic event or events in the present in the form of vivid intrusive memories, flashbacks, or nightmares, which are typically accompanied by strong and overwhelming emotions such as fear or horror and strong physical sensations, or feelings of being overwhelmed or immersed in the same intense emotions that were experienced during the traumatic event; 
**2) avoidance of thoughts and memories of the event or events, or avoidance of activities, situations, or people reminiscent of the event or events
**3) persistent perceptions of heightened current threat, for example as indicated by hypervigilance or an enhanced startle reaction to stimuli such as unexpected noises. 
*The symptoms must persist for at least several weeks and cause significant impairment in personal, family, social, educational, occupational or other important areas of functioning. 

Additionally, ICD 11 includes a category called &quot;Complex post-traumatic stress disorder,&quot; which is described as:
*A disorder that may develop following exposure to an event or series of events of an extremely threatening or horrific nature, most commonly prolonged or repetitive events from which escape is difficult or impossible (e.g., torture, slavery, genocide campaigns, prolonged domestic violence, repeated childhood sexual or physical abuse). 
*The disorder is characterized by the core symptoms of PTSD; that is, all diagnostic requirements for PTSD have been met at some point during the course of the disorder. 
*In addition, Complex PTSD is characterized by:
**1) severe and pervasive problems in affect regulation; 
**2) persistent beliefs about oneself as diminished, defeated or worthless, accompanied by deep and pervasive feelings of shame, guilt or failure related to the traumatic event
**3) persistent difficulties in sustaining relationships and in feeling close to others. The disturbance causes significant impairment in personal, family, social, educational, occupational or other important areas of functioning.
{{blockquotebottom}}

===='''Changes in DSM-5'''====
* The diagnostic criteria for post-traumatic stress disorder changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here] and [[w:DSM-5|here]].


=== Developmental sensitivities ===
{{collapse top| Click for more information}}
*Diagnostic thresholds have been lowered for children and adolescents to account for development.
*Separate and additional criteria have been added for children age 6 or younger.
*Child sexual abuse has been found to have a substantial effect on the development of PTSD.&lt;ref&gt;{{cite journal|last1=Paolucci|first1=Elizabeth ODDONE|last2=Genuis|first2=Mark L.|last3=Violato|first3=Claudio|title=A Meta-Analysis of the Published Research on the Effects of Child Sexual Abuse|journal=The Journal of Psychology|date=January 2001|volume=135|issue=1|pages=17–36|doi=10.1080/00223980109603677}}&lt;/ref&gt;
*Children with higher exposure to trauma, less social support, and other major life events are more likely to have continued PTSD symptoms 7 months after a trauma.&lt;ref name=&quot;:2&quot;&gt;{{cite journal |last1=La Greca|first1=A|last2=Silverman|first2=WK|last3=Vernberg|first3=EM|last4=Prinstein|first4=MJ|title=Symptoms of posttraumatic stress in children after Hurricane Andrew: a prospective study.|journal=Journal of consulting and clinical psychology|date=August 1996|volume=64|issue=4|pages=712-23|pmid=8803361}}&lt;/ref&gt; 10 months after a trauma, however, only experience of a major life event remained predictive of continuing PTSD symptoms.&lt;ref name=&quot;:2&quot; /&gt; 
*Lack of social support, specifically lack of support by a teacher, was predictive of higher PTSD symptoms among children who had gone through an environmental trauma.&lt;ref name=&quot;:2&quot; /&gt;  
*Children who reported using blame and anger as strategies for coping had higher levels of PTSD symptoms 10 months after a trauma.&lt;ref name=&quot;:2&quot; /&gt;  
*Negative affect in children before a traumatic event was predictive of development of PTSD symptoms after a traumatic event.&lt;ref&gt;{{cite journal|last1=Weems|first1=CF|last2=Pina|first2=AA|last3=Costa|first3=NM|last4=Watts|first4=SE|last5=Taylor|first5=LK|last6=Cannon|first6=MF|title=Predisaster trait anxiety and negative affect predict posttraumatic stress in youths after hurricane Katrina.|journal=Journal of consulting and clinical psychology|date=February 2007|volume=75|issue=1|pages=154-9|doi=10.1037/0022-006X.75.1.154|pmid=17295574}}&lt;/ref&gt;  

[[wikipedia:Posttraumatic_stress_disorder|Posttraumatic stress disorder]] is now more sensitive to development in that diagnostic thresholds have been lowered for children and adolescents. Furthermore, separate and additional criteria have been added for children age 6 years of age or younger.
{{collapse bottom}}

=== Base rates of PTSD in different clinical settings and populations ===
This section describes the demographic settings of the populations sampled, base rates of PTSD diagnoses, country/region sampled, and the diagnostic methods that were used. Using this information, clinicians will be able to anchor the most appropriate rate of PTSD that they are likely to see in their clinical practice.
* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']] 

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Demography
! Setting
! Base Rate
! Diagnostic Method
|-
| United States, nationally representative, age 18 and older
| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Kessler|first=Ronald C.|last2=Berglund|first2=Patricia|last3=Demler|first3=Olga|last4=Jin|first4=Robert|last5=Merikangas|first5=Kathleen R.|last6=Walters|first6=Ellen E.|date=2005-06-01|title=Lifetime Prevalence and Age-of-Onset Distributions of DSM-IV Disorders in the National Comorbidity Survey Replication|url=http://archpsyc.jamanetwork.com/article.aspx?doi=10.1001/archpsyc.62.6.593|journal=Archives of General Psychiatry|language=en|volume=62|issue=6|doi=10.1001/archpsyc.62.6.593|issn=0003-990X}}&lt;/ref&gt;
| 6.8%
| National Comorbidity Survey - Replication
|-
| Netherlands, nationally representative, age 18-80
| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=de Vries|first=Giel-Jan|last2=Olff|first2=Miranda|date=2009-08-01|title=The lifetime prevalence of traumatic events and posttraumatic stress disorder in the Netherlands|url=http://onlinelibrary.wiley.com/doi/10.1002/jts.20429/abstract|journal=Journal of Traumatic Stress|language=en|volume=22|issue=4|pages=259–267|doi=10.1002/jts.20429|issn=1573-6598}}&lt;/ref&gt;
| 7.4%
| Composite International Diagnostic Interview (CIDI)
|-
| United States, nationally representative, age 18 and older
| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Pietrzak|first=Robert H.|last2=Goldstein|first2=Risë B.|last3=Southwick|first3=Steven M.|last4=Grant|first4=Bridget F.|title=Prevalence and Axis I comorbidity of full and partial posttraumatic stress disorder in the United States: Results from Wave 2 of the National Epidemiologic Survey on Alcohol and Related Conditions|url=http://linkinghub.elsevier.com/retrieve/pii/S0887618510002288|journal=Journal of Anxiety Disorders|volume=25|issue=3|pages=456–465|doi=10.1016/j.janxdis.2010.11.010}}&lt;/ref&gt;
| 6.4%
| Wave 2 National Epidemiologic Survey on Alcohol and Related Conditions
|-
| Northern Ireland, representative sample, age 18 and older
| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Bunting|first=Brendan P.|last2=Ferry|first2=Finola R.|last3=Murphy|first3=Samuel D.|last4=O'Neill|first4=Siobhan M.|last5=Bolton|first5=David|date=2013-02-01|title=Trauma Associated With Civil Conflict and Posttraumatic Stress Disorder: Evidence From the Northern Ireland Study of Health and Stress|url=http://onlinelibrary.wiley.com/doi/10.1002/jts.21766/abstract|journal=Journal of Traumatic Stress|language=en|volume=26|issue=1|pages=134–141|doi=10.1002/jts.21766|issn=1573-6598}}&lt;/ref&gt;
| 8.8%
| Northern Ireland Study of Health and Stress
|-
| South Africa, nationally representative sample, age 18 and older
| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Atwoli|first=Lukoye|last2=Stein|first2=Dan J.|last3=Williams|first3=David R.|last4=Mclaughlin|first4=Katie A.|last5=Petukhova|first5=Maria|last6=Kessler|first6=Ronald C.|last7=Koenen|first7=Karestan C.|date=2013-07-03|title=Trauma and posttraumatic stress disorder in South Africa: analysis from the South African Stress and Health Study|url=https://doi.org/10.1186/1471-244X-13-182|journal=BMC Psychiatry|volume=13|pages=182|doi=10.1186/1471-244x-13-182|issn=1471-244X}}&lt;/ref&gt;
| 2.3%
| South African Stress and Health Study, using the Composite International Diagnostic Interview (CIDI)
|-
| U.S. Army and Marine Soldiers Deployed to Iraq and Afghanistan
| U.S. Service Members&lt;ref&gt;{{Cite journal|last=Hoge|first=Charles W.|last2=Castro|first2=Carl A.|last3=Messer|first3=Stephen C.|last4=McGurk|first4=Dennis|last5=Cotting|first5=Dave I.|last6=Koffman|first6=Robert L.|date=2004-07-01|title=Combat Duty in Iraq and Afghanistan, Mental Health Problems, and Barriers to Care|url=http://dx.doi.org/10.1056/NEJMoa040603|journal=New England Journal of Medicine|volume=351|issue=1|pages=13–22|doi=10.1056/nejmoa040603|issn=0028-4793|pmid=15229303}}&lt;/ref&gt;
|11.5% - 19.5% ♦
| PTSD Checklist
|-
|United States, nationally representative, ages 13-18
|Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Merikangas|first=Kathleen Ries|last2=He|first2=Jian-ping|last3=Burstein|first3=Marcy|last4=Swanson|first4=Sonja A.|last5=Avenevoli|first5=Shelli|last6=Cui|first6=Lihong|last7=Benjet|first7=Corina|last8=Georgiades|first8=Katholiki|last9=Swendsen|first9=Joel|title=Lifetime Prevalence of Mental Disorders in U.S. Adolescents: Results from the National Comorbidity Survey Replication–Adolescent Supplement (NCS-A)|url=https://doi.org/10.1016/j.jaac.2010.05.017|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=49|issue=10|pages=980–989|doi=10.1016/j.jaac.2010.05.017}}&lt;/ref&gt;
|5.0%
|National Comorbidity Survey Replication—Adolescent Supplement (NCS-A)
|}
♦ '''Note''': These rates were using broad PTSD Checklist scoring criteria of being scored positive if subjects reported at least one intrusion symptom, three avoidance symptoms, and two hyperarousal symptom that were categorized as at the moderate level. The 11.5% is for soldiers returning from deployment in Iraq, 19.5% is for soldiers returning from Afghanistan. Another common practice is to use a strict cutoff of 50 on the PCL, above which someone screens positive for PTSD. With this cutoff, rates are 6.2% and 12.9% for Service Members returned from Afghanistan and Iraqi, respectively.

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==

=== Psychometric properties of screening for PTSD ===
The following section contains a list of screening and diagnostic instruments for PTSD. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here.]]

* '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]'''''
{| class=&quot;wikitable&quot;
|-
! Measure 
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Highly recommended
|-
|[https://www.ptsd.va.gov/professional/assessment/adult-sr/ptsd-checklist.asp PCL (PTSD Checklist for DSM-5)]
|Self-Report 
|adult
|5-10 minutes
|X
|-
|[https://www.appi.org/products/structured-clinical-interview-for-dsm-5-scid-5 SCID-IV (Structured Clinical Interview for DSM-IV)]
|Clinician Administered Interview
|adult
|1-2 hours 
|X
|-
|[https://adaa.org/sites/default/files/Yusko%20_210.pdf PDS (Post-traumatic Diagnosis Scale)]
|Self-Report
|adult
|10-20 minutes
|X
|-
|[https://www.ptsd.va.gov/professional/assessment/adult-int/caps.asp CAPS (Clinician Administered PTSD Scale)]
|Clinician Administered Interview 
|adult and child versions available 
|40-60 minutes 
|X
|-
|[https://www.reactionindex.com/index.php/ UCLA PTSD Reaction Index for DSM-5]
|Clinician Administered
|child, school age, adolescents
|20-30 minutes
|Yes
|-
|[https://www.aacap.org/App_Themes/AACAP/docs/resource_centers/resources/misc/child_ptsd_symptom_scale.pdf CPSS (Children's PTSD Symptom Scale)]
|Self-Report
|child
|10-20 minutes
|XX (new rec)
|-
|[https://medicine.tulane.edu/infant-institute Young Child PTSD Checklist (YCPC)]
|Parent Report
|child
|13 items
|Yes
|}

'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

=== Likelihood ratios and AUCs of screening measures for PTSD ===

* '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' 

{| class=&quot;wikitable sortable&quot;
|-
! Screening Measure (Primary Reference) !! AUC (sample size) !! DLR+ (score) !! DLR- (score) !! Clinical Generalizability
!Download
|-            
| ||  ||  ||  || 
|
|-
| ||  ||  ||  || 
|
|- 
| ||  ||  ||  || 
|
|-
| ||  ||  ||  || 
|
|-
| ||  ||  ||  || 
|
|-
|}

=== Interpreting PTSD screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]
* Also see the page on [https://en.wikipedia.org/wiki/Likelihood_ratios_in_diagnostic_testing likelihood ratios in diagnostic testing] for more information

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==
===Gold standard diagnostic interviews===
* For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

===Recommended diagnostic interviews for PTSD===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! colspan=&quot;10&quot; |Diagnostic instruments for PTSD
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| Children’s PTSD Inventory (CPTSD-I)
| Diagnostic Interview
| 6-18 years
| 50 items
| G
| 
| G
| 
| 
|
|-
| 
| 
| 
| 
| 
| 
| 
| 
| 
|

|-
| 
|
|
|
| 
| 
| 
| 
| 
|
|-
| 
|
| 
| 
| 
| 
| 
| 
| 
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Severity interviews for PTSD===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| [https://www.kennedykrieger.org/sites/default/files/library/documents/faculty/ksads-dsm-5-screener.pdf Schedule for Affective Disorders and Schizophrenia for School Age Children (K-SADS)]
| Diagnostic Interview
| 6-18 years
| 18 PTSD items
| G
| 
| G
| 
| 
|
|-
| Anxiety Disorders Interview Schedule, Child Version (ADIS-C)
| Diagnostic Interview
| 7-17 years
| 26 PTSD items
| G
| 
| G
| 
| 
|

|-
| Children’s Interview for Psychiatric Symptoms (ChIPS)
|Diagnostic Interview
|6 – 18 years
|31 PTSD items
| G
| 
| G
| 
| 
|
|-
| Diagnostic Infant and Preschool Assessment (DIPA)
|Diagnostic Interview
| Age 6 and younger
| 46 PTSD items
| G
| 
| G
| 
| 
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable
==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a brief overview of treatment options for PTSD and list of process and outcome measures for PTSD. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]].

=== Process measures ===

* Information coming soon

=== Outcome and severity measures ===
This table includes clinically significant benchmarks for PTSD specific outcome measures

* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
* ''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
| colspan=&quot;7&quot; |'''Clinically significant change benchmarks with common instruments for PTSD'''
|-
| colspan=&quot;7&quot; style=&quot;font-size:110%; text-align:center;&quot; span |&lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; |&lt;b&gt; Measure&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; |&lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; |&lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | Primary Care PTSD Screen
| style=&quot;text-align:center;&quot; | 1.0
| style=&quot;text-align:center;&quot; | 3.1
| style=&quot;text-align:center;&quot; | 2.0
| style=&quot;text-align:center;&quot; | 1.0
| style=&quot;text-align:center;&quot; | .8
| style=&quot;text-align:center;&quot; | .5
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | PTSD Checklist Scores 
| style=&quot;text-align:center;&quot; | 28.8
| style=&quot;text-align:center;&quot; | 40.8
| style=&quot;text-align:center;&quot; | 34.9
| style=&quot;text-align:center;&quot; | 4.6
| style=&quot;text-align:center;&quot; | 3.8
| style=&quot;text-align:center;&quot; | 2.3
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | Clinician Administered PTSD Scale
| style=&quot;text-align:center;&quot; | 28.8
| style=&quot;text-align:center;&quot; | 40.8
| style=&quot;text-align:center;&quot; | 34.9
| style=&quot;text-align:center;&quot; | 8.3
| style=&quot;text-align:center;&quot; | 7.0
| style=&quot;text-align:center;&quot; | 4.2
|}
=== Treatment ===
* Please refer to the Wikipedia page on [[wikipedia:Posttraumatic_stress_disorder|PTSD]] for more information on available treatment for PTSD or go to the [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/post-traumatic-stress-disorder/ Effective Child Therapy] page for for a curated resource on effective treatments for PTSD.

{{collapse top| Click here for more information}}
'''&lt;big&gt;Behavioral interventions&lt;/big&gt;'''

Recommended (have ''significant benefit'') &lt;ref name=&quot;:0&quot;&gt;Department of Veterans Affairs and Department of Defense. . (2010). VA/DoD clinical practice guidelines: management of post-traumatic stress. Washington, D.C.: Veterans Health Administration, Department of Defense.&lt;/ref&gt;:

# Brief Cognitive Behavioral Therapy (CBT; 4-5 sessions)
#* This includes stress inoculation training, trauma-focused therapy including components of cognitive restructuring, Cognitive Processing Therapy (CPT), imaginal, virtual, and in-vivo exposure as in Prolonged Exposure psychotherapy (PE)&lt;sup&gt;2&lt;/sup&gt;
#* Patient education is recommended as part of psychotherapy for patients and family members
# EMDR may help both acute and chronic PTSD, especially individuals who have trouble with prolonged exposure or have trouble verbalizing their trauma.&lt;ref name=&quot;:1&quot;&gt;Ursano, R. J., Bell, C., Eth, S., Friedman, M., Norwood, A., Pfefferbaum, B., . . . McIntyre, J. S. (2004). ''Practice guideline for the treatment of patients with acute stress disorder and posttraumatic stress disorder'': American Psychiatric Publ
&lt;/ref&gt; 
## Long term gains require further study.

Treatments with weaker evidence (have ''some benefit'') &lt;ref name=&quot;:0&quot; /&gt;:

# Patient education,
# Imagery rehearsal therapy,
# Psychodynamic therapy,
# Hypnosis,
# Relaxation techniques,  
# and Group therapy.

Treatment with ''unknown benefit'' &lt;ref name=&quot;:0&quot; /&gt;: 

# Web-based CBT,
# Acceptance and commitment therapy,
# and Dialectical Behavioral Therapy. 

&lt;big&gt;'''Medication'''&lt;/big&gt;

*SSRIs are more effective than placebo in treating PTSD.&lt;ref name=&quot;:1&quot; /&gt;
*There is no evidence to support a medication to prevent the development of PTSD.&lt;ref name=&quot;:0&quot; /&gt;
**Imipramine, propranolol, prazosin, other antidepressants, anticonvulsants, and atypical antipsychotics have ''unknown benefit.''
**Strongly recommend against the use of benzodiazepines&lt;sup&gt;2&lt;/sup&gt; and typical antipsychotics since they have ''no benefit and potential harm.''

{{collapse bottom}}

== '''External resources''' ==

=== For professionals ===
* [http://www.som.uq.edu.au/ptsd Post Traumatic Stress Disorder Information Resource] from [http://www.uq.edu.au/ The University of Queensland School of Medicine]
* [http://www.apa.org/ptsd-guideline/ APA practice parameters for assessment and treatment for PTSD (Updated 2017)]
* [http://www.ptsd.va.gov/professional/index.asp Resources for professionals] from [http://www.ptsd.va.gov/index.asp VA National PTSD Center]
* [https://www.integration.samhsa.gov/clinical-practice/trauma-informed SAMHSA's Trauma-Informed Care Resources and Training]
* [https://www.integration.samhsa.gov/clinical-practice/trauma-informed SAMHSA's Resources on Intimate Partner Violence (IPV) for Clinicians, Patients, Families, and Organizations]

=== For caregivers ===
* [http://www.nctsn.org/sites/default/files/assets/pdfs/tips_for_parents_media_final.pdf Tips for parents on shooting media coverage]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/parents_guidelines_for_helping_teens_after_the_recent_attacks.pdf Parent guidelines to helping youths after a shooting]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/helping_teens_with_traumatic_grief_caregivers_final.pdf Caregiver tips for helping teens with traumatic grief]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/10things_schoolage_ctg.pdf Caregiver tips for helping school-age children with traumatic grief]
*[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/post-traumatic-stress-disorder/ EffectiveChildTherapy.Org information on PTSD]
*[https://sccap53.org Society of Clinical Child and Adolescent Psychology]

=== For educators ===
* [http://www.nctsn.org/sites/default/files/assets/pdfs/pfa_for_schools_appendix_a_teachers.pdf Teacher tips for providing psychological first aid]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/helping_youth_after_community_violence_educators.pdf Educator tips for helping youths after a community trauma]

=== For public ===
* [https://complicatedgrief.columbia.edu/for-the-public/resources/ Resources for grief and loss]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/psychological_information_sheet_two_pager.pdf Psychological impacts of recent shootings]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/youth_journalists.pdf Tips to talking to youths about a shooting]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/talking_to_children_about_the_shooting.pdf Tips to talking to children about a shooting]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/helping_young_children_heal_crisis.pdf Helping young children heal after a crisis]
* [http://www.nctsn.org/content/psychological-first-aid-schoolspfa Psychological first aid for schools]
* [http://www.nctsn.org/content/psychological-first-aid Resources and manuals on psychological first aid]
* [http://www.ptsd.va.gov/public/index.asp Resources for the public] from [http://www.ptsd.va.gov/index.asp VA National PTSD Center]
* [https://curlie.org/Health/Mental_Health/Disorders/Anxiety/Post-traumatic_Stress Posttraumatic stress disorder] at Curlie (based on DMOZ)

=='''References'''==
{{collapse top|Click here for references}}
{{Reflist|30em}}

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==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
* For background information on what assessment portfolios are, click the link in the heading above.
* Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Posttraumatic stress disorder (disorder portfolio)/extended version|here]].

== [[Evidence based assessment/Preparation phase|'''Preparation phase''']] ==
=== Diagnostic criteria for post traumatic stress disorder ===

===='''ICD-11 Diagnostic Criteria'''====
{{blockquotetop}}
===='''ICD-11 Criteria'''====
*Post-traumatic stress disorder (PTSD) is a syndrome that develops following exposure to an extremely threatening or horrific event or series of events that is characterized by all of the following: 
**1) re-experiencing the traumatic event or events in the present in the form of vivid intrusive memories, flashbacks, or nightmares, which are typically accompanied by strong and overwhelming emotions such as fear or horror and strong physical sensations, or feelings of being overwhelmed or immersed in the same intense emotions that were experienced during the traumatic event; 
**2) avoidance of thoughts and memories of the event or events, or avoidance of activities, situations, or people reminiscent of the event or events
**3) persistent perceptions of heightened current threat, for example as indicated by hypervigilance or an enhanced startle reaction to stimuli such as unexpected noises. 
*The symptoms must persist for at least several weeks and cause significant impairment in personal, family, social, educational, occupational or other important areas of functioning. 

Additionally, ICD 11 includes a category called &quot;Complex post-traumatic stress disorder,&quot; which is described as:
*A disorder that may develop following exposure to an event or series of events of an extremely threatening or horrific nature, most commonly prolonged or repetitive events from which escape is difficult or impossible (e.g., torture, slavery, genocide campaigns, prolonged domestic violence, repeated childhood sexual or physical abuse). 
*The disorder is characterized by the core symptoms of PTSD; that is, all diagnostic requirements for PTSD have been met at some point during the course of the disorder. 
*In addition, Complex PTSD is characterized by:
**1) severe and pervasive problems in affect regulation; 
**2) persistent beliefs about oneself as diminished, defeated or worthless, accompanied by deep and pervasive feelings of shame, guilt or failure related to the traumatic event
**3) persistent difficulties in sustaining relationships and in feeling close to others. The disturbance causes significant impairment in personal, family, social, educational, occupational or other important areas of functioning.
{{blockquotebottom}}

===='''Changes in DSM-5'''====
* The diagnostic criteria for post-traumatic stress disorder changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here] and [[w:DSM-5|here]].


=== Developmental sensitivities ===
{{collapse top| Click for more information}}
*Diagnostic thresholds have been lowered for children and adolescents to account for development.
*Separate and additional criteria have been added for children age 6 or younger.
*Child sexual abuse has been found to have a substantial effect on the development of PTSD.&lt;ref&gt;{{cite journal|last1=Paolucci|first1=Elizabeth ODDONE|last2=Genuis|first2=Mark L.|last3=Violato|first3=Claudio|title=A Meta-Analysis of the Published Research on the Effects of Child Sexual Abuse|journal=The Journal of Psychology|date=January 2001|volume=135|issue=1|pages=17–36|doi=10.1080/00223980109603677}}&lt;/ref&gt;
*Children with higher exposure to trauma, less social support, and other major life events are more likely to have continued PTSD symptoms 7 months after a trauma.&lt;ref name=&quot;:2&quot;&gt;{{cite journal |last1=La Greca|first1=A|last2=Silverman|first2=WK|last3=Vernberg|first3=EM|last4=Prinstein|first4=MJ|title=Symptoms of posttraumatic stress in children after Hurricane Andrew: a prospective study.|journal=Journal of consulting and clinical psychology|date=August 1996|volume=64|issue=4|pages=712-23|pmid=8803361}}&lt;/ref&gt; 10 months after a trauma, however, only experience of a major life event remained predictive of continuing PTSD symptoms.&lt;ref name=&quot;:2&quot; /&gt; 
*Lack of social support, specifically lack of support by a teacher, was predictive of higher PTSD symptoms among children who had gone through an environmental trauma.&lt;ref name=&quot;:2&quot; /&gt;  
*Children who reported using blame and anger as strategies for coping had higher levels of PTSD symptoms 10 months after a trauma.&lt;ref name=&quot;:2&quot; /&gt;  
*Negative affect in children before a traumatic event was predictive of development of PTSD symptoms after a traumatic event.&lt;ref&gt;{{cite journal|last1=Weems|first1=CF|last2=Pina|first2=AA|last3=Costa|first3=NM|last4=Watts|first4=SE|last5=Taylor|first5=LK|last6=Cannon|first6=MF|title=Predisaster trait anxiety and negative affect predict posttraumatic stress in youths after hurricane Katrina.|journal=Journal of consulting and clinical psychology|date=February 2007|volume=75|issue=1|pages=154-9|doi=10.1037/0022-006X.75.1.154|pmid=17295574}}&lt;/ref&gt;  

[[wikipedia:Posttraumatic_stress_disorder|Posttraumatic stress disorder]] is now more sensitive to development in that diagnostic thresholds have been lowered for children and adolescents. Furthermore, separate and additional criteria have been added for children age 6 years of age or younger.
{{collapse bottom}}

=== Base rates of PTSD in different clinical settings and populations ===
This section describes the demographic settings of the populations sampled, base rates of PTSD diagnoses, country/region sampled, and the diagnostic methods that were used. Using this information, clinicians will be able to anchor the most appropriate rate of PTSD that they are likely to see in their clinical practice.
* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']] 

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Demography
! Setting
! Base Rate
! Diagnostic Method
|-
| United States, nationally representative, age 18 and older
| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Kessler|first=Ronald C.|last2=Berglund|first2=Patricia|last3=Demler|first3=Olga|last4=Jin|first4=Robert|last5=Merikangas|first5=Kathleen R.|last6=Walters|first6=Ellen E.|date=2005-06-01|title=Lifetime Prevalence and Age-of-Onset Distributions of DSM-IV Disorders in the National Comorbidity Survey Replication|url=http://archpsyc.jamanetwork.com/article.aspx?doi=10.1001/archpsyc.62.6.593|journal=Archives of General Psychiatry|language=en|volume=62|issue=6|doi=10.1001/archpsyc.62.6.593|issn=0003-990X}}&lt;/ref&gt;
| 6.8%
| National Comorbidity Survey - Replication
|-
| Netherlands, nationally representative, age 18-80
| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=de Vries|first=Giel-Jan|last2=Olff|first2=Miranda|date=2009-08-01|title=The lifetime prevalence of traumatic events and posttraumatic stress disorder in the Netherlands|url=http://onlinelibrary.wiley.com/doi/10.1002/jts.20429/abstract|journal=Journal of Traumatic Stress|language=en|volume=22|issue=4|pages=259–267|doi=10.1002/jts.20429|issn=1573-6598}}&lt;/ref&gt;
| 7.4%
| Composite International Diagnostic Interview (CIDI)
|-
| United States, nationally representative, age 18 and older
| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Pietrzak|first=Robert H.|last2=Goldstein|first2=Risë B.|last3=Southwick|first3=Steven M.|last4=Grant|first4=Bridget F.|title=Prevalence and Axis I comorbidity of full and partial posttraumatic stress disorder in the United States: Results from Wave 2 of the National Epidemiologic Survey on Alcohol and Related Conditions|url=http://linkinghub.elsevier.com/retrieve/pii/S0887618510002288|journal=Journal of Anxiety Disorders|volume=25|issue=3|pages=456–465|doi=10.1016/j.janxdis.2010.11.010}}&lt;/ref&gt;
| 6.4%
| Wave 2 National Epidemiologic Survey on Alcohol and Related Conditions
|-
| Northern Ireland, representative sample, age 18 and older
| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Bunting|first=Brendan P.|last2=Ferry|first2=Finola R.|last3=Murphy|first3=Samuel D.|last4=O'Neill|first4=Siobhan M.|last5=Bolton|first5=David|date=2013-02-01|title=Trauma Associated With Civil Conflict and Posttraumatic Stress Disorder: Evidence From the Northern Ireland Study of Health and Stress|url=http://onlinelibrary.wiley.com/doi/10.1002/jts.21766/abstract|journal=Journal of Traumatic Stress|language=en|volume=26|issue=1|pages=134–141|doi=10.1002/jts.21766|issn=1573-6598}}&lt;/ref&gt;
| 8.8%
| Northern Ireland Study of Health and Stress
|-
| South Africa, nationally representative sample, age 18 and older
| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Atwoli|first=Lukoye|last2=Stein|first2=Dan J.|last3=Williams|first3=David R.|last4=Mclaughlin|first4=Katie A.|last5=Petukhova|first5=Maria|last6=Kessler|first6=Ronald C.|last7=Koenen|first7=Karestan C.|date=2013-07-03|title=Trauma and posttraumatic stress disorder in South Africa: analysis from the South African Stress and Health Study|url=https://doi.org/10.1186/1471-244X-13-182|journal=BMC Psychiatry|volume=13|pages=182|doi=10.1186/1471-244x-13-182|issn=1471-244X}}&lt;/ref&gt;
| 2.3%
| South African Stress and Health Study, using the Composite International Diagnostic Interview (CIDI)
|-
| U.S. Army and Marine Soldiers Deployed to Iraq and Afghanistan
| U.S. Service Members&lt;ref&gt;{{Cite journal|last=Hoge|first=Charles W.|last2=Castro|first2=Carl A.|last3=Messer|first3=Stephen C.|last4=McGurk|first4=Dennis|last5=Cotting|first5=Dave I.|last6=Koffman|first6=Robert L.|date=2004-07-01|title=Combat Duty in Iraq and Afghanistan, Mental Health Problems, and Barriers to Care|url=http://dx.doi.org/10.1056/NEJMoa040603|journal=New England Journal of Medicine|volume=351|issue=1|pages=13–22|doi=10.1056/nejmoa040603|issn=0028-4793|pmid=15229303}}&lt;/ref&gt;
|11.5% - 19.5% ♦
| PTSD Checklist
|-
|United States, nationally representative, ages 13-18
|Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Merikangas|first=Kathleen Ries|last2=He|first2=Jian-ping|last3=Burstein|first3=Marcy|last4=Swanson|first4=Sonja A.|last5=Avenevoli|first5=Shelli|last6=Cui|first6=Lihong|last7=Benjet|first7=Corina|last8=Georgiades|first8=Katholiki|last9=Swendsen|first9=Joel|title=Lifetime Prevalence of Mental Disorders in U.S. Adolescents: Results from the National Comorbidity Survey Replication–Adolescent Supplement (NCS-A)|url=https://doi.org/10.1016/j.jaac.2010.05.017|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=49|issue=10|pages=980–989|doi=10.1016/j.jaac.2010.05.017}}&lt;/ref&gt;
|5.0%
|National Comorbidity Survey Replication—Adolescent Supplement (NCS-A)
|}
♦ '''Note''': These rates were using broad PTSD Checklist scoring criteria of being scored positive if subjects reported at least one intrusion symptom, three avoidance symptoms, and two hyperarousal symptom that were categorized as at the moderate level. The 11.5% is for soldiers returning from deployment in Iraq, 19.5% is for soldiers returning from Afghanistan. Another common practice is to use a strict cutoff of 50 on the PCL, above which someone screens positive for PTSD. With this cutoff, rates are 6.2% and 12.9% for Service Members returned from Afghanistan and Iraqi, respectively.

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==

=== Psychometric properties of screening for PTSD ===
The following section contains a list of screening and diagnostic instruments for PTSD. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here.]]

* '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]'''''
{| class=&quot;wikitable&quot;
|-
! Measure 
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Highly recommended
|-
|[https://www.ptsd.va.gov/professional/assessment/adult-sr/ptsd-checklist.asp PCL (PTSD Checklist for DSM-5)]
|Self-Report 
|adult
|5-10 minutes
|X
|-
|[https://www.appi.org/products/structured-clinical-interview-for-dsm-5-scid-5 SCID-IV (Structured Clinical Interview for DSM-IV)]
|Clinician Administered Interview
|adult
|1-2 hours 
|X
|-
|[https://adaa.org/sites/default/files/Yusko%20_210.pdf PDS (Post-traumatic Diagnosis Scale)]
|Self-Report
|adult
|10-20 minutes
|X
|-
|[https://www.ptsd.va.gov/professional/assessment/adult-int/caps.asp CAPS (Clinician Administered PTSD Scale)]
|Clinician Administered Interview 
|adult and child versions available 
|40-60 minutes 
|X
|-
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''For Children and Adolescents Specifically'''
|-
|[https://www.reactionindex.com/index.php/ UCLA PTSD Reaction Index for DSM-5]
|Clinician Administered
|child, school age, adolescents
|20-30 minutes
|Yes
|-
|[https://www.aacap.org/App_Themes/AACAP/docs/resource_centers/resources/misc/child_ptsd_symptom_scale.pdf CPSS (Children's PTSD Symptom Scale)]
|Self-Report
|child
|10-20 minutes
|XX (new rec)
|-
|[https://medicine.tulane.edu/infant-institute Young Child PTSD Checklist (YCPC)]
|Parent Report
|child
|13 items
|Yes
|}

'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

=== Likelihood ratios and AUCs of screening measures for PTSD ===

* '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' 

{| class=&quot;wikitable sortable&quot;
|-
! Screening Measure (Primary Reference) !! AUC (sample size) !! DLR+ (score) !! DLR- (score) !! Clinical Generalizability
!Download
|-            
| ||  ||  ||  || 
|
|-
| ||  ||  ||  || 
|
|- 
| ||  ||  ||  || 
|
|-
| ||  ||  ||  || 
|
|-
| ||  ||  ||  || 
|
|-
|}

=== Interpreting PTSD screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]
* Also see the page on [https://en.wikipedia.org/wiki/Likelihood_ratios_in_diagnostic_testing likelihood ratios in diagnostic testing] for more information

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==
===Gold standard diagnostic interviews===
* For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

===Recommended diagnostic interviews for PTSD===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! colspan=&quot;10&quot; |Diagnostic instruments for PTSD
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| Children’s PTSD Inventory (CPTSD-I)
| Diagnostic Interview
| 6-18 years
| 50 items
| G
| 
| G
| 
| 
|
|-
| 
| 
| 
| 
| 
| 
| 
| 
| 
|

|-
| 
|
|
|
| 
| 
| 
| 
| 
|
|-
| 
|
| 
| 
| 
| 
| 
| 
| 
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Severity interviews for PTSD===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| [https://www.kennedykrieger.org/sites/default/files/library/documents/faculty/ksads-dsm-5-screener.pdf Schedule for Affective Disorders and Schizophrenia for School Age Children (K-SADS)]
| Diagnostic Interview
| 6-18 years
| 18 PTSD items
| G
| 
| G
| 
| 
|
|-
| Anxiety Disorders Interview Schedule, Child Version (ADIS-C)
| Diagnostic Interview
| 7-17 years
| 26 PTSD items
| G
| 
| G
| 
| 
|

|-
| Children’s Interview for Psychiatric Symptoms (ChIPS)
|Diagnostic Interview
|6 – 18 years
|31 PTSD items
| G
| 
| G
| 
| 
|
|-
| Diagnostic Infant and Preschool Assessment (DIPA)
|Diagnostic Interview
| Age 6 and younger
| 46 PTSD items
| G
| 
| G
| 
| 
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable
==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a brief overview of treatment options for PTSD and list of process and outcome measures for PTSD. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]].

=== Process measures ===

* Information coming soon

=== Outcome and severity measures ===
This table includes clinically significant benchmarks for PTSD specific outcome measures

* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
* ''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
| colspan=&quot;7&quot; |'''Clinically significant change benchmarks with common instruments for PTSD'''
|-
| colspan=&quot;7&quot; style=&quot;font-size:110%; text-align:center;&quot; span |&lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; |&lt;b&gt; Measure&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; |&lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; |&lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | Primary Care PTSD Screen
| style=&quot;text-align:center;&quot; | 1.0
| style=&quot;text-align:center;&quot; | 3.1
| style=&quot;text-align:center;&quot; | 2.0
| style=&quot;text-align:center;&quot; | 1.0
| style=&quot;text-align:center;&quot; | .8
| style=&quot;text-align:center;&quot; | .5
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | PTSD Checklist Scores 
| style=&quot;text-align:center;&quot; | 28.8
| style=&quot;text-align:center;&quot; | 40.8
| style=&quot;text-align:center;&quot; | 34.9
| style=&quot;text-align:center;&quot; | 4.6
| style=&quot;text-align:center;&quot; | 3.8
| style=&quot;text-align:center;&quot; | 2.3
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | Clinician Administered PTSD Scale
| style=&quot;text-align:center;&quot; | 28.8
| style=&quot;text-align:center;&quot; | 40.8
| style=&quot;text-align:center;&quot; | 34.9
| style=&quot;text-align:center;&quot; | 8.3
| style=&quot;text-align:center;&quot; | 7.0
| style=&quot;text-align:center;&quot; | 4.2
|}
=== Treatment ===
* Please refer to the Wikipedia page on [[wikipedia:Posttraumatic_stress_disorder|PTSD]] for more information on available treatment for PTSD or go to the [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/post-traumatic-stress-disorder/ Effective Child Therapy] page for for a curated resource on effective treatments for PTSD.

{{collapse top| Click here for more information}}
'''&lt;big&gt;Behavioral interventions&lt;/big&gt;'''

Recommended (have ''significant benefit'') &lt;ref name=&quot;:0&quot;&gt;Department of Veterans Affairs and Department of Defense. . (2010). VA/DoD clinical practice guidelines: management of post-traumatic stress. Washington, D.C.: Veterans Health Administration, Department of Defense.&lt;/ref&gt;:

# Brief Cognitive Behavioral Therapy (CBT; 4-5 sessions)
#* This includes stress inoculation training, trauma-focused therapy including components of cognitive restructuring, Cognitive Processing Therapy (CPT), imaginal, virtual, and in-vivo exposure as in Prolonged Exposure psychotherapy (PE)&lt;sup&gt;2&lt;/sup&gt;
#* Patient education is recommended as part of psychotherapy for patients and family members
# EMDR may help both acute and chronic PTSD, especially individuals who have trouble with prolonged exposure or have trouble verbalizing their trauma.&lt;ref name=&quot;:1&quot;&gt;Ursano, R. J., Bell, C., Eth, S., Friedman, M., Norwood, A., Pfefferbaum, B., . . . McIntyre, J. S. (2004). ''Practice guideline for the treatment of patients with acute stress disorder and posttraumatic stress disorder'': American Psychiatric Publ
&lt;/ref&gt; 
## Long term gains require further study.

Treatments with weaker evidence (have ''some benefit'') &lt;ref name=&quot;:0&quot; /&gt;:

# Patient education,
# Imagery rehearsal therapy,
# Psychodynamic therapy,
# Hypnosis,
# Relaxation techniques,  
# and Group therapy.

Treatment with ''unknown benefit'' &lt;ref name=&quot;:0&quot; /&gt;: 

# Web-based CBT,
# Acceptance and commitment therapy,
# and Dialectical Behavioral Therapy. 

&lt;big&gt;'''Medication'''&lt;/big&gt;

*SSRIs are more effective than placebo in treating PTSD.&lt;ref name=&quot;:1&quot; /&gt;
*There is no evidence to support a medication to prevent the development of PTSD.&lt;ref name=&quot;:0&quot; /&gt;
**Imipramine, propranolol, prazosin, other antidepressants, anticonvulsants, and atypical antipsychotics have ''unknown benefit.''
**Strongly recommend against the use of benzodiazepines&lt;sup&gt;2&lt;/sup&gt; and typical antipsychotics since they have ''no benefit and potential harm.''

{{collapse bottom}}

== '''External resources''' ==

=== For professionals ===
* [http://www.som.uq.edu.au/ptsd Post Traumatic Stress Disorder Information Resource] from [http://www.uq.edu.au/ The University of Queensland School of Medicine]
* [http://www.apa.org/ptsd-guideline/ APA practice parameters for assessment and treatment for PTSD (Updated 2017)]
* [http://www.ptsd.va.gov/professional/index.asp Resources for professionals] from [http://www.ptsd.va.gov/index.asp VA National PTSD Center]
* [https://www.integration.samhsa.gov/clinical-practice/trauma-informed SAMHSA's Trauma-Informed Care Resources and Training]
* [https://www.integration.samhsa.gov/clinical-practice/trauma-informed SAMHSA's Resources on Intimate Partner Violence (IPV) for Clinicians, Patients, Families, and Organizations]

=== For caregivers ===
* [http://www.nctsn.org/sites/default/files/assets/pdfs/tips_for_parents_media_final.pdf Tips for parents on shooting media coverage]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/parents_guidelines_for_helping_teens_after_the_recent_attacks.pdf Parent guidelines to helping youths after a shooting]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/helping_teens_with_traumatic_grief_caregivers_final.pdf Caregiver tips for helping teens with traumatic grief]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/10things_schoolage_ctg.pdf Caregiver tips for helping school-age children with traumatic grief]
*[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/post-traumatic-stress-disorder/ EffectiveChildTherapy.Org information on PTSD]
*[https://sccap53.org Society of Clinical Child and Adolescent Psychology]

=== For educators ===
* [http://www.nctsn.org/sites/default/files/assets/pdfs/pfa_for_schools_appendix_a_teachers.pdf Teacher tips for providing psychological first aid]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/helping_youth_after_community_violence_educators.pdf Educator tips for helping youths after a community trauma]

=== For public ===
* [https://complicatedgrief.columbia.edu/for-the-public/resources/ Resources for grief and loss]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/psychological_information_sheet_two_pager.pdf Psychological impacts of recent shootings]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/youth_journalists.pdf Tips to talking to youths about a shooting]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/talking_to_children_about_the_shooting.pdf Tips to talking to children about a shooting]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/helping_young_children_heal_crisis.pdf Helping young children heal after a crisis]
* [http://www.nctsn.org/content/psychological-first-aid-schoolspfa Psychological first aid for schools]
* [http://www.nctsn.org/content/psychological-first-aid Resources and manuals on psychological first aid]
* [http://www.ptsd.va.gov/public/index.asp Resources for the public] from [http://www.ptsd.va.gov/index.asp VA National PTSD Center]
* [https://curlie.org/Health/Mental_Health/Disorders/Anxiety/Post-traumatic_Stress Posttraumatic stress disorder] at Curlie (based on DMOZ)

=='''References'''==
{{collapse top|Click here for references}}
{{Reflist|30em}}

[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]
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{{medical disclaimer}}
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==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
* For background information on what assessment portfolios are, click the link in the heading above.
* Want even 'more' information about this topic? There's an extended version of this page [[Evidence-based assessment/Posttraumatic stress disorder (disorder portfolio)/extended version|here]].

== [[Evidence based assessment/Preparation phase|'''Preparation phase''']] ==
=== Diagnostic criteria for post traumatic stress disorder ===

===='''ICD-11 Diagnostic Criteria'''====
{{blockquotetop}}
===='''ICD-11 Criteria'''====
*Post-traumatic stress disorder (PTSD) is a syndrome that develops following exposure to an extremely threatening or horrific event or series of events that is characterized by all of the following: 
**1) re-experiencing the traumatic event or events in the present in the form of vivid intrusive memories, flashbacks, or nightmares, which are typically accompanied by strong and overwhelming emotions such as fear or horror and strong physical sensations, or feelings of being overwhelmed or immersed in the same intense emotions that were experienced during the traumatic event; 
**2) avoidance of thoughts and memories of the event or events, or avoidance of activities, situations, or people reminiscent of the event or events
**3) persistent perceptions of heightened current threat, for example as indicated by hypervigilance or an enhanced startle reaction to stimuli such as unexpected noises. 
*The symptoms must persist for at least several weeks and cause significant impairment in personal, family, social, educational, occupational or other important areas of functioning. 

Additionally, ICD 11 includes a category called &quot;Complex post-traumatic stress disorder,&quot; which is described as:
*A disorder that may develop following exposure to an event or series of events of an extremely threatening or horrific nature, most commonly prolonged or repetitive events from which escape is difficult or impossible (e.g., torture, slavery, genocide campaigns, prolonged domestic violence, repeated childhood sexual or physical abuse). 
*The disorder is characterized by the core symptoms of PTSD; that is, all diagnostic requirements for PTSD have been met at some point during the course of the disorder. 
*In addition, Complex PTSD is characterized by:
**1) severe and pervasive problems in affect regulation; 
**2) persistent beliefs about oneself as diminished, defeated or worthless, accompanied by deep and pervasive feelings of shame, guilt or failure related to the traumatic event
**3) persistent difficulties in sustaining relationships and in feeling close to others. The disturbance causes significant impairment in personal, family, social, educational, occupational or other important areas of functioning.
{{blockquotebottom}}

===='''Changes in DSM-5'''====
* The diagnostic criteria for post-traumatic stress disorder changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here] and [[w:DSM-5|here]].


=== Developmental sensitivities ===
{{collapse top| Click for more information}}
*Diagnostic thresholds have been lowered for children and adolescents to account for development.
*Separate and additional criteria have been added for children age 6 or younger.
*Child sexual abuse has been found to have a substantial effect on the development of PTSD.&lt;ref&gt;{{cite journal|last1=Paolucci|first1=Elizabeth ODDONE|last2=Genuis|first2=Mark L.|last3=Violato|first3=Claudio|title=A Meta-Analysis of the Published Research on the Effects of Child Sexual Abuse|journal=The Journal of Psychology|date=January 2001|volume=135|issue=1|pages=17–36|doi=10.1080/00223980109603677}}&lt;/ref&gt;
*Children with higher exposure to trauma, less social support, and other major life events are more likely to have continued PTSD symptoms 7 months after a trauma.&lt;ref name=&quot;:2&quot;&gt;{{cite journal |last1=La Greca|first1=A|last2=Silverman|first2=WK|last3=Vernberg|first3=EM|last4=Prinstein|first4=MJ|title=Symptoms of posttraumatic stress in children after Hurricane Andrew: a prospective study.|journal=Journal of consulting and clinical psychology|date=August 1996|volume=64|issue=4|pages=712-23|pmid=8803361}}&lt;/ref&gt; 10 months after a trauma, however, only experience of a major life event remained predictive of continuing PTSD symptoms.&lt;ref name=&quot;:2&quot; /&gt; 
*Lack of social support, specifically lack of support by a teacher, was predictive of higher PTSD symptoms among children who had gone through an environmental trauma.&lt;ref name=&quot;:2&quot; /&gt;  
*Children who reported using blame and anger as strategies for coping had higher levels of PTSD symptoms 10 months after a trauma.&lt;ref name=&quot;:2&quot; /&gt;  
*Negative affect in children before a traumatic event was predictive of development of PTSD symptoms after a traumatic event.&lt;ref&gt;{{cite journal|last1=Weems|first1=CF|last2=Pina|first2=AA|last3=Costa|first3=NM|last4=Watts|first4=SE|last5=Taylor|first5=LK|last6=Cannon|first6=MF|title=Predisaster trait anxiety and negative affect predict posttraumatic stress in youths after hurricane Katrina.|journal=Journal of consulting and clinical psychology|date=February 2007|volume=75|issue=1|pages=154-9|doi=10.1037/0022-006X.75.1.154|pmid=17295574}}&lt;/ref&gt;  

[[wikipedia:Posttraumatic_stress_disorder|Posttraumatic stress disorder]] is now more sensitive to development in that diagnostic thresholds have been lowered for children and adolescents. Furthermore, separate and additional criteria have been added for children age 6 years of age or younger.
{{collapse bottom}}

=== Base rates of PTSD in different clinical settings and populations ===
This section describes the demographic settings of the populations sampled, base rates of PTSD diagnoses, country/region sampled, and the diagnostic methods that were used. Using this information, clinicians will be able to anchor the most appropriate rate of PTSD that they are likely to see in their clinical practice.
* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']] 

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Demography
! Setting
! Base Rate
! Diagnostic Method
|-
| United States, nationally representative, age 18 and older
| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Kessler|first=Ronald C.|last2=Berglund|first2=Patricia|last3=Demler|first3=Olga|last4=Jin|first4=Robert|last5=Merikangas|first5=Kathleen R.|last6=Walters|first6=Ellen E.|date=2005-06-01|title=Lifetime Prevalence and Age-of-Onset Distributions of DSM-IV Disorders in the National Comorbidity Survey Replication|url=http://archpsyc.jamanetwork.com/article.aspx?doi=10.1001/archpsyc.62.6.593|journal=Archives of General Psychiatry|language=en|volume=62|issue=6|doi=10.1001/archpsyc.62.6.593|issn=0003-990X}}&lt;/ref&gt;
| 6.8%
| National Comorbidity Survey - Replication
|-
| Netherlands, nationally representative, age 18-80
| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=de Vries|first=Giel-Jan|last2=Olff|first2=Miranda|date=2009-08-01|title=The lifetime prevalence of traumatic events and posttraumatic stress disorder in the Netherlands|url=http://onlinelibrary.wiley.com/doi/10.1002/jts.20429/abstract|journal=Journal of Traumatic Stress|language=en|volume=22|issue=4|pages=259–267|doi=10.1002/jts.20429|issn=1573-6598}}&lt;/ref&gt;
| 7.4%
| Composite International Diagnostic Interview (CIDI)
|-
| United States, nationally representative, age 18 and older
| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Pietrzak|first=Robert H.|last2=Goldstein|first2=Risë B.|last3=Southwick|first3=Steven M.|last4=Grant|first4=Bridget F.|title=Prevalence and Axis I comorbidity of full and partial posttraumatic stress disorder in the United States: Results from Wave 2 of the National Epidemiologic Survey on Alcohol and Related Conditions|url=http://linkinghub.elsevier.com/retrieve/pii/S0887618510002288|journal=Journal of Anxiety Disorders|volume=25|issue=3|pages=456–465|doi=10.1016/j.janxdis.2010.11.010}}&lt;/ref&gt;
| 6.4%
| Wave 2 National Epidemiologic Survey on Alcohol and Related Conditions
|-
| Northern Ireland, representative sample, age 18 and older
| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Bunting|first=Brendan P.|last2=Ferry|first2=Finola R.|last3=Murphy|first3=Samuel D.|last4=O'Neill|first4=Siobhan M.|last5=Bolton|first5=David|date=2013-02-01|title=Trauma Associated With Civil Conflict and Posttraumatic Stress Disorder: Evidence From the Northern Ireland Study of Health and Stress|url=http://onlinelibrary.wiley.com/doi/10.1002/jts.21766/abstract|journal=Journal of Traumatic Stress|language=en|volume=26|issue=1|pages=134–141|doi=10.1002/jts.21766|issn=1573-6598}}&lt;/ref&gt;
| 8.8%
| Northern Ireland Study of Health and Stress
|-
| South Africa, nationally representative sample, age 18 and older
| Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Atwoli|first=Lukoye|last2=Stein|first2=Dan J.|last3=Williams|first3=David R.|last4=Mclaughlin|first4=Katie A.|last5=Petukhova|first5=Maria|last6=Kessler|first6=Ronald C.|last7=Koenen|first7=Karestan C.|date=2013-07-03|title=Trauma and posttraumatic stress disorder in South Africa: analysis from the South African Stress and Health Study|url=https://doi.org/10.1186/1471-244X-13-182|journal=BMC Psychiatry|volume=13|pages=182|doi=10.1186/1471-244x-13-182|issn=1471-244X}}&lt;/ref&gt;
| 2.3%
| South African Stress and Health Study, using the Composite International Diagnostic Interview (CIDI)
|-
| U.S. Army and Marine Soldiers Deployed to Iraq and Afghanistan
| U.S. Service Members&lt;ref&gt;{{Cite journal|last=Hoge|first=Charles W.|last2=Castro|first2=Carl A.|last3=Messer|first3=Stephen C.|last4=McGurk|first4=Dennis|last5=Cotting|first5=Dave I.|last6=Koffman|first6=Robert L.|date=2004-07-01|title=Combat Duty in Iraq and Afghanistan, Mental Health Problems, and Barriers to Care|url=http://dx.doi.org/10.1056/NEJMoa040603|journal=New England Journal of Medicine|volume=351|issue=1|pages=13–22|doi=10.1056/nejmoa040603|issn=0028-4793|pmid=15229303}}&lt;/ref&gt;
|11.5% - 19.5% ♦
| PTSD Checklist
|-
|United States, nationally representative, ages 13-18
|Non-clinical: Population based&lt;ref&gt;{{Cite journal|last=Merikangas|first=Kathleen Ries|last2=He|first2=Jian-ping|last3=Burstein|first3=Marcy|last4=Swanson|first4=Sonja A.|last5=Avenevoli|first5=Shelli|last6=Cui|first6=Lihong|last7=Benjet|first7=Corina|last8=Georgiades|first8=Katholiki|last9=Swendsen|first9=Joel|title=Lifetime Prevalence of Mental Disorders in U.S. Adolescents: Results from the National Comorbidity Survey Replication–Adolescent Supplement (NCS-A)|url=https://doi.org/10.1016/j.jaac.2010.05.017|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|volume=49|issue=10|pages=980–989|doi=10.1016/j.jaac.2010.05.017}}&lt;/ref&gt;
|5.0%
|National Comorbidity Survey Replication—Adolescent Supplement (NCS-A)
|}
♦ '''Note''': These rates were using broad PTSD Checklist scoring criteria of being scored positive if subjects reported at least one intrusion symptom, three avoidance symptoms, and two hyperarousal symptom that were categorized as at the moderate level. The 11.5% is for soldiers returning from deployment in Iraq, 19.5% is for soldiers returning from Afghanistan. Another common practice is to use a strict cutoff of 50 on the PCL, above which someone screens positive for PTSD. With this cutoff, rates are 6.2% and 12.9% for Service Members returned from Afghanistan and Iraqi, respectively.

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==

=== Psychometric properties of screening for PTSD ===
The following section contains a list of screening and diagnostic instruments for PTSD. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here.]]

* '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]'''''
{| class=&quot;wikitable&quot;
|-
! Measure 
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Highly recommended
|-
|[https://www.ptsd.va.gov/professional/assessment/adult-sr/ptsd-checklist.asp PCL (PTSD Checklist for DSM-5)]
|Self-Report 
|adult
|5-10 minutes
|X
|-
|[https://adaa.org/sites/default/files/Yusko%20_210.pdf PDS (Post-traumatic Diagnosis Scale)]
|Self-Report
|adult
|10-20 minutes
|X
|-
|[https://www.ptsd.va.gov/professional/assessment/adult-int/caps.asp CAPS (Clinician Administered PTSD Scale)]
|Clinician Administered Interview 
|adult and child versions available 
|40-60 minutes 
|X
|-
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''For Children and Adolescents Specifically'''
|-
|[https://www.reactionindex.com/index.php/ UCLA PTSD Reaction Index for DSM-5]
|Clinician Administered
|child, school age, adolescents
|20-30 minutes
|Yes
|-
|[https://www.aacap.org/App_Themes/AACAP/docs/resource_centers/resources/misc/child_ptsd_symptom_scale.pdf CPSS (Children's PTSD Symptom Scale)]
|Self-Report
|child
|10-20 minutes
|XX (new rec)
|-
|[https://medicine.tulane.edu/infant-institute Young Child PTSD Checklist (YCPC)]
|Parent Report
|child
|13 items
|Yes
|}

'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

=== Likelihood ratios and AUCs of screening measures for PTSD ===

* '''''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments click here.]''''' 

{| class=&quot;wikitable sortable&quot;
|-
! Screening Measure (Primary Reference) !! AUC (sample size) !! DLR+ (score) !! DLR- (score) !! Clinical Generalizability
!Download
|-            
| ||  ||  ||  || 
|
|-
| ||  ||  ||  || 
|
|- 
| ||  ||  ||  || 
|
|-
| ||  ||  ||  || 
|
|-
| ||  ||  ||  || 
|
|-
|}

=== Interpreting PTSD screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]
* Also see the page on [https://en.wikipedia.org/wiki/Likelihood_ratios_in_diagnostic_testing likelihood ratios in diagnostic testing] for more information

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==
===Gold standard diagnostic interviews===
* For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

===Recommended diagnostic interviews for PTSD===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! colspan=&quot;10&quot; |Diagnostic instruments for PTSD
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| Children’s PTSD Inventory (CPTSD-I)
| Diagnostic Interview
| 6-18 years
| 50 items
| G
| 
| G
| 
| 
|
|-
| 
| 
| 
| 
| 
| 
| 
| 
| 
|

|-
| 
|
|
|
| 
| 
| 
| 
| 
|
|-
| 
|
| 
| 
| 
| 
| 
| 
| 
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Severity interviews for PTSD===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| [https://www.kennedykrieger.org/sites/default/files/library/documents/faculty/ksads-dsm-5-screener.pdf Schedule for Affective Disorders and Schizophrenia for School Age Children (K-SADS)]
| Diagnostic Interview
| 6-18 years
| 18 PTSD items
| G
| 
| G
| 
| 
|
|-
| Anxiety Disorders Interview Schedule, Child Version (ADIS-C)
| Diagnostic Interview
| 7-17 years
| 26 PTSD items
| G
| 
| G
| 
| 
|

|-
| Children’s Interview for Psychiatric Symptoms (ChIPS)
|Diagnostic Interview
|6 – 18 years
|31 PTSD items
| G
| 
| G
| 
| 
|
|-
| Diagnostic Infant and Preschool Assessment (DIPA)
|Diagnostic Interview
| Age 6 and younger
| 46 PTSD items
| G
| 
| G
| 
| 
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable
==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a brief overview of treatment options for PTSD and list of process and outcome measures for PTSD. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]].

=== Process measures ===

* Information coming soon

=== Outcome and severity measures ===
This table includes clinically significant benchmarks for PTSD specific outcome measures

* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
* ''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
| colspan=&quot;7&quot; |'''Clinically significant change benchmarks with common instruments for PTSD'''
|-
| colspan=&quot;7&quot; style=&quot;font-size:110%; text-align:center;&quot; span |&lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; |&lt;b&gt; Measure&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; |&lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; |&lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | Primary Care PTSD Screen
| style=&quot;text-align:center;&quot; | 1.0
| style=&quot;text-align:center;&quot; | 3.1
| style=&quot;text-align:center;&quot; | 2.0
| style=&quot;text-align:center;&quot; | 1.0
| style=&quot;text-align:center;&quot; | .8
| style=&quot;text-align:center;&quot; | .5
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | PTSD Checklist Scores 
| style=&quot;text-align:center;&quot; | 28.8
| style=&quot;text-align:center;&quot; | 40.8
| style=&quot;text-align:center;&quot; | 34.9
| style=&quot;text-align:center;&quot; | 4.6
| style=&quot;text-align:center;&quot; | 3.8
| style=&quot;text-align:center;&quot; | 2.3
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; | Clinician Administered PTSD Scale
| style=&quot;text-align:center;&quot; | 28.8
| style=&quot;text-align:center;&quot; | 40.8
| style=&quot;text-align:center;&quot; | 34.9
| style=&quot;text-align:center;&quot; | 8.3
| style=&quot;text-align:center;&quot; | 7.0
| style=&quot;text-align:center;&quot; | 4.2
|}
=== Treatment ===
* Please refer to the Wikipedia page on [[wikipedia:Posttraumatic_stress_disorder|PTSD]] for more information on available treatment for PTSD or go to the [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/post-traumatic-stress-disorder/ Effective Child Therapy] page for for a curated resource on effective treatments for PTSD.

{{collapse top| Click here for more information}}
'''&lt;big&gt;Behavioral interventions&lt;/big&gt;'''

Recommended (have ''significant benefit'') &lt;ref name=&quot;:0&quot;&gt;Department of Veterans Affairs and Department of Defense. . (2010). VA/DoD clinical practice guidelines: management of post-traumatic stress. Washington, D.C.: Veterans Health Administration, Department of Defense.&lt;/ref&gt;:

# Brief Cognitive Behavioral Therapy (CBT; 4-5 sessions)
#* This includes stress inoculation training, trauma-focused therapy including components of cognitive restructuring, Cognitive Processing Therapy (CPT), imaginal, virtual, and in-vivo exposure as in Prolonged Exposure psychotherapy (PE)&lt;sup&gt;2&lt;/sup&gt;
#* Patient education is recommended as part of psychotherapy for patients and family members
# EMDR may help both acute and chronic PTSD, especially individuals who have trouble with prolonged exposure or have trouble verbalizing their trauma.&lt;ref name=&quot;:1&quot;&gt;Ursano, R. J., Bell, C., Eth, S., Friedman, M., Norwood, A., Pfefferbaum, B., . . . McIntyre, J. S. (2004). ''Practice guideline for the treatment of patients with acute stress disorder and posttraumatic stress disorder'': American Psychiatric Publ
&lt;/ref&gt; 
## Long term gains require further study.

Treatments with weaker evidence (have ''some benefit'') &lt;ref name=&quot;:0&quot; /&gt;:

# Patient education,
# Imagery rehearsal therapy,
# Psychodynamic therapy,
# Hypnosis,
# Relaxation techniques,  
# and Group therapy.

Treatment with ''unknown benefit'' &lt;ref name=&quot;:0&quot; /&gt;: 

# Web-based CBT,
# Acceptance and commitment therapy,
# and Dialectical Behavioral Therapy. 

&lt;big&gt;'''Medication'''&lt;/big&gt;

*SSRIs are more effective than placebo in treating PTSD.&lt;ref name=&quot;:1&quot; /&gt;
*There is no evidence to support a medication to prevent the development of PTSD.&lt;ref name=&quot;:0&quot; /&gt;
**Imipramine, propranolol, prazosin, other antidepressants, anticonvulsants, and atypical antipsychotics have ''unknown benefit.''
**Strongly recommend against the use of benzodiazepines&lt;sup&gt;2&lt;/sup&gt; and typical antipsychotics since they have ''no benefit and potential harm.''

{{collapse bottom}}

== '''External resources''' ==

=== For professionals ===
* [http://www.som.uq.edu.au/ptsd Post Traumatic Stress Disorder Information Resource] from [http://www.uq.edu.au/ The University of Queensland School of Medicine]
* [http://www.apa.org/ptsd-guideline/ APA practice parameters for assessment and treatment for PTSD (Updated 2017)]
* [http://www.ptsd.va.gov/professional/index.asp Resources for professionals] from [http://www.ptsd.va.gov/index.asp VA National PTSD Center]
* [https://www.integration.samhsa.gov/clinical-practice/trauma-informed SAMHSA's Trauma-Informed Care Resources and Training]
* [https://www.integration.samhsa.gov/clinical-practice/trauma-informed SAMHSA's Resources on Intimate Partner Violence (IPV) for Clinicians, Patients, Families, and Organizations]

=== For caregivers ===
* [http://www.nctsn.org/sites/default/files/assets/pdfs/tips_for_parents_media_final.pdf Tips for parents on shooting media coverage]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/parents_guidelines_for_helping_teens_after_the_recent_attacks.pdf Parent guidelines to helping youths after a shooting]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/helping_teens_with_traumatic_grief_caregivers_final.pdf Caregiver tips for helping teens with traumatic grief]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/10things_schoolage_ctg.pdf Caregiver tips for helping school-age children with traumatic grief]
*[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/post-traumatic-stress-disorder/ EffectiveChildTherapy.Org information on PTSD]
*[https://sccap53.org Society of Clinical Child and Adolescent Psychology]

=== For educators ===
* [http://www.nctsn.org/sites/default/files/assets/pdfs/pfa_for_schools_appendix_a_teachers.pdf Teacher tips for providing psychological first aid]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/helping_youth_after_community_violence_educators.pdf Educator tips for helping youths after a community trauma]

=== For public ===
* [https://complicatedgrief.columbia.edu/for-the-public/resources/ Resources for grief and loss]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/psychological_information_sheet_two_pager.pdf Psychological impacts of recent shootings]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/youth_journalists.pdf Tips to talking to youths about a shooting]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/talking_to_children_about_the_shooting.pdf Tips to talking to children about a shooting]
* [http://www.nctsn.org/sites/default/files/assets/pdfs/helping_young_children_heal_crisis.pdf Helping young children heal after a crisis]
* [http://www.nctsn.org/content/psychological-first-aid-schoolspfa Psychological first aid for schools]
* [http://www.nctsn.org/content/psychological-first-aid Resources and manuals on psychological first aid]
* [http://www.ptsd.va.gov/public/index.asp Resources for the public] from [http://www.ptsd.va.gov/index.asp VA National PTSD Center]
* [https://curlie.org/Health/Mental_Health/Disorders/Anxiety/Post-traumatic_Stress Posttraumatic stress disorder] at Curlie (based on DMOZ)

=='''References'''==
{{collapse top|Click here for references}}
{{Reflist|30em}}

[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]
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==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
* For background information on what assessment portfolios are, click the link in the heading above.
* Want more information? There's an extended version of this page [[Evidence-based assessment/Simple phobia/Extended version|here]].

== [[Evidence based assessment/Preparation phase|'''Preparation phase''']] ==
=== Diagnostic criteria for phobic anxiety disorders ===
{{collapse top| Click here for ICD-11 and DSM-5 Diagnostic Information}}
{{blockquotetop}}
'''ICD-11 Diagnostic Criteria'''

Specific phobia is characterized by a marked and excessive fear or anxiety that consistently occurs when exposed to one or more specific objects or situations (e.g., proximity to certain animals, flying, heights, closed spaces, sight of blood or injury) and that is out of proportion to actual danger. The phobic objects or situations are avoided or else endured with intense fear or anxiety. Symptoms persist for at least several months and are sufficiently severe to result in significant distress or significant impairment in personal, family, social, educational, occupational, or other important areas of functioning.

'''Inclusions'''

*Simple phobia

*Acrophobia

*Claustrophobia

'''Exclusions'''

*Body dysmorphic disorder

*Hypochondriasis

'''Changes in DSM-5'''

The diagnostic criteria for simple phobia changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here] and [[w:DSM-5|here]].
{{blockquotebottom}}
{{Collapse bottom}}

=== Base rates of simple phobia in different populations and clinical settings ===
This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of simple phobia that they are likely to see in their clinical practice.
* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']] 
{| class=&quot;wikitable&quot;
|-
! Demography 
! Setting !! Base Rate !! Diagnostic Method
|-
| California&lt;ref name=&quot;VegaEtAl1998&quot; /&gt; 
| Mexican-American Prevalence and Service Survey (MAPSS) – adults 18+, all specific phobias || 7.4% || CIDI/DSM-III-R
|-
| All of US&lt;ref name=&quot;KesslerEtAl2005a&quot; /&gt; 
| NCS replication, adults 18+, 12-month prevalence || 8.7% || CIDI/DSM-IV
|-
| All of US&lt;ref name=&quot;KesslerEtAl2005b&quot; /&gt; 
| National Comorbidity Survey (NCS); non-institutionalized adults between 18-54, all specific phobias || 11.3% || CIDI/DSM-III-R
|-
| All of US&lt;ref name=&quot;StinsonEtAl2007&quot; /&gt; 
| National Epidemiologic Survey on Alcohol and Related Conditions (NESARC), adults 18+, all specific phobias || 9.4% || AUDADIS-IV/DSM-IV
|-
| New Haven, CT&lt;ref name=&quot;RobinsEtAl1984&quot; /&gt; 
| Adults 18+, all specific phobias || 7.8% || Clinical interview/DSM-III
|-
| Baltimore, MD&lt;ref name=&quot;RobinsEtAl1984&quot; /&gt;
| Adults 18+, all specific phobias || 23.3% || Clinical interview/DSM-III
|-
| St. Louis, MI&lt;ref name=&quot;RobinsEtAl1984&quot; /&gt; 
| Adults 18+, all specific phobias || 11.1% || Clinical interview/DSM-III
|-
| All of US&lt;ref name=&quot;GrantEtAl2004&quot; /&gt; 
| National Epidemiologic Survey on Alcohol and Related Conditions (NESARC), adults 18+, all Specific Phobias || 7.14% || AUDADIS-IV/DSM-IV
|-
|| All of US 
| Different age groups, all specific phobias || Community Prevalence: 7-9%

Children: 5%

13- to 17- year olds: 16%

Older Adults: 3-5%

Note: Females are more frequently affected than males at a rate of 2:1
 | DSM-V
|-
|Varied&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Rettew|first=David C.|last2=Lynch|first2=Alicia Doyle|last3=Achenbach|first3=Thomas M.|last4=Dumenci|first4=Levent|last5=Ivanova|first5=Masha Y.|date=2009-09|title=Meta-analyses of agreement between diagnoses made from clinical evaluations and standardized diagnostic interviews|url=http://dx.doi.org/10.1002/mpr.289|journal=International Journal of Methods in Psychiatric Research|language=en|volume=18|issue=3|pages=169–184|doi=10.1002/mpr.289|issn=1049-8931}}&lt;/ref&gt;
|Outpatient clinic (DAU)
|6% (specific)
|Varied
|-
|Varied&lt;ref name=&quot;:13&quot;&gt;{{Cite journal|last=Rettew|first=David C.|last2=Lynch|first2=Alicia Doyle|last3=Achenbach|first3=Thomas M.|last4=Dumenci|first4=Levent|last5=Ivanova|first5=Masha Y.|date=2009-09|title=Meta-analyses of agreement between diagnoses made from clinical evaluations and standardized diagnostic interviews|url=http://dx.doi.org/10.1002/mpr.289|journal=International Journal of Methods in Psychiatric Research|language=en|volume=18|issue=3|pages=169–184|doi=10.1002/mpr.289|issn=1049-8931}}&lt;/ref&gt;
|Outpatient clinic (SDI)
|15% (specific)
|Varied
|-
|Varied&lt;ref name=&quot;:3&quot;&gt;Kessler, R. C., Avenevoli, S., Costello, E. J., Georgiades, K., Green, J. G., Gruber, M. J., . . . Merikangas, K. R. (2012). Prevalence, persistence, and sociodemographic correlates of DSM-IV disorders in the National Comorbidity Survey Replication Adolescent Supplement. Archives of General Psychiatry, 69(4), 372-380. doi:10.1001/archgenpsychiatry.2011.160&lt;/ref&gt;
|General Population
|19% specific
|Varied
|}

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==
=== Psychometric properties of screening instruments for simple phobia ===
The following section contains a list of screening and diagnostic instruments for simple phobia. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. 

* '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]'''''

{| class=&quot;wikitable sortable&quot;
! colspan=&quot;5&quot; |Screening measures for simple phobia
|-
! Measure 
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Where to access
|-
|Fear Survey Schedule for Children-Revised (FSSC-R)
|Questionnaire (self-report)
|7-16 years
|19-20 minutes
|[https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470713334.app3 Yes (article PDF link)]
|-
|Fear of Spiders Questionnaire (FSQ)
|Questionnaire (self-report)
|13 years-adult
|5 minutes
|[https://www.sciencedirect.com/science/article/pii/000579169400072T Yes (article PDF link)]
|-
|Disgust Scale
|Questionnaire (self-report) 
|16 years-adult
|8 minutes
|[http://people.stern.nyu.edu/jhaidt/disgustscale.html Yes (Questionnaire homepage and PDF)]
|-
|Revised Children’s Anxiety and Depression Scale (RCADS)
|Questionnaire (Child)
|6-18
|12 minutes
|
*[http://www.corc.uk.net/outcome-experience-measures/revised-childrens-anxiety-and-depression-scale-and-subscales/ RCADS homepage]

'''PDFs for RCADS'''

*[https://mfr.osf.io/render?url=https://osf.io/s3fu2/?action=download%26mode=render RCADS Child Self-reported (8-18 years)]
*[https://mfr.osf.io/render?url=https://osf.io/fp9mk/?action=download%26mode=render RCADS Parent-reported]
*[https://mfr.osf.io/render?url=https://osf.io/vy7ta/?action=download%26mode=render Child Scoring Aid]
*[https://mfr.osf.io/render?url=https://osf.io/t4bz6/?action=download%26mode=render Parent Scoring Aid]

'''Subscales'''

*[https://mfr.osf.io/render?url=https://osf.io/ectpf/?action=download%26mode=render Social Anxiety/Phobia Self-reported]

*[https://mfr.osf.io/render?url=https://osf.io/pu6rn/?action=download%26mode=render Social Anxiety/Phobia Parent-reported]
*[https://mfr.osf.io/render?url=https://osf.io/4gc8d/?action=download%26mode=render Panic Self-Report]
*[https://mfr.osf.io/render?url=https://osf.io/nhcsu/?action=download%26mode=render Panic Parent-Report]

'''Translations'''

'''[https://mfr.osf.io/render?url=https://osf.io/qsjh9/?action=download%26mode=render User Guide]'''
|}
'''Note:''' Reliability and validity are included in the extended version (link). This table includes measures with Good or Excellent ratings.

=== Likelihood ratios and AUCs of screening measures for simple phobia ===
* ''For a list of the likelihood ratios for more broadly reaching screening instruments,'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments '''''click here.''''']
{| class=&quot;wikitable&quot;
|-
! Screening Measure (Primary Reference) !! AUC and Sample Size !! LR+ (Score) !! LR- (Score) !! Clinical generalizability
|-
| Specific Phobia of Vomiting Inventory (SPOVI)&lt;ref name=&quot;VealeEtAl2012&quot; /&gt;|| 0.944 (N = 185) || 24.3 (10+) || 0.03 (&lt;10) || High: able to distinguish between phobics and controls 
|-
| The Claustrophobia Scale (CS) - Anxiety Subscale (Rachman and Taylor, 1993) &lt;ref name=&quot;Ost2007&quot; /&gt;|| --- (N = 285) || 49.0 (24+) || 0.0002 (&lt;24) || High: able to distinguish between phobics and controls
|-
| The Claustrophobia Scale (CS) - Avoidance  Subscale (Rachman and Taylor, 1993)&lt;ref name=&quot;Ost2007&quot; /&gt;|| --- (N = 285) || 19.2 (9+) || 0.0004 (&lt;9) || High: able to distinguish between phobics and controls
|}
'''Search terms:''' [specific phobia] AND [sensitivity OR specificity] in Google Scholar and PsycINFO

=== Interpreting specific phobia screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==

===Gold standard diagnostic interviews===
For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

=== Recommended diagnostic instruments for simple phobia ===
{| class=&quot;wikitable sortable&quot;
! colspan=&quot;5&quot; |Diagnostic instruments for simple phobia
|-
!Measure
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Where to access
|-
|Anxiety Disorders Interview Schedule for Children (ADIS-C)&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|date=2001-08-01|title=Test-Retest Reliability of Anxiety Symptoms and Diagnoses With the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Versions|url=https://www.sciencedirect.com/science/article/pii/S0890856709603427|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|language=en|volume=40|issue=8|pages=937–944|doi=10.1097/00004583-200108000-00016|issn=0890-8567}}&lt;/ref&gt;
|Child
|6 years-adult
|90 minutes
|
* [https://books.google.com/books/about/Anxiety_Disorders_Interview_Schedule_for.html?id=xpR6V3rboxwC Purchase]
|-
|Anxiety Disorders Interview Schedule for Children (ADIS-P)&lt;ref name=&quot;:1&quot; /&gt;
|Parent
|6 years-adult
|90 minutes
|
* [https://books.google.com/books/about/Anxiety_Disorders_Interview_Schedule_for.html?id=xpR6V3rboxwC Purchase]
|-
|Diagnostic Interview Schedule for Children and Adolescents (DICA)
|Interview (clinician)
|6-17 years
|1-2 hours
|
|}
'''Note:''' Reliability and validity are included in the extended version (link). This table includes measures with Good or Excellent ratings.

==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a list of process and outcome measures for simple phobia. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. 
=== Process measures ===
{{blockquotetop}}
Two types of ''behavioral approach tests'' (BAT) can be used to observe patients in typically avoided situations. 

# A '''progressive BAT''' gradually exposes the patient to a fear-inducing situation in a step-by-step manner, and responses to each step are recorded.
# A '''selective BAT''' allows the clinician to select one or more challenges from the patient’s hierarchy, and the patient is to complete each challenge to induce a phobic response and rate the inducing fear.
{{blockquotebottom}}
=== Outcome and severity measures ===
This table includes clinically significant benchmarks for simple phobia specific outcome measures
* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
*''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable&quot;
! colspan=&quot;8&quot; |Clinically significant change benchmarks (based on published norms)
|-
! rowspan=&quot;2&quot; |&lt;big&gt;Measure&lt;/big&gt;
! rowspan=&quot;2&quot; |&lt;big&gt;Subscale&lt;/big&gt;!! colspan=&quot;3&quot; |&lt;big&gt;Cut-off scores&lt;/big&gt;!! colspan=&quot;3&quot; |'''&lt;big&gt;Critical Change&lt;/big&gt;'''
'''&lt;big&gt;(unstandardized scores)&lt;/big&gt;'''
|-
!&lt;big&gt;A&lt;/big&gt; 
!&lt;big&gt;B&lt;/big&gt;
!&lt;big&gt;C&lt;/big&gt;
!&lt;big&gt;95%&lt;/big&gt;
!&lt;big&gt;90%&lt;/big&gt;
!&lt;big&gt;SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/big&gt;
|-
| Dental Cognitions Questionnaire (1995 Norms)&lt;ref name=&quot;deJonghEtAl1995&quot; /&gt;
| || 9.1 || 16.4 || 41.2 || 4.2 || 3.6 || 2.2 
|-
| The Claustrophobia Questionnaire (2001 Norms)  - Total&lt;ref name=&quot;RadomskyEtAl02001&quot; /&gt;
| || 18.6 || 67.7 || 41.2 || 11. || 9.4 || 5.7
|-
| The Claustrophobia Questionnaire (2001 Norms) - Suffocation&lt;ref name=&quot;RadomskyEtAl02001&quot; /&gt;
| || 7.0 || 24.9 || 16.2 || 5.0 || 4.3 || 2.6 
|-
| The Claustrophobia Questionnaire (2001 Norms) - Restriction&lt;ref name=&quot;RadomskyEtAl02001&quot; /&gt;
| || 8.4 || 45.5 || 24.3 || 6.9 || 5.8 || 3.5
|-
|Spider Phobia Questionnaire (1996 Norms)&lt;ref name=&quot;MulkensEtAl1996&quot; /&gt;
| || 15.1 || 20.7 || 17.3 || 3.0 || 2.5 || 1.5
|-
|Anxiety Disorder Interview Schedule (ADIS) 
| || 5.9 || 4.4 || 5.2 || 0.2 || 0.2 || 0.1
|-
|Fear Survey Schedule for Children-Revised (FSSC-R) 
| || 77.8 || 159 || 118.4 || 6.3 || 5.3 || 3.2
|}

'''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean.
'''Search terms:''' [specific phobia] AND [adults] AND [clinical significance OR outcomes] in Google Scholar and PsycINFO

=== Treatment === 
Two treatments of specific phobia treatment include ''in-vivo exposure'' and ''virtual reality therapy''. The former is most effective in specific phobias by hierarchically exposing the client to the fear-inducing stimulus and measuring anxiety response. The latter therapy is most effective in driving and height fears by using computer-generated, interactive virtual environments that the clinician manipulates.

* Please refer to the page on [[wikipedia:Phobia|simple phobia]] for more information on available treatment or go to [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/fear-worry-and-anxiety/ Effective Child Therapy] for a curated resource on effective treatments for simple phobia.
*For information on conducting Exposure Therapy for anxiety disordered youth, see [https://www.bravepracticeforkids.com/ www.BravePracticeForKids.com]

=='''External Resources'''==

# [http://apps.who.int/classifications/icd10/browse/2010/en#/F40 ICD-10 diagnostic criteria]
# [https://en.wikiversity.org/w/index.php?title=Helping_Give_Away_Psychological_Science/Resources/Annotated_List_of_Where_and_How_to_Find_a_Therapist Find-a-Therapist] (a curated list of find-a-therapist websites where you can find a provider)
# [https://www.nimh.nih.gov/health/topics/anxiety-disorders/index.shtml NIMH] entry about anxiety disorders
# [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/fear-worry-and-anxiety/ Effective Child Therapy](guide about anxiety symptoms, treatment, and more)
# OMIM (Online Mendelian Inheritance in Man) for simple phobia
## [https://www.omim.org/entry/608251?search=phobia&amp;highlight=phobic%20phobia 608251]
# [https://emedicine.medscape.com/article/288016-overview eMedicine entry about phobic disorders]
#[https://sccap53.org Society of Clinical Child and Adolescent Psychology]
#[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/fear-worry-and-anxiety/ EffectiveChildTherapy.Org information on fear, worry, and anxiety]
#For information on conducting Exposure Therapy for anxiety disordered youth, see [https://www.bravepracticeforkids.com/ www.BravePracticeForKids.com]

== '''References''' ==
{{collapse top| Click here for references}}
{{Reflist|2|refs=
&lt;ref name=VegaEtAl1998&gt;{{cite journal|last1=Vega|first1=WA|last2=Kolody|first2=B|last3=Aguilar-Gaxiola|first3=S|last4=Alderete|first4=E|last5=Catalano|first5=R|last6=Caraveo-Anduaga|first6=J|title=Lifetime prevalence of DSM-III-R psychiatric disorders among urban and rural Mexican Americans in California.|journal=Archives of general psychiatry|date=September 1998|volume=55|issue=9|pages=771-8|pmid=9736002}}&lt;/ref&gt;

&lt;ref name=KesslerEtAl2005a&gt;{{cite journal|last1=Kessler|first1=RC|last2=Berglund|first2=P|last3=Demler|first3=O|last4=Jin|first4=R|last5=Merikangas|first5=KR|last6=Walters|first6=EE|title=Lifetime prevalence and age-of-onset distributions of DSM-IV disorders in the National Comorbidity Survey Replication.|journal=Archives of general psychiatry|date=June 2005|volume=62|issue=6|pages=593-602|pmid=15939837}}&lt;/ref&gt;

&lt;ref name=KesslerEtAl2005b&gt;{{cite journal|last1=Kessler|first1=RC|last2=Chiu|first2=WT|last3=Demler|first3=O|last4=Merikangas|first4=KR|last5=Walters|first5=EE|title=Prevalence, severity, and comorbidity of 12-month DSM-IV disorders in the National Comorbidity Survey Replication.|journal=Archives of general psychiatry|date=June 2005|volume=62|issue=6|pages=617-27|pmid=15939839}}&lt;/ref&gt;

&lt;ref name=StinsonEtAl2007&gt;{{cite journal|last1=Stinson|first1=FS|last2=Dawson|first2=DA|last3=Patricia Chou|first3=S|last4=Smith|first4=S|last5=Goldstein|first5=RB|last6=June Ruan|first6=W|last7=Grant|first7=BF|title=The epidemiology of DSM-IV specific phobia in the USA: results from the National Epidemiologic Survey on Alcohol and Related Conditions.|journal=Psychological medicine|date=July 2007|volume=37|issue=7|pages=1047-59|pmid=17335637}}&lt;/ref&gt;

&lt;ref name=RobinsEtAl1984&gt;{{cite journal|last1=Robins|first1=LN|last2=Helzer|first2=JE|last3=Weissman|first3=MM|last4=Orvaschel|first4=H|last5=Gruenberg|first5=E|last6=Burke JD|first6=Jr|last7=Regier|first7=DA|title=Lifetime prevalence of specific psychiatric disorders in three sites.|journal=Archives of general psychiatry|date=October 1984|volume=41|issue=10|pages=949-58|pmid=6332590}}&lt;/ref&gt;

&lt;ref name=GrantEtAl2004&gt;{{cite journal|last1=Grant|first1=BF|last2=Stinson|first2=FS|last3=Dawson|first3=DA|last4=Chou|first4=SP|last5=Dufour|first5=MC|last6=Compton|first6=W|last7=Pickering|first7=RP|last8=Kaplan|first8=K|title=Prevalence and co-occurrence of substance use disorders and independent mood and anxiety disorders: results from the National Epidemiologic Survey on Alcohol and Related Conditions.|journal=Archives of general psychiatry|date=August 2004|volume=61|issue=8|pages=807-16|pmid=15289279}}&lt;/ref&gt;

&lt;ref name=Ost2007&gt;{{cite journal|last1=Ost|first1=LG|title=The claustrophobia scale: a psychometric evaluation.|journal=Behaviour research and therapy|date=May 2007|volume=45|issue=5|pages=1053-64|pmid=17303070}}&lt;/ref&gt;

&lt;ref name=VealeEtAl2012&gt;{{cite journal|last1=Veale|first1=David|last2=Ellison|first2=Nell|last3=Boschen|first3=Mark J.|last4=Costa|first4=Ana|last5=Whelan|first5=Chantelle|last6=Muccio|first6=Francesca|last7=Henry|first7=Kareina|title=Development of an Inventory to Measure Specific Phobia of Vomiting (Emetophobia)|journal=Cognitive Therapy and Research|date=18 December 2012|volume=37|issue=3|pages=595–604|doi=10.1007/s10608-012-9495-y}}&lt;/ref&gt;

&lt;ref name=deJonghEtAl1995&gt;{{cite journal|last1=de Jongh|first1=A|last2=Muris|first2=P|last3=Schoenmakers|first3=N|last4=ter Horst|first4=G|title=Negative cognitions of dental phobics: reliability and validity of the dental cognitions questionnaire.|journal=Behaviour research and therapy|date=June 1995|volume=33|issue=5|pages=507-15|pmid=7598671}}&lt;/ref&gt;

&lt;ref name=MulkensEtAl1996&gt;{{cite journal|last1=Mulkens|first1=SA|last2=de Jong|first2=PJ|last3=Merckelbach|first3=H|title=Disgust and spider phobia.|journal=Journal of abnormal psychology|date=August 1996|volume=105|issue=3|pages=464-8|pmid=8772018}}&lt;/ref&gt;

&lt;ref name=RadomskyEtAl02001&gt;{{cite journal|last1=Radomsky|first1=AS|last2=Rachman|first2=S|last3=Thordarson|first3=DS|last4=McIsaac|first4=HK|last5=Teachman|first5=BA|title=The Claustrophobia Questionnaire.|journal=Journal of anxiety disorders|date=2001|volume=15|issue=4|pages=287-97|pmid=11474815}}&lt;/ref&gt;

}}
{{collapse bottom}}
[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]</text>
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{{medical disclaimer}}
{{:{{BASEPAGENAME}}/Sidebar}}

==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
* For background information on what assessment portfolios are, click the link in the heading above.
* Want more information? There's an extended version of this page [[Evidence-based assessment/Simple phobia/Extended version|here]].

== [[Evidence based assessment/Preparation phase|'''Preparation phase''']] ==
=== Diagnostic criteria for phobic anxiety disorders ===
{{collapse top| Click here for ICD-11 and DSM-5 Diagnostic Information}}
{{blockquotetop}}
'''ICD-11 Diagnostic Criteria'''

Specific phobia is characterized by a marked and excessive fear or anxiety that consistently occurs when exposed to one or more specific objects or situations (e.g., proximity to certain animals, flying, heights, closed spaces, sight of blood or injury) and that is out of proportion to actual danger. The phobic objects or situations are avoided or else endured with intense fear or anxiety. Symptoms persist for at least several months and are sufficiently severe to result in significant distress or significant impairment in personal, family, social, educational, occupational, or other important areas of functioning.

'''Inclusions'''

*Simple phobia

*Acrophobia

*Claustrophobia

'''Exclusions'''

*Body dysmorphic disorder

*Hypochondriasis

'''Changes in DSM-5'''

The diagnostic criteria for simple phobia changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here] and [[w:DSM-5|here]].
{{blockquotebottom}}
{{Collapse bottom}}

=== Base rates of simple phobia in different populations and clinical settings ===
This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of simple phobia that they are likely to see in their clinical practice.
* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']] 
{| class=&quot;wikitable&quot;
|-
! Demography 
! Setting !! Base Rate !! Diagnostic Method
|-
| California&lt;ref name=&quot;VegaEtAl1998&quot; /&gt; 
| Mexican-American Prevalence and Service Survey (MAPSS) – adults 18+, all specific phobias || 7.4% || CIDI/DSM-III-R
|-
| All of US&lt;ref name=&quot;KesslerEtAl2005a&quot; /&gt; 
| NCS replication, adults 18+, 12-month prevalence || 8.7% || CIDI/DSM-IV
|-
| All of US&lt;ref name=&quot;KesslerEtAl2005b&quot; /&gt; 
| National Comorbidity Survey (NCS); non-institutionalized adults between 18-54, all specific phobias || 11.3% || CIDI/DSM-III-R
|-
| All of US&lt;ref name=&quot;StinsonEtAl2007&quot; /&gt; 
| National Epidemiologic Survey on Alcohol and Related Conditions (NESARC), adults 18+, all specific phobias || 9.4% || AUDADIS-IV/DSM-IV
|-
| New Haven, CT&lt;ref name=&quot;RobinsEtAl1984&quot; /&gt; 
| Adults 18+, all specific phobias || 7.8% || Clinical interview/DSM-III
|-
| Baltimore, MD&lt;ref name=&quot;RobinsEtAl1984&quot; /&gt;
| Adults 18+, all specific phobias || 23.3% || Clinical interview/DSM-III
|-
| St. Louis, MI&lt;ref name=&quot;RobinsEtAl1984&quot; /&gt; 
| Adults 18+, all specific phobias || 11.1% || Clinical interview/DSM-III
|-
| All of US&lt;ref name=&quot;GrantEtAl2004&quot; /&gt; 
| National Epidemiologic Survey on Alcohol and Related Conditions (NESARC), adults 18+, all Specific Phobias || 7.14% || AUDADIS-IV/DSM-IV
|-
|| All of US 
| Different age groups, all specific phobias || Community Prevalence: 7-9%

Children: 5%

13- to 17- year olds: 16%

Older Adults: 3-5%

Note: Females are more frequently affected than males at a rate of 2:1
 | DSM-V
|-
|Varied&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Rettew|first=David C.|last2=Lynch|first2=Alicia Doyle|last3=Achenbach|first3=Thomas M.|last4=Dumenci|first4=Levent|last5=Ivanova|first5=Masha Y.|date=2009-09|title=Meta-analyses of agreement between diagnoses made from clinical evaluations and standardized diagnostic interviews|url=http://dx.doi.org/10.1002/mpr.289|journal=International Journal of Methods in Psychiatric Research|language=en|volume=18|issue=3|pages=169–184|doi=10.1002/mpr.289|issn=1049-8931}}&lt;/ref&gt;
|Outpatient clinic (DAU)
|6% (specific)
|Varied
|-
|Varied&lt;ref name=&quot;:13&quot;&gt;{{Cite journal|last=Rettew|first=David C.|last2=Lynch|first2=Alicia Doyle|last3=Achenbach|first3=Thomas M.|last4=Dumenci|first4=Levent|last5=Ivanova|first5=Masha Y.|date=2009-09|title=Meta-analyses of agreement between diagnoses made from clinical evaluations and standardized diagnostic interviews|url=http://dx.doi.org/10.1002/mpr.289|journal=International Journal of Methods in Psychiatric Research|language=en|volume=18|issue=3|pages=169–184|doi=10.1002/mpr.289|issn=1049-8931}}&lt;/ref&gt;
|Outpatient clinic (SDI)
|15% (specific)
|Varied
|-
|Varied&lt;ref name=&quot;:3&quot;&gt;Kessler, R. C., Avenevoli, S., Costello, E. J., Georgiades, K., Green, J. G., Gruber, M. J., . . . Merikangas, K. R. (2012). Prevalence, persistence, and sociodemographic correlates of DSM-IV disorders in the National Comorbidity Survey Replication Adolescent Supplement. Archives of General Psychiatry, 69(4), 372-380. doi:10.1001/archgenpsychiatry.2011.160&lt;/ref&gt;
|General Population
|19% specific
|Varied
|}

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==
=== Psychometric properties of screening instruments for simple phobia ===
The following section contains a list of screening and diagnostic instruments for simple phobia. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. 

* '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]'''''

{| class=&quot;wikitable sortable&quot;
! colspan=&quot;5&quot; |Screening measures for simple phobia
|-
! Measure 
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Where to access
|-
|Fear Survey Schedule for Children-Revised (FSSC-R)
|Questionnaire (self-report)
|7-16 years
|19-20 minutes
|[https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470713334.app3 Yes (article PDF link)]
|-
|Fear of Spiders Questionnaire (FSQ)
|Questionnaire (self-report)
|13 years-adult
|5 minutes
|[https://www.sciencedirect.com/science/article/pii/000579169400072T Yes (article PDF link)]
|-
|Disgust Scale
|Questionnaire (self-report) 
|16 years-adult
|8 minutes
|[http://people.stern.nyu.edu/jhaidt/disgustscale.html Yes (Questionnaire homepage and PDF)]
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''For Children and Adolescents Specifically'''
|-
|Revised Children’s Anxiety and Depression Scale (RCADS)
|Questionnaire (Child)
|6-18
|12 minutes
|
*[http://www.corc.uk.net/outcome-experience-measures/revised-childrens-anxiety-and-depression-scale-and-subscales/ RCADS homepage]

'''PDFs for RCADS'''

*[https://mfr.osf.io/render?url=https://osf.io/s3fu2/?action=download%26mode=render RCADS Child Self-reported (8-18 years)]
*[https://mfr.osf.io/render?url=https://osf.io/fp9mk/?action=download%26mode=render RCADS Parent-reported]
*[https://mfr.osf.io/render?url=https://osf.io/vy7ta/?action=download%26mode=render Child Scoring Aid]
*[https://mfr.osf.io/render?url=https://osf.io/t4bz6/?action=download%26mode=render Parent Scoring Aid]

'''Subscales'''

*[https://mfr.osf.io/render?url=https://osf.io/ectpf/?action=download%26mode=render Social Anxiety/Phobia Self-reported]

*[https://mfr.osf.io/render?url=https://osf.io/pu6rn/?action=download%26mode=render Social Anxiety/Phobia Parent-reported]
*[https://mfr.osf.io/render?url=https://osf.io/4gc8d/?action=download%26mode=render Panic Self-Report]
*[https://mfr.osf.io/render?url=https://osf.io/nhcsu/?action=download%26mode=render Panic Parent-Report]

'''Translations'''

'''[https://mfr.osf.io/render?url=https://osf.io/qsjh9/?action=download%26mode=render User Guide]'''
|}
'''Note:''' Reliability and validity are included in the extended version (link). This table includes measures with Good or Excellent ratings.

=== Likelihood ratios and AUCs of screening measures for simple phobia ===
* ''For a list of the likelihood ratios for more broadly reaching screening instruments,'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments '''''click here.''''']
{| class=&quot;wikitable&quot;
|-
! Screening Measure (Primary Reference) !! AUC and Sample Size !! LR+ (Score) !! LR- (Score) !! Clinical generalizability
|-
| Specific Phobia of Vomiting Inventory (SPOVI)&lt;ref name=&quot;VealeEtAl2012&quot; /&gt;|| 0.944 (N = 185) || 24.3 (10+) || 0.03 (&lt;10) || High: able to distinguish between phobics and controls 
|-
| The Claustrophobia Scale (CS) - Anxiety Subscale (Rachman and Taylor, 1993) &lt;ref name=&quot;Ost2007&quot; /&gt;|| --- (N = 285) || 49.0 (24+) || 0.0002 (&lt;24) || High: able to distinguish between phobics and controls
|-
| The Claustrophobia Scale (CS) - Avoidance  Subscale (Rachman and Taylor, 1993)&lt;ref name=&quot;Ost2007&quot; /&gt;|| --- (N = 285) || 19.2 (9+) || 0.0004 (&lt;9) || High: able to distinguish between phobics and controls
|}
'''Search terms:''' [specific phobia] AND [sensitivity OR specificity] in Google Scholar and PsycINFO

=== Interpreting specific phobia screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==

===Gold standard diagnostic interviews===
For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

=== Recommended diagnostic instruments for simple phobia ===
{| class=&quot;wikitable sortable&quot;
! colspan=&quot;5&quot; |Diagnostic instruments for simple phobia
|-
!Measure
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Where to access
|-
|Anxiety Disorders Interview Schedule for Children (ADIS-C)&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|date=2001-08-01|title=Test-Retest Reliability of Anxiety Symptoms and Diagnoses With the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Versions|url=https://www.sciencedirect.com/science/article/pii/S0890856709603427|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|language=en|volume=40|issue=8|pages=937–944|doi=10.1097/00004583-200108000-00016|issn=0890-8567}}&lt;/ref&gt;
|Child
|6 years-adult
|90 minutes
|
* [https://books.google.com/books/about/Anxiety_Disorders_Interview_Schedule_for.html?id=xpR6V3rboxwC Purchase]
|-
|Anxiety Disorders Interview Schedule for Children (ADIS-P)&lt;ref name=&quot;:1&quot; /&gt;
|Parent
|6 years-adult
|90 minutes
|
* [https://books.google.com/books/about/Anxiety_Disorders_Interview_Schedule_for.html?id=xpR6V3rboxwC Purchase]
|-
|Diagnostic Interview Schedule for Children and Adolescents (DICA)
|Interview (clinician)
|6-17 years
|1-2 hours
|
|}
'''Note:''' Reliability and validity are included in the extended version (link). This table includes measures with Good or Excellent ratings.

==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a list of process and outcome measures for simple phobia. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. 
=== Process measures ===
{{blockquotetop}}
Two types of ''behavioral approach tests'' (BAT) can be used to observe patients in typically avoided situations. 

# A '''progressive BAT''' gradually exposes the patient to a fear-inducing situation in a step-by-step manner, and responses to each step are recorded.
# A '''selective BAT''' allows the clinician to select one or more challenges from the patient’s hierarchy, and the patient is to complete each challenge to induce a phobic response and rate the inducing fear.
{{blockquotebottom}}
=== Outcome and severity measures ===
This table includes clinically significant benchmarks for simple phobia specific outcome measures
* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
*''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable&quot;
! colspan=&quot;8&quot; |Clinically significant change benchmarks (based on published norms)
|-
! rowspan=&quot;2&quot; |&lt;big&gt;Measure&lt;/big&gt;
! rowspan=&quot;2&quot; |&lt;big&gt;Subscale&lt;/big&gt;!! colspan=&quot;3&quot; |&lt;big&gt;Cut-off scores&lt;/big&gt;!! colspan=&quot;3&quot; |'''&lt;big&gt;Critical Change&lt;/big&gt;'''
'''&lt;big&gt;(unstandardized scores)&lt;/big&gt;'''
|-
!&lt;big&gt;A&lt;/big&gt; 
!&lt;big&gt;B&lt;/big&gt;
!&lt;big&gt;C&lt;/big&gt;
!&lt;big&gt;95%&lt;/big&gt;
!&lt;big&gt;90%&lt;/big&gt;
!&lt;big&gt;SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/big&gt;
|-
| Dental Cognitions Questionnaire (1995 Norms)&lt;ref name=&quot;deJonghEtAl1995&quot; /&gt;
| || 9.1 || 16.4 || 41.2 || 4.2 || 3.6 || 2.2 
|-
| The Claustrophobia Questionnaire (2001 Norms)  - Total&lt;ref name=&quot;RadomskyEtAl02001&quot; /&gt;
| || 18.6 || 67.7 || 41.2 || 11. || 9.4 || 5.7
|-
| The Claustrophobia Questionnaire (2001 Norms) - Suffocation&lt;ref name=&quot;RadomskyEtAl02001&quot; /&gt;
| || 7.0 || 24.9 || 16.2 || 5.0 || 4.3 || 2.6 
|-
| The Claustrophobia Questionnaire (2001 Norms) - Restriction&lt;ref name=&quot;RadomskyEtAl02001&quot; /&gt;
| || 8.4 || 45.5 || 24.3 || 6.9 || 5.8 || 3.5
|-
|Spider Phobia Questionnaire (1996 Norms)&lt;ref name=&quot;MulkensEtAl1996&quot; /&gt;
| || 15.1 || 20.7 || 17.3 || 3.0 || 2.5 || 1.5
|-
|Anxiety Disorder Interview Schedule (ADIS) 
| || 5.9 || 4.4 || 5.2 || 0.2 || 0.2 || 0.1
|-
|Fear Survey Schedule for Children-Revised (FSSC-R) 
| || 77.8 || 159 || 118.4 || 6.3 || 5.3 || 3.2
|}

'''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean.
'''Search terms:''' [specific phobia] AND [adults] AND [clinical significance OR outcomes] in Google Scholar and PsycINFO

=== Treatment === 
Two treatments of specific phobia treatment include ''in-vivo exposure'' and ''virtual reality therapy''. The former is most effective in specific phobias by hierarchically exposing the client to the fear-inducing stimulus and measuring anxiety response. The latter therapy is most effective in driving and height fears by using computer-generated, interactive virtual environments that the clinician manipulates.

* Please refer to the page on [[wikipedia:Phobia|simple phobia]] for more information on available treatment or go to [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/fear-worry-and-anxiety/ Effective Child Therapy] for a curated resource on effective treatments for simple phobia.
*For information on conducting Exposure Therapy for anxiety disordered youth, see [https://www.bravepracticeforkids.com/ www.BravePracticeForKids.com]

=='''External Resources'''==

# [http://apps.who.int/classifications/icd10/browse/2010/en#/F40 ICD-10 diagnostic criteria]
# [https://en.wikiversity.org/w/index.php?title=Helping_Give_Away_Psychological_Science/Resources/Annotated_List_of_Where_and_How_to_Find_a_Therapist Find-a-Therapist] (a curated list of find-a-therapist websites where you can find a provider)
# [https://www.nimh.nih.gov/health/topics/anxiety-disorders/index.shtml NIMH] entry about anxiety disorders
# [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/fear-worry-and-anxiety/ Effective Child Therapy](guide about anxiety symptoms, treatment, and more)
# OMIM (Online Mendelian Inheritance in Man) for simple phobia
## [https://www.omim.org/entry/608251?search=phobia&amp;highlight=phobic%20phobia 608251]
# [https://emedicine.medscape.com/article/288016-overview eMedicine entry about phobic disorders]
#[https://sccap53.org Society of Clinical Child and Adolescent Psychology]
#[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/fear-worry-and-anxiety/ EffectiveChildTherapy.Org information on fear, worry, and anxiety]
#For information on conducting Exposure Therapy for anxiety disordered youth, see [https://www.bravepracticeforkids.com/ www.BravePracticeForKids.com]

== '''References''' ==
{{collapse top| Click here for references}}
{{Reflist|2|refs=
&lt;ref name=VegaEtAl1998&gt;{{cite journal|last1=Vega|first1=WA|last2=Kolody|first2=B|last3=Aguilar-Gaxiola|first3=S|last4=Alderete|first4=E|last5=Catalano|first5=R|last6=Caraveo-Anduaga|first6=J|title=Lifetime prevalence of DSM-III-R psychiatric disorders among urban and rural Mexican Americans in California.|journal=Archives of general psychiatry|date=September 1998|volume=55|issue=9|pages=771-8|pmid=9736002}}&lt;/ref&gt;

&lt;ref name=KesslerEtAl2005a&gt;{{cite journal|last1=Kessler|first1=RC|last2=Berglund|first2=P|last3=Demler|first3=O|last4=Jin|first4=R|last5=Merikangas|first5=KR|last6=Walters|first6=EE|title=Lifetime prevalence and age-of-onset distributions of DSM-IV disorders in the National Comorbidity Survey Replication.|journal=Archives of general psychiatry|date=June 2005|volume=62|issue=6|pages=593-602|pmid=15939837}}&lt;/ref&gt;

&lt;ref name=KesslerEtAl2005b&gt;{{cite journal|last1=Kessler|first1=RC|last2=Chiu|first2=WT|last3=Demler|first3=O|last4=Merikangas|first4=KR|last5=Walters|first5=EE|title=Prevalence, severity, and comorbidity of 12-month DSM-IV disorders in the National Comorbidity Survey Replication.|journal=Archives of general psychiatry|date=June 2005|volume=62|issue=6|pages=617-27|pmid=15939839}}&lt;/ref&gt;

&lt;ref name=StinsonEtAl2007&gt;{{cite journal|last1=Stinson|first1=FS|last2=Dawson|first2=DA|last3=Patricia Chou|first3=S|last4=Smith|first4=S|last5=Goldstein|first5=RB|last6=June Ruan|first6=W|last7=Grant|first7=BF|title=The epidemiology of DSM-IV specific phobia in the USA: results from the National Epidemiologic Survey on Alcohol and Related Conditions.|journal=Psychological medicine|date=July 2007|volume=37|issue=7|pages=1047-59|pmid=17335637}}&lt;/ref&gt;

&lt;ref name=RobinsEtAl1984&gt;{{cite journal|last1=Robins|first1=LN|last2=Helzer|first2=JE|last3=Weissman|first3=MM|last4=Orvaschel|first4=H|last5=Gruenberg|first5=E|last6=Burke JD|first6=Jr|last7=Regier|first7=DA|title=Lifetime prevalence of specific psychiatric disorders in three sites.|journal=Archives of general psychiatry|date=October 1984|volume=41|issue=10|pages=949-58|pmid=6332590}}&lt;/ref&gt;

&lt;ref name=GrantEtAl2004&gt;{{cite journal|last1=Grant|first1=BF|last2=Stinson|first2=FS|last3=Dawson|first3=DA|last4=Chou|first4=SP|last5=Dufour|first5=MC|last6=Compton|first6=W|last7=Pickering|first7=RP|last8=Kaplan|first8=K|title=Prevalence and co-occurrence of substance use disorders and independent mood and anxiety disorders: results from the National Epidemiologic Survey on Alcohol and Related Conditions.|journal=Archives of general psychiatry|date=August 2004|volume=61|issue=8|pages=807-16|pmid=15289279}}&lt;/ref&gt;

&lt;ref name=Ost2007&gt;{{cite journal|last1=Ost|first1=LG|title=The claustrophobia scale: a psychometric evaluation.|journal=Behaviour research and therapy|date=May 2007|volume=45|issue=5|pages=1053-64|pmid=17303070}}&lt;/ref&gt;

&lt;ref name=VealeEtAl2012&gt;{{cite journal|last1=Veale|first1=David|last2=Ellison|first2=Nell|last3=Boschen|first3=Mark J.|last4=Costa|first4=Ana|last5=Whelan|first5=Chantelle|last6=Muccio|first6=Francesca|last7=Henry|first7=Kareina|title=Development of an Inventory to Measure Specific Phobia of Vomiting (Emetophobia)|journal=Cognitive Therapy and Research|date=18 December 2012|volume=37|issue=3|pages=595–604|doi=10.1007/s10608-012-9495-y}}&lt;/ref&gt;

&lt;ref name=deJonghEtAl1995&gt;{{cite journal|last1=de Jongh|first1=A|last2=Muris|first2=P|last3=Schoenmakers|first3=N|last4=ter Horst|first4=G|title=Negative cognitions of dental phobics: reliability and validity of the dental cognitions questionnaire.|journal=Behaviour research and therapy|date=June 1995|volume=33|issue=5|pages=507-15|pmid=7598671}}&lt;/ref&gt;

&lt;ref name=MulkensEtAl1996&gt;{{cite journal|last1=Mulkens|first1=SA|last2=de Jong|first2=PJ|last3=Merckelbach|first3=H|title=Disgust and spider phobia.|journal=Journal of abnormal psychology|date=August 1996|volume=105|issue=3|pages=464-8|pmid=8772018}}&lt;/ref&gt;

&lt;ref name=RadomskyEtAl02001&gt;{{cite journal|last1=Radomsky|first1=AS|last2=Rachman|first2=S|last3=Thordarson|first3=DS|last4=McIsaac|first4=HK|last5=Teachman|first5=BA|title=The Claustrophobia Questionnaire.|journal=Journal of anxiety disorders|date=2001|volume=15|issue=4|pages=287-97|pmid=11474815}}&lt;/ref&gt;

}}
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{{:{{BASEPAGENAME}}/Sidebar}}

==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
* For background information on what assessment portfolios are, click the link in the heading above.
* Want more information? There's an extended version of this page [[Evidence-based assessment/Simple phobia/Extended version|here]].

== [[Evidence based assessment/Preparation phase|'''Preparation phase''']] ==
=== Diagnostic criteria for phobic anxiety disorders ===
{{collapse top| Click here for ICD-11 and DSM-5 Diagnostic Information}}
{{blockquotetop}}
'''ICD-11 Diagnostic Criteria'''

Specific phobia is characterized by a marked and excessive fear or anxiety that consistently occurs when exposed to one or more specific objects or situations (e.g., proximity to certain animals, flying, heights, closed spaces, sight of blood or injury) and that is out of proportion to actual danger. The phobic objects or situations are avoided or else endured with intense fear or anxiety. Symptoms persist for at least several months and are sufficiently severe to result in significant distress or significant impairment in personal, family, social, educational, occupational, or other important areas of functioning.

'''Inclusions'''

*Simple phobia

*Acrophobia

*Claustrophobia

'''Exclusions'''

*Body dysmorphic disorder

*Hypochondriasis

'''Changes in DSM-5'''

The diagnostic criteria for simple phobia changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here] and [[w:DSM-5|here]].
{{blockquotebottom}}
{{Collapse bottom}}

=== Base rates of simple phobia in different populations and clinical settings ===
This section describes the demographic setting of the population(s) sampled, base rates of diagnosis, country/region sampled and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of simple phobia that they are likely to see in their clinical practice.
* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']] 
{| class=&quot;wikitable&quot;
|-
! Demography 
! Setting !! Base Rate !! Diagnostic Method
|-
| California&lt;ref name=&quot;VegaEtAl1998&quot; /&gt; 
| Mexican-American Prevalence and Service Survey (MAPSS) – adults 18+, all specific phobias || 7.4% || CIDI/DSM-III-R
|-
| All of US&lt;ref name=&quot;KesslerEtAl2005a&quot; /&gt; 
| NCS replication, adults 18+, 12-month prevalence || 8.7% || CIDI/DSM-IV
|-
| All of US&lt;ref name=&quot;KesslerEtAl2005b&quot; /&gt; 
| National Comorbidity Survey (NCS); non-institutionalized adults between 18-54, all specific phobias || 11.3% || CIDI/DSM-III-R
|-
| All of US&lt;ref name=&quot;StinsonEtAl2007&quot; /&gt; 
| National Epidemiologic Survey on Alcohol and Related Conditions (NESARC), adults 18+, all specific phobias || 9.4% || AUDADIS-IV/DSM-IV
|-
| New Haven, CT&lt;ref name=&quot;RobinsEtAl1984&quot; /&gt; 
| Adults 18+, all specific phobias || 7.8% || Clinical interview/DSM-III
|-
| Baltimore, MD&lt;ref name=&quot;RobinsEtAl1984&quot; /&gt;
| Adults 18+, all specific phobias || 23.3% || Clinical interview/DSM-III
|-
| St. Louis, MI&lt;ref name=&quot;RobinsEtAl1984&quot; /&gt; 
| Adults 18+, all specific phobias || 11.1% || Clinical interview/DSM-III
|-
| All of US&lt;ref name=&quot;GrantEtAl2004&quot; /&gt; 
| National Epidemiologic Survey on Alcohol and Related Conditions (NESARC), adults 18+, all Specific Phobias || 7.14% || AUDADIS-IV/DSM-IV
|-
|| All of US 
| Different age groups, all specific phobias || Community Prevalence: 7-9%

Children: 5%

13- to 17- year olds: 16%

Older Adults: 3-5%

Note: Females are more frequently affected than males at a rate of 2:1
 | DSM-V
|-
|Varied&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Rettew|first=David C.|last2=Lynch|first2=Alicia Doyle|last3=Achenbach|first3=Thomas M.|last4=Dumenci|first4=Levent|last5=Ivanova|first5=Masha Y.|date=2009-09|title=Meta-analyses of agreement between diagnoses made from clinical evaluations and standardized diagnostic interviews|url=http://dx.doi.org/10.1002/mpr.289|journal=International Journal of Methods in Psychiatric Research|language=en|volume=18|issue=3|pages=169–184|doi=10.1002/mpr.289|issn=1049-8931}}&lt;/ref&gt;
|Outpatient clinic (DAU)
|6% (specific)
|Varied
|-
|Varied&lt;ref name=&quot;:13&quot;&gt;{{Cite journal|last=Rettew|first=David C.|last2=Lynch|first2=Alicia Doyle|last3=Achenbach|first3=Thomas M.|last4=Dumenci|first4=Levent|last5=Ivanova|first5=Masha Y.|date=2009-09|title=Meta-analyses of agreement between diagnoses made from clinical evaluations and standardized diagnostic interviews|url=http://dx.doi.org/10.1002/mpr.289|journal=International Journal of Methods in Psychiatric Research|language=en|volume=18|issue=3|pages=169–184|doi=10.1002/mpr.289|issn=1049-8931}}&lt;/ref&gt;
|Outpatient clinic (SDI)
|15% (specific)
|Varied
|-
|Varied&lt;ref name=&quot;:3&quot;&gt;Kessler, R. C., Avenevoli, S., Costello, E. J., Georgiades, K., Green, J. G., Gruber, M. J., . . . Merikangas, K. R. (2012). Prevalence, persistence, and sociodemographic correlates of DSM-IV disorders in the National Comorbidity Survey Replication Adolescent Supplement. Archives of General Psychiatry, 69(4), 372-380. doi:10.1001/archgenpsychiatry.2011.160&lt;/ref&gt;
|General Population
|19% specific
|Varied
|}

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==
=== Psychometric properties of screening instruments for simple phobia ===
The following section contains a list of screening and diagnostic instruments for simple phobia. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here]]. 

* '''''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments click here.]'''''

{| class=&quot;wikitable sortable&quot;
! colspan=&quot;5&quot; |Screening measures for simple phobia
|-
! Measure 
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Where to access
|-
|Fear Survey Schedule for Children-Revised (FSSC-R)
|Questionnaire (self-report)
|7-16 years
|19-20 minutes
|[https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470713334.app3 Yes (article PDF link)]
|-
|Fear of Spiders Questionnaire (FSQ)
|Questionnaire (self-report)
|13 years-adult
|5 minutes
|[https://www.sciencedirect.com/science/article/pii/000579169400072T Yes (article PDF link)]
|-
|Disgust Scale
|Questionnaire (self-report) 
|16 years-adult
|8 minutes
|[http://people.stern.nyu.edu/jhaidt/disgustscale.html Yes (Questionnaire homepage and PDF)]
|-
|Revised Children’s Anxiety and Depression Scale (RCADS)
|Questionnaire (Child)
|6-18
|12 minutes
|
*[http://www.corc.uk.net/outcome-experience-measures/revised-childrens-anxiety-and-depression-scale-and-subscales/ RCADS homepage]

'''PDFs for RCADS'''

*[https://mfr.osf.io/render?url=https://osf.io/s3fu2/?action=download%26mode=render RCADS Child Self-reported (8-18 years)]
*[https://mfr.osf.io/render?url=https://osf.io/fp9mk/?action=download%26mode=render RCADS Parent-reported]
*[https://mfr.osf.io/render?url=https://osf.io/vy7ta/?action=download%26mode=render Child Scoring Aid]
*[https://mfr.osf.io/render?url=https://osf.io/t4bz6/?action=download%26mode=render Parent Scoring Aid]

'''Subscales'''

*[https://mfr.osf.io/render?url=https://osf.io/ectpf/?action=download%26mode=render Social Anxiety/Phobia Self-reported]

*[https://mfr.osf.io/render?url=https://osf.io/pu6rn/?action=download%26mode=render Social Anxiety/Phobia Parent-reported]
*[https://mfr.osf.io/render?url=https://osf.io/4gc8d/?action=download%26mode=render Panic Self-Report]
*[https://mfr.osf.io/render?url=https://osf.io/nhcsu/?action=download%26mode=render Panic Parent-Report]

'''Translations'''

'''[https://mfr.osf.io/render?url=https://osf.io/qsjh9/?action=download%26mode=render User Guide]'''
|}
'''Note:''' Reliability and validity are included in the extended version (link). This table includes measures with Good or Excellent ratings.

=== Likelihood ratios and AUCs of screening measures for simple phobia ===
* ''For a list of the likelihood ratios for more broadly reaching screening instruments,'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments '''''click here.''''']
{| class=&quot;wikitable&quot;
|-
! Screening Measure (Primary Reference) !! AUC and Sample Size !! LR+ (Score) !! LR- (Score) !! Clinical generalizability
|-
| Specific Phobia of Vomiting Inventory (SPOVI)&lt;ref name=&quot;VealeEtAl2012&quot; /&gt;|| 0.944 (N = 185) || 24.3 (10+) || 0.03 (&lt;10) || High: able to distinguish between phobics and controls 
|-
| The Claustrophobia Scale (CS) - Anxiety Subscale (Rachman and Taylor, 1993) &lt;ref name=&quot;Ost2007&quot; /&gt;|| --- (N = 285) || 49.0 (24+) || 0.0002 (&lt;24) || High: able to distinguish between phobics and controls
|-
| The Claustrophobia Scale (CS) - Avoidance  Subscale (Rachman and Taylor, 1993)&lt;ref name=&quot;Ost2007&quot; /&gt;|| --- (N = 285) || 19.2 (9+) || 0.0004 (&lt;9) || High: able to distinguish between phobics and controls
|}
'''Search terms:''' [specific phobia] AND [sensitivity OR specificity] in Google Scholar and PsycINFO

=== Interpreting specific phobia screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==

===Gold standard diagnostic interviews===
For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

=== Recommended diagnostic instruments for simple phobia ===
{| class=&quot;wikitable sortable&quot;
! colspan=&quot;5&quot; |Diagnostic instruments for simple phobia
|-
!Measure
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Where to access
|-
|Anxiety Disorders Interview Schedule for Children (ADIS-C)&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|date=2001-08-01|title=Test-Retest Reliability of Anxiety Symptoms and Diagnoses With the Anxiety Disorders Interview Schedule for DSM-IV: Child and Parent Versions|url=https://www.sciencedirect.com/science/article/pii/S0890856709603427|journal=Journal of the American Academy of Child &amp; Adolescent Psychiatry|language=en|volume=40|issue=8|pages=937–944|doi=10.1097/00004583-200108000-00016|issn=0890-8567}}&lt;/ref&gt;
|Child
|6 years-adult
|90 minutes
|
* [https://books.google.com/books/about/Anxiety_Disorders_Interview_Schedule_for.html?id=xpR6V3rboxwC Purchase]
|-
|Anxiety Disorders Interview Schedule for Children (ADIS-P)&lt;ref name=&quot;:1&quot; /&gt;
|Parent
|6 years-adult
|90 minutes
|
* [https://books.google.com/books/about/Anxiety_Disorders_Interview_Schedule_for.html?id=xpR6V3rboxwC Purchase]
|-
|Diagnostic Interview Schedule for Children and Adolescents (DICA)
|Interview (clinician)
|6-17 years
|1-2 hours
|
|}
'''Note:''' Reliability and validity are included in the extended version (link). This table includes measures with Good or Excellent ratings.

==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a list of process and outcome measures for simple phobia. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information of differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]]. 
=== Process measures ===
{{blockquotetop}}
Two types of ''behavioral approach tests'' (BAT) can be used to observe patients in typically avoided situations. 

# A '''progressive BAT''' gradually exposes the patient to a fear-inducing situation in a step-by-step manner, and responses to each step are recorded.
# A '''selective BAT''' allows the clinician to select one or more challenges from the patient’s hierarchy, and the patient is to complete each challenge to induce a phobic response and rate the inducing fear.
{{blockquotebottom}}
=== Outcome and severity measures ===
This table includes clinically significant benchmarks for simple phobia specific outcome measures
* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
*''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable&quot;
! colspan=&quot;8&quot; |Clinically significant change benchmarks (based on published norms)
|-
! rowspan=&quot;2&quot; |&lt;big&gt;Measure&lt;/big&gt;
! rowspan=&quot;2&quot; |&lt;big&gt;Subscale&lt;/big&gt;!! colspan=&quot;3&quot; |&lt;big&gt;Cut-off scores&lt;/big&gt;!! colspan=&quot;3&quot; |'''&lt;big&gt;Critical Change&lt;/big&gt;'''
'''&lt;big&gt;(unstandardized scores)&lt;/big&gt;'''
|-
!&lt;big&gt;A&lt;/big&gt; 
!&lt;big&gt;B&lt;/big&gt;
!&lt;big&gt;C&lt;/big&gt;
!&lt;big&gt;95%&lt;/big&gt;
!&lt;big&gt;90%&lt;/big&gt;
!&lt;big&gt;SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/big&gt;
|-
| Dental Cognitions Questionnaire (1995 Norms)&lt;ref name=&quot;deJonghEtAl1995&quot; /&gt;
| || 9.1 || 16.4 || 41.2 || 4.2 || 3.6 || 2.2 
|-
| The Claustrophobia Questionnaire (2001 Norms)  - Total&lt;ref name=&quot;RadomskyEtAl02001&quot; /&gt;
| || 18.6 || 67.7 || 41.2 || 11. || 9.4 || 5.7
|-
| The Claustrophobia Questionnaire (2001 Norms) - Suffocation&lt;ref name=&quot;RadomskyEtAl02001&quot; /&gt;
| || 7.0 || 24.9 || 16.2 || 5.0 || 4.3 || 2.6 
|-
| The Claustrophobia Questionnaire (2001 Norms) - Restriction&lt;ref name=&quot;RadomskyEtAl02001&quot; /&gt;
| || 8.4 || 45.5 || 24.3 || 6.9 || 5.8 || 3.5
|-
|Spider Phobia Questionnaire (1996 Norms)&lt;ref name=&quot;MulkensEtAl1996&quot; /&gt;
| || 15.1 || 20.7 || 17.3 || 3.0 || 2.5 || 1.5
|-
|Anxiety Disorder Interview Schedule (ADIS) 
| || 5.9 || 4.4 || 5.2 || 0.2 || 0.2 || 0.1
|-
|Fear Survey Schedule for Children-Revised (FSSC-R) 
| || 77.8 || 159 || 118.4 || 6.3 || 5.3 || 3.2
|}

'''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean.
'''Search terms:''' [specific phobia] AND [adults] AND [clinical significance OR outcomes] in Google Scholar and PsycINFO

=== Treatment === 
Two treatments of specific phobia treatment include ''in-vivo exposure'' and ''virtual reality therapy''. The former is most effective in specific phobias by hierarchically exposing the client to the fear-inducing stimulus and measuring anxiety response. The latter therapy is most effective in driving and height fears by using computer-generated, interactive virtual environments that the clinician manipulates.

* Please refer to the page on [[wikipedia:Phobia|simple phobia]] for more information on available treatment or go to [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/fear-worry-and-anxiety/ Effective Child Therapy] for a curated resource on effective treatments for simple phobia.
*For information on conducting Exposure Therapy for anxiety disordered youth, see [https://www.bravepracticeforkids.com/ www.BravePracticeForKids.com]

=='''External Resources'''==

# [http://apps.who.int/classifications/icd10/browse/2010/en#/F40 ICD-10 diagnostic criteria]
# [https://en.wikiversity.org/w/index.php?title=Helping_Give_Away_Psychological_Science/Resources/Annotated_List_of_Where_and_How_to_Find_a_Therapist Find-a-Therapist] (a curated list of find-a-therapist websites where you can find a provider)
# [https://www.nimh.nih.gov/health/topics/anxiety-disorders/index.shtml NIMH] entry about anxiety disorders
# [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/fear-worry-and-anxiety/ Effective Child Therapy](guide about anxiety symptoms, treatment, and more)
# OMIM (Online Mendelian Inheritance in Man) for simple phobia
## [https://www.omim.org/entry/608251?search=phobia&amp;highlight=phobic%20phobia 608251]
# [https://emedicine.medscape.com/article/288016-overview eMedicine entry about phobic disorders]
#[https://sccap53.org Society of Clinical Child and Adolescent Psychology]
#[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/fear-worry-and-anxiety/ EffectiveChildTherapy.Org information on fear, worry, and anxiety]
#For information on conducting Exposure Therapy for anxiety disordered youth, see [https://www.bravepracticeforkids.com/ www.BravePracticeForKids.com]

== '''References''' ==
{{collapse top| Click here for references}}
{{Reflist|2|refs=
&lt;ref name=VegaEtAl1998&gt;{{cite journal|last1=Vega|first1=WA|last2=Kolody|first2=B|last3=Aguilar-Gaxiola|first3=S|last4=Alderete|first4=E|last5=Catalano|first5=R|last6=Caraveo-Anduaga|first6=J|title=Lifetime prevalence of DSM-III-R psychiatric disorders among urban and rural Mexican Americans in California.|journal=Archives of general psychiatry|date=September 1998|volume=55|issue=9|pages=771-8|pmid=9736002}}&lt;/ref&gt;

&lt;ref name=KesslerEtAl2005a&gt;{{cite journal|last1=Kessler|first1=RC|last2=Berglund|first2=P|last3=Demler|first3=O|last4=Jin|first4=R|last5=Merikangas|first5=KR|last6=Walters|first6=EE|title=Lifetime prevalence and age-of-onset distributions of DSM-IV disorders in the National Comorbidity Survey Replication.|journal=Archives of general psychiatry|date=June 2005|volume=62|issue=6|pages=593-602|pmid=15939837}}&lt;/ref&gt;

&lt;ref name=KesslerEtAl2005b&gt;{{cite journal|last1=Kessler|first1=RC|last2=Chiu|first2=WT|last3=Demler|first3=O|last4=Merikangas|first4=KR|last5=Walters|first5=EE|title=Prevalence, severity, and comorbidity of 12-month DSM-IV disorders in the National Comorbidity Survey Replication.|journal=Archives of general psychiatry|date=June 2005|volume=62|issue=6|pages=617-27|pmid=15939839}}&lt;/ref&gt;

&lt;ref name=StinsonEtAl2007&gt;{{cite journal|last1=Stinson|first1=FS|last2=Dawson|first2=DA|last3=Patricia Chou|first3=S|last4=Smith|first4=S|last5=Goldstein|first5=RB|last6=June Ruan|first6=W|last7=Grant|first7=BF|title=The epidemiology of DSM-IV specific phobia in the USA: results from the National Epidemiologic Survey on Alcohol and Related Conditions.|journal=Psychological medicine|date=July 2007|volume=37|issue=7|pages=1047-59|pmid=17335637}}&lt;/ref&gt;

&lt;ref name=RobinsEtAl1984&gt;{{cite journal|last1=Robins|first1=LN|last2=Helzer|first2=JE|last3=Weissman|first3=MM|last4=Orvaschel|first4=H|last5=Gruenberg|first5=E|last6=Burke JD|first6=Jr|last7=Regier|first7=DA|title=Lifetime prevalence of specific psychiatric disorders in three sites.|journal=Archives of general psychiatry|date=October 1984|volume=41|issue=10|pages=949-58|pmid=6332590}}&lt;/ref&gt;

&lt;ref name=GrantEtAl2004&gt;{{cite journal|last1=Grant|first1=BF|last2=Stinson|first2=FS|last3=Dawson|first3=DA|last4=Chou|first4=SP|last5=Dufour|first5=MC|last6=Compton|first6=W|last7=Pickering|first7=RP|last8=Kaplan|first8=K|title=Prevalence and co-occurrence of substance use disorders and independent mood and anxiety disorders: results from the National Epidemiologic Survey on Alcohol and Related Conditions.|journal=Archives of general psychiatry|date=August 2004|volume=61|issue=8|pages=807-16|pmid=15289279}}&lt;/ref&gt;

&lt;ref name=Ost2007&gt;{{cite journal|last1=Ost|first1=LG|title=The claustrophobia scale: a psychometric evaluation.|journal=Behaviour research and therapy|date=May 2007|volume=45|issue=5|pages=1053-64|pmid=17303070}}&lt;/ref&gt;

&lt;ref name=VealeEtAl2012&gt;{{cite journal|last1=Veale|first1=David|last2=Ellison|first2=Nell|last3=Boschen|first3=Mark J.|last4=Costa|first4=Ana|last5=Whelan|first5=Chantelle|last6=Muccio|first6=Francesca|last7=Henry|first7=Kareina|title=Development of an Inventory to Measure Specific Phobia of Vomiting (Emetophobia)|journal=Cognitive Therapy and Research|date=18 December 2012|volume=37|issue=3|pages=595–604|doi=10.1007/s10608-012-9495-y}}&lt;/ref&gt;

&lt;ref name=deJonghEtAl1995&gt;{{cite journal|last1=de Jongh|first1=A|last2=Muris|first2=P|last3=Schoenmakers|first3=N|last4=ter Horst|first4=G|title=Negative cognitions of dental phobics: reliability and validity of the dental cognitions questionnaire.|journal=Behaviour research and therapy|date=June 1995|volume=33|issue=5|pages=507-15|pmid=7598671}}&lt;/ref&gt;

&lt;ref name=MulkensEtAl1996&gt;{{cite journal|last1=Mulkens|first1=SA|last2=de Jong|first2=PJ|last3=Merckelbach|first3=H|title=Disgust and spider phobia.|journal=Journal of abnormal psychology|date=August 1996|volume=105|issue=3|pages=464-8|pmid=8772018}}&lt;/ref&gt;

&lt;ref name=RadomskyEtAl02001&gt;{{cite journal|last1=Radomsky|first1=AS|last2=Rachman|first2=S|last3=Thordarson|first3=DS|last4=McIsaac|first4=HK|last5=Teachman|first5=BA|title=The Claustrophobia Questionnaire.|journal=Journal of anxiety disorders|date=2001|volume=15|issue=4|pages=287-97|pmid=11474815}}&lt;/ref&gt;

}}
{{collapse bottom}}
[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]</text>
      <sha1>eryswl9fc5n67qcf0rc8nh1zuva9czu</sha1>
    </revision>
  </page>
  <page>
    <title>User:ThaniosAkro/sandbox</title>
    <ns>2</ns>
    <id>219126</id>
    <revision>
      <id>2411818</id>
      <parentid>2411774</parentid>
      <timestamp>2022-08-04T12:26:58Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* A simple example: */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23305" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===
{{RoundBoxTop|theme=2}}
Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;

&lt;math&gt;\frac{x^2 - y^2}{N}&lt;/math&gt; exactly.

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}((x+y), N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}((x-y), N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121
| -4050

|-
| 12
| 144
| -4027

|-
| 35
| 1225
| -2946

|-
| 36
| 1296
| -2875

|-
| 37
| 1369
| -2802

|-
| 55
| 3025
| -1146

|-
| 56
| 3136
| -1035

|-
| 57
| 3249
| -922

|-
| 60
| 3600
| -571

|-
| 61
| 3721
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75

|-
| 65
| 4225
| 54
|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107

|-
| 64
| 4096
| -75
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;


The congruences:
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121
| -4050

|-
| 61
| 3721
| -450
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv 1822500 \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;
&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

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===Adding 2 congruences===
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If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
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===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

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Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
{{RoundBoxBottom}}

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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
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=====For speed=====
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======Many comparisons======
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If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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======Divide by 2======
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

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To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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{{RoundBoxBottom}}
{{RoundBoxBottom}}

=====For clarity=====
{{RoundBoxTop|theme=7}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}

===tables===
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{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

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[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
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&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
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&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


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[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
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&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
{{RoundBoxTop|theme=5}}
Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>50hyne2xck01ho27oipvqi9p7muzzy6</sha1>
    </revision>
    <revision>
      <id>2411823</id>
      <parentid>2411818</parentid>
      <timestamp>2022-08-04T12:53:59Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* A simple example: */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23589" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===
{{RoundBoxTop|theme=2}}
Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;

&lt;math&gt;\frac{x^2 - y^2}{N}&lt;/math&gt; exactly.

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}((x+y), N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}((x-y), N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121
| -4050

|-
| 12
| 144
| -4027

|-
| 35
| 1225
| -2946

|-
| 36
| 1296
| -2875

|-
| 37
| 1369
| -2802

|-
| 55
| 3025
| -1146

|-
| 56
| 3136
| -1035

|-
| 57
| 3249
| -922

|-
| 60
| 3600
| -571

|-
| 61
| 3721
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75

|-
| 65
| 4225
| 54
|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107

|-
| 64
| 4096
| -75
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;


The congruences:
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121
| -4050

|-
| 61
| 3721
| -450
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 182250 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

===Adding 2 congruences===
{{RoundBoxTop|theme=2}}
If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
{{RoundBoxBottom}}


===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

{{RoundBoxTop|theme=2}}
Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
{{RoundBoxTop|theme=5}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
{{RoundBoxTop|theme=4}}
=====For speed=====
{{RoundBoxTop|theme=7}}
======Many comparisons======
{{RoundBoxTop|theme=8}}
If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
======Divide by 2======
{{RoundBoxTop|theme=8}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

{{RoundBoxTop|theme=8}}
To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}
{{RoundBoxBottom}}

=====For clarity=====
{{RoundBoxTop|theme=7}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}

===tables===
{{RoundBoxTop|theme=1}}

{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
{{RoundBoxTop|theme=5}}
Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>7ysilm8iwo9jwbvbsgha2fcvaftpr4u</sha1>
    </revision>
    <revision>
      <id>2411827</id>
      <parentid>2411823</parentid>
      <timestamp>2022-08-04T13:04:06Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* A simple example: */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23590" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===
{{RoundBoxTop|theme=2}}
Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;

&lt;math&gt;\frac{x^2 - y^2}{N}&lt;/math&gt; exactly.

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}((x+y), N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}((x-y), N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121
| -4050

|-
| 12
| 144
| -4027

|-
| 35
| 1225
| -2946

|-
| 36
| 1296
| -2875

|-
| 37
| 1369
| -2802

|-
| 55
| 3025
| -1146

|-
| 56
| 3136
| -1035

|-
| 57
| 3249
| -922

|-
| 60
| 3600
| -571

|-
| 61
| 3721
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75

|-
| 65
| 4225
| 54
|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107

|-
| 64
| 4096
| -75
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;


The congruences:
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121
| -4050

|-
| 61
| 3721
| -450
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

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===Adding 2 congruences===
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If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
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===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

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Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
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=====For speed=====
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======Many comparisons======
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If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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======Divide by 2======
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

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To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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{{RoundBoxBottom}}
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=====For clarity=====
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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===tables===
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{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

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[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
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&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
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&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


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[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
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&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
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Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>0qbbzu59mnrn6vjzwbnta0lt68o48rr</sha1>
    </revision>
    <revision>
      <id>2411828</id>
      <parentid>2411827</parentid>
      <timestamp>2022-08-04T13:28:27Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* A simple example: */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="24081" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===
{{RoundBoxTop|theme=2}}
Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;

&lt;math&gt;\frac{x^2 - y^2}{N}&lt;/math&gt; exactly.

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}((x+y), N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}((x-y), N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;


One congruence produced a square for y:
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;
|-
| 70
| 4900
| 729
|}
:&lt;math&gt;4900 \equiv 729 \pmod{N}&lt;/math&gt;
:&lt;math&gt;70^2 \equiv 27^2 \pmod{N}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(70-27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(70+27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(97, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121
| -4050

|-
| 12
| 144
| -4027

|-
| 35
| 1225
| -2946

|-
| 36
| 1296
| -2875

|-
| 37
| 1369
| -2802

|-
| 55
| 3025
| -1146

|-
| 56
| 3136
| -1035

|-
| 57
| 3249
| -922

|-
| 60
| 3600
| -571

|-
| 61
| 3721
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75

|-
| 65
| 4225
| 54
|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107

|-
| 64
| 4096
| -75
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;


The congruences:
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121
| -4050

|-
| 61
| 3721
| -450
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

===Adding 2 congruences===
{{RoundBoxTop|theme=2}}
If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
{{RoundBoxBottom}}


===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

{{RoundBoxTop|theme=2}}
Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
{{RoundBoxTop|theme=5}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
{{RoundBoxTop|theme=4}}
=====For speed=====
{{RoundBoxTop|theme=7}}
======Many comparisons======
{{RoundBoxTop|theme=8}}
If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
======Divide by 2======
{{RoundBoxTop|theme=8}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

{{RoundBoxTop|theme=8}}
To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}
{{RoundBoxBottom}}

=====For clarity=====
{{RoundBoxTop|theme=7}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}

===tables===
{{RoundBoxTop|theme=1}}

{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
{{RoundBoxTop|theme=5}}
Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>a4gtgd20189x4ph2rix3881a0c6ly0s</sha1>
    </revision>
    <revision>
      <id>2411886</id>
      <parentid>2411828</parentid>
      <timestamp>2022-08-04T19:22:55Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* A simple example: */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23962" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===
{{RoundBoxTop|theme=2}}
Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;

&lt;math&gt;\frac{x^2 - y^2}{N}&lt;/math&gt; exactly.

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}((x+y), N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}((x-y), N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;


One congruence produced a square for y:
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;
|-
| 70
| 4900
| 729
|}
:&lt;math&gt;4900 \equiv 729 \pmod{N}&lt;/math&gt;
:&lt;math&gt;70^2 \equiv 27^2 \pmod{N}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(70-27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(70+27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(97, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107 **

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121 **
| -4050

|-
| 12
| 144
| -4027

|-
| 60
| 3600
| -571

|-
| 61
| 3721 **
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75 **

|-
| 65
| 4225
| 54
|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107 **

|-
| 64
| 4096
| -75 **
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;


The congruences:
{| class=&quot;wikitable&quot;
|- 
!   x !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121 **
| -4050

|-
| 61
| 3721 **
| -450
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

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===Adding 2 congruences===
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If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
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===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

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Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

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&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
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=====For speed=====
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======Many comparisons======
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If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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======Divide by 2======
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

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To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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=====For clarity=====
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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===tables===
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{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

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[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
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&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
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&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


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[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
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&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
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Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>52usvtg6jblmbyk998bsvfmyk4cq9b8</sha1>
    </revision>
    <revision>
      <id>2411951</id>
      <parentid>2411886</parentid>
      <timestamp>2022-08-04T21:30:06Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* A simple example: */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="24014" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===
{{RoundBoxTop|theme=2}}
Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;

&lt;math&gt;\frac{x^2 - y^2}{N}&lt;/math&gt; exactly.

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}((x+y), N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}((x-y), N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;


One congruence produced a square for y:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;
|-
| 70
| 4900
| 729
|}
:&lt;math&gt;4900 \equiv 729 \pmod{N}&lt;/math&gt;
:&lt;math&gt;70^2 \equiv 27^2 \pmod{N}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(70-27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(70+27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(97, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107 **

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121 **
| -4050

|-
| 12
| 144
| -4027

|-
| 60
| 3600
| -571

|-
| 61
| 3721 **
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75 **

|-
| 65
| 4225
| 54
|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107 **

|-
| 64
| 4096
| -75 **
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;


The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121 **
| -4050

|-
| 61
| 3721 **
| -450
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

===Adding 2 congruences===
{{RoundBoxTop|theme=2}}
If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
{{RoundBoxBottom}}


===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

{{RoundBoxTop|theme=2}}
Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
{{RoundBoxTop|theme=5}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
{{RoundBoxTop|theme=4}}
=====For speed=====
{{RoundBoxTop|theme=7}}
======Many comparisons======
{{RoundBoxTop|theme=8}}
If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
======Divide by 2======
{{RoundBoxTop|theme=8}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

{{RoundBoxTop|theme=8}}
To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}
{{RoundBoxBottom}}

=====For clarity=====
{{RoundBoxTop|theme=7}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}

===tables===
{{RoundBoxTop|theme=1}}

{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
{{RoundBoxTop|theme=5}}
Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>3dvy2c07gi3mpzstg6hnau6d8lcbfgm</sha1>
    </revision>
    <revision>
      <id>2411954</id>
      <parentid>2411951</parentid>
      <timestamp>2022-08-04T21:43:45Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* A simple example: */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="24422" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===
{{RoundBoxTop|theme=2}}
Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;

&lt;math&gt;\frac{x^2 - y^2}{N}&lt;/math&gt; exactly.

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}((x+y), N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}((x-y), N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;


One congruence produced a square for y:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;
|-
| 70
| 4900
| 729
|}
:&lt;math&gt;4900 \equiv 729 \pmod{N}&lt;/math&gt;
:&lt;math&gt;70^2 \equiv 27^2 \pmod{N}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(70-27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(70+27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(97, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107 **

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121 **
| -4050

|-
| 12
| 144
| -4027

|-
| 60
| 3600
| -571

|-
| 61
| 3721 **
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75 **

|-
| 65
| 4225
| 54
|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107 **

|-
| 64
| 4096
| -75 **
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;


The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121 **
| -4050

|-
| 61
| 3721 **
| -450
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce positive 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 64
| 4096
| -75

|-
| 65
| 4225
| 54

|-
| 66
| 4356
| 185

|-
| 88
| 7744
| 3573

|-
| 89
| 7921
| 3750

|-
| 90
| 8100
| 3929

|-
| 144
| 20736
| 16565

|-
| 145
| 21025
| 16854

|-
| 146
| 21316
| 17145

|}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

===Adding 2 congruences===
{{RoundBoxTop|theme=2}}
If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
{{RoundBoxBottom}}


===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

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Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
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=====For speed=====
{{RoundBoxTop|theme=7}}
======Many comparisons======
{{RoundBoxTop|theme=8}}
If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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======Divide by 2======
{{RoundBoxTop|theme=8}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

{{RoundBoxTop|theme=8}}
To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}
{{RoundBoxBottom}}

=====For clarity=====
{{RoundBoxTop|theme=7}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}

===tables===
{{RoundBoxTop|theme=1}}

{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

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[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
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&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


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[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
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&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
{{RoundBoxTop|theme=5}}
Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>e182yblg6l413vdh00gyhc2t3yfvk52</sha1>
    </revision>
    <revision>
      <id>2411957</id>
      <parentid>2411954</parentid>
      <timestamp>2022-08-04T21:58:44Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* A simple example: */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25084" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===
{{RoundBoxTop|theme=2}}
Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;

&lt;math&gt;\frac{x^2 - y^2}{N}&lt;/math&gt; exactly.

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}((x+y), N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}((x-y), N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;


One congruence produced a square for y:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;
|-
| 70
| 4900
| 729
|}
:&lt;math&gt;4900 \equiv 729 \pmod{N}&lt;/math&gt;
:&lt;math&gt;70^2 \equiv 27^2 \pmod{N}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(70-27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(70+27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(97, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107 **

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121 **
| -4050

|-
| 12
| 144
| -4027

|-
| 60
| 3600
| -571

|-
| 61
| 3721 **
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75 **

|-
| 65
| 4225
| 54
|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107 **

|-
| 64
| 4096
| -75 **
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;


The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121 **
| -4050

|-
| 61
| 3721 **
| -450
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce positive 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 64
| 4096
| -75

|-
| 65
| 4225
| 54

|-
| 66
| 4356
| 185

|-
| 88
| 7744
| 3573

|-
| 89
| 7921
| 3750

|-
| 90
| 8100
| 3929

|-
| 144
| 20736
| 16565

|-
| 145
| 21025
| 16854

|-
| 146
| 21316
| 17145

|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 65
| 4225
| 54

|-
| 89
| 7921
| 3750
|}

:&lt;math&gt;4225 \equiv 54 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4225\cdot 7921 \equiv 54 \cdot 3750 \pmod{N}&lt;/math&gt;

:&lt;math&gt;33466225 \equiv 202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;5785^2 \equiv 450^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(5785-450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(5335, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(5785+450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(6235, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

===Adding 2 congruences===
{{RoundBoxTop|theme=2}}
If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
{{RoundBoxBottom}}


===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

{{RoundBoxTop|theme=2}}
Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
{{RoundBoxTop|theme=5}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
{{RoundBoxTop|theme=4}}
=====For speed=====
{{RoundBoxTop|theme=7}}
======Many comparisons======
{{RoundBoxTop|theme=8}}
If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
======Divide by 2======
{{RoundBoxTop|theme=8}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

{{RoundBoxTop|theme=8}}
To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}
{{RoundBoxBottom}}

=====For clarity=====
{{RoundBoxTop|theme=7}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}

===tables===
{{RoundBoxTop|theme=1}}

{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
{{RoundBoxTop|theme=5}}
Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>gj3whb9sib7hv8nkk6k7au580ge5psw</sha1>
    </revision>
    <revision>
      <id>2411961</id>
      <parentid>2411957</parentid>
      <timestamp>2022-08-04T22:19:14Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* A simple example: */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25773" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===
{{RoundBoxTop|theme=2}}
Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;

&lt;math&gt;\frac{x^2 - y^2}{N}&lt;/math&gt; exactly.

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}((x+y), N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}((x-y), N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;


One congruence produced a square for y:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;
|-
| 70
| 4900
| 729
|}
:&lt;math&gt;4900 \equiv 729 \pmod{N}&lt;/math&gt;
:&lt;math&gt;70^2 \equiv 27^2 \pmod{N}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(70-27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(70+27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(97, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107 **

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121 **
| -4050

|-
| 12
| 144
| -4027

|-
| 60
| 3600
| -571

|-
| 61
| 3721 **
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75 **

|-
| 65
| 4225
| 54
|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107 **

|-
| 64
| 4096
| -75 **
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;


The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121 **
| -4050

|-
| 61
| 3721 **
| -450
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce positive 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 64
| 4096
| -75

|-
| 65
| 4225
| 54

|-
| 66
| 4356
| 185

|-
| 88
| 7744
| 3573

|-
| 89
| 7921
| 3750

|-
| 90
| 8100
| 3929

|-
| 144
| 20736
| 16565

|-
| 145
| 21025
| 16854

|-
| 146
| 21316
| 17145

|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 65
| 4225
| 54

|-
| 89
| 7921
| 3750
|}

:&lt;math&gt;4225 \equiv 54 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4225\cdot 7921 \equiv 54 \cdot 3750 \pmod{N}&lt;/math&gt;

:&lt;math&gt;33466225 \equiv 202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;5785^2 \equiv 450^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(5785-450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(5335, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(5785+450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(6235, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 89
| 7921
| 3750

|-
| 145
| 21025
| 16854
|}

:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;21025 \equiv 16854 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921\cdot 21025 \equiv 3750 \cdot 16854 \pmod{N}&lt;/math&gt;

:&lt;math&gt;166539025  \equiv 63202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;12905^2 \equiv 7950^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(12905-7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(4955, 4171)&lt;/math&gt;
&lt;math&gt;= 1.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(12905+7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(20855, 4171)&lt;/math&gt;
&lt;math&gt;= 4171.&lt;/math&gt;

This congruence produced the trivial factors of &lt;math&gt;4171.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

===Adding 2 congruences===
{{RoundBoxTop|theme=2}}
If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
{{RoundBoxBottom}}


===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

{{RoundBoxTop|theme=2}}
Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
{{RoundBoxTop|theme=5}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
{{RoundBoxTop|theme=4}}
=====For speed=====
{{RoundBoxTop|theme=7}}
======Many comparisons======
{{RoundBoxTop|theme=8}}
If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
======Divide by 2======
{{RoundBoxTop|theme=8}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

{{RoundBoxTop|theme=8}}
To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}
{{RoundBoxBottom}}

=====For clarity=====
{{RoundBoxTop|theme=7}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}

===tables===
{{RoundBoxTop|theme=1}}

{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
{{RoundBoxTop|theme=5}}
Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>n9wefb4dg97ck0nrq68idgvhjlm04o8</sha1>
    </revision>
    <revision>
      <id>2411965</id>
      <parentid>2411961</parentid>
      <timestamp>2022-08-04T22:38:09Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* A simple example: */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25881" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===
{{RoundBoxTop|theme=2}}
Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;

&lt;math&gt;\frac{x^2 - y^2}{N}&lt;/math&gt; exactly.

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}((x+y), N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}((x-y), N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;

=====Right hand side exact square=====
One congruence produced a square for y:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;
|-
| 70
| 4900
| 729
|}
:&lt;math&gt;4900 \equiv 729 \pmod{N}&lt;/math&gt;
:&lt;math&gt;70^2 \equiv 27^2 \pmod{N}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(70-27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(70+27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(97, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


=====Right hand side negative=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107 **

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121 **
| -4050

|-
| 12
| 144
| -4027

|-
| 60
| 3600
| -571

|-
| 61
| 3721 **
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75 **

|-
| 65
| 4225
| 54
|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107 **

|-
| 64
| 4096
| -75 **
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;


The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121 **
| -4050

|-
| 61
| 3721 **
| -450
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

=====Right hand side positive=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce positive 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 64
| 4096
| -75

|-
| 65
| 4225
| 54

|-
| 66
| 4356
| 185

|-
| 88
| 7744
| 3573

|-
| 89
| 7921
| 3750

|-
| 90
| 8100
| 3929

|-
| 144
| 20736
| 16565

|-
| 145
| 21025
| 16854

|-
| 146
| 21316
| 17145

|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 65
| 4225
| 54

|-
| 89
| 7921
| 3750
|}

:&lt;math&gt;4225 \equiv 54 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4225\cdot 7921 \equiv 54 \cdot 3750 \pmod{N}&lt;/math&gt;

:&lt;math&gt;33466225 \equiv 202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;5785^2 \equiv 450^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(5785-450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(5335, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(5785+450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(6235, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 89
| 7921
| 3750

|-
| 145
| 21025
| 16854
|}

:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;21025 \equiv 16854 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921\cdot 21025 \equiv 3750 \cdot 16854 \pmod{N}&lt;/math&gt;

:&lt;math&gt;166539025  \equiv 63202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;12905^2 \equiv 7950^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(12905-7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(4955, 4171)&lt;/math&gt;
&lt;math&gt;= 1.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(12905+7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(20855, 4171)&lt;/math&gt;
&lt;math&gt;= 4171.&lt;/math&gt;

This congruence produced the trivial factors of &lt;math&gt;4171.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

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===Adding 2 congruences===
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If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
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===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

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Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

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&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
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=====For speed=====
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======Many comparisons======
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If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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======Divide by 2======
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

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To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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=====For clarity=====
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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===tables===
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{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

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[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
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&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
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&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


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[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
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&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
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Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>gx129ckt0n465jlsqr1qmh67d5ko7bh</sha1>
    </revision>
    <revision>
      <id>2412084</id>
      <parentid>2411965</parentid>
      <timestamp>2022-08-05T10:50:37Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* Factors of integer N */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25823" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===

Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;
&lt;math&gt;N\mid (x^2 - y^2).&lt;/math&gt;

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}(x+y, N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}(x-y, N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;

=====Right hand side exact square=====
One congruence produced a square for y:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;
|-
| 70
| 4900
| 729
|}
:&lt;math&gt;4900 \equiv 729 \pmod{N}&lt;/math&gt;
:&lt;math&gt;70^2 \equiv 27^2 \pmod{N}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(70-27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(70+27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(97, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


=====Right hand side negative=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107 **

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121 **
| -4050

|-
| 12
| 144
| -4027

|-
| 60
| 3600
| -571

|-
| 61
| 3721 **
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75 **

|-
| 65
| 4225
| 54
|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107 **

|-
| 64
| 4096
| -75 **
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;


The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121 **
| -4050

|-
| 61
| 3721 **
| -450
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

=====Right hand side positive=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce positive 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 64
| 4096
| -75

|-
| 65
| 4225
| 54

|-
| 66
| 4356
| 185

|-
| 88
| 7744
| 3573

|-
| 89
| 7921
| 3750

|-
| 90
| 8100
| 3929

|-
| 144
| 20736
| 16565

|-
| 145
| 21025
| 16854

|-
| 146
| 21316
| 17145

|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 65
| 4225
| 54

|-
| 89
| 7921
| 3750
|}

:&lt;math&gt;4225 \equiv 54 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4225\cdot 7921 \equiv 54 \cdot 3750 \pmod{N}&lt;/math&gt;

:&lt;math&gt;33466225 \equiv 202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;5785^2 \equiv 450^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(5785-450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(5335, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(5785+450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(6235, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 89
| 7921
| 3750

|-
| 145
| 21025
| 16854
|}

:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;21025 \equiv 16854 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921\cdot 21025 \equiv 3750 \cdot 16854 \pmod{N}&lt;/math&gt;

:&lt;math&gt;166539025  \equiv 63202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;12905^2 \equiv 7950^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(12905-7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(4955, 4171)&lt;/math&gt;
&lt;math&gt;= 1.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(12905+7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(20855, 4171)&lt;/math&gt;
&lt;math&gt;= 4171.&lt;/math&gt;

This congruence produced the trivial factors of &lt;math&gt;4171.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

===Adding 2 congruences===
{{RoundBoxTop|theme=2}}
If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
{{RoundBoxBottom}}


===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

{{RoundBoxTop|theme=2}}
Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
{{RoundBoxTop|theme=5}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
{{RoundBoxTop|theme=4}}
=====For speed=====
{{RoundBoxTop|theme=7}}
======Many comparisons======
{{RoundBoxTop|theme=8}}
If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
======Divide by 2======
{{RoundBoxTop|theme=8}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

{{RoundBoxTop|theme=8}}
To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}
{{RoundBoxBottom}}

=====For clarity=====
{{RoundBoxTop|theme=7}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}

===tables===
{{RoundBoxTop|theme=1}}

{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
{{RoundBoxTop|theme=5}}
Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>hwbbmep5vi5ttbbqmlydu3gc6rjvqad</sha1>
    </revision>
    <revision>
      <id>2412085</id>
      <parentid>2412084</parentid>
      <timestamp>2022-08-05T10:53:16Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Right hand side negative */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25890" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===

Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;
&lt;math&gt;N\mid (x^2 - y^2).&lt;/math&gt;

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}(x+y, N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}(x-y, N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;

=====Right hand side exact square=====
One congruence produced a square for y:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;
|-
| 70
| 4900
| 729
|}
:&lt;math&gt;4900 \equiv 729 \pmod{N}&lt;/math&gt;
:&lt;math&gt;70^2 \equiv 27^2 \pmod{N}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(70-27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(70+27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(97, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


=====Right hand side negative=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107 **

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121 **
| -4050

|-
| 12
| 144
| -4027

|-
| 60
| 3600
| -571

|-
| 61
| 3721 **
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75 **

|-
| 65
| 4225
| 54
|}

======Non-trivial result 1======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107 **

|-
| 64
| 4096
| -75 **
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

======Non-trivial result 2======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121 **
| -4050

|-
| 61
| 3721 **
| -450
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

=====Right hand side positive=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce positive 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 64
| 4096
| -75

|-
| 65
| 4225
| 54

|-
| 66
| 4356
| 185

|-
| 88
| 7744
| 3573

|-
| 89
| 7921
| 3750

|-
| 90
| 8100
| 3929

|-
| 144
| 20736
| 16565

|-
| 145
| 21025
| 16854

|-
| 146
| 21316
| 17145

|}

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 65
| 4225
| 54

|-
| 89
| 7921
| 3750
|}

:&lt;math&gt;4225 \equiv 54 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4225\cdot 7921 \equiv 54 \cdot 3750 \pmod{N}&lt;/math&gt;

:&lt;math&gt;33466225 \equiv 202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;5785^2 \equiv 450^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(5785-450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(5335, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(5785+450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(6235, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 89
| 7921
| 3750

|-
| 145
| 21025
| 16854
|}

:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;21025 \equiv 16854 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921\cdot 21025 \equiv 3750 \cdot 16854 \pmod{N}&lt;/math&gt;

:&lt;math&gt;166539025  \equiv 63202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;12905^2 \equiv 7950^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(12905-7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(4955, 4171)&lt;/math&gt;
&lt;math&gt;= 1.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(12905+7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(20855, 4171)&lt;/math&gt;
&lt;math&gt;= 4171.&lt;/math&gt;

This congruence produced the trivial factors of &lt;math&gt;4171.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

===Adding 2 congruences===
{{RoundBoxTop|theme=2}}
If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
{{RoundBoxBottom}}


===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

{{RoundBoxTop|theme=2}}
Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
{{RoundBoxTop|theme=5}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
{{RoundBoxTop|theme=4}}
=====For speed=====
{{RoundBoxTop|theme=7}}
======Many comparisons======
{{RoundBoxTop|theme=8}}
If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
======Divide by 2======
{{RoundBoxTop|theme=8}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

{{RoundBoxTop|theme=8}}
To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}
{{RoundBoxBottom}}

=====For clarity=====
{{RoundBoxTop|theme=7}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}

===tables===
{{RoundBoxTop|theme=1}}

{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
{{RoundBoxTop|theme=5}}
Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>dxlm232b6apkp3h84orxq219mg4mfmu</sha1>
    </revision>
    <revision>
      <id>2412086</id>
      <parentid>2412085</parentid>
      <timestamp>2022-08-05T10:55:28Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Right hand side positive */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25950" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===

Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;
&lt;math&gt;N\mid (x^2 - y^2).&lt;/math&gt;

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}(x+y, N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}(x-y, N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;

=====Right hand side exact square=====
One congruence produced a square for y:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;
|-
| 70
| 4900
| 729
|}
:&lt;math&gt;4900 \equiv 729 \pmod{N}&lt;/math&gt;
:&lt;math&gt;70^2 \equiv 27^2 \pmod{N}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(70-27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(70+27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(97, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


=====Right hand side negative=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107 **

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121 **
| -4050

|-
| 12
| 144
| -4027

|-
| 60
| 3600
| -571

|-
| 61
| 3721 **
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75 **

|-
| 65
| 4225
| 54
|}

======Non-trivial result 1======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107 **

|-
| 64
| 4096
| -75 **
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

======Non-trivial result 2======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121 **
| -4050

|-
| 61
| 3721 **
| -450
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

=====Right hand side positive=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce positive 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 64
| 4096
| -75

|-
| 65
| 4225
| 54

|-
| 66
| 4356
| 185

|-
| 88
| 7744
| 3573

|-
| 89
| 7921
| 3750

|-
| 90
| 8100
| 3929

|-
| 144
| 20736
| 16565

|-
| 145
| 21025
| 16854

|-
| 146
| 21316
| 17145

|}

======Non-trivial result======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 65
| 4225
| 54

|-
| 89
| 7921
| 3750
|}

:&lt;math&gt;4225 \equiv 54 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4225\cdot 7921 \equiv 54 \cdot 3750 \pmod{N}&lt;/math&gt;

:&lt;math&gt;33466225 \equiv 202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;5785^2 \equiv 450^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(5785-450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(5335, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(5785+450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(6235, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

======Trivial result======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 89
| 7921
| 3750

|-
| 145
| 21025
| 16854
|}

:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;21025 \equiv 16854 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921\cdot 21025 \equiv 3750 \cdot 16854 \pmod{N}&lt;/math&gt;

:&lt;math&gt;166539025  \equiv 63202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;12905^2 \equiv 7950^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(12905-7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(4955, 4171)&lt;/math&gt;
&lt;math&gt;= 1.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(12905+7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(20855, 4171)&lt;/math&gt;
&lt;math&gt;= 4171.&lt;/math&gt;

This congruence produced the trivial factors of &lt;math&gt;4171.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

===Adding 2 congruences===
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If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
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===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

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Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

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&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
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=====For speed=====
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======Many comparisons======
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If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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======Divide by 2======
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

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To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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{{RoundBoxBottom}}
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=====For clarity=====
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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{{RoundBoxBottom}}

===tables===
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{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

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[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
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&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
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&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


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[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
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&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
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Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>ib16ya1b3qmmzjryoi1swfg9sc8ne11</sha1>
    </revision>
    <revision>
      <id>2412087</id>
      <parentid>2412086</parentid>
      <timestamp>2022-08-05T11:31:34Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Non-trivial result 1 */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25971" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===

Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;
&lt;math&gt;N\mid (x^2 - y^2).&lt;/math&gt;

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}(x+y, N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}(x-y, N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;

=====Right hand side exact square=====
One congruence produced a square for y:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;
|-
| 70
| 4900
| 729
|}
:&lt;math&gt;4900 \equiv 729 \pmod{N}&lt;/math&gt;
:&lt;math&gt;70^2 \equiv 27^2 \pmod{N}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(70-27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(70+27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(97, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


=====Right hand side negative=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107 **

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121 **
| -4050

|-
| 12
| 144
| -4027

|-
| 60
| 3600
| -571

|-
| 61
| 3721 **
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75 **

|-
| 65
| 4225
| 54
|}

======Non-trivial result 1======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107 **

|-
| 64
| 4096
| -75 &lt;math&gt;\ \ \ \ &lt;/math&gt;**
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

======Non-trivial result 2======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121 **
| -4050

|-
| 61
| 3721 **
| -450
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

=====Right hand side positive=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce positive 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 64
| 4096
| -75

|-
| 65
| 4225
| 54

|-
| 66
| 4356
| 185

|-
| 88
| 7744
| 3573

|-
| 89
| 7921
| 3750

|-
| 90
| 8100
| 3929

|-
| 144
| 20736
| 16565

|-
| 145
| 21025
| 16854

|-
| 146
| 21316
| 17145

|}

======Non-trivial result======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 65
| 4225
| 54

|-
| 89
| 7921
| 3750
|}

:&lt;math&gt;4225 \equiv 54 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4225\cdot 7921 \equiv 54 \cdot 3750 \pmod{N}&lt;/math&gt;

:&lt;math&gt;33466225 \equiv 202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;5785^2 \equiv 450^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(5785-450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(5335, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(5785+450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(6235, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

======Trivial result======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 89
| 7921
| 3750

|-
| 145
| 21025
| 16854
|}

:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;21025 \equiv 16854 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921\cdot 21025 \equiv 3750 \cdot 16854 \pmod{N}&lt;/math&gt;

:&lt;math&gt;166539025  \equiv 63202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;12905^2 \equiv 7950^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(12905-7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(4955, 4171)&lt;/math&gt;
&lt;math&gt;= 1.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(12905+7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(20855, 4171)&lt;/math&gt;
&lt;math&gt;= 4171.&lt;/math&gt;

This congruence produced the trivial factors of &lt;math&gt;4171.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

===Adding 2 congruences===
{{RoundBoxTop|theme=2}}
If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
{{RoundBoxBottom}}


===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

{{RoundBoxTop|theme=2}}
Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
{{RoundBoxTop|theme=5}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
{{RoundBoxTop|theme=4}}
=====For speed=====
{{RoundBoxTop|theme=7}}
======Many comparisons======
{{RoundBoxTop|theme=8}}
If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
======Divide by 2======
{{RoundBoxTop|theme=8}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

{{RoundBoxTop|theme=8}}
To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}
{{RoundBoxBottom}}

=====For clarity=====
{{RoundBoxTop|theme=7}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}

===tables===
{{RoundBoxTop|theme=1}}

{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
{{RoundBoxTop|theme=5}}
Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>p30vayc1j8skccesxmqewh8r7cp5nte</sha1>
    </revision>
    <revision>
      <id>2412088</id>
      <parentid>2412087</parentid>
      <timestamp>2022-08-05T11:37:32Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Non-trivial result 2 */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25990" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===

Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;
&lt;math&gt;N\mid (x^2 - y^2).&lt;/math&gt;

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}(x+y, N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}(x-y, N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;

=====Right hand side exact square=====
One congruence produced a square for y:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;
|-
| 70
| 4900
| 729
|}
:&lt;math&gt;4900 \equiv 729 \pmod{N}&lt;/math&gt;
:&lt;math&gt;70^2 \equiv 27^2 \pmod{N}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(70-27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(70+27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(97, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


=====Right hand side negative=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107 **

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121 **
| -4050

|-
| 12
| 144
| -4027

|-
| 60
| 3600
| -571

|-
| 61
| 3721 **
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75 **

|-
| 65
| 4225
| 54
|}

======Non-trivial result 1======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107 **

|-
| 64
| 4096
| -75 &lt;math&gt;\ \ \ \ &lt;/math&gt;**
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

======Non-trivial result 2======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121 
| -4050 !!

|-
| 61
| 3721 
| -450 &lt;math&gt;\ \ &lt;/math&gt;!!
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

=====Right hand side positive=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce positive 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 64
| 4096
| -75

|-
| 65
| 4225
| 54

|-
| 66
| 4356
| 185

|-
| 88
| 7744
| 3573

|-
| 89
| 7921
| 3750

|-
| 90
| 8100
| 3929

|-
| 144
| 20736
| 16565

|-
| 145
| 21025
| 16854

|-
| 146
| 21316
| 17145

|}

======Non-trivial result======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 65
| 4225
| 54

|-
| 89
| 7921
| 3750
|}

:&lt;math&gt;4225 \equiv 54 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4225\cdot 7921 \equiv 54 \cdot 3750 \pmod{N}&lt;/math&gt;

:&lt;math&gt;33466225 \equiv 202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;5785^2 \equiv 450^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(5785-450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(5335, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(5785+450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(6235, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

======Trivial result======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 89
| 7921
| 3750

|-
| 145
| 21025
| 16854
|}

:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;21025 \equiv 16854 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921\cdot 21025 \equiv 3750 \cdot 16854 \pmod{N}&lt;/math&gt;

:&lt;math&gt;166539025  \equiv 63202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;12905^2 \equiv 7950^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(12905-7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(4955, 4171)&lt;/math&gt;
&lt;math&gt;= 1.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(12905+7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(20855, 4171)&lt;/math&gt;
&lt;math&gt;= 4171.&lt;/math&gt;

This congruence produced the trivial factors of &lt;math&gt;4171.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

===Adding 2 congruences===
{{RoundBoxTop|theme=2}}
If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
{{RoundBoxBottom}}


===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

{{RoundBoxTop|theme=2}}
Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
{{RoundBoxTop|theme=5}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
{{RoundBoxTop|theme=4}}
=====For speed=====
{{RoundBoxTop|theme=7}}
======Many comparisons======
{{RoundBoxTop|theme=8}}
If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
======Divide by 2======
{{RoundBoxTop|theme=8}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

{{RoundBoxTop|theme=8}}
To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}
{{RoundBoxBottom}}

=====For clarity=====
{{RoundBoxTop|theme=7}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}

===tables===
{{RoundBoxTop|theme=1}}

{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
{{RoundBoxTop|theme=5}}
Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>j3x4xrevz3vpmaw96swn0s7iaffd2fi</sha1>
    </revision>
    <revision>
      <id>2412089</id>
      <parentid>2412088</parentid>
      <timestamp>2022-08-05T11:40:06Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Right hand side positive */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25968" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===

Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;
&lt;math&gt;N\mid (x^2 - y^2).&lt;/math&gt;

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}(x+y, N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}(x-y, N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;

=====Right hand side exact square=====
One congruence produced a square for y:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;
|-
| 70
| 4900
| 729
|}
:&lt;math&gt;4900 \equiv 729 \pmod{N}&lt;/math&gt;
:&lt;math&gt;70^2 \equiv 27^2 \pmod{N}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(70-27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(70+27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(97, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


=====Right hand side negative=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107 **

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121 **
| -4050

|-
| 12
| 144
| -4027

|-
| 60
| 3600
| -571

|-
| 61
| 3721 **
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75 **

|-
| 65
| 4225
| 54
|}

======Non-trivial result 1======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107 **

|-
| 64
| 4096
| -75 &lt;math&gt;\ \ \ \ &lt;/math&gt;**
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

======Non-trivial result 2======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121 
| -4050 !!

|-
| 61
| 3721 
| -450 &lt;math&gt;\ \ &lt;/math&gt;!!
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

=====Right hand side positive=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce positive 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 65
| 4225
| 54

|-
| 66
| 4356
| 185

|-
| 88
| 7744
| 3573

|-
| 89
| 7921
| 3750

|-
| 90
| 8100
| 3929

|-
| 144
| 20736
| 16565

|-
| 145
| 21025
| 16854

|-
| 146
| 21316
| 17145

|}

======Non-trivial result======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 65
| 4225
| 54

|-
| 89
| 7921
| 3750
|}

:&lt;math&gt;4225 \equiv 54 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4225\cdot 7921 \equiv 54 \cdot 3750 \pmod{N}&lt;/math&gt;

:&lt;math&gt;33466225 \equiv 202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;5785^2 \equiv 450^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(5785-450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(5335, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(5785+450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(6235, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

======Trivial result======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 89
| 7921
| 3750

|-
| 145
| 21025
| 16854
|}

:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;21025 \equiv 16854 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921\cdot 21025 \equiv 3750 \cdot 16854 \pmod{N}&lt;/math&gt;

:&lt;math&gt;166539025  \equiv 63202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;12905^2 \equiv 7950^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(12905-7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(4955, 4171)&lt;/math&gt;
&lt;math&gt;= 1.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(12905+7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(20855, 4171)&lt;/math&gt;
&lt;math&gt;= 4171.&lt;/math&gt;

This congruence produced the trivial factors of &lt;math&gt;4171.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

===Adding 2 congruences===
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If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
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===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

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Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
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&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

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&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
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=====For speed=====
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======Many comparisons======
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If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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======Divide by 2======
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

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To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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{{RoundBoxBottom}}
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=====For clarity=====
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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{{RoundBoxBottom}}

===tables===
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{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

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[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
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&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
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&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


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[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
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&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
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Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>32r0drz4gwc0a65sk1zc1c6gfqo5jq9</sha1>
    </revision>
    <revision>
      <id>2412090</id>
      <parentid>2412089</parentid>
      <timestamp>2022-08-05T11:50:14Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Non-trivial result */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25995" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===

Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;
&lt;math&gt;N\mid (x^2 - y^2).&lt;/math&gt;

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}(x+y, N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}(x-y, N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;

=====Right hand side exact square=====
One congruence produced a square for y:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;
|-
| 70
| 4900
| 729
|}
:&lt;math&gt;4900 \equiv 729 \pmod{N}&lt;/math&gt;
:&lt;math&gt;70^2 \equiv 27^2 \pmod{N}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(70-27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(70+27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(97, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


=====Right hand side negative=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107 **

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121 **
| -4050

|-
| 12
| 144
| -4027

|-
| 60
| 3600
| -571

|-
| 61
| 3721 **
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75 **

|-
| 65
| 4225
| 54
|}

======Non-trivial result 1======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107 **

|-
| 64
| 4096
| -75 &lt;math&gt;\ \ \ \ &lt;/math&gt;**
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

======Non-trivial result 2======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121 
| -4050 !!

|-
| 61
| 3721 
| -450 &lt;math&gt;\ \ &lt;/math&gt;!!
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

=====Right hand side positive=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce positive 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 65
| 4225
| 54

|-
| 66
| 4356
| 185

|-
| 88
| 7744
| 3573

|-
| 89
| 7921
| 3750

|-
| 90
| 8100
| 3929

|-
| 144
| 20736
| 16565

|-
| 145
| 21025
| 16854

|-
| 146
| 21316
| 17145

|}

======Non-trivial result======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 65
| 4225
| 54 &lt;math&gt;\ \ \ &lt;/math&gt;**

|-
| 89
| 7921
| 3750 **!!
|}

:&lt;math&gt;4225 \equiv 54 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4225\cdot 7921 \equiv 54 \cdot 3750 \pmod{N}&lt;/math&gt;

:&lt;math&gt;33466225 \equiv 202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;5785^2 \equiv 450^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(5785-450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(5335, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(5785+450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(6235, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

======Trivial result======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 89
| 7921
| 3750

|-
| 145
| 21025
| 16854
|}

:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;21025 \equiv 16854 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921\cdot 21025 \equiv 3750 \cdot 16854 \pmod{N}&lt;/math&gt;

:&lt;math&gt;166539025  \equiv 63202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;12905^2 \equiv 7950^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(12905-7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(4955, 4171)&lt;/math&gt;
&lt;math&gt;= 1.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(12905+7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(20855, 4171)&lt;/math&gt;
&lt;math&gt;= 4171.&lt;/math&gt;

This congruence produced the trivial factors of &lt;math&gt;4171.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

===Adding 2 congruences===
{{RoundBoxTop|theme=2}}
If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
{{RoundBoxBottom}}


===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

{{RoundBoxTop|theme=2}}
Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
{{RoundBoxTop|theme=5}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
{{RoundBoxTop|theme=4}}
=====For speed=====
{{RoundBoxTop|theme=7}}
======Many comparisons======
{{RoundBoxTop|theme=8}}
If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
======Divide by 2======
{{RoundBoxTop|theme=8}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

{{RoundBoxTop|theme=8}}
To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}
{{RoundBoxBottom}}

=====For clarity=====
{{RoundBoxTop|theme=7}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}

===tables===
{{RoundBoxTop|theme=1}}

{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
{{RoundBoxTop|theme=5}}
Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>elgzlbxw2gg2l3a1kim0q9coa7gb4bu</sha1>
    </revision>
    <revision>
      <id>2412093</id>
      <parentid>2412090</parentid>
      <timestamp>2022-08-05T11:53:29Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Trivial result */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="26018" xml:space="preserve">
&lt;math&gt;3&lt;/math&gt; cube roots of &lt;math&gt;W&lt;/math&gt; 

&lt;math&gt;W = 0.828 + 2.035\cdot i&lt;/math&gt; 

&lt;math&gt;w_0 = 1.2 + 0.5\cdot i&lt;/math&gt; 

&lt;math&gt;w_1 = \frac{-1.2 - 0.5\sqrt{3}}{2} + \frac{1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_2 = \frac{-1.2 + 0.5\sqrt{3}}{2} + \frac{- 1.2\sqrt{3} - 0.5}{2}\cdot i&lt;/math&gt; 

&lt;math&gt;w_0^3 = w_1^3 = w_2^3 = W&lt;/math&gt; 

&lt;math&gt;&lt;/math&gt; 


&lt;math&gt;&lt;/math&gt;


&lt;math&gt;y = x^3 - x&lt;/math&gt;


&lt;math&gt;y = x^3&lt;/math&gt;


&lt;math&gt;y = x^3 + x&lt;/math&gt;
===allEqual===

&lt;math&gt;y = f(x) = x^3&lt;/math&gt;

&lt;math&gt;y = f(-x)&lt;/math&gt;

&lt;math&gt;y = f(x) = x^3 + x&lt;/math&gt;

&lt;math&gt;x = p&lt;/math&gt;

&lt;math&gt;y = f(x) =  (x-5)^3 - 4(x-5) + 7&lt;/math&gt;

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;

==Law of addition==
===Factors of integer N===

Several modern methods for determining the factors of a given integer attempt
to create two congruent squares modulo integer 
&lt;math&gt;N.&lt;/math&gt;

&lt;math&gt;x^2 \equiv y^2 \pmod{N}&lt;/math&gt;

This means that the difference between the two squares is exactly divisible by &lt;math&gt;N:&lt;/math&gt;
&lt;math&gt;N\mid (x^2 - y^2).&lt;/math&gt;

Integer &lt;math&gt;N&lt;/math&gt; always contains the factors &lt;math&gt;N,1,&lt;/math&gt; called trivial factors.

If &lt;math&gt;N&lt;/math&gt; contains two non-trivial factors
&lt;math&gt;p,q,&lt;/math&gt; then:

&lt;math&gt;\frac{(x+y)(x-y)}{p \cdot q}.&lt;/math&gt;

With a little luck &lt;math&gt;p\mid (x+y)&lt;/math&gt; and &lt;math&gt;q\mid (x-y)&lt;/math&gt; in which case:

&lt;math&gt;p = \text{igcd}(x+y, N)&lt;/math&gt; and &lt;math&gt;q = \text{igcd}(x-y, N)&lt;/math&gt; where

&quot;&lt;math&gt;\text{igcd}&lt;/math&gt;&quot; is function &quot;&lt;math&gt;\text{integer greatest common divisor.}&lt;/math&gt;&quot;
====A simple example:====

We will use quadratic congruences to calculate factors of &lt;math&gt;N = 4171&lt;/math&gt; for &lt;math&gt;164 \ge x \ge 1.&lt;/math&gt;

=====Right hand side exact square=====
One congruence produced a square for y:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;
|-
| 70
| 4900
| 729
|}
:&lt;math&gt;4900 \equiv 729 \pmod{N}&lt;/math&gt;
:&lt;math&gt;70^2 \equiv 27^2 \pmod{N}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(70-27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(70+27, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(97, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;


Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


=====Right hand side negative=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce negative 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 7
| 49
| -4122

|-
| 8
| 64
| -4107 **

|-
| 9
| 81
| -4090

|-
| 10
| 100
| -4071

|-
| 11
| 121 **
| -4050

|-
| 12
| 144
| -4027

|-
| 60
| 3600
| -571

|-
| 61
| 3721 **
| -450

|-
| 62
| 3844
| -327

|-
| 63
| 3969
| -202

|-
| 64
| 4096
| -75 **

|-
| 65
| 4225
| 54
|}

======Non-trivial result 1======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 8
| 64
| -4107 **

|-
| 64
| 4096
| -75 &lt;math&gt;\ \ \ \ &lt;/math&gt;**
|}

:&lt;math&gt;64 \equiv -4107 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4096 \equiv -75 \pmod{N}&lt;/math&gt;
:&lt;math&gt;64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}&lt;/math&gt;
:&lt;math&gt;262144 \equiv 308025 \pmod{N}&lt;/math&gt;
:&lt;math&gt;512^2 \equiv 555^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(555-512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(43, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(555+512, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(1067, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

======Non-trivial result 2======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 11
| 121 
| -4050 !!

|-
| 61
| 3721 
| -450 &lt;math&gt;\ \ &lt;/math&gt;!!
|}

:&lt;math&gt;121 \equiv -4050 \pmod{N}&lt;/math&gt;
:&lt;math&gt;3721 \equiv -450 \pmod{N}&lt;/math&gt;
:&lt;math&gt;121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}&lt;/math&gt;

:&lt;math&gt;450241 \equiv 1822500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;671^2 \equiv 1350^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(1350-671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(679, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(1350+671, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(2021, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;

=====Right hand side positive=====
Table below contains a sample of values of 
&lt;math&gt;x&lt;/math&gt; that produce positive 
&lt;math&gt;y:&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 65
| 4225
| 54

|-
| 66
| 4356
| 185

|-
| 88
| 7744
| 3573

|-
| 89
| 7921
| 3750

|-
| 90
| 8100
| 3929

|-
| 144
| 20736
| 16565

|-
| 145
| 21025
| 16854

|-
| 146
| 21316
| 17145

|}

======Non-trivial result======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 65
| 4225
| 54 &lt;math&gt;\ \ \ &lt;/math&gt;**

|-
| 89
| 7921
| 3750 **!!
|}

:&lt;math&gt;4225 \equiv 54 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;4225\cdot 7921 \equiv 54 \cdot 3750 \pmod{N}&lt;/math&gt;

:&lt;math&gt;33466225 \equiv 202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;5785^2 \equiv 450^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(5785-450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(5335, 4171)&lt;/math&gt;
&lt;math&gt;= 97.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(5785+450, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(6235, 4171)&lt;/math&gt;
&lt;math&gt;= 43.&lt;/math&gt;

Non-trivial factors of &lt;math&gt;4171&lt;/math&gt; are &lt;math&gt;43,97.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

======Trivial result======

The congruences:
{| class=&quot;wikitable&quot;
|- 
!   &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2&lt;/math&gt;  !! &lt;math&gt;y = x^2 - N&lt;/math&gt;

|-
| 89
| 7921
| 3750 **!!

|-
| 145
| 21025
| 16854 &lt;math&gt;\ &lt;/math&gt;!!
|}

:&lt;math&gt;7921 \equiv 3750 \pmod{N}&lt;/math&gt;
:&lt;math&gt;21025 \equiv 16854 \pmod{N}&lt;/math&gt;
:&lt;math&gt;7921\cdot 21025 \equiv 3750 \cdot 16854 \pmod{N}&lt;/math&gt;

:&lt;math&gt;166539025  \equiv 63202500 \pmod{N}&lt;/math&gt;
:&lt;math&gt;12905^2 \equiv 7950^2 \pmod{4171}&lt;/math&gt;

&lt;math&gt;p = \text{igcd}(12905-7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(4955, 4171)&lt;/math&gt;
&lt;math&gt;= 1.&lt;/math&gt;

&lt;math&gt;q = \text{igcd}(12905+7950, 4171)&lt;/math&gt;
&lt;math&gt;= \text{igcd}(20855, 4171)&lt;/math&gt;
&lt;math&gt;= 4171.&lt;/math&gt;

This congruence produced the trivial factors of &lt;math&gt;4171.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
&lt;/syntaxhighlight&gt;

===Adding 2 congruences===
{{RoundBoxTop|theme=2}}
If &lt;math&gt;A \equiv B \pmod{N}, &lt;/math&gt; and
&lt;math&gt;C \equiv D \pmod{N}, &lt;/math&gt; then:
&lt;math&gt;A+C \equiv B+D \pmod{N}.&lt;/math&gt;

Proof:

&lt;math&gt;A-B = K_1\cdot N&lt;/math&gt;, therefore &lt;math&gt;A = B + K_1\cdot N&lt;/math&gt;
and &lt;math&gt;C = D + K_2\cdot N&lt;/math&gt;

&lt;math&gt;(A+C) - (B+D) = B + K_1\cdot N + D + K_2\cdot N  - B - D = N(K_1 + K_2)&lt;/math&gt; which is exactly divisible by N.
{{RoundBoxBottom}}


===Quadratic Congruences===
A quadratic congruence is a congruence that contains at least one exact square,
for example:

&lt;math&gt;x^2 \equiv y \pmod{N}&lt;/math&gt; or &lt;math&gt;x^2 \equiv y^2 \pmod{N}.&lt;/math&gt;

Initially, let us consider the congruence: &lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

If &lt;math&gt;y = x^2 - N,&lt;/math&gt; then:

&lt;math&gt;x^2 \equiv y \pmod{N}.&lt;/math&gt;

Proof: &lt;math&gt;x^2 - y  = x^2 - (x^2 - N) = N&lt;/math&gt; which is exactly divisible by
&lt;math&gt;N.&lt;/math&gt;

Consider an example with real numbers.

Let &lt;math&gt;N = 257&lt;/math&gt; and &lt;math&gt;26 \ge x \ge 6.&lt;/math&gt;
&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  6 | -221
  7 | -208
  8 | -193
  9 | -176
 10 | -157
 11 | -136
 12 | -113
 13 |  -88
 14 |  -61
 15 |  -32
 16 |   -1
 17 |   32
 18 |   67
 19 |  104
 20 |  143
 21 |  184
 22 |  227
 23 |  272
 24 |  319
 25 |  368
 26 |  419
&lt;/syntaxhighlight&gt;
A cursory glance at the values of &lt;math&gt;x^2 - N&lt;/math&gt; indicates that the value 
&lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5.&lt;/math&gt;

Proof:
&lt;math&gt;N \equiv 2 \pmod{5}&lt;/math&gt; therefore &lt;math&gt;N - 2 = k5&lt;/math&gt; or 
&lt;math&gt;N = 5k + 2.&lt;/math&gt;

The table shows all possible values of &lt;math&gt;x\ %\ 5:&lt;/math&gt;
&lt;syntaxhighlight&gt;
   x   |       x^2       |        y = x^2 - N
------ | --------------- | -----------------------------------------------
5p + 0 | 25pp            | 25pp            - (5k+2) = 25pp       - 5k -  2
5p + 1 | 25pp + 10p +  1 | 25pp + 10p +  1 - (5k+2) = 25pp + 10p - 5k -  1
5p + 2 | 25pp + 20p +  4 | 25pp + 20p +  4 - (5k+2) = 25pp + 20p - 5k +  2
5p + 3 | 25pp + 30p +  9 | 25pp + 30p +  9 - (5k+2) = 25pp + 30p - 5k +  7
5p + 4 | 25pp + 40p + 16 | 25pp + 40p + 16 - (5k+2) = 25pp + 40p - 5k + 14
&lt;/syntaxhighlight&gt;
As you can see, the value &lt;math&gt;y = x^2 - N&lt;/math&gt; is never exactly divisible by
&lt;math&gt;5.&lt;/math&gt;

If you look closely, you will see also that it is never exactly divisible by
&lt;math&gt;3.&lt;/math&gt;

Why is this? An interesting question that leads us to the topic of quadratic residues.
====Quadratic Residues====
Consider all the congruences for prime number &lt;math&gt;5:&lt;/math&gt;

&lt;math&gt;x^2 \equiv y \pmod{5}&lt;/math&gt; for &lt;math&gt;5 &gt; x \ge 0.&lt;/math&gt;

&lt;syntaxhighlight&gt;
 x |   x^2   | (x^2) % 5
---|---------|-----------
 0 |    0    | 0
 1 |    1    | 1
 2 |    4    | 4
 3 |    9    | 4
 4 |   16    | 1
 &lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;5&lt;/math&gt; are &lt;math&gt;0,1,4.&lt;/math&gt;

Values &lt;math&gt;2,3&lt;/math&gt; are not quadratic residues of &lt;math&gt;5.&lt;/math&gt; These values are quadratic non-residues.

To calculate the quadratic residues of a small prime &lt;math&gt;p:&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code:
def quadResidues(p) :
    L1 = []
    for	v in range (p&gt;&gt;1, -1, -1) :
      	L1 += [(v*v) % p]
    return L1

print (quadResidues(11))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
[3, 5, 9, 4, 1, 0]
&lt;/syntaxhighlight&gt;
Quadratic residues of &lt;math&gt;11&lt;/math&gt; are &lt;math&gt;0,1,3,4,5,9.&lt;/math&gt;

The method presented here answers the question, &quot;What are the quadratic residues of p?&quot;

If &lt;math&gt;p&lt;/math&gt; is a very large prime, the question is often,
&quot;Is r a quadratic residue of p?&quot; The answer is found in advanced number theory.

Let us return to quadratic residues mod &lt;math&gt;N = 257.&lt;/math&gt;

&lt;math&gt;N\ %\ 5 = 2,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is not a quadratic residue of
&lt;math&gt;5.&lt;/math&gt; This is why &lt;math&gt;x^2 - N&lt;/math&gt; is never divisible by &lt;math&gt;5&lt;/math&gt;
exactly.

&lt;math&gt;N\ %\ 11 = 4,&lt;/math&gt; therefore &lt;math&gt;N&lt;/math&gt; is a quadratic residue of
&lt;math&gt;11&lt;/math&gt; and a value of &lt;math&gt;x&lt;/math&gt; that satisfies the congruence
&lt;math&gt;x^2 \equiv 4 \pmod{257}&lt;/math&gt; has form &lt;math&gt;11p \pm 2.&lt;/math&gt;

From the table above:

&lt;syntaxhighlight&gt;
N = 257
  x | x^2 - N
----|--------
  9 | -176
 13 |  -88
 20 |  143
 24 |  319
&lt;/syntaxhighlight&gt;

These &lt;math&gt;4&lt;/math&gt; values of &lt;math&gt;x^2 - N&lt;/math&gt; are exactly divisible by
&lt;math&gt;11.&lt;/math&gt;

&lt;math&gt;x = 9&lt;/math&gt; is &lt;math&gt;11\cdot 1 - 2.&lt;/math&gt;

&lt;math&gt;x = 13&lt;/math&gt; is &lt;math&gt;11\cdot 1 + 2.&lt;/math&gt;

&lt;math&gt;x = 20&lt;/math&gt; is &lt;math&gt;11\cdot 2 - 2.&lt;/math&gt;

&lt;math&gt;x = 24&lt;/math&gt; is &lt;math&gt;11\cdot 2 + 2.&lt;/math&gt;
=====Products=====
This section uses prime number &lt;math&gt;41&lt;/math&gt; as an example.

Using &lt;code&gt;quadResidues(p)&lt;/code&gt; quadratic residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40]
&lt;/syntaxhighlight&gt;
Quadratic non-residues of &lt;math&gt;41&lt;/math&gt; are:
&lt;syntaxhighlight&gt;
qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38]
&lt;/syntaxhighlight&gt;
======of 2 residues======
A simple test to verify that the product of 2 residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (index1, len(qr41)) :
    	v2 = qr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 residues is a residue. Advanced math proves that this is true for all primes.

======of 2 non-residues======
A simple test to verify that the product of 2 non-residues is a residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (0, len(qnr41)) :
    v1 = qnr41[index1]
    for index2 in range (index1, len(qnr41)) :
	    v2 = qnr41[index2]
	    residue = (v1*v2) % 41
	    if residue not in qr41 : print ('residue',residue,'not quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes.

======of residue and non-residue======
A simple test to verify that the product of residue and non-residue is non-residue:
&lt;syntaxhighlight lang=python&gt;
# Python code.
for index1 in range (1, len(qr41)) :
    v1 = qr41[index1]
    for index2 in range (0, len(qnr41)) :
        v2 = qnr41[index2]
        residue = (v1*v2) % 41
        if residue not in qnr41 : print ('residue',residue,'quadratic.')
&lt;/syntaxhighlight&gt;
This test shows that, at least for prime number &lt;math&gt;41,&lt;/math&gt; the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes.

{{RoundBoxTop|theme=2}}
Some authors may consider &lt;math&gt;0&lt;/math&gt; as not a legitimate residue.

&lt;math&gt;0&lt;/math&gt; is not included as a residue in the test above.
{{RoundBoxBottom}}

{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Examples===
{{RoundBoxTop|theme=5}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;math&gt;39x^2 + 64y^2 - 2496 = 0&lt;/math&gt;

&lt;math&gt;64x^2 + 39y^2 - 2496 = 0&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

====Techniques====
{{RoundBoxTop|theme=4}}
=====For speed=====
{{RoundBoxTop|theme=7}}
======Many comparisons======
{{RoundBoxTop|theme=8}}
If your code contains many numerical comparisons, it may be tempting to put:

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
If all values 
&lt;code&gt;a,b,c,d,e,f,g,h&lt;/code&gt; are equal and non-zero, processing the above statement takes time.
For greater speed, put &lt;math&gt;0&lt;/math&gt; and the value most likely to be non-zero at beginning of comparison:
&lt;syntaxhighlight lang=python&gt;
# python code.
if 0 == f == a == b == c == d == e == g == h :
    pass
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
======Divide by 2======
{{RoundBoxTop|theme=8}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
Division by 2 seems simple enough:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b / 2
&lt;/syntaxhighlight&gt;
Divisions are time consuming. If b is a large Decimal number, the following code is faster:
&lt;syntaxhighlight lang=python&gt;
# python code.
a = D('0.5') * b
&lt;/syntaxhighlight&gt;
If b is &lt;code&gt;type int,&lt;/code&gt; right shift is faster than multiplication by &lt;code&gt;0.5:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
a = b &gt;&gt; 1
&lt;/syntaxhighlight&gt;
Also, right shift preserves precision of &lt;code&gt;type int:&lt;/code&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; a = b/2 ; a
6.172839450617284e+27
&gt;&gt;&gt; a = b &gt;&gt; 1 ; a
6172839450617283945061728394
&lt;/syntaxhighlight&gt;

{{RoundBoxTop|theme=8}}
To preserve rightmost bit:
&lt;syntaxhighlight lang=python&gt;
# python code.
&gt;&gt;&gt; b = 12345678901234567890123456789
&gt;&gt;&gt; rightbit = b &amp; 1 ; rightbit 
1
&gt;&gt;&gt; b &gt;&gt;= 1 ; b
6172839450617283945061728394
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}
{{RoundBoxBottom}}

=====For clarity=====
{{RoundBoxTop|theme=7}}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

{{RoundBoxBottom}}
{{RoundBoxBottom}}

===tables===
{{RoundBoxTop|theme=1}}

{| class=&quot;wikitable&quot;
|- 
!  || No equal roots !! 2 equal roots  !! 3 equal roots  !! 4 equal roots  !! 2 pairs of equal roots
|-
| Cubic: 1(a), 2(a)
| different
| different
| different
| same
| different
|-
| Quadratic: 1(b), 2(b)
| different
| different
| same, 1root
| null
| same, 2roots
|-
| Linear: 1(c), 2(c)
| different
| same
| null
| null
| null
|}


See [[Cubic_function#Function_as_product_of_linear_function_and_quadratic | Function_as_product_of_linear_function_and_quadratic]] above.

To calculate all roots:


&lt;syntaxhighlight lang=python&gt;
# python code.
a,b,c,d = 1,-3,-9,-5

# Associated quadratic:
p = -1
A = a
B = A*p + b
C = B*p + c

# Associated linear function:
a1 = A
b1 = a1*p + B

print ('x3 =', -b1/a1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x3 = 5.0
&lt;/syntaxhighlight&gt;
Roots of cubic function &lt;math&gt;f(x) = x^3 - 3x^2 - 9x - 5&lt;/math&gt; are &lt;math&gt;-1, -1, 5.&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
# python code.

&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;
====Introduction to floats====
{{RoundBoxTop|theme=5}}
Although integers are great for many situations, they have a serious limitation, integers are [[Wikipedia:Natural number|whole numbers]]. This means that they do not include all [[Wikipedia:Real number|real numbers]]. A ''real number'' is a value that represents a quantity along a continuous line&lt;ref&gt;[[Wikipedia:Real number]]&lt;/ref&gt;, which means that it can have fractions in decimal forms. &lt;code&gt;4.5&lt;/code&gt;, &lt;code&gt;1.25&lt;/code&gt;, and &lt;code&gt;0.75&lt;/code&gt; are all real numbers. In computer science, real numbers are represented as floats. To test if a number is float, we can use the &lt;code&gt;isinstance&lt;/code&gt; built-in function.
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; isinstance(4.5, float)
True
&gt;&gt;&gt; isinstance(1.25, float)
True
&gt;&gt;&gt; isinstance(0.75, float)
True
&gt;&gt;&gt; isinstance(3.14159, float)
True
&gt;&gt;&gt; isinstance(2.71828, float)
True
&gt;&gt;&gt; isinstance(1.0, float)
True
&gt;&gt;&gt; isinstance(271828, float)
False
&gt;&gt;&gt; isinstance(0, float)
False
&gt;&gt;&gt; isinstance(0.0, float)
True
&lt;/syntaxhighlight&gt;


As a general rule of thumb, floats have a ''[[Wikipedia:Decimal mark|decimal point]]'' and integers do not have a ''decimal point''. So even though &lt;code&gt;4&lt;/code&gt; and &lt;code&gt;4.0&lt;/code&gt; are the same number, &lt;code&gt;4&lt;/code&gt; is an integer while &lt;code&gt;4.0&lt;/code&gt; is a float.

The basic arithmetic operations used for integers will also work for floats. (Bitwise operators will not work with floats.)
&lt;syntaxhighlight lang=python&gt;
&gt;&gt;&gt; 4.0 + 2.0
6.0
&gt;&gt;&gt; -1.0 + 4.5
3.5
&gt;&gt;&gt; 1.75 - 1.5
0.25
&gt;&gt;&gt; 4.13 - 1.1
3.03
&gt;&gt;&gt; 4.5 // 1.0
4.0
&gt;&gt;&gt; 4.5 / 1.0
4.5
&gt;&gt;&gt; 4.5 % 1.0
0.5
&gt;&gt;&gt; 7.75 * 0.25
1.9375
&gt;&gt;&gt; 0.5 * 0.5
0.25
&gt;&gt;&gt; 1.5 ** 2.0
2.25
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}</text>
      <sha1>b28o5lehgk877rmzl03ei4mi0evu163</sha1>
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  </page>
  <page>
    <title>JCCAP FDF/2017</title>
    <ns>0</ns>
    <id>224042</id>
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      <contributor>
        <username>Ncharamut</username>
        <id>2824970</id>
      </contributor>
      <comment>/* Addresses */ removed redundancy in dr/phd</comment>
      <model>wikitext</model>
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=Inaugural Meeting of the JCCAP Future Directions Forum (FDF)=
The conference included four keynote themes. Each had a speaker who delivered a talk focused on a topic from a recent Future Directions featured article in the Journal of Clinical Child and Adolescent Psychology. After each talk, the audience broke into small groups for facilitated discussion led by other content experts. An innovation at the meeting was to build Wikiversity pages to gather together ideas and links from the talks and discussions. 

The four themes in 2017 were:

*Evidence Based Assessment
*Suicide
*Bipolar Disorder
*Conduct Problems

== '''Addresses''' ==

=== '''''Future Directions Address 1: Future Directions of Assessment''''' ===
'''Presented by Eric Youngstrom, Ph.D.'''

====Description====
Eric Youngstrom did the keynote, using the [[Evidence based assessment/Vignettes/Lea|vignette of Lea]] as an example to illustrate steps and principles in applying [[Evidence based assessment|evidence-based assessment]]. 

The talk lays out a way of reorganizing the sequence of assessment to maximize efficiency and reduce costs. The choice of when to assess and what tool to use is guided by focusing on three different phases of the clinical encounter: a Prediction phase (focused on risk assessment, screening, and integrating the results to produce a dashboard of probable hypotheses), a Prescription phase (where assessment finalizes diagnoses and case formulation and guides treatment selection) and a Process phase (where the emphasis shifts to measuring progress, process measures such as session attendance, homework, or mediational mechanisms, as well as outcome evaluation). By optimizing the order of assessments and using the best of the free measures whenever available, the approach can yield large improvements in accuracy, and may also improve satisfaction and outcome, while adding little time or expense to the evaluation process.

The barriers to implementation include a lack of familiarity with the model, and difficulty finding the tools and the supporting information.

Recent initiatives that are addressing these barriers include:

# Multiple societies contributing small grant support to build Wikipedia pages that describe the best free assessment resources for common presenting problems and diagnoses (a list is available [[Evidence based assessment/Measures|here]]). These pages provide information for the general public, and they have been developed using small teams mixing students and content experts. They are working to include links to PDFs of the measure, or sometimes even online scoring tools that automate the process.
# A second initiative, with support from SCCAP, has also built out a set of sister pages on Wikiversity that are intended for an audience of clinicians and trainees. They include more information about scoring, norms, psychometrics, and examples of interpretation.
# A new student service organization, [[Helping Give Away Psychological Science|Helping Give Away Psychological Science (HGAPS)]] has been founded with the purpose of helping undergraduate and graduate students learn the editing skills, etiquette, and critical thinking needed to make successful edits on Wikipedia and Wikiversity pages. HGAPS will work in partnership with SCCAP and other professional societies to help extend the quantity and quality of information about psychology that reaches the public.

=== '''''Future Directions Address 2: Future Directions of Suicide and Self-Injury''''' ===
'''Presented by [[wikipedia:Matthew_Nock|Matthew Nock, Ph.D.]]'''

====Description====
In this talk Dr. Matt Nock discusses the future directions in suicide research and how we can use technology to understand, predict, and prevent suicidal behavior. He begins by mentioning the slow progress made in the field when it comes to studying suicidal behaviors and then goes into predictors of these behaviors. However, there are big gaps in the understanding, prediction, and prevention of suicidal behaviors. These gaps include needing methods to combine known risk factors, needing objective markers of suicide risk, and needing data on imminent risk. Dr. Nock discusses how machine learning (ML) can be applied to medical data in order to create &quot;risk scores&quot;. ML can also be used to discover new risk factors. The next part of his talk was to discuss the need for objective markers (i.e., not self-report) of suicide risk. In doing so, he studied using an implicit association test (IAT) as an objective measure and found that IAT did incrementally add to the prediction of self-harm. IAT can also be delivered via technology such as a phone or computer, which makes using this method easy to administer. Lastly, he covered the need for data on imminent risk. To address this, Dr. Nock talks about &quot;digital phenotyping&quot; which is defined as &quot;moment-by-moment quantification of the individual-level phenotype in situ using data from personal digital devices&quot;. Using a smartphone app, this technique can be utilized to monitor suicidal ideation in real time and decreases the amount of recall bias while predicting the behavior in context. Overall, Dr. Nock provides logic for using technology (especially smartphones) as a way of providing real-time interventions for those with suicidal behaviors/ideation.   

=== '''''Future Directions Address 3: Future Directions of Bipolar Spectrum Disorders''''' ===
'''Presented by Mary Fristad, Ph.D.'''

====Description====
Dr. Fristad presented on the future directions of research on youth with bipolar spectrum disorders. She began by giving a brief history and overview of bipolar disorder in youth including influential studies in the field such as the Course of Bipolar Youth (COBY) Study&lt;ref&gt;{{Cite journal|last=Birmaher|first=Boris|last2=Axelson|first2=David|last3=Goldstein|first3=Benjamin|last4=Strober|first4=Michael|last5=Gill,|first5=Mary Kay|last6=Hunt|first6=Jeffrey|last7=Houck|first7=Patricia|last8=Ha|first8=Wonho|last9=Iyengar|first9=Satish|date=2009-07|title=Four-Year Longitudinal Course of Children and Adolescents With Bipolar Spectrum Disorders: The Course and Outcome of Bipolar Youth (COBY) Study|url=http://psychiatryonline.org/doi/abs/10.1176/appi.ajp.2009.08101569|journal=American Journal of Psychiatry|language=en|volume=166|issue=7|pages=795–804|doi=10.1176/appi.ajp.2009.08101569|issn=0002-953X}}&lt;/ref&gt; and the Bipolar Offspring Study (BIOS)&lt;ref&gt;{{Cite web|url=https://www.pediatricbipolar.pitt.edu/research/bios-family-study|title=BIOS Family Study {{!}} CABS {{!}} University of Pittsburgh|website=www.pediatricbipolar.pitt.edu|access-date=2022-06-22}}&lt;/ref&gt;. She also provided statistics on the prevalence of bipolar in youth as well as the developmental course of the disorder. Dr. Fristad discusses the current treatments of bipolar in youth. Additionally, she talks about how many of the known prognoses and courses no longer fit what researchers are seeing. There are new findings that bipolar may be a progressive disease with increased cortical volume as well as two early variants: one with a poor prognosis and the other that is developmentally limited with a good prognosis. Using smartphones we can asses individuals by using ecological momentary assessment and treat by using intervention prompts. She also discusses the promising literature surrounding nutritional interventions including broad-spectrum nutrients, Omega 3, whole food diets, and Vitamin D. The possible mechanisms surrounding these interventions are inflammation, microbiome imbalance-gut dysbiosis, oxidative stress, and impaired mitochondrial output. Dr. Fristad finishes by discussing the questions around the treatment of bipolar disorder that still need to be looked at as well as future directions in assessment of bipolar disorder in youth. 

=== '''''Future Directions Address 4: Future Directions of Conduct Problems''''' ===
'''Presented by Daniel Shaw, Ph.D.'''

====Description====
Dr. Shaw was the keynote speaker who focused on the development and prevention of early conduct problems. He discussed the developmental course of conduct problems (CP) and the need to begin intervention in early childhood; even as early as age 2. He spoke on the malleable nature of conduct problems before the age of 5. Dr. Shaw also discussed risk factors for conduct problems including negative emotionality, inhibitory control, fearlessness, deceitful-callous/unemotional behavior. He also mentioned that parenting factors and attachment style can influence the development of CP. Lastly he discussed future directions in research on the development of conduct problems. Some areas for research include child sex as there is little research done on conduct problems in girls and poverty as there are many risk factors that are associated with living in poverty that need to be explored. In highlighting interventions for conduct problems, Dr. Shaw discussed how most new interventions focus on promoting positive parent-child relationships and general child outcomes rather than CP per se, using the attachment theory as a basis, The biggest challenge with this he says is that interventions emphasizing similar types of parenting would not be relevant for all kinds of parents of children with CP, despite the strong evidence that a diverse number of children benefit from this.

== '''Workshops''' ==

=== '''''Strategies for Improving Writing Clarity''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====
People tend to be drawn to and understand information best when it is communicated to them in the form of a narrative or “story” rather than a list of facts. However, researchers rarely receive formal training on leveraging narrative tools when writing about their academic work. In this workshop, Dr. Andres De Los Reyes describes evidence-based strategies for consistently applying narrative structure to academic work, with a focus on preparing manuscripts for submission to peer-reviewed academic journals. 

=== '''''Strategies for Writing Training Grants''''' ===
'''Presented by''' '''Andres De Los Reyes, Ph.D.'''

===== Description =====
This workshop serves as a companion to our workshop on writing clarity (“Strategies for Improving Writing Clarity”). In this workshop, Dr. Andres De Los Reyes describes narrative tools for writing clear and successful training grant applications for submission to funding agencies. The examples used during the workshop refer to applications submitted to the National Institutes of Health. However, the principles covered in this workshop apply generally to how one uses narrative tools to construct funding applications. He begins with discussing steps one should take when writing a training grant such as determining the &quot;big picture&quot; of your research and why it is important, what is known in the field, and how will your research contribute the field. 

=== '''''Strategies for Boosting Writing Productivity''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====
In this workshop Dr. De Los Reyes offered evidence-based strategies to boost writing productivity.  He made an argument on the importance for those in academia to be excellent writers.  However, being an excellent writer comes with challenges.  He offered strategies of overcoming dreaded writers’ block, while discussing five strategies based on research for becoming a successful writer. Dr. De Los Reyes offered various strategies to transform ideas and thoughts into action before ending the workshop.  

=== '''''Strategies for the Peer-Review Process''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====
Publishing academic work often involves submitting scholarly manuscripts to peer-reviewed journals. A key component of the publishing process involves receiving commentary about your work from peers in your field, and satisfactorily responding to such commentary. Despite it being a core feature of the publishing process, researchers rarely receive formal training on responding to peer review commentary. In this workshop, Dr. Andres De Los Reyes describes evidence-based strategies for responding to peer review commentary, including strategies for how to compose cover letters for responding to such commentary.  

== '''Ceremony for the ''Future Directions Launch Award''''' ==

=== Sunhye (Sunny) Bai, Ph.D. ===

* Award Recipient in the area of child and adolescent assessment
* Received Ph.D. from University of California, Los Angeles

==== About the award recipient ====
Sunhye (Sunny) received the 2017 Future Directions Launch Award in Assessment. After receiving her Ph.D. in Clinical and Health Psychology at the University of California, Los Angeles in 2018, she currently holds a tenure-track position as Assistant Professor of Human Development and Family Studies at the Pennsylvania State University. Her lab focuses on daily family processes that shape adolescent development, with a focus on family-based risk and protective factors for youth internalizing problems. Learn more about Sunny's lab here: [https://Bai-lab.weebly.com bai-lab.weebly.com]

Watch the YouTube recording of the remarks [https://www.youtube.com/watch?v=ylp_PjbWOJ4&amp;feature=youtu.be&amp;ab_channel=JCCAPFutureDirectionsForum here].

=== Taylor A. Burke, Ph.D. ===
* Award Recipient in the area of suicide and self-injury among children and adolescents
* Received Ph.D. from Temple University

==== About the award recipient ====
Taylor is a recipient of the 2017 Future Directions Launch Award in Suicide and Self- Injury. Taylor is a clinical psychologist and Assistant Professor at Harvard Medical School and Massachusetts General Hospital. She specializes in the prediction and prevention of self-injurious thoughts and behaviors (SITBs) among adolescents and young adults. Dr. Burke uses novel methodologies and computational approaches to improve the identification of individuals at risk to better intervene and prevent SITBs. Dr. Burke earned her BA in psychology at Duke University and her PhD in clinical psychology at Temple University. She subsequently completed a pre-doctoral clinical psychology internship and an NIMH-funded T32 post-doctoral fellowship in child mental health at the Alpert Medical School of Brown University. Dr. Burke holds a five-year NIMH career development award that focuses on using passive mobile sensing, adolescent sleep and physical activity assessment, and advanced computational approaches to idiographic modeling to develop proximal risk models for increases in suicidal ideation. She also has other ongoing research supported by the American Foundation for Suicide Prevention and the NIMH focused on leveraging computer vision to enhance suicide risk screening in pediatric health care settings. Learn more about Taylor's work here: https://connects.catalyst.harvard.edu/Profiles/display/Person/200999

Watch the YouTube recording of the remarks [https://www.youtube.com/watch?v=mPw9U4CG52Q&amp;feature=youtu.be&amp;ab_channel=JCCAPFutureDirectionsForum here]. 

=== Tess K. Drazdowski, Ph.D. ===

* Award Recipient in the area of conduct problems among children and adolescents
* Received Ph.D. from Virginia Commonwealth University

==== About the award recipient ====
Tess received the 2017 Future Directions Launch Award in Conduct Problems.  After receiving her Ph.D. in Clinical and Developmental Psychology at Virginia Commonwealth University in 2016, she currently holds a position as Research Scientist and Licensed Psychologist at the Oregon Social Learning Center. She is interested in investigating how to improve access to evidence-based practices for youth and young adults with substance use and mental health symptoms, particularly for those with justice system involvement. Most recently, her research has concentrated on the prevention and intervention for the misuse of prescription drugs, cannabis use, and polysubstance use in primarily young adults. Learn more about her work here: [https://www.researchgate.net/profile/Tess&amp;#x20;Drazdowski www.researchgate.net/profile/Tess_Drazdowski]

Watch the YouTube recording of the remarks [https://www.youtube.com/watch?v=LVo6Bk35CvA&amp;feature=youtu.be&amp;ab_channel=JCCAPFutureDirectionsForum here].

== References ==</text>
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    <title>Journal of Clinical Child and Adolescent Psychology Future Directions Forum (JCCAP FDF)</title>
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&lt;!-- {{Helping Give Away Psychological Science Banner}} --&gt;

{{Wikipedia2|Society of Clinical Child and Adolescent Psychology}}

This is the page for the '''Journal of Clinical Child and Adolescent Psychology's (JCCAP) Future Directions Forum (FDF)'''. It is held annually in June. Starting in 2018, ''The Forum'' will be held in Washington, DC, on the top floor of the American Psychological Association’s home office (www.spireeventsdc.com). Specifically, the ''JCCAP'' ''Future Directions Forum'' gives early-career scientists the tools to learn about emerging trends in child and adolescent mental health (CAMH), and succeed as academics within the “new frontiers” of interdisciplinary team science approaches to research. ''The'' ''Forum'' typically follows the format of 3-4 addresses on future directions topics as well as several workshops designed to help early-career scientists build their toolkit to be successful. Additionally, ''The Forum'' offers other programming such as grant consults, pocket labs, poster presentations, and the ''Future Directions Launch Award''. The pages below will give a breakdown of the topics discussed during each year's Forum as well as provide YouTube links to recorded workshops and addresses.

== FDF Conference Coverage Pages ==
{| style=&quot;border-spacing: 2px; border: 1px solid darkgray;&quot;
! style=&quot;width: 140px;&quot; | [[JCCAP FDF/2017|'''&lt;u&gt;&lt;big&gt;2017 JCCAP FDF&lt;/big&gt;&lt;/u&gt;''']]
! style=&quot;width: 150px;&quot; | [[JCCAP FDF/2018|'''&lt;u&gt;&lt;big&gt;2018 JCCAP FDF&lt;/big&gt;&lt;/u&gt;''']]
! style=&quot;width: 140px;&quot; | [[JCCAP FDF/2019|'''&lt;u&gt;&lt;big&gt;2019 JCCAP FDF&lt;/big&gt;&lt;/u&gt;''']]
! style=&quot;width: 140px;&quot; | [[JCCAP FDF/2020|'''&lt;u&gt;&lt;big&gt;2020 JCCAP FDF&lt;/big&gt;&lt;/u&gt;''']]
![[JCCAP FDF/2021|'''&lt;big&gt;&lt;u&gt;2021 JCCAP FDF&lt;/u&gt;&lt;/big&gt;''']]
!'''[[JCCAP FDF/2022|&lt;u&gt;2022 JCCAP FDF&lt;/u&gt;]]'''
|}

== What is special about this conference? ==
The field of child and adolescent mental health (CAMH) increasingly leverages knowledge from multiple disciplines to understand, assess, prevent, and treat the mental health concerns of children and adolescents. As a field, CAMH thrives when its constituent disciplines—Psychology, Psychiatry, Pediatrics, Public Health, Public Policy, Social Work, Education, Nursing, and Neuroscience, among others—build bridges to work with one another in an effort to improve the mental health of children and adolescents worldwide. However, these bridges do not build themselves. As a leading journal in CAMH, the ''Journal of Clinical Child and Adolescent Psychology'' (''JCCAP'') is in a unique position to leverage its status in the field to build bridges among CAMH disciplines. In fact, ''JCCAP'' publishes content with a large, interdisciplinary outreach. Regularly invited articles (i.e., ''Future Directions'') seek to unite affiliated disciplines in CAMH, in that they focus on topics of relevance to the diverse disciplines that comprise CAMH. In an effort to further strengthen these connections, 2017 marked the launch of the ''JCCAP'' ''Future Directions Forum''. This annual event showcases interdisciplinary work in CAMH, provides professional development training to early career scientists in CAMH, and raises public awareness of the “best and brightest” of this scientists-in-training. Specifically, the ''JCCAP'' ''Future Directions Forum'' gives early-career scientists the tools to learn about emerging trends in CAMH, and succeed as academics within the “new frontiers” of interdisciplinary team science approaches to research. The ''JCCAP'' ''Future Directions Forum'' achieves these goals in three ways:
# Through invited addresses and panel discussions focused on the content of ''Future Directions'' articles, showcase emerging areas of interdisciplinary work in the field of CAMH
# Through workshops and online resources, provide professional development training to all early career scientists who attend ''The'' ''Forum''
# Through research presentations, early career awards, and social media outreach raise international awareness of CAMH’s most promising early career scientists

== Format ==

=== Day 1: Professional Development Training ===
''The Forum'' includes an opening day completely dedicated to professional development training. ''The Forum'' registration fee includes a “professional development workshop day” consisting of interactive sessions and panel discussions. Specifically, on Day 1, attendees receive professional development training that consists of live, interactive workshops on such topics as scientific writing style and productivity, peer review, and preparing training grant applications. Day 1 also includes panel discussions on issues of grant funding. Panelists consist of program officers from federal funding agencies with funding priorities that intersect with topics covered at the forum, such as the National Institutes of Health, National Science Foundation, and Institute of Education Sciences.

Every year, new attendees of the forum receive exposure to the interactive workshops described previously, as well as attend panel discussions on grant funding. However, returning attendees have the opportunity to continue their education via online resources, and more intensive, personalized on-site sessions at the forum. Using this graduated scaling of professional development, the ''JCCAP Future Directions Forum'' seeks to build lasting relationships with its early career attendees, and ensure a growing and sustained attendance at the forum from year-to-year.

=== Day 2: Showcasing ''JCCAP''’s ''Future Directions'' Content ===
'''''Addresses:''''' At each forum, leaders in the field who recently published ''Future Directions'' articles at ''JCCAP'' give formal addresses on the “next steps” of scientific research in their area of expertise. We hold these addresses on Day 2 of the forum. These addresses are based on the content of ''Future Directions'' articles written by the speakers.

'''''Panel Discussions''':'' Following each ''Future Directions Address'', panel discussions serve as “brainstorming sessions” for new science that builds on research covered in the address. Attendees “break out” into one of several discussions moderated by faculty with expertise in the ''Future Directions'' content. These interactive discussions are organized by areas linked to the ''Future Directions Addresses'' (e.g., neuroscience, treatment, cross-cultural research). In advance of the forum, panel leaders identify publicly available resources for new research (e.g., public datasets, funding announcements). In turn, these panel discussions yield “deliverables” in that attendees develop both specific research aims, and concrete strategies for securing data or funding to test the aims. This format sparks “omnidirectional” discussions among panel leaders and attendees about new science along topics linked to the forum addresses. At our panel discussions, forum attendees make the future of science happen!

=== Days 1-2: Early Career Scientists: Celebrating Promising Research in Mental Health ===
'''''Poster Session Socials:''''' Before the forum, early career scientists submit abstracts of their research. At the forum, they make poster presentations of this research during catered social events. In advance of the forum, presenters are prompted via email to upload a single-slide ''Powerpoint'' file of their presentation. At the forum, we integrate these files into our own digital system to have them ready to present during the poster session socials. Thus, all poster presentations are digitally presented, saving presenters both time and money. Poster session socials take place during the evenings of the forum. Further, these events provide early career scientists with opportunities for “one-on-one” time with scientists who gave addresses during the forum.

'''''Future Directions Launch Award:''''' Before the forum, a committee of internationally recognized scientists selects several promising early career scientists who are presenting their research at the forum to receive a widely-publicized award: the ''Future Directions Launch Award''. The award recognizes early career scientists conducting research in the ''Future Directions Address'' topics covered that year (i.e., one awardee for each ''Future Directions Address'' topic). At an award ceremony held during the forum, awardees give TED Talk-style presentations about their work. We stream these presentations live, store them, and link them to a major media dissemination platform (e.g., YouTube). The ''Future Directions Launch Award'' serves as a capstone to the academic training of its recipients—a signal to the field and larger public that they are ready to enter academia and begin independent research careers.

== Benefits to society ==
Over the last 50 years, the field of CAMH has made enormous contributions to society. Scientists in the field produced accurate, reliable diagnostic tools, allowing clinicians to identify children and adolescents in need of care. CAMH scientists developed effective, evidence-based treatments, allowing children and adolescents to work through their challenges and learn coping skills. Now, the field needs a new generation of scientists to solve the field’s most pressing problems. For instance, although some children and adolescents benefit from receiving mental health treatments, roughly 1/3 to 1/2 of all those treated receive little-to-no benefits. Why do these problems persist? A key issue involves poor understanding of the factors that cause children and adolescents to develop mental health concerns. As a result, the field has a lot to learn about why mental health treatments “work,” and for whom these treatments work best. The next generation of scientists in the field need to solve these problems. However, these problems require the collective action of all disciplines affiliated with CAMH. The CAMH field needs the next generation of scientists in CAMH to learn to build bridges among its constituent disciplines. Further, beyond the “book smarts” they acquire in the classroom, early career scientists need the “street smarts” to succeed in academia. Few of them take a class in writing great journal articles, communicating ideas during a job interview, or writing successful grant proposals. If society’s early career scientists do not “make it,” then progress in CAMH stalls, and society would settle for mental health treatments that help only a fraction of those who receive them. The ''JCCAP'' ''Future Directions Forum'' meets these needs by providing early career scientists exposure to interdisciplinary research in CAMH, and the fuel to become great academics!

== Location  ==
The ''JCCAP'' ''Future Directions Forum'' is held annually in June. Starting in 2018, the Forum will be held in Washington, DC, on the top floor of the [https://apabuildings.buildingengines.com/geofire/BDPW?conid=1613209322&amp;id_site=0&amp;id_client_site_rel=0 American Psychological Association’s home office]. 2020 and 2021 Forums were held online due to [[COVID-19]]. The 2022 Forum will be held in a different format as a webinar series throughout the summer and fall of 2022. For more information on this year's Forum, please visit the website [https://jccapfuturedirectionsforum.weebly.com/ here]. 

== Further information ==
Each year's Forum has a page that lists the address and workshops with links to view the recording for those we were able to record. Each year's page also has the Launch Award winners with links to view their remarks.  

Thanks to Dr. De Los Reyes for the vision and leadership to build the conference; to SCCAP, the American Psychological Association, and the University of Maryland for the support and resources to make the inaugural conference possible, and to the team of research assistants, students, and colleagues that did a wonderful job making the FDF such a success!</text>
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{{Wikipedia2|Society of Clinical Child and Adolescent Psychology}}

This is the page for the '''Journal of Clinical Child and Adolescent Psychology's (JCCAP) Future Directions Forum (FDF)'''. It is held annually in June. Starting in 2018, ''The Forum'' will be held in Washington, DC, on the top floor of the American Psychological Association’s home office (www.spireeventsdc.com). Specifically, the ''JCCAP'' ''Future Directions Forum'' gives early-career scientists the tools to learn about emerging trends in child and adolescent mental health (CAMH), and succeed as academics within the “new frontiers” of interdisciplinary team science approaches to research. ''The'' ''Forum'' typically follows the format of 3-4 addresses on future directions topics as well as several workshops designed to help early-career scientists build their toolkit to be successful. Additionally, ''The Forum'' offers other programming such as grant consults, pocket labs, poster presentations, and the ''Future Directions Launch Award''. The pages below will give a breakdown of the topics discussed during each year's Forum as well as provide YouTube links to recorded workshops and addresses.

== FDF Conference Coverage Pages ==
{| style=&quot;border-spacing: 2px; border: 1px solid darkgray;&quot;
! style=&quot;width: 140px;&quot; | [[JCCAP FDF/2017|'''&lt;u&gt;&lt;big&gt;2017 JCCAP FDF&lt;/big&gt;&lt;/u&gt;''']]
! style=&quot;width: 150px;&quot; | [[JCCAP FDF/2018|'''&lt;u&gt;&lt;big&gt;2018 JCCAP FDF&lt;/big&gt;&lt;/u&gt;''']]
! style=&quot;width: 140px;&quot; | [[JCCAP FDF/2019|'''&lt;u&gt;&lt;big&gt;2019 JCCAP FDF&lt;/big&gt;&lt;/u&gt;''']]
! style=&quot;width: 140px;&quot; | [[JCCAP FDF/2020|'''&lt;u&gt;&lt;big&gt;2020 JCCAP FDF&lt;/big&gt;&lt;/u&gt;''']]
![[JCCAP FDF/2021|'''&lt;big&gt;&lt;u&gt;2021 JCCAP FDF&lt;/u&gt;&lt;/big&gt;''']]
! style=&quot;width: 140px;&quot; | [[JCCAP FDF/2022|'''&lt;u&gt;&lt;big&gt;2022 JCCAP FDF&lt;/big&gt;&lt;/u&gt;''']]
|}

== What is special about this conference? ==
The field of child and adolescent mental health (CAMH) increasingly leverages knowledge from multiple disciplines to understand, assess, prevent, and treat the mental health concerns of children and adolescents. As a field, CAMH thrives when its constituent disciplines—Psychology, Psychiatry, Pediatrics, Public Health, Public Policy, Social Work, Education, Nursing, and Neuroscience, among others—build bridges to work with one another in an effort to improve the mental health of children and adolescents worldwide. However, these bridges do not build themselves. As a leading journal in CAMH, the ''Journal of Clinical Child and Adolescent Psychology'' (''JCCAP'') is in a unique position to leverage its status in the field to build bridges among CAMH disciplines. In fact, ''JCCAP'' publishes content with a large, interdisciplinary outreach. Regularly invited articles (i.e., ''Future Directions'') seek to unite affiliated disciplines in CAMH, in that they focus on topics of relevance to the diverse disciplines that comprise CAMH. In an effort to further strengthen these connections, 2017 marked the launch of the ''JCCAP'' ''Future Directions Forum''. This annual event showcases interdisciplinary work in CAMH, provides professional development training to early career scientists in CAMH, and raises public awareness of the “best and brightest” of this scientists-in-training. Specifically, the ''JCCAP'' ''Future Directions Forum'' gives early-career scientists the tools to learn about emerging trends in CAMH, and succeed as academics within the “new frontiers” of interdisciplinary team science approaches to research. The ''JCCAP'' ''Future Directions Forum'' achieves these goals in three ways:
# Through invited addresses and panel discussions focused on the content of ''Future Directions'' articles, showcase emerging areas of interdisciplinary work in the field of CAMH
# Through workshops and online resources, provide professional development training to all early career scientists who attend ''The'' ''Forum''
# Through research presentations, early career awards, and social media outreach raise international awareness of CAMH’s most promising early career scientists

== Format ==

=== Day 1: Professional Development Training ===
''The Forum'' includes an opening day completely dedicated to professional development training. ''The Forum'' registration fee includes a “professional development workshop day” consisting of interactive sessions and panel discussions. Specifically, on Day 1, attendees receive professional development training that consists of live, interactive workshops on such topics as scientific writing style and productivity, peer review, and preparing training grant applications. Day 1 also includes panel discussions on issues of grant funding. Panelists consist of program officers from federal funding agencies with funding priorities that intersect with topics covered at the forum, such as the National Institutes of Health, National Science Foundation, and Institute of Education Sciences.

Every year, new attendees of the forum receive exposure to the interactive workshops described previously, as well as attend panel discussions on grant funding. However, returning attendees have the opportunity to continue their education via online resources, and more intensive, personalized on-site sessions at the forum. Using this graduated scaling of professional development, the ''JCCAP Future Directions Forum'' seeks to build lasting relationships with its early career attendees, and ensure a growing and sustained attendance at the forum from year-to-year.

=== Day 2: Showcasing ''JCCAP''’s ''Future Directions'' Content ===
'''''Addresses:''''' At each forum, leaders in the field who recently published ''Future Directions'' articles at ''JCCAP'' give formal addresses on the “next steps” of scientific research in their area of expertise. We hold these addresses on Day 2 of the forum. These addresses are based on the content of ''Future Directions'' articles written by the speakers.

'''''Panel Discussions''':'' Following each ''Future Directions Address'', panel discussions serve as “brainstorming sessions” for new science that builds on research covered in the address. Attendees “break out” into one of several discussions moderated by faculty with expertise in the ''Future Directions'' content. These interactive discussions are organized by areas linked to the ''Future Directions Addresses'' (e.g., neuroscience, treatment, cross-cultural research). In advance of the forum, panel leaders identify publicly available resources for new research (e.g., public datasets, funding announcements). In turn, these panel discussions yield “deliverables” in that attendees develop both specific research aims, and concrete strategies for securing data or funding to test the aims. This format sparks “omnidirectional” discussions among panel leaders and attendees about new science along topics linked to the forum addresses. At our panel discussions, forum attendees make the future of science happen!

=== Days 1-2: Early Career Scientists: Celebrating Promising Research in Mental Health ===
'''''Poster Session Socials:''''' Before the forum, early career scientists submit abstracts of their research. At the forum, they make poster presentations of this research during catered social events. In advance of the forum, presenters are prompted via email to upload a single-slide ''Powerpoint'' file of their presentation. At the forum, we integrate these files into our own digital system to have them ready to present during the poster session socials. Thus, all poster presentations are digitally presented, saving presenters both time and money. Poster session socials take place during the evenings of the forum. Further, these events provide early career scientists with opportunities for “one-on-one” time with scientists who gave addresses during the forum.

'''''Future Directions Launch Award:''''' Before the forum, a committee of internationally recognized scientists selects several promising early career scientists who are presenting their research at the forum to receive a widely-publicized award: the ''Future Directions Launch Award''. The award recognizes early career scientists conducting research in the ''Future Directions Address'' topics covered that year (i.e., one awardee for each ''Future Directions Address'' topic). At an award ceremony held during the forum, awardees give TED Talk-style presentations about their work. We stream these presentations live, store them, and link them to a major media dissemination platform (e.g., YouTube). The ''Future Directions Launch Award'' serves as a capstone to the academic training of its recipients—a signal to the field and larger public that they are ready to enter academia and begin independent research careers.

== Benefits to society ==
Over the last 50 years, the field of CAMH has made enormous contributions to society. Scientists in the field produced accurate, reliable diagnostic tools, allowing clinicians to identify children and adolescents in need of care. CAMH scientists developed effective, evidence-based treatments, allowing children and adolescents to work through their challenges and learn coping skills. Now, the field needs a new generation of scientists to solve the field’s most pressing problems. For instance, although some children and adolescents benefit from receiving mental health treatments, roughly 1/3 to 1/2 of all those treated receive little-to-no benefits. Why do these problems persist? A key issue involves poor understanding of the factors that cause children and adolescents to develop mental health concerns. As a result, the field has a lot to learn about why mental health treatments “work,” and for whom these treatments work best. The next generation of scientists in the field need to solve these problems. However, these problems require the collective action of all disciplines affiliated with CAMH. The CAMH field needs the next generation of scientists in CAMH to learn to build bridges among its constituent disciplines. Further, beyond the “book smarts” they acquire in the classroom, early career scientists need the “street smarts” to succeed in academia. Few of them take a class in writing great journal articles, communicating ideas during a job interview, or writing successful grant proposals. If society’s early career scientists do not “make it,” then progress in CAMH stalls, and society would settle for mental health treatments that help only a fraction of those who receive them. The ''JCCAP'' ''Future Directions Forum'' meets these needs by providing early career scientists exposure to interdisciplinary research in CAMH, and the fuel to become great academics!

== Location  ==
The ''JCCAP'' ''Future Directions Forum'' is held annually in June. Starting in 2018, the Forum will be held in Washington, DC, on the top floor of the [https://apabuildings.buildingengines.com/geofire/BDPW?conid=1613209322&amp;id_site=0&amp;id_client_site_rel=0 American Psychological Association’s home office]. 2020 and 2021 Forums were held online due to [[COVID-19]]. The 2022 Forum will be held in a different format as a webinar series throughout the summer and fall of 2022. For more information on this year's Forum, please visit the website [https://jccapfuturedirectionsforum.weebly.com/ here]. 

== Further information ==
Each year's Forum has a page that lists the address and workshops with links to view the recording for those we were able to record. Each year's page also has the Launch Award winners with links to view their remarks.  

Thanks to Dr. De Los Reyes for the vision and leadership to build the conference; to SCCAP, the American Psychological Association, and the University of Maryland for the support and resources to make the inaugural conference possible, and to the team of research assistants, students, and colleagues that did a wonderful job making the FDF such a success!</text>
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=== '''''Future Directions Address 1: &quot;Evidence-Based Youth Psychotherapy in the Mental Health Ecosystem&quot;''''' ===

'''Presented by John Weisz, Ph.D.'''
==== Description ====
Five decades of randomized trials research have produced dozens of evidence-based psychotherapies for youths. These psychotherapies produce respectable effects in traditional efficacy trials, but the effects shrink markedly when tested in practice contexts with clinically referred youths and compared to standard clinical care. Dr. Weisz considers why this might be the case, examines relevant research literature, and recommends strategies for future research.

=== '''''Future Directions Address 2: &quot;Future Directions in Research on Structural Stigma and Sexual Orientation Disparities in Mental Health Among Youth&quot;''''' ===

'''Presented by Mark Hatzenbuehler, Ph.D.'''
==== Description ====
In 2011, the National Academy of Sciences released a landmark report on health disparities in LGBT youth, hypothesizing these early life-course disparities to be the result of stigma. Still, research within the field of clinical psychology has left the connection between the structural stigma that LGBT youth face and early presentations of psychopathology largely unexplored - Hatzenbuehler aims to change this.  

In his address, Hatzenbuehler illustrates stigma as a multi-level construct, including individual stigma (such as self-stigmatizing behaviors), interpersonal stigma (such as abuse, rejection, and overt/covert discrimination), and structural stigma (including state policies and institutional practice). Hatzenbuehler then walks the audience through the methodology of his research on structural stigma, as well as the critical implications of his findings. As Hatzenbuehler explains, there is a significant interaction between state-level policies that target LGB youth and rates of clinical presentation of dysthymia, GAD, PTSD, and psychological commodities.Moving on from discussing work that establishes a correlation between state policy and clinical presentation of psychopathology, Hatzenbuehler shifts to detail attempts to establish causation and directionality. Hatzenbuehler emphasizes the need for the field to capitalize on naturally occurring changes in policy, pulling data from a study that examines how prior exposure to stigma as an adolescent can affect an individual's current physiological stress response to stimuli presented in a lab environment.  

Hatzenbuehler then addresses the future directions in this field of research, mentioning the necessity of the following: 

* Testing the ability of previous research on structural stigma to be generalized to transgender youth
* Adopting a life-course approach to the study of structural stigma 
* Developing and validating new methods of structural stigma measurement, particularly ones that are relevant to youth
* Examining structural stigma within the context of intersectionality 
* Examining how structural stigma might impair the efficacy of psychological interventions aimed at improving the mental health of LGBT populations

Hatzenbuehler concludes his discussion by addressing the implications of his research and possible interventions, outlining the need for psychological research to influence informed public policies aimed at reducing stigma and bettering mental health outcomes. 

Watch YouTube video recording of the address [https://youtu.be/HRhglLNBcZ0 here].

=== '''''Future Directions Address 3: &quot;Future Directions in Sleep and Developmental Psychopathology&quot;''''' ===

'''Presented by Lisa Meltzer, Ph.D.'''
==== Description ====
By the age of 18, children and adolescents will have spent 40% of their lives sleeping, but very few psychologists spend 40% of their training learning about sleep. In her discussion of the interaction between sleep patterns and psychopathology, Meltzer emphasizes the pressing need for the incorporation of research relating to sleep in the field of child and adolescent clinical psychology.

Meltzer begins by walking her audience through the basic science of sleep cycles, sleep homeostasis, circadian rhythm, and the changes that occur in sleep patterns throughout development. Meltzer points out that the DSM lists sleep is a diagnostic, a symptom, and a negative outcome for many disorders, underlining the complex relationship between sleep and psychopathology in order to set the stage for her discussion of the future directions within the field. 

In her discussion, Meltzer outlines the need to establish direction in the relationship between sleep and developmental psychopathology. As Meltzer explains, prospective studies have been used to help establish direction in this relationship, suggesting a degree of bidirectionality, but with a possible stronger effect on psychopathology by sleep patterns, rather than vise versa. Meltzer also explains the necessity for the field to more closely determine what goes on in the brain during sleep in order to best consider all dimensions of the relationship, drawing from existing studies to posit slow wave activity and sleep spindle density as possible markers for psychopathology. Finally, Meltzer returns to the classic nature and nurture debate to demonstrate the need for sleep psychology to examine the overlap between the heritability of insomnia, depression, and anxiety in order to between identify whether sleep problems or manifestations of psychopathology should be treated first when kids are struggling with both.  

Metlzer concludes her address by briefly touching on several more future directions for inquiry within the field of sleep as it relates psychopathology, including the role of night to night variability in sleep patterns, contribution of circadian rhythm, individual vulnerabilities to sleep loss, and the occurrence of co-morbid sleep/psych issues pertaining to pharmaceutical treatment, as they relate to the presentation of clinical psychological abnormality in children and adolescents. 

Watch YouTube video recording of the address [https://youtu.be/EzGNFy88SKM here]. 

=== '''''Future Directions Address 4: &quot;Future Directions for Clinical Research, Services, and Training: Evidence-Based Assessment Across Informants, Cultures, and Dimensional Hierarchies&quot;''''' ===

'''Presented by Thomas M. Achenbach, Ph.D.'''
==== Description ====
In order to successfully treat psychopathological issues, you first need to know what you are treating.  

Using the hypothetical example of a young boy who is referred to clinical psychological services for “ADHD-like” behavioral issues by a teacher, Achenbach demonstrates the need for multi-informant comparisons that are not targeted toward a specific disorder or set of diagnostic criteria when assessing children for psychiatric illness. Achenbach emphasizes the value of obtaining comprehensive reports from parents, not only in respect to the child, but also pertaining to the parents’ own psychological wellbeing. Though parental assessment is often overlooked in child clinical psychology, Achenbach argues such practices can help communicate about their relationship (with each other, themselves, and the child), identify the parents need for help, allow parents to see similarities and differences between child and parents problems, and ultimately to foster a therapeutic alliance to work on child and parent issues. Achenbach then details the methods of the Multicultural Family Assessment Module (MFAM), the formulaic and individualized approach of the MFAM to evaluating progress and outcome, and the application of the MFAM for use in assessment by masters-level clinicians without statistical knowledge and training. 

Achenbach additionally discusses the benefits of incorporating multicultural norms, also detailing how to construct multicultural norms within different batteries in order to best optimize psychological assessment. Based on data from many societies, software can score individual data points in relation to societal norms in order to evaluate clients’ responses in relation to an appropriate context. Achenbach then outlines the future directions within the field of child psychiatric assessment that clinicians and research training should aim to achieve. These aims include: increasing providers’ use of multi-informant multicultural EBA, organizing clinical services around the use of EBA in all cases, and basing service decisions on intake/progress/outcome EBA. 

Achenbach concludes his address by emphasizing the importance of personalized treatment, and not lumping clients into rigid (and often inefficient) diagnostic boxes, as this practice does not honor the dynamic characteristics of human nature and interpersonal interactions. 

Watch YouTube video recording of the address [https://youtu.be/CXcVfwKsbqQ here].

== '''Workshops''' ==

=== '''''Strategies for Improving Writing Clarity''''' ===

'''Presented by Andres De Los Reyes, Ph.D.'''
===== Description =====
People tend to be drawn to and understand information best when it is communicated to them in the form of a narrative or “story” rather than a list of facts. However, researchers rarely receive formal training on leveraging narrative tools when writing about their academic work. In this workshop, Dr. Andres De Los Reyes describes evidence-based strategies for consistently applying narrative structure to academic work, with a focus on preparing manuscripts for submission to peer-reviewed academic journals. This includes his description of the and-but-therefore approach to writing and how this narrative structure can be utilized in academic writing as a way to make scientific information more interesting to consume and memorable to the audience.

Watch the YouTube recording of the workshop [https://youtu.be/XzVWz15TrzI here]. 

=== '''''Disseminating Science Through Wikipedia and Wikiversity''''' ===

'''Presented by [[wikipedia:Eric_Youngstrom|Eric A. Youngstrom]], Ph.D.'''

===== Description =====
Currently, many evidence based practices are behind pay walls and not easily accessed by those without resources to paid subscriptions. The goal of this workshop was to share with individuals a way in which evidence based practices could be made freely available to anyone who has access to the internet with a focus on psychological science. Wikipedia and Wikiversity are two powerful tools for disseminating knowledge to diverse audiences, including scientists and other key stakeholders (e.g., parents, practitioners, policymakers). This workshop showed ways to leverage these tools for disseminating knowledge, reaching a wider audience and accelerating collaboration. In this workshop, Dr. Eric Youngstrom provided attendees with the know-how for using Wikipedia and Wikiversity, with a focus on the use of these platforms for increasing access to knowledge and resources relevant to evidence-based assessments and treatments of child and adolescent mental health. Dr. Youngstrom also shared his initiative to make psychological science more accessible by creating a non-profit called [[Helping Give Away Psychological Science]] (HGAPS) where students, professors, and other professionals in the field come together with the aim of making evidence-based psychological information more accessible. His team also discussed new student projects, funding opportunities, and travel awards.

=== '''''Networking at Conferences''''' ===

'''Presented by Deborah A.G. Drabick, Ph.D. and Matthew D. Lerner, Ph.D.'''

===== Description =====
To an early career scientist, attending professional meetings can be an overwhelming experience, with many opportunities to not only learn new things but also connect with like-minded scholars in the field. In this workshop, Drs. Deborah Drabick and Matthew Lerner demystify the process of networking at conferences, and provide attendees with concrete tools for developing and maintaining professional relationships with conference attendees. The presenters discuss how to approach a researcher who you are interested in building a connection with and how to maximize the time you have to speak with them and to make yourself stand out. Drs. Drabick and Lerner also discuss how to have a meaningful discussion that can carry into a long-term professional relationship such as collaborations on research projects. It is also important to be mindful of how you are carrying yourself and what your interactions look like as you never know who is around and watching. Having a go to question can be a good way to start the conversation such as asking what the current state of that person's research is or discussing the current state of the literature in their field of study. Always make sure you know about the person's research before approaching them and consider how their work can inform yours. It is also important to attend their session if they are speaking at the conference or event so you can be informed on their current work and research goals.

=== '''''Preparing Your First Grant as a Principal Investigator''''' ===

'''Presented by Joshua M. Langberg, Ph.D. and Tara S. Peris, Ph.D.'''

===== Description =====
Submitting your first grant as a Principal Investigator (PI) can appear on the surface to be a daunting task, with many expectations, requirements, and complicated forms. In this workshop, Drs. Joshua Langberg and Tara Peris leverage their years of experience with extramural funding to explain the grant submission process, and provide attendees with concrete tools for submitting successful grant applications. This includes different stages of applying for grants such as picking a topic, collecting preliminary data, forming your team and budget, and telling your story. In picking a topic, usually, you start by thinking of a topic related to research you conducted during graduate school or a fellowship where as one may be a newer area you are wanting to explore. From there, you should explore the literature and think about the trends in the field on these topics. Most grants require preliminary data to show there is promise on your research topic, this means you need to gather some data to prove your topic is worthwhile. Showing this data can be done by a small pilot study or open trial. Next, you need to strategically select your team that will conduct the study. This may include needing a senior researcher with many years of experience in the field, individuals with experience in the contexts you are working in (i.e., schools), or individuals who are experts in statistical analyses. The next step is planning the budget and deciding who will serve as a Co-PI and who will serve as a consultant, will you hire graduate students or project coordinators and what will their salaries be, what are the costs of running the study, etc. Lastly, you need to tell your story. This should be done similarly to the why in a manuscript and describe how this study will change the field or what implications this research will have on the field. Additionally, Drs. Langberg and Peris describe how you should select which institutions to apply for grants at, how to align your research goals to their mission, how many grants you should apply for, what the process is like, and the timeline you should expect.   

=== '''''Strategies for Writing Training Grants''''' ===

'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====
This workshop serves as a companion to our workshop on writing clarity (“Strategies for Improving Writing Clarity”). In this workshop, Dr. Andres De Los Reyes describes narrative tools for writing clear and successful training grant applications for submission to funding agencies. The examples used during the workshop refer to applications submitted to the National Institutes of Health. However, the principles covered in this workshop apply generally to how one uses narrative tools to construct funding applications. Similar to the advice given in “Strategies for Improving Writing Clarity”, Dr. De Los Reyes focuses on how to tell a story about your research when writing a training grant. 

=== '''''Work-Life Balance''''' ===

'''Presented by Sarah J. Racz, Ph.D. and Joshua M. Langberg, Ph.D.'''

===== Description =====
Sometimes it feels like everyone in our field is “always on task” and unable to “unplug”. But is that a realistic view of how we balance our work lives with our lives outside of work? In this workshop, Drs. Joshua Langberg and Sarah Racz discuss the competing demands placed on us across our various work, family, and social spheres; and strategies to manage these demands in the necessary pursuit of healthy, balanced lives. They begin with practical advice such as knowing what your priorities are and how you operationalize them. Then map your schedule so you have a clear picture of your time and your commitments. From there you will need to make choices out of the time items you have left on your priorities and what time you have left in the day. By having routine and structure for each day, you are able to maximize your time and make conscious choices of what you are spending your day doing. They also recommend being flexible, but organized when creating a work-life balance as sometimes things in your schedule may shift and you need to quickly adjust. Being organized may look like keeping a calendar and/or to do list as well as using family management apps. It is also important to communicate with others in your household about your schedule and set up a plan that works for all of you. Lastly, it is important to advocate for yourself and to know your rights and responsibilities if you are a working parent when it comes to parental leave and time off.  

=== '''''Starting a Lab''''' ===

'''Presented by [[wikipedia:Eric_Youngstrom|Eric A. Youngstrom]], Ph.D. and Matthew D. Lerner, Ph.D.'''

===== Description =====
In graduate school there is no course on how to start your own lab once you graduate. This workshop gives you the tools you need to start a new lab. Drs. Matthew Lerner and Eric Youngstrom both established their own laboratories in a variety of academic settings that interface with multiple campus units including Departments of Psychology, Psychiatry, and Pediatrics. On the basis of these experiences, attendees will learn from the experts on key considerations with starting a lab, including selecting and purchasing equipment, hiring competent staff, recruiting graduate students, and progressing along the tenure track.

=== '''''Job Search and Job Negotiations''''' ===

'''Presented by Deborah A.G. Drabick, Ph.D. and Tara S. Peris, Ph.D.'''

===== Description =====
In this workshop Drs. Deborah Drabick and Tara Peris provide advice when it comes to searching for jobs and negotiating job offers. They give practical advice for determining when to go on the job market, where are the best places to search for job openings, how to write a good cover letter that will catch an employer's attention, and where to find resources that will assist in the preparation for a job interview. It is always good to monitor list-servs from APA, ABCT, and other relevant groups for job postings as well as checking APA psyc careers and following up via word of mouth. When applying for jobs, you should have your CV up to date and a cover letter ready that details your experiences and qualifications. They also detail how to advocate for yourself, negotiate salary and benefits, and how to get a lab startup package when given a job offer. When negotiating you can negotiate on salary, title, space, start-up funds, time between moving from your current place to your new job, protected time, parking, childcare, and housing. You should always ask for more than what you want since you are negotiating and will not get everything you ask for. Lastly, the remind you to always be gracious when interviewing and negotiating job offers as well as to send thank you notes to those you interact with during the process.

=== '''''Responding to Peer Review Commentary''''' ===

'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====
Publishing academic work often involves submitting scholarly manuscripts to peer-reviewed journals. A key component of the publishing process involves receiving commentary about your work from peers in your field and satisfactorily responding to such commentary. Despite it being a core feature of the publishing process, researchers rarely receive formal training on responding to peer review commentary. In this workshop, Dr. Andres De Los Reyes describes evidence-based strategies for responding to peer review commentary, including strategies for how to compose cover letters for responding to such commentary. Along these lines, Dr. De Los Reyes details how to respond to peer reviews line by line to ensure all comments have been addressed. Additionally, he outlines how to respectfully respond to and address comments from a reviewer that you may not agree with. 

=== '''''Preparing a Training Grant''''' ===

'''Presented by Deborah A.G. Drabick, Ph.D. and Tara S. Peris, Ph.D.'''

===== Description =====
Submitting a training grant involves considering multiple factors that focus on not only a proposed study but also a concrete plan for developing the skills needed to execute this study. By construction, these applications carry many expectations, requirements, and complicated forms. In this workshop, Drs. Deborah Drabick and Tara Peris leverage their years of experience with extramural funding to clarify the process of submitting a training grant, and provide attendees with concrete tools for submitting successful training grant applications. Training grants are important because they offer opportunities to graduate students and post-docs that may not otherwise be available and provide additional mentorship and consultation. There are several types of training grants that typically fall into either F grant class or K grant class. Based on the type of grant you are applying for, the information needed to apply will vary; however, picking your topic and telling your story will always be required. In doing this, you should describe what you are interested in researching, the current state of the field, and how your research will address gaps you have identified in the current research. Grants are typically reviewed on a scale of 1-9 in 5 areas which are then compiled into an overall impact score. Reviewers then meet and discuss the top 50% of the grant applications and make a decision on who will receive funding.

=== '''''Job Options in Academia''''' ===

'''Presented by [[wikipedia:Eric_Youngstrom|Eric A. Youngstrom]], Ph.D. and Matthew D. Lerner, Ph.D.'''

===== Description =====
Graduate training in fields relevant to child and adolescent mental health (e.g., Education, Psychiatry, Psychology, and Social Work) prepares trainees for careers in a variety of policy, research, and practice settings. How does a trainee learn about these opportunities and maximize their chances for landing jobs in one or more of these settings? Drs. Matthew Lerner and Eric Youngstrom  provide attendees with a broad overview of the job options available in academia, with a specific focus on strategies for crafting the training and scholarly records that make someone a compelling candidate for these job options.

=== '''''Preparing Your First Grant as a Principal Investigator''''' ===

'''Presented by Joshua M. Langberg, Ph.D.'''

===== Description =====
Submitting your first grant as a Principal Investigator (PI) can appear on the surface to be a daunting task, with many expectations, requirements, and complicated forms. In this workshop, Dr. Joshua Langberg leverages his years of experience with extramural funding to demystify the grant submission process, and provide attendees with concrete tools for submitting successful grant applications.

== '''Ceremony for the ''Future Directions Launch Award''''' ==
=== Emily M. Becker-Haimes Ph.D. ===
* Received Ph.D. from University of Miami

====== About the award recipient ======
Emily received the 2018 Future Directions Launch Award in Treatment. After receiving her Ph.D. in Clinical Psychology from University of Miami in 2016, she completed a postdoctoral fellowship at the Penn Center for Mental Health. She is currently an Assistant Professor of Psychiatry at University of Pennsylvania and is the Clinical Director of the Pediatric Anxiety Treatment Program at Hall-Mercer (PATCH). Dr. Becker-Haimes’ research dually focuses on treatment optimization for pediatric anxiety and related disorders and studying how to optimize the implementation of evidence-based treatments such as cognitive behavioral therapy into community settings. Learn more about Emily's work here: https://hosting.med.upenn.edu/cmh/people/emily-becker-haimes

Watch the YouTube video recording of the remarks [https://www.youtube.com/watch?v=omFQjCL_Zi4 here].

=== Spencer C. Evans ===
* Received Ph.D. University of Kansas

====== About the award recipient ======
Spencer was a recipient of 2018 Future Directions Launch Award in Assessment.  After receiving his Ph.D. in Clinical Child Psychology from University of Kansas in 2017, he held a Postdoctoral Fellow position in the Department of Psychology at Harvard University. He is currently a tenure-track Assistant Professor in the Department of Psychology at University of Miami. His lab focuses on irritability and aggressive behavior in children and adolescents, including interests in developmental processes, assessment, and intervention. Learn more about Spencer's lab here: [https://www.evans-lab.org www.evans-lab.org]

Watch the YouTube video recording of the remarks [https://www.youtube.com/watch?v=rD_okfirfc8 here].

=== Maggi Price ===
* Received Ph.D. from Boston College
====== About the award recipient ======
Maggi received the 2018 Future Directions Launch Award in Stigma. After receiving her Ph.D. in Counseling Psychology from Boston College in 2018, she completed a Postdoctoral Fellowship in the Department of Psychology at Harvard University. Maggi is currently a tenure-track Assistant Professor in the School of Social Work at Boston College. Her lab focuses on stigma, trauma, and youth mental health, with an emphasis on understanding and effectively reducing health inequities for youth with stigmatized identities (e.g., transgender youth, youth of Color). Learn more about Maggi's research and lab here: [https://www.affirmlab.org www.affirmlab.org]

Watch the YouTube video recording of the remarks [https://www.youtube.com/watch?v=hMJVVgruERU here].</text>
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      <text bytes="26535" xml:space="preserve">== '''Addresses''' ==

=== '''''Future Directions Address 1: &quot;Future Directions in Adversity and Mental Health&quot;''''' ===

'''Presented by Kate McLaughlin, Ph.D.'''

==== Description ====
Research has shown that experiences of childhood adversity are more common than previously thought among children in the US. Furthermore, children who have experienced more than six instances of childhood adversity are five times more likely than their peers to develop at least one clinical manifestation of psychopathology by adulthood. Yet, many current approaches to child psychopathology have a tendency to lump vastly different experiences of adversity together - a practice that has likely served to further complicate efforts to identify an underlying neurological mechanism linking childhood adversity and trauma. 

In her address, McLaughlin identifies level of threat and level of deprivation as two distinct contextual axes of childhood adversity, influencing underlying neurological mechanisms and eventual manifestation of psychopathology in distinctly different ways. 

The threat exposure axis pertains to the harm or threat of harm that a child may experience during their developmental years. As McLaughlin discusses, threat exposure influences neural systems involved in fear learning and salience processing. This adaptive response to growing up in a dangerous environment can be essential for the child’s survival in the short term, but as children grow up these changes  can result in increased emotional reactivity and poor emotion regulation. McLaughlin details how her lab used brain maps to identify and track possible neurological mechanisms that explain a significant portion of the relationship between childhood trauma and the general psychopathology (p) factor.

Pulling in additional lab data, McLaughlin explains that sustained childhood exposure to threat results in children having a harder time distinguishing between what is safe and what is dangerous, and likely perceiving a broader set of stimuli to be potentially dangerous compared to children who have never experienced trauma. McLaughlin states that a key additional implication of this data is that the poor cross talk between the amygdala and hippocampus that occurs as a result of trauma serves as a direct predictor of higher levels of psychopathology across the board. 

McLaughlin then shifts the focus of her discussion to discuss possible mechanisms linking the p factor and deprivation during key developmental years. McLaughlin begins by stating that children with inadequate exposure to caregivers are not exposed to many of the stimuli necessary for optimal learning and development, including language, nurturing, and complex interpersonal stimulation. It is hypothesized that the result of this deprivation is an exaggerated experience of synaptic pruning, wherein unused neural circuits are destroyed. While synaptic pruning is an important component of neural development, exaggerated synaptic pruning can lead to a dramatic reduction in the density of areas of the brain relating to language processing, social cognition, attention, executive function, and working memory, describing how these data apply to children growing up in poverty in the US. 

Watch the YouTube video recording of the address [https://www.youtube.com/watch?v=ceZUPrRHxlo here].

=== '''''Future Directions Address 2: &quot;Future Directions in Mediators of Treatment&quot;''''' ===

'''Presented by [[wikipedia:Philip_C._Kendall|Philip Kendall, Ph.D]].'''

==== Description ====
How do psychological therapies work? How can we enhance treatment to improve outcomes? Questions of mediation lie at the heart of these inquiries. In this address, Dr. Philip Kendall delineates some of the issues confronting tests of treatment mediation in youth mental health and suggests future directions in research on addressing these issues.  

Kendall begins his address by defining predictors, moderators, and mediators within the context of clinical psychology. In contrast to both predictors and moderators, a mediator is a variable that has implications for possible mechanisms. A change in a mediating variable will tell you something about the possible mechanism of change in outcome. Importantly, mediators have temporal precedence and can help explain observed outcomes within a clinical or experimental setting. As Kendall points out, there are very few instances of mediation in child and adolescent outcome studies. In order to identify a mediating variable, one has to test each possible variable in order to understand all of the potential mediating factors. Additionally, there is a need for full temporal precedence in measurement and identification of a mediator. In identification and analysis of mediating variables, weekly study is ideal. However, surveying participants at such a high frequency over the course of a trial can be challenging, since responses tend to decrease in complexity and frequency over time if a significant incentive is not offered. Because of this tendency, psychologists must balance the intrusive quality of such measurements against the minimum amount of inquiries necessary to properly identify and examine a mediating variable. As Kendall mentions, burst-style surveying presents a possible solution to upholding this balance. 

Kendall discusses in detail the limitations of prior group-based approaches, pointing out the dangers of assuming that change over time is equivalent across participants, that change occurs as a result of the same mechanism, and that change is linear. Kendall emphasizes the need for more frequent measurement in mediator analysis, highlighting the use of dynamical systems approaches to test potential mechanisms of mediation.  

Kendall concludes his discussion by touching on several potential novel study designs to best facilitate medication, emphasizing Bayesian approaches. To summarize his discussion, Kendall leaves the audience with several mechanisms of incorporating mediation analysis into their research: implementing high-frequency data collection methods, establishing temporal precedence, and considering the use of person-centered approaches.

Watch the YouTube video recording of the address [https://www.youtube.com/watch?v=0w0ul3YH17o here].

=== '''''Future Directions Address 3: &quot;Future Directions in Immunology and Mental Health&quot;''''' ===

'''Presented by Greg Miller, Ph.D.'''

==== Description ====
An established body of research has identified significant disparities in health outcomes between racial and ethnic groups, patterning linearly by socioeconomic status (SES). These disparities start early in life, continue throughout the life course, and cannot be explained by genetics, lifestyle choices, or lack of access to healthcare alone. 

In this address, Miller seeks to identify both the biological mechanisms linking childhood disadvantage with risk for health problems across the lifespan, as well as potential protective/vulnerability factors that could be used to assess health risk, with the end goal of using these results to inform better policy and practice. 

Miller begins by conceptualizing the immune system as a key interface between the social environment and physical/mental health. In his discussion of the immune interface, Miller places special focus on non-resolving, low-grade inflammation as an identified underlying factor in many chronic health conditions that pattern by SES.

As Miller explains, childhood socioeconomic disadvantages put kids at greater risk of negative social and physical exposures. These experiences profoundly shape our our brains develop, and how our brain/body system learns to engage with stressors. Kids who live in low SES environments display an exaggerated immune response compared with their peers. This exaggerated immune response hangs around in stem and progenitor cells, perpetuating a cycle of negative health outcomes throughout the lifespan. 

However, as Miller notes, having a nurturing mother has been shown to almost completely negate the pro-inflammatory effects of SES, pointing toward a great potential for parenting interventions to stop this pro-inflammatory cycle. Starting as early as pregnancy, Miller discusses multiple novel horizons for parenting and child development interventions focused on decreasing the gap in health outcomes by racial/ethnic groups and SES. 

Watch the YouTube video recording of the address [https://www.youtube.com/watch?v=kgu3lnyZNmA&amp;feature=youtu.be here].

=== '''''Future Directions Address 4: &quot;Future Directions in Parent-Child Separation&quot;''''' ===

''' Presented by Kathryn Humphreys, Ph.D.'''

==== Description ====
Modern societies have long since struggled to understand how to best provide care for orphaned/neglected children. As issues involving child/caregiver separation become increasingly present in the political and cultural spotlight amidst immigration debates, the relationship between early childhood adversity by way of neglect and later manifestation of psychopathology has become especially relevant. 

As Humphreys emphasizes, the presence of a nurturing relationship between a caregiver and a child is absolutely essential for normal development. In almost every domain of psychopathology there exists a correlation between early live separation/neglect and later risk of psychiatric illness, suggesting that there is something about the caregiver/child separation itself that is traumatizing. 

Throughout her discussion, Humphreys incorporates data to back up her claim that children who are separated from their caregivers (such as immigrants at the US/Mexico border) are hit with a &quot;double whammy&quot; of trauma, both from losing the presence of a committed caregiver, and from having to navigate adverse conditions without the presence of that caregiver. Pre- and post-separation adverse experiences can compound the effect of this double trauma, leading to increased rates of attachment disorders, internalizing/externalizing disorders, and callous-unemotional traits in these children. 

Given the importance of attachment processes in early life, Humphreys discusses how timing and sensitive developmental periods may be particularly relevant in the study of parent-child separations. In addition to discussing multiple neurological mechanisms of stress stemming from neglect, Humphreys frames adolescence as a second sensitive period in development, in addition to the first five years of life. As Humphreys explains, the plasticity associated with adolescence, taken in conjunction with the face that development does not end with infancy, points to the potential for interventions focused on building healthy caregiver/child relationships during adolescence and late childhood to impact the later physical and mental health of children who experienced early-live neglect in a positive manner. 

Watch the YouTube video recording of the address [https://www.youtube.com/watch?v=fq_OHfGyF0o&amp;feature=youtu.be here].

== '''Workshops''' ==

=== '''''Strategies for Improving Writing Clarity''''' ===

'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====
People tend to be drawn to and understand information best when it is communicated to them in the form of a narrative or “story” rather than a list of facts. However, researchers rarely receive formal training on leveraging narrative tools when writing about their academic work. In this workshop, Dr. Andres De Los Reyes describes evidence-based strategies for consistently applying narrative structure to academic work, with a focus on preparing manuscripts for submission to peer-reviewed academic journals. This includes his description of the and-but-therefore approach to writing and how this narrative structure can be utilized in academic writing as a way to make scientific information more interesting to consume and memorable to the audience.

=== '''''Job Options in Academia''''' ===

'''Presented by Susan White, Ph.D and Matthew Lerner, Ph.D. '''

===== Description =====
Graduate training in fields relevant to child and adolescent mental health (e.g., Education, Psychiatry, Psychology, and Social Work) prepares trainees for careers in a variety of policy, research, and practice settings. While there are many options one can take with a career in mental health, academic jobs are among the most common and include traditional academic settings such as R1 research universities and research positions in a medical school, the government, or a research center. Another big consideration is whether or not to go for a job with tenure track (which is like a probationary period that typically lasts about 6 years) especially because these positions are very difficult to come by. There are also considerations to be made on whether to take a position that requires/allows teaching and service work like serving on an editorial board. Research finds that the best predictor of success in academia is: department reputation. Drs. White and Lerner give advice on the timeline of when to apply and resources to use on your applications such as your advisor, organization, and practice. They also provide brief discussion on job interviews which mostly consist of job talks in academia. Lastly, the give advice on negotiating a job offer with the most important piece of advice being to '''get everything in writing''' when negotiating a position.  

=== '''''Preparing a Training Grant''''' ===

'''Presented by Deborah A.G. Drabick, Ph.D. and Tara Peris, Ph.D. '''

===== Description =====
Submitting a training grant involves considering multiple factors that focus on not only a proposed study but also a concrete plan for developing the skills needed to execute this study. By construction, these applications carry many expectations, requirements, and complicated forms. In this workshop, Drs. Deborah Drabick and Tara Peris leverage their years of experience with extramural funding to clarify the process of submitting a training grant, and provide attendees with concrete tools for submitting successful training grant applications. Training grants are important because they offer opportunities to graduate students and post-docs that may not otherwise be available and provide additional mentorship and consultation. There are several types of training grants that typically fall into either F grant class or K grant class. Based on the type of grant you are applying for, the information needed to apply will vary; however, picking your topic and telling your story will always be required. In doing this, you should describe what you are interested in researching, the current state of the field, and how your research will address gaps you have identified in the current research. Grants are typically reviewed on a scale of 1-9 in 5 areas which are then compiled into an overall impact score. Reviewers then meet and discuss the top 50% of the grant applications and make a decision on who will receive funding. 

=== '''''Responding to Peer Review''''' ===

'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====
Publishing academic work often involves submitting scholarly manuscripts to peer-reviewed journals. A key component of the publishing process involves receiving commentary about your work from peers in your field and satisfactorily responding to such commentary. Despite it being a core feature of the publishing process, researchers rarely receive formal training on responding to peer review commentary. In this workshop, Dr. Andres De Los Reyes describes evidence-based strategies for responding to peer review commentary, including strategies for how to compose cover letters for responding to such commentary. Along these lines, Dr. De Los Reyes details how to respond to peer reviews line by line to ensure all comments have been addressed. Additionally, he outlines how to respectfully respond to and address comments from a reviewer that you may not agree with.

=== '''''Job Search and Negotiation''''' ===

''' Presented by Deborah A.G. Drabick, Ph.D. and Tara Peris, Ph.D. '''

===== Description =====
In this workshop Drs. Deborah Drabick and Tara Peris provide advice when it comes to searching for jobs and negotiating job offers. They give practical advice for determining when to go on the job market, where are the best places to search for job openings, how to write a good cover letter that will catch an employer's attention, and where to find resources that will assist in the preparation for a job interview. It is always good to monitor list-servs from APA, ABCT, and other relevant groups for job postings as well as checking APA psyc careers and following up via word of mouth. When applying for jobs, you should have your CV up to date and a cover letter ready that details your experiences and qualifications. They also detail how to advocate for yourself, negotiate salary and benefits, and how to get a lab startup package when given a job offer. When negotiating you can negotiate on salary, title, space, start-up funds, time between moving from your current place to your new job, protected time, parking, childcare, and housing. You should always ask for more than what you want since you are negotiating and will not get everything you ask for. Lastly, the remind you to always be gracious when interviewing and negotiating job offers as well as to send thank you notes to those you interact with during the process. 

=== '''''Preparing a Grant Post-Ph.D''''' ===

''' Presented by Joshua Langberg, Ph.D. and Susan White, Ph.D. '''

===== Description =====
Submitting your first grant as a Ph.D. can appear on the surface to be a daunting task, with many expectations, requirements, and complicated forms. In this workshop, we leverage years of experience with extramural funding to explain the grant submission process, and provide attendees with concrete tools for submitting successful applications via multiple post-Ph.D. mechanisms, including project grants and K Series, F Series, and T Series applications. K Series grants are most often used to fund early career research, F Series are individual fellowships, and T Series are institutional training grants. Each grant will have different elements specific to that grant; however, each will require picking a topic/telling your story, showing preliminary data, forming a team, and budgeting. It is important to establish a timeline for writing the grant and submitting materials to ensure that everything is submitted on time. You should ideally have multiple grants under review at one time to maximize the chance of having a successfully funded grant. Once you have written one, it is easy to tweak it to meet the requirements of other grant applications. Lastly, ask others in the field for examples of grants they have written and ask them to review yours before submitting. 

=== '''''Strategies for Developing a Research Program''''' ===

''' Presented by Andres De Los Reyes, Ph.D. '''

===== Description =====
Our first two writing workshops dealt with applying narrative tools to academic work and responding to peer review commentary, with the key goal of publishing a single journal article. How might you use these tools to connect separate articles together into a larger story? In research, our larger stories are the “research programs” we build from years of work and multiple articles. These are the stories we take with us “on the road” when interviewing for jobs and applying for grants. In this workshop, we discover how narrative devices commonly used in filmmaking actually help us weave related but distinct articles together into the “story” of an entire body of work.

=== '''''Networking at Conferences''''' ===

''' Presented by Deborah A.G. Drabick, Ph.D. and Matthew Lerner, Ph.D. '''

===== Description =====
To an early career scientist, attending professional meetings can be an overwhelming experience, with many opportunities to not only learn new things but also connect with like-minded scholars in the field. In this workshop, Drs. Deborah Drabick and Matthew Lerner demystify the process of networking at conferences, and provide attendees with concrete tools for developing and maintaining professional relationships with conference attendees. The presenters discuss how to approach a researcher who you are interested in building a connection with and how to maximize the time you have to speak with them and to make yourself stand out. Drs. Drabick and Lerner also discuss how to have a meaningful discussion that can carry into a long-term professional relationship such as collaborations on research projects. It is also important to be mindful of how you are carrying yourself and what your interactions look like as you never know who is around and watching. Having a go to question can be a good way to start the conversation such as asking what the current state of that person's research is or discussing the current state of the literature in their field of study. Always make sure you know about the person's research before approaching them and consider how their work can inform yours. It is also important to attend their session if they are speaking at the conference or event so you can be informed on their current work and research goals.

=== '''''Work-Life Balance''''' ===

'''Presented by Joshua Langberg, Ph.D. and Sarah Racz, Ph.D.'''

===== Description =====
Sometimes it feels like everyone in our field is “always on task” and unable to “unplug”. But is that a realistic view of how we balance our work lives with our lives outside of work? In this workshop, Drs. Joshua Langberg and Sarah Racz discuss the competing demands placed on us across our various work, family, and social spheres; and strategies to manage these demands in the necessary pursuit of healthy, balanced lives. They begin with practical advice such as knowing what your priorities are and how you operationalize them. Then map your schedule so you have a clear picture of your time and your commitments. From there you will need to make choices out of the time items you have left on your priorities and what time you have left in the day. By having routine and structure for each day, you are able to maximize your time and make conscious choices of what you are spending your day doing. They also recommend being flexible, but organized when creating a work-life balance as sometimes things in your schedule may shift and you need to quickly adjust. Being organized may look like keeping a calendar and/or to do list as well as using family management apps. It is also important to communicate with others in your household about your schedule and set up a plan that works for all of you. Lastly, it is important to advocate for yourself and to know your rights and responsibilities if you are a working parent when it comes to parental leave and time off.

== '''Ceremony for the ''Future Directions Launch Award''''' ==

=== John L. Cooley ===

* Received Ph.D. from the University of Kansas

====== About the award recipient ======
John is a recipient of the 2019 Future Directions Launch Award in Adversity.  After receiving his Ph.D. in Clinical Child Psychology from University of Kansas in 2018, he worked as a NIMH T32 postdoctoral fellow in the Department of Psychiatry at the University of Colorado Anschutz Medical Campus. He is currently a tenure-track Assistant Professor in the Department of Psychological Sciences at Texas Tech University, where his lab is guided by two overarching questions: “Why are some children and adolescents more impacted by peer victimization/bullying than others?” and “How can we address the mental health needs of peer-victimized/bullied youth?&quot; More specifically, John’s lab is focused on investigating a) factors that influence risk for peer victimization/bullying and their associated negative outcomes, b) methods for identifying victims of peer aggression in need of intervention, and c) prevention and intervention approaches. Learn more about John's lab here: [https://www.peerrelationslab.com www.peerrelationslab.com]

Watch the YouTube video recording of the remarks [https://www.youtube.com/watch?v=Jq6k0URJ1es here].

=== Erin Kang ===

* Ph.D. Candidate at Stony Brook University at the time of the award

====== About the award recipient ======
Erin received the 2019 Future Directions Launch Award in Treatment. She earned her Ph.D. in Clinical Psychology in 2020 at Stony Brook University under the mentorship of Dr. Matthew Lerner. She is currently a tenure-track Assistant Professor in the Psychology Department at Montclair State University, where her lab focuses on understanding how the processing of social information shapes, and is shaped by, social experience in autistic youth and those with related neurodevelopmental disorders. This focus on social plasticity, or capacity to learn from and adapt to their complex social environments, includes the role of social experiences, affective processing, and neural plasticity that underlie this capacity. Learn more about Erin's lab here: [https://www.erinkanglab.com www.erinkanglab.com]

Watch the YouTube video recording of the remarks [https://www.youtube.com/watch?v=oaK14hoO3UI here].

=== Nicole Lorenzo ===
* Ph.D. candidate at Florida International University at the time of the award

====== About the award recipient ======
Nicole received the 2019 Future Directions Launch Award in Treatment. After receiving her Ph.D. in Clinical Science at Florida International University in 2019, she worked as a post-doctoral fellow at the University of Maryland, College Park. Currently, she is a tenure-track Assistant Professor in the psychology department at American University. Her research focuses on the transactional processes involved in parent-child interactions, examining how these processes develop and the impact of factors like temperament and parent mental health. This developmental work informs her intervention work which seeks to understand how we can refine and individualize treatment targets to develop early intervention programs that are accessible and scalable for families and providers, particularly those from underserved and underrepresented backgrounds. Learn more about Nicole's work here: [https://www.researchgate.net/profile/Nicole&amp;#x20;Lorenzo www.researchgate.net/profile/Nicole_Lorenzo] 

Watch the YouTube video recording of the remarks [https://www.youtube.com/watch?v=3T53yHtdcqY here].</text>
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      <text bytes="47346" xml:space="preserve">𝑓(◯)

{| + 𝑓(◯)
|-
|| ◯
| style=&quot;background:#000 ;&quot; | &lt;span style=&quot;color:#fff; font-size: 11px; &quot;&gt; &lt;center&gt;⟁ ◁ #fff ○ #000 △ ⟁ &lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#fff ;&quot; | &lt;span style=&quot;color:#000; font-size: 11px; &quot;&gt; &lt;center&gt;⟁ ◁ #000 ○ #fff △ ⟁&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:gray ;&quot; | &lt;span style=&quot;color:pink; font-size: 11px; &quot;&gt; &lt;center&gt;⟁ ◁ pink ○ gray △ ⟁&lt;/span&gt;
|-
|| 人
| style=&quot;background:#000;&quot; | &lt;span style=&quot;color:#fff; font-size: 111px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#fff ;&quot; | &lt;span style=&quot;color:#000; font-size: 111px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:gray;&quot; | &lt;span style=&quot;color:pink; font-size: 111px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; || [http://www.amp-what.com/unicode/search/triangle 
|-
|| 1
| style=&quot;background:#111;&quot; | &lt;span style=&quot;color:#ff0; font-size: 11px; &quot;&gt; &lt;center&gt;◁ #ff0 ○ #111△&lt;/center&gt;&lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#111;&quot; | &lt;span style=&quot;color:#F0f; font-size: 11px; &quot;&gt; &lt;center&gt;◁ #f0f ○ #111△&lt;/center&gt;&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#111;&quot; | &lt;span style=&quot;color:#0ff; font-size: 11px; &quot;&gt; &lt;center&gt;◁ #0ff ○ #111 △&lt;/center&gt;&lt;/span&gt;
|-
|| 人
| style=&quot;background:#111;&quot; | &lt;span style=&quot;color:#ff0; font-size: 111px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#111;&quot; | &lt;span style=&quot;color:#f0f; font-size: 111px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#111;&quot; | &lt;span style=&quot;color:#0ff; font-size: 111px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; || 
|-
|| ⟁
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#f00; font-size: 11px; &quot;&gt; &lt;center&gt;◁ #ff0 ○ #333 △&lt;/center&gt;&lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#0f0; font-size: 11px; &quot;&gt; &lt;center&gt;◁ #f0f ○ #333 △&lt;/center&gt;&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#00f; font-size: 11px; &quot;&gt; &lt;center&gt;◁ #0ff ○ #333 △&lt;/center&gt;&lt;/span&gt;
|-
|| 人
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#f00; font-size: 111px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#0f0; font-size: 111px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#00f; font-size: 111px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; || 
|-
|| ⟁
| style=&quot;background:#555;&quot; | &lt;span style=&quot;color:black; font-size: 11px; &quot;&gt; ◁ black ≡ #000 △ &lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#555;&quot; | &lt;span style=&quot;color:indigo; font-size: 9px; &quot;&gt; ◁ indigo ≡ #4B0082△&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#555;&quot; | &lt;span style=&quot;color:midnightblue; font-size: 11px; &quot;&gt; ◁ midnightblue ≡ #191970△&lt;/span&gt;
|-
|| 人
| style=&quot;background:#555;&quot; | &lt;span style=&quot;color:black; font-size: 111px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#555;&quot; | &lt;span style=&quot;color:indigo; font-size: 111px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#555;&quot; | &lt;span style=&quot;color:midnightblue; font-size: 111px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; || 
|-
|| ⟁
| style=&quot;background:#777;&quot; | &lt;span style=&quot;color:#fff9c4; font-size: 11px; &quot;&gt; ◁ #fff9c4 ○ #777 △ &lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#777;&quot; | &lt;span style=&quot;color:#FFCDD2; font-size: 11px; &quot;&gt; ◁ #FFCDD2 ○ #777 △ &lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#777;&quot; | &lt;span style=&quot;color:#BBDEFB; font-size: 11px; &quot;&gt; ◁ #BBDEFB ○ #777 △ &lt;/span&gt;
|-
|| 人
| style=&quot;background:#777;&quot; | &lt;span style=&quot;color:#fff9c4; font-size: 111px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; ||  
||
| style=&quot;background:#777;&quot; | &lt;span style=&quot;color:#FFCDD2; font-size: 111px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#777;&quot; | &lt;span style=&quot;color:#BBDEFB; font-size: 111px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; ||  
|-
|| ⟁
| style=&quot;background:#999;&quot; | &lt;span style=&quot;color:#ffe0b2; font-size: 11px; &quot;&gt; ◁ #ffe0b2 ○ #999 △ &lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#999;&quot; | &lt;span style=&quot;color:#e1bee7; font-size: 11px; &quot;&gt; ◁ #e1bee7 ○ #999 △ &lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#999;&quot; | &lt;span style=&quot;color:#C8E6C9; font-size: 11px; &quot;&gt; ◁ #C8E6C9 ○ #999 △ &lt;/span&gt;
|-
|| 人
| style=&quot;background:#999;&quot; | &lt;span style=&quot;color:#ffe0b2; font-size: 111px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#999;&quot; | &lt;span style=&quot;color:#e1bee7; font-size: 111px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#999;&quot; | &lt;span style=&quot;color:#C8E6C9; font-size: 111px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
|-
|| ⟁
| style=&quot;background:#aaa;&quot; | &lt;span style=&quot;color:#efefef; font-size: 11px; &quot;&gt; ◁ #efefef ○ #aaa △ &lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#aaa;&quot; | &lt;span style=&quot;color:#222; font-size: 11px; &quot;&gt; ◁ #e1bee7 ○ #aaa △ &lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#aaa;&quot; | &lt;span style=&quot;color:#777; font-size: 11px; &quot;&gt; ◁ #ababab ○ #aaa △ &lt;/span&gt;
|-
|| 人
| style=&quot;background:#aaa;&quot; | &lt;span style=&quot;color:#efefef ; font-size: 33px;&quot;&gt;&lt;center&gt;  ⚞人⚟ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#aaa;&quot; | &lt;span style=&quot;color:#222; font-size: 33px;&quot;&gt;&lt;center&gt; ⚞人⚟ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#aaa;&quot; | &lt;span style=&quot;color:#777; font-size: 33px;&quot;&gt;&lt;center&gt; ⚞人⚟ &lt;/center&gt;&lt;/span&gt; || 
|}


{| + font-size: 25px;
|-
|| 3
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#fff9c4&quot;&gt; ◁ ○&lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#FFCDD2&quot;&gt; ◁ ○&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#BBDEFB&quot;&gt; ◁  ○&lt;/span&gt;
|-
|| 人
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#fff9c4; font-size: 25px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#FFCDD2; font-size: 25px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#BBDEFB; font-size: 25px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
|}
{| + font-size: 10px;
|-
|| 4
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#fff9c4&quot;&gt; ◁ ○&lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#FFCDD2&quot;&gt; ◁ ○&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#BBDEFB&quot;&gt; ◁  ○&lt;/span&gt;
|-
|| 人
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#fff9c4; font-size: 10px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#FFCDD2; font-size: 10px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#BBDEFB; font-size: 10px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
|}

= Salem Accord =
{| + font-size: 222px; primary
|-
|| ◯
| style=&quot;background:#000 ;&quot; | &lt;span style=&quot;color:#fff &quot;&gt; &lt;center&gt;◁ #fff ○ #000 △ ⟁ &lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#fff ;&quot; | &lt;span style=&quot;color:#000 &quot;&gt; &lt;center&gt;◁ #000 ○ #fff △ ⧋&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:gray ;&quot; | &lt;span style=&quot;color:pink &quot;&gt; &lt;center&gt;◁ pink ○ gray △ 📐&lt;/span&gt;
|-
|| 人
| style=&quot;background:#000;&quot; | &lt;span style=&quot;color:#fff; font-size: 77px;&quot;&gt;&lt;center&gt; ⟁ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#fff ;&quot; | &lt;span style=&quot;color:#000; font-size: 77px;&quot;&gt;&lt;center&gt; ⧋ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:gray;&quot; | &lt;span style=&quot;color:pink; font-size: 77px;&quot;&gt;&lt;center&gt; 📐 &lt;/center&gt;&lt;/span&gt; || [http://www.amp-what.com/unicode/search/triangle 
|-
|| 1
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#ff0&quot;&gt; &lt;center&gt;◁ #ff0 ○ #333 △&lt;/center&gt;&lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#F0f&quot;&gt; &lt;center&gt;◁ #f0f ○ #333 △&lt;/center&gt;&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#0ff&quot;&gt; &lt;center&gt;◁ #0ff ○ #333 △&lt;/center&gt;&lt;/span&gt;
|-
|| 人
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#ff0; font-size: 277px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#f0f; font-size: 277px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#0ff; font-size: 277px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
|-
|| 2
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:pink&quot;&gt; ◁ pink ○ #808080 △&lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:midnightblue&quot;&gt; ◁ midnightblue ○ #808080 △&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:indigo&quot;&gt; ◁ indigo ○ #808080 △&lt;/span&gt;
|-
|| 人
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:pink; font-size: 33px;&quot;&gt;&lt;center&gt; ⚞🧿⚟ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:midnightblue; font-size: 33px;&quot;&gt;&lt;center&gt; ⚞🧿⚟ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:indigo; font-size: 33px;&quot;&gt;&lt;center&gt; ⚞🧿⚟ &lt;/center&gt;&lt;/span&gt; || 
|}


or ∨↯∧ &amp; + 🔥 十 ⨁ ⨂ ❂ ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○ ⌂ +


{| + font-size: 100px;
|-
|| ◯ 
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:white&quot;&gt; ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○&lt;/span&gt;
|| 人 
|-
||
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:salmon; font-size: 46px;&quot;&gt;&lt;center&gt; ∨↯∧ &lt;/center&gt;&lt;/span&gt; || 
|}

{| + font-size: 222px; primary
|-
|| 1
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#fff9c4&quot;&gt; ◁ #fff9c4 ○&lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#FFCDD2&quot;&gt; ◁ #FFCDD2 ○&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#BBDEFB&quot;&gt; ◁ #BBDEFB ○&lt;/span&gt;
|-
|| 人
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#fff9c4; font-size: 222px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#FFCDD2; font-size: 222px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#BBDEFB; font-size: 222px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
|}

{| + font-size: 111px; secondary
|-
|| 2
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#ffe0b2&quot;&gt; ◁ #ffe0b2 ○&lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#e1bee7&quot;&gt; ◁ #e1bee7 ○&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#C8E6C9&quot;&gt; ◁ #C8E6C9 ○&lt;/span&gt;
|-
|| 人
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#ffe0b2; font-size: 222px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#e1bee7; font-size: 222px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#C8E6C9; font-size: 222px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
|}


{| + font-size: 25px;
|-
|| 3
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#fff9c4&quot;&gt; ◁ ○&lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#FFCDD2&quot;&gt; ◁ ○&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#BBDEFB&quot;&gt; ◁  ○&lt;/span&gt;
|-
|| 人
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#fff9c4; font-size: 25px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#FFCDD2; font-size: 25px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#BBDEFB; font-size: 25px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
|}
{| + font-size: 10px;
|-
|| 4
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#fff9c4&quot;&gt; ◁ ○&lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#FFCDD2&quot;&gt; ◁ ○&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#BBDEFB&quot;&gt; ◁  ○&lt;/span&gt;
|-
|| 人
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#fff9c4; font-size: 10px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#FFCDD2; font-size: 10px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#BBDEFB; font-size: 10px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
|}
{| + font-size: 100px;
|-
|| 5
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#fff9c4&quot;&gt; ◁ ○&lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#FFCDD2&quot;&gt; ◁ ○&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#BBDEFB&quot;&gt; ◁  ○&lt;/span&gt;
|-
|| 人
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#fff9c4; font-size: 10px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#FFCDD2; font-size: 10px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#BBDEFB; font-size: 10px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
|}
{| + font-size: 10px;
|-
|| 6
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#fff9c4&quot;&gt; ◁ ○&lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#FFCDD2&quot;&gt; ◁ ○&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#BBDEFB&quot;&gt; ◁  ○&lt;/span&gt;
|-
|| 人
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#fff9c4; font-size: 10px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#FFCDD2; font-size: 10px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#BBDEFB; font-size: 10px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
|}

== 0 ==

{|
|-
|| [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] 
|-
|| [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] 
|-
|| [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家]
||  小
|-
|| [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘]
|}
 
 meritorium . meritorious : merit .or.iou.us
{{User alternative account|VeronicaJeanAnderson}}

{| + font-size: 100px;
|-
|| 人 
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:white&quot;&gt; ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:white&quot;&gt; ◁ ○&lt;/span&gt;
|| 人
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:white&quot;&gt; ◁ ○&lt;/span&gt;
|| 人 
|| 人 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:white&quot;&gt; ◁  ○&lt;/span&gt;
|| 人
|-
||
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:salmon; font-size: 100px;&quot;&gt;&lt;center&gt; ∨↯∧ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#fff9c4; font-size: 100px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#FFCDD2; font-size: 100px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
||
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#BBDEFB; font-size: 100px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
|-
||
|-
||
|}

== or ∨↯∧ &amp; + 🔥 十 ⨁ ⨂ ❂ ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○ ⌂ + .us ==

==⌂==


{| + font-size: 100px;
|-
|| 人 
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:white&quot;&gt; ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○&lt;/span&gt;
|| 人 
|-
||
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:salmon; font-size: 100px;&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; || 
|-
||
|-
||
|}

 [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家][https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘]
 
 meritorium . meritorious : merit .or.iou.us
{{User alternative account|VeronicaJeanAnderson}}
==✪⚪⬤🔥◍⚫⨁	👀	≡ odd → +1	[ { ( East	⚫🔴⚪○💮⭕	West ) } ]	iff even ⇒ ÷2==
∨⚡\🗲↯/ϟ∧  ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/
{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
|+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟
|-
|| ✪⚪⬤🔥◍⚫⨁
|| [https://www.twitch.tv/archie97305 👀]
| style=&quot;background:pink;&quot; | &lt;span style=&quot;color:#808080&quot;&gt; ≡ odd → +1 &lt;/span&gt;
| style=&quot;background:pink;&quot; | &lt;span style=&quot;color:#808080&quot;&gt; [ { ( East &lt;/span&gt;
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; ⚫🔴⚪○💮⭕&lt;/span&gt;
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:pink&quot;&gt; West ) } ] &lt;/span&gt;
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:pink&quot;&gt;  iff even ⇒ ÷2 &lt;/span&gt;
|-
|| Primary
|| [https://www.amnesty.org/en/ 1]
|| ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪
|| ||  ○人∘🧿⚪⬤◍⚫ ||
|| ⨁⚉⨂❂✧✷✸✡✵
|-
|| Secondary
|| [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2]
|-
|| Tertiary
|| [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3]
|-
|| Quaternary
|| [http://localhost:8080/ 4]
|-
|| Quinary
|| [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5]
|-
|| Senary
|| [https://en.wikiversity.org/wiki/User:Archie97305 6]
|-
|| Septenary
|| [https://maritimearchaeological.org/beeswax-wreck/ 7]
|-
|| Octenary
|| [https://www.youtube.com/freecodecamp 8]
|-
|| nonary
|| [https://www.freecodecamp.org/ 9]
|-
||Base Name
||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&amp;Number=84101 `]
|-
|| binary
|| 2
|}

== ya ==
 ◯ ○ ∘ ⬤
 大 【 ヤ / や 】 (ya)
 人 【 ジン、 ニン / ひと 】(jean、he toe)
 大家 (ya ya)

 Ya (hiragana: や, katakana: ヤ)
 ヤフー
 屋 - Wiktionaryhttps://en.wiktionary.org › wiki › 屋
 Semantic compound of 尸+至. 尸 does not represent the radical for death, but is a pictogram depicting a cloth draped. 至 means &quot;dead end&quot;.
 Home (家)
 A home, or domicile, is a space used as a permanent or semi-permanent residence for one or many humans. It is a fully or semi sheltered space and can have both interior and exterior aspects to it.
 
 Ya (や)
 Kana
 Ya is one of the Japanese kana, each of which represents one mora. The hiragana is written in three strokes, while 
 the katakana is written in two. Both represent. Their shapes have origins in the character 也. Wikipedia
 hiragana origin: 也
 spelling kana: 大和のヤ Yamato no &quot;ya&quot;
 transliteration: ya
 unicode: U+3084, U+30E4

 What is the pronunciation of Ya line?
In historical kana orthography, it is written as &quot;yau&quot;, &quot;say&quot;, and &quot;eu&quot;, and read as &quot;yo&quot;, &quot;you&quot;, and &quot;yo&quot;, respectively. Even in modern times, &quot;saying&quot; and &quot;going&quot; are sometimes pronounced as &quot;yu&quot; and &quot;yuku . &quot; From the above, it can be said that Ya line is the yoon of that line .

 [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家][https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘]

== g ==

{| + 
|-
|| g 
|| ⌂
|| 時
|-
|| nono
|| ⌂
|| ◁
|| 前
|| の
|| ノ
|| 名
|| &quot;Salmon&quot;
|| #fa8072
||
||
||
||
||
|| rgb(250, 128, 114)
|| ::
|| hsl(6, 93%, 71%)
|-
|| &quot;のノ&quot;
|| ⌂
|| 	&lt;span style=&quot;color:salmon;&quot;&gt;&lt;center&gt; ∅ &lt;/center&gt;&lt;/span&gt; 
| style=&quot;background:#000;&quot;|&lt;span style=&quot;color:salmon;&quot;&gt;&lt;center&gt; #000 &lt;/center&gt;&lt;/span&gt; 
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:salmon;&quot;&gt;&lt;center&gt; #808080 &lt;/center&gt;&lt;/span&gt; 
| style=&quot;background:#fff;&quot; | &lt;span style=&quot;color:salmon;&quot;&gt;&lt;center&gt; #fff &lt;/center&gt;&lt;/span&gt; 
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:salmon;&quot;&gt;&lt;center&gt; #333 &lt;/center&gt;&lt;/span&gt; 

| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#fa8072;&quot;&gt;&lt;center&gt; #fa8072 &lt;/center&gt;&lt;/span&gt; 
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#fa8072ff;&quot;&gt;&lt;center&gt; #fa8072ff &lt;/center&gt;&lt;/span&gt; 
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#fa8072aa;&quot;&gt;&lt;center&gt; #fa8072aa &lt;/center&gt;&lt;/span&gt; 
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#fa807280;&quot;&gt;&lt;center&gt; #fa807280 &lt;/center&gt;&lt;/span&gt; 
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#fa807233;&quot;&gt;&lt;center&gt; #fa807233 &lt;/center&gt;&lt;/span&gt; 
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#fa807200;&quot;&gt;&lt;center&gt; #fa807200 &lt;/center&gt;&lt;/span&gt; 
|-
|| know no ノノ
|| ⌂
|| &lt;span style=&quot;color:salmon; font-size: 100px;&quot;&gt; ☂ &lt;/span&gt;
| style=&quot;background:#000;&quot; | &lt;span style=&quot;color:salmon; font-size: 100px;&quot;&gt; ☂ &lt;/span&gt;
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:salmon; font-size: 100px;&quot;&gt; ☂ &lt;/span&gt;
| style=&quot;background:#fff;&quot; | &lt;span style=&quot;color:salmon; font-size: 100px;&quot;&gt; ☂ &lt;/span&gt;
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:salmon; font-size: 100px;&quot;&gt; ☂ &lt;/span&gt;

| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#fa8072; font-size: 100px;&quot;&gt; ☂ &lt;/span&gt;
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#fa8072ff; font-size: 100px;&quot;&gt; ☂ &lt;/span&gt;
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#fa8072aa; font-size: 100px;&quot;&gt; ☂ &lt;/span&gt;
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#fa807280; font-size: 100px;&quot;&gt; ☂ &lt;/span&gt;
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#fa807233; font-size: 100px;&quot;&gt; ☂ &lt;/span&gt;
| style=&quot;background:#333;&quot; | &lt;span style=&quot;color:#fa807200; font-size: 100px;&quot;&gt; ☂ &lt;/span&gt;
|}

{| + 
|-
|| hsl(0-33, 93%, 71%)
|| &lt;span style=&quot;color:hsl(0, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(1, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(2, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(3, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt;  
|| &lt;span style=&quot;color:hsl(4, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(5, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(7, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(8, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(9, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(10, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(11, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(12, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(13, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(14, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(15, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(16, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(17, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(18, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(19, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(20, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(21, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(22, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(23, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(24, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(25, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(26, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(27, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(28, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(29, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(30, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(31, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(32, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(33, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|}

{| + 
|-
|| hsl(6-606, 93%, 71%)
|| &lt;span style=&quot;color:hsl(6, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(36, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(66, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(96, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt;  
|| &lt;span style=&quot;color:hsl(126, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(156, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(186, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(216, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(246, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(276, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(306, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(336, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(366, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(396, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(426, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(456, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(486, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(516, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(546, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(576, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(606, 93%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|}

{| + 
|-
|| hsl(6, 93%, 0-100%)
|| &lt;span style=&quot;color:hsl(6, 93%, 0%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 93%, 10%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 93%, 20%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 93%, 30%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt;  
|| &lt;span style=&quot;color:hsl(6, 93%, 40%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 93%, 50%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 93%, 60%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 93%, 70%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 93%, 80%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 93%, 90%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 93%, 100%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|}

{| + 
|-
|| hsl(6, 0-100%, 71%)
|| &lt;span style=&quot;color:hsl(6, 0%, 71%);&quot;&gt;&lt;center&gt; ⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 10%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 20%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 30%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 40%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 50%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt;  
|| &lt;span style=&quot;color:hsl(6, 60%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 70%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt;  
|| &lt;span style=&quot;color:hsl(6, 80%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 90%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|| &lt;span style=&quot;color:hsl(6, 100%, 71%);&quot;&gt;&lt;center&gt;⌂ &lt;/center&gt;&lt;/span&gt; 
|}

 meritorium . meritorious : merit .or.iou.us
{{User alternative account|VeronicaJeanAnderson}}
∨⚡\🗲↯/ϟ∧  ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/
{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
|+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟
|-
|| ✪⚪⬤🔥◍⚫⨁
|| [https://www.twitch.tv/archie97305 👀]
| style=&quot;background:pink;&quot; | &lt;span style=&quot;color:#808080&quot;&gt; ≡ odd → +1 &lt;/span&gt;
| style=&quot;background:pink;&quot; | &lt;span style=&quot;color:#808080&quot;&gt; [ { ( East &lt;/span&gt;
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; ⚫🔴⚪○💮⭕&lt;/span&gt;
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:pink&quot;&gt; West ) } ] &lt;/span&gt;
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:pink&quot;&gt;  iff even ⇒ ÷2 &lt;/span&gt;
|-
|| Primary
|| [https://www.amnesty.org/en/ 1]
|-
|| Secondary
|| [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2]
|-
|| Tertiary
|| [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3]
|-
|| Quaternary
|| [http://localhost:8080/ 4]
|-
|| Quinary
|| [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5]
|-
|| Senary
|| [https://en.wikiversity.org/wiki/User:Archie97305 6]
|-
|| Septenary
|| [https://maritimearchaeological.org/beeswax-wreck/ 7]
|-
|| Octenary
|| [https://www.youtube.com/freecodecamp 8]
|-
|| nonary
|| [https://www.freecodecamp.org/ 9]
|-
||Base Name
||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&amp;Number=84101 `]
|-
|| binary
|| 2
|-
||ternary
||3
|-
||quaternary
||4
|-
||quinary
||5
|-
||senary
||6
|-
||septenary
||7
|-
||octal
||8
|-
||nonary
|-
||decimal
|-
||undenary
|-
||duodecimal
|-
||hexadecimal
||16
|-
||vigesimal
||20
|-
||sexagesimal 
||60
|}


 How do you want your water served when you get here? https://pubs.usgs.gov/sir/2005/5168/pdf/sir2005-5168.pdf Robert Lee Stinson %VOX

 &quot;tautology club says hi&quot;

 w 11am &quot;Naturalist Society for the Humane Treatment of Monsters&quot; from dnd game on twitter [https://www.youtube.com/watch?v=n7uNA5fO1iI rice ex in CA] https://www.oregonwild.org/about/blog/oregon-grizzly-country https://therevelator.org/yellowstone-grizzlies-unbearable-divides/ https://developer.mozilla.org/en-US/docs/Web/CSS/color_value/hsl

 https://www.researchgate.net/about
 Amare, Nicole &amp; Manning, A.. (2012). Seeing typeface personality: Emotional responses to form as tone. IEEE International Professional Communication Conference. 1-9. 10.1109/IPCC.2012.6408605. Various studies have correlated specific visual characteristics of typefaces with specific overall emotional effects: curvilinear forms and open letter shapes generally feel “friendly” but also “formal” or “informal,” depending on other factors; large contrasts in stroke widths, cap height, and aspect ratio generally feel “interesting,” but also “attractive” or “aggressive,” depending on other factors; low-variety and low-contrast forms generally feel “professional” but also “reliable” or “boring.” Although the current findings on typeface personality are useful, they have not indicated a systematic explanation for why specific physical typeface forms have the specific emotion effects that they do. This paper will report results of an empirical study in which 102 participants indicated their immediate emotional responses to each of 36 distinct typeface designs. Results support correlation between specific typeface features (variety vs. contrast vs. pattern) and specific emotional parameters (amusement vs. agitation vs. focus), explaining findings of previous studies, suggesting various classroom approaches to purpose-driven typeface selection.


{{User alternative account|VeronicaJeanAnderson}}

{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
|+ ᐪgenki-ness; +, -tachi . .   .
|-
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; [ { ( A B E ) } ] &lt;/span&gt;
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; [ { ( [https://www.twitch.tv/archie97305 👀] ) } ]&lt;/span&gt;
|-
|| Primary
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; index.html&lt;/span&gt;
|| notepad/atom (atom is deprecated)
|-
|| Secondary
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; vue &lt;/span&gt;
|| [https://www.vim.org/ vim] [https://github.com/vim/vim-win32-installer/releases installer]
|-
|| Tertiary
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; css &lt;/span&gt;
|| global css @ 
|| gg css @ 
|| NPC css @
|-
|| Quaternary
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; pug &lt;/span&gt;
|-
|| Quinary
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; (direct object) &lt;/span&gt;
|-
|| Senary
|| b
|}

 https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/sandbox

 trying to create a 1 -&gt; 2 -&gt; 3 -&gt; 4 -&gt; 5 -&gt; 6 system in the apartment here that can be copied from site to site using artistic threads to help a Nice And Proper NAP-er navigate between properties with ease while maintaining adequate supportive care that we all require to enable us to focus on whatever catches our fancy.

 sun; natural light; breathe; BGs
 carbs; hygiene; laundry away
 bedroom; needles; blood; garbage out
 kitchen/nutritional/study
 social/outreach/linking worlds
 back porch 


0 -- Computer Science, information and general works


{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
|+ ᐪgenki-ness; +, -tachi . .   .
|-
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; [ { ( T O P ) } ] &lt;/span&gt;
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; [ ℳ ] &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; { ¢ } &lt;/span&gt;
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; ( ৳ ) &lt;/span&gt;
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; [ { ( I.n C.ase of E.mergency ) } ] &lt;/span&gt;
| style=&quot;background:teal;&quot; | &lt;span style=&quot;color:lime&quot;&gt; ᐪ l i p s c h i t z &lt;/span&gt;
|| [https://www.youtube.com/watch?v=qrrz54UtkCc ᐪ]
|-
|| Primary
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:black&quot;&gt; physical&lt;/span&gt;
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; emotional&lt;/span&gt;
| style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; social&lt;/span&gt;
|| This reflects health enough to communicate with people intimately enough to address real immediate issues
| style=&quot;background:#FFFFE6;&quot; | &lt;span style=&quot;color:teal&quot;&gt; ^ torikomu &lt;/span&gt;
||[https://www.youtube.com/watch?v=YxvBPH4sArQ ^]
|-
|| Secondary
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:black&quot;&gt; occupational&lt;/span&gt;
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:black&quot;&gt; intellectual&lt;/span&gt;
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:black&quot;&gt; environmental&lt;/span&gt;
|| This reflects living somewhere promoting healthy reasoning
| style=&quot;background:#FFE6E6;&quot; | &lt;span style=&quot;color: teal&quot;&gt; | kaizen | &lt;/span&gt;
|| |
|-
|| Tertiary
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; spiritual&lt;/span&gt;
| style=&quot;background:#BFBFBF;&quot; | &lt;span style=&quot;color:white&quot;&gt; factual &lt;/span&gt;
| style=&quot;background:#F2F2F2;&quot; | &lt;span style=&quot;color:black&quot;&gt; nutritional&lt;/span&gt;
|| This reflects healthy mindful every habits
|  style=&quot;background:#E6EAFF;&quot; | &lt;span style=&quot;color:teal&quot;&gt; . genkiness . .&lt;/span&gt;
|| .
|-
|| Quaternary
| style=&quot;background:#FFE6FB;&quot; | &lt;span style=&quot;color:black&quot;&gt; generational&lt;/span&gt;
| style=&quot;background:#E6FFFF;&quot; | &lt;span style=&quot;color:black&quot;&gt; miscellaneal&lt;/span&gt;
| style=&quot;background:#F2E0CE;&quot; | &lt;span style=&quot;color:black&quot;&gt; punctuational&lt;/span&gt;
|| This reflects having it all together enough to enjoy the holidays
| style=&quot;background:#FFF2E6;&quot; | &lt;span style=&quot;color:lime&quot;&gt; # goblin &lt;/span&gt;
|| #
|-
|| Quinary
| style=&quot;background:#F9F9F9;&quot; | &lt;span style=&quot;color:pink&quot;&gt; (direct object) &lt;/span&gt;
| style=&quot;background:white;&quot; | &lt;span style=&quot;color:black&quot;&gt; {verb} &lt;/span&gt;
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; [noun] &lt;/span&gt;
|| This reflects deliberate professional progress
| style=&quot;background:#F2E6FF;&quot; | &lt;span style=&quot;color:lime&quot;&gt; / tsugu /&lt;/span&gt;
|| /
|-
|| Senary
|| b
|| 〇
|| x
|| This reflects influencing others
| style=&quot;background:#E6FFEA;&quot; | &lt;span style=&quot;color:lime&quot;&gt; @ g @ g @ &lt;/span&gt;
|| [https://www.youtube.com/watch?v=SYnVYJDxu2Q @] 
|}

== 100 -- Philosophy and psychology ==
 How can I use color to manipulate behavior and improve communication?
===named===
==== Re⋮Beccaδ#639 ====
===== rebeccapurple :: #663399 =====
 https://meyerweb.com/eric/thoughts/2014/06/19/rebeccapurple/

====black====
====white====
====græy====
====pink====
====indigo====
====midnightblue====
===hex===
====#fff====
====#fff====
{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
|+
|-
| style=&quot;background:black;&quot; | &lt;span style=&quot;color:white&quot;&gt; [ white { on black ⚞🧿⚟ #fff on #000 ⚞🧿⚟  }  ] &lt;/span&gt;
|-
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:pink&quot;&gt; [ pink { on 50% grey ⚞🧿⚟ #ffc0cb on #808080 ⚞🧿⚟ } ] &lt;/span&gt;
|-
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#191970&quot;&gt; [ midnightblue { on 50% grey ⚞🧿⚟ #191970 on #808080 ⚞🧿⚟ } ] &lt;/span&gt;
|-
| style=&quot;background:#808080;&quot; | &lt;span style=&quot;color:#4b0082&quot;&gt; [ indigo { on 50% grey ⚞🧿⚟ #4b0082 on #808080 ⚞🧿⚟ } ] &lt;/span&gt;
|}

===cmyk===
 https://colordesigner.io/convert/cmyktohex
====gg on cmyk(0,0,0,33.3) w|materializecss.com====

 https://materializecss.com/color.html
{| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;
|+
|-
| style=&quot;background:#ababab&quot; | &lt;span style=&quot;color:#fff9c4&quot;&gt; [ gg_yellow { on cmyk(0,0,0,33.3) ⚞🧿⚟ #fff9c4 on #ababab⚞🧿⚟ } ] &lt;/span&gt;
|-
| style=&quot;background:#ababab;&quot; | &lt;span style=&quot;color:#ffe0b2&quot;&gt; [ gg_orange { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffe0b2 on #ababab⚞🧿⚟ } ] &lt;/span&gt;
|-
| style=&quot;background:#ababab;&quot; | &lt;span style=&quot;color:#ffcdd2&quot;&gt; [ gg_red { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffcdd2 on #ababab⚞🧿⚟ } ] &lt;/span&gt;
|-
| style=&quot;background:#ababab;&quot; | &lt;span style=&quot;color:#e1bee7&quot;&gt; [ gg_purple { on cmyk(0,0,0,33.3) ⚞🧿⚟ #e1bee7 on #ababab⚞🧿⚟ } ] &lt;/span&gt;
|-
| style=&quot;background:#ababab;&quot; | &lt;span style=&quot;color:#bbdefb&quot;&gt; [ gg_blue { on cmyk(0,0,0,33.3) ⚞🧿⚟ #bbdefb on #ababab⚞🧿⚟ } ] &lt;/span&gt;
|-
| style=&quot;background:#ababab;&quot; | &lt;span style=&quot;color:#c8e6c9&quot;&gt; [ gg_green { on cmyk(0,0,0,33.3) ⚞🧿⚟ #c8e6c9 on #ababab⚞🧿⚟ } ] &lt;/span&gt;
|-
| style=&quot;background:#ababab;&quot; | &lt;span style=&quot;color:#efefef&quot;&gt; [ gg_white { on cmyk(0,0,0,33.3) ⚞🧿⚟ #efefef on #ababab⚞🧿⚟ } ] &lt;/span&gt;
|-
| style=&quot;background:#ababab;&quot; | &lt;span style=&quot;color:#111&quot;&gt; [ gg_black { on cmyk(0,0,0,33.3) ⚞🧿⚟ #111 on #ababab ⚞🧿⚟ } ] &lt;/span&gt;
|-
| style=&quot;background:#ababab;&quot; | &lt;span style=&quot;color:#808080&quot;&gt; [ gg_grey { on cmyk(0,0,0,33.3) ⚞🧿⚟ #808080 on #ababab ⚞🧿⚟ } ] &lt;/span&gt;
|-
| style=&quot;background:#ababab;&quot; | &lt;span style=&quot;color:#f8bbd0&quot;&gt; [ gg_pink { on cmyk(0,0,0,33.3) ⚞🧿⚟ #f8bbd0 on #ababab ⚞🧿⚟ } ] &lt;/span&gt;
|-
| style=&quot;background:#ababab;&quot; | &lt;span style=&quot;color:#b2ebf2&quot;&gt; [ gg_cyan { on cmyk(0,0,0,33.3) ⚞🧿⚟ #b2ebf2 on #ababab ⚞🧿⚟ } ] &lt;/span&gt;
|-
| style=&quot;background:#ababab;&quot; | &lt;span style=&quot;color:#d7ccc8&quot;&gt; [ gg_brown { on cmyk(0,0,0,33.3) ⚞🧿⚟ #d7ccc8 on #ababab )⚞🧿⚟} ] &lt;/span&gt;
|}

===rgba===

=== TrumPutin-ism ===

 Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about.

== 200 -- Religion ==
 Royal We

 1000 things I did 1992-2022 other than lie my way onto the supreme court to overturn Roe v Wade
{|
|-
|| ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ
|| ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ
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|| ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ
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|-
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|}

== 300 -- Social sciences ==
 https://wattention.com/traditional-rice-harvesting-in-japan/
 https://www.wwoofjapan.com/home/index.php?lang=en

== 400 -- Language ==
=== Programming ===
==== .png ====
==== Esperanto ====
==== HTML ====
==== PUG ====

== 500 -- Pure Science ==
== 600 -- Technology ==
=== local hosts===
 [http://localhost:8080/ 8080]
 file:///D:/index.html

=== Roland SP 404MKII ===
 https://www.roland.com/global/products/sp-404mk2/
 https://www.roland.com/global/support/by_product/sp-404mk2/owners_manuals/

 @https://static.roland.com/manuals/sp-404mk2_app/eng/19610757.html

=== VIM ===

 https://vim-adventures.com/

== 700 -- Arts and recreation ==
== 800 -- Literature ==
== 900 -- History and geography ==
 https://geology.com/stories/13/bear-areas/
=== pre-2030 ===
2022 &quot;booked&quot; by Hillsboro Police for sending email addressing &quot;Christian Hate&quot; and &quot;Spiritual War&quot; along with &quot;exorcisms&quot; and &quot;Halloween Hysteria&quot; in Marion County, OR where Salem Police Department abdicated from protecting some children in Salem from 2016-2021.

2021 Kaiser Permanente promised cash settlement to mitigate their abdication in Marion County. KP lawyer with intimate details about my vagina: terrence .j . loeber@kp.org

2012 &quot;unliked&quot; on FB by some Nazarene peers after openly questioning Alex Jones' allegation that Sandy Hook didn't happen and asking for compassion for parents who were called actors while they grieved publicly through no choice of their own.

2011 Lupron given by KP for menorrhagia as alternative to b/c pills first rxd in 86. How many women who have &quot;mostly&quot; been on b/c pills from 87-11 are obese? Why no menorrhagia while immersed in Japan? How close to a traditional Japanese diet can I get in the Willamette Valley and how close to no meat will my body allow?

=== TrumPutish War Against Humanity ===

 Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about.

=== Ring of Fire ===
=== Post &quot;Roe v Wade&quot; ===
 Who did Roe v Wade protect?

 Why would a Nazarene raised pro-life support an &quot;underground&quot; network post Roe v Wade?

=== Our Contemporary &quot;Underground Railroad&quot; needs a submarine? ===
 Why did Portland, OR close the Shanghai Tunnels recently?

 Human Trafficking through Astoria, OR has been going on &quot;forever&quot;. How do we align an &quot;underground railroad&quot; with contemporary supports?

== 10 -- A &amp; + ==
== 11 -- B * x ==
== 12 -- C f(◯) ==
== 13 -- D  Δ δ ƍ ≜ 𝜟 𝝳 ==
== 14 -- E 🐘 𓃰 ==
== 15 -- F ==
== 16 -- G ==
== 17 -- H ==
== 18 -- I ==
== 19 -- J ==
== 20 -- K ==
== 21 -- L ==
== 22 -- M ==
== 23 -- N ==
== 24 -- O ==
== 25 -- P ==
== 26 -- Q ==
== 27 -- R ==
== 28 -- S ==
== 29 -- T ==
== 30 -- U ==
== 31 -- V ==
== 32 -- W == 
== 33 -- X ==
== 34 -- Y ==
== 35 -- Z ==</text>
      <sha1>cso9h29ohcw0vc2a1zln4m006cuq2xt</sha1>
    </revision>
  </page>
  <page>
    <title>JCCAP FDF/2020</title>
    <ns>0</ns>
    <id>262457</id>
    <revision>
      <id>2411874</id>
      <parentid>2411405</parentid>
      <timestamp>2022-08-04T17:57:01Z</timestamp>
      <contributor>
        <username>Ncharamut</username>
        <id>2824970</id>
      </contributor>
      <comment>/* Addresses */ fixed typo</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="29479" xml:space="preserve">== '''Addresses''' ==

=== '''''Future Directions Address 1: Father Inclusion, Engagement, Retention, and Positive Outcomes in Child and Adolescent Research''''' ===
'''Presented by Greg Fabiano, Ph.D.'''

====Description====
In this address, Dr. Greg Fabiano outlines future directions in the next generation of father-focused studies in the child and adolescent psychology literature, with an emphasis on improving the study of the parameters of inclusion, engagement, retention, and measurement of outcomes. Research shows that parents may play a key role in child resilience. It is important to focus on fathers in research because father involvement lags behind mother involvement and father involvement in childhood has been a long-standing national priority. Research has found father involvement to contribute to language development and early literacy skills as well as the development of appropriate socialization skills. However, few studies have looked at father involvement in child treatment. The format most research studies are in, may actually discourage fathers to participate. The COACHES program works to address this gap in research by focusing on father involvement in the intervention. Results show that fathers are satisfied with the outcomes of the program (100%) and satisfied with the treatment process (80%). This research shows initial promise for interventions that include fathers. &lt;ref&gt;{{Cite journal|last=Fabiano|first=Gregory A.|last2=Caserta|first2=Abigail|date=2018-09-03|title=Future Directions in Father Inclusion, Engagement, Retention, and Positive Outcomes in Child and Adolescent Research|url=https://doi.org/10.1080/15374416.2018.1485106|journal=Journal of Clinical Child &amp; Adolescent Psychology|volume=47|issue=5|pages=847–862|doi=10.1080/15374416.2018.1485106|issn=1537-4416|pmid=30110185}}&lt;/ref&gt;

Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=EogVEgi3n80&amp;feature=youtu.be here]

=== '''''Future Directions Address 2: Research and Intervention with Youths in Poverty''''' ===
'''Presented by Martha Wadsworth, Ph.D.'''

====Description====
In this address, Dr. Martha Wadsworth integrates theory and empirical findings about understanding and fostering the process of resilience and adaptation in children and families who live in poverty. She starts off by defining what &quot;normal&quot; is and how it is culturally defined. The stress response system gathers information from the environment which it then uses to guide the development of the individual in a way that matches the environment. There are many ways in which our bodies develop and respond to the environment. Because development is extremely dependent on environment, children that grow up in dangerous and unpredictable environments, develop in a way suited to handle those environments and usually do not develop behaviors that are well suited to a more enriched environment. We develop abilities that match our environment rather than abilities that may be useful in other environments. Dr. Wadsworth proposes that focusing on skills that are strengths may be more beneficial than focusing on deficits. Cultural identity should also play a large role in interventions and can be used as a &quot;coping resource&quot; for youth in poverty. By building a strong identity, children can gain coping skills. Interventions for youth in poverty should begin by focusing on stress and coping. Next, the intervention can focus on cultural identity and finish with collective empowerment. Overall, a multifaceted approach is necessary to treatment youth in poverty. &lt;ref&gt;{{Cite journal|last=Wadsworth|first=Martha E.|last2=Ahlkvist|first2=Jarl A.|last3=McDonald|first3=Ashley|last4=Tilghman-Osborne|first4=Emile M.|date=2018-11-02|title=Future Directions in Research and Intervention with Youths in Poverty|url=https://doi.org/10.1080/15374416.2018.1485108|journal=Journal of Clinical Child &amp; Adolescent Psychology|volume=47|issue=6|pages=1023–1038|doi=10.1080/15374416.2018.1485108|issn=1537-4416|pmc=PMC6348127|pmid=30052089}}&lt;/ref&gt; 

Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=Mj3bgzgXI6g here]

=== '''''Future Directions Address 3: Examination of Brain Networks in Neurodevelopmental Disorders''''' ===
'''Presented by Lucina Uddin, Ph.D.'''

====Description====
In this address, Dr. Lucina Uddin discusses future directions for neuroscience researchers examining brain networks in neurodevelopmental disorders, highlighting gaps in the current literature. Cognitive flexibility is important in individuals with autism spectrum disorder (ASD). Research has shown atypical patterns of brain dynamics often seen in those with neurodevelopmental disorders which is characterized by cognitive inflexibility. Dr. Uddin and her lab have conducted several studies looking at brain networks in those with neurodevelopmental disorders. They concluded that ASD and ADHD must be considered in the context of phenotypic heterogeneity. There are also associated difficulties in parsing heterogeneity in neurodevelopmental disorders. She suggests that researchers use a novel data-driven approach to inform diagnostic nosology. There are new studies looking at whether bilingualism can help with executive functioning in individuals with ASD. &lt;ref&gt;{{Cite journal|last=Uddin|first=Lucina Q.|last2=Karlsgodt|first2=Katherine H.|date=2018-05-04|title=Future Directions for Examination of Brain Networks in Neurodevelopmental Disorders|url=https://doi.org/10.1080/15374416.2018.1443461|journal=Journal of Clinical Child &amp; Adolescent Psychology|volume=47|issue=3|pages=483–497|doi=10.1080/15374416.2018.1443461|issn=1537-4416|pmc=PMC6842321|pmid=29634380}}&lt;/ref&gt;  

Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=CoSM8C2ucRE here]

=== '''''Future Directions Address 4: the Treatment of Youth Mental Health''''' ===
'''Presented by Bruce Chorpita, Ph.D.'''

====Description====
In this address, Dr. Bruce Chorpita discusses mental health care systems and presents ideas and examples of methods that may preserve the strengths of the two major paradigms in children’s mental health, evidence-based treatments, and individualized care models, but that also have the potential to extend their applicability and impact. While over 20% of the world experiences mental health concerns at some point, most don't receive services and those who do are typically not receiving evidence-based services. There is also an insufficient workforce to meet society's needs. The future of research needs to make sure that the evidence-base treatments match diverse populations, are continuing in development, and are dynamic. We need to &quot;chunk and filter&quot; treatments per individual who is receiving them. When modular design, feedback, and expert process managing were combined for treatment of youth, they performed better than evidence-based treatments and usual care. The MATCH program performed well in multiple research clinical trials. In the future, treatment engagement, handling a patient with a crisis, and ethnic and cultural diversity need to be addressed in treatments. Chunking, filtering, and discovering are the best ways to empower individuals to act on best ideas first and self correct. &lt;ref&gt;{{Cite journal|last=Chorpita|first=Bruce F.|last2=Daleiden|first2=Eric L.|date=2014-03-01|title=Structuring the Collaboration of Science and Service in Pursuit of a Shared Vision|url=https://doi.org/10.1080/15374416.2013.828297|journal=Journal of Clinical Child &amp; Adolescent Psychology|volume=43|issue=2|pages=323–338|doi=10.1080/15374416.2013.828297|issn=1537-4416|pmid=23981145}}&lt;/ref&gt;

Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=bnKkc2iyeLo here].

== '''Workshops''' ==

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====
Applying to doctoral programs marks an important life milestone for you and other undergraduate majors and post-baccalaureate trainees. Importantly, some of the considerations for choosing where to receive undergraduate training (e.g., faculty-to-student ratio, quality of institution) take a &quot;back seat&quot; to the key factor in doctoral training that most impacts your career: Identifying the person who will serve as your mentor. Undergraduate programs rarely offer formal instruction in choosing doctoral mentors, and some of the factors you might consider could vary from year-to-year and by mentor. Overall, when selecting a mentor you should consider who fits your needs and learning style. Because the quality of your doctoral training is mostly impacted by your mentor, you should think about who is going to fit best with your goals. When searching for mentors to apply to, you should first identify researchers who study topics similar to your interests. You can do this by reading research articles and looking for ones that align with your &quot;burning question&quot;. Other factors you should consider when selecting mentors include: the size of their lab, the time they devote to their mentees, the area in which their research encompasses, and the resources they have available. To determine what to look for in these factors, it is important to know your values and learning style. 

Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=OXpO0FqAY9E here].

=== '''''Responding to Peer Review Commentary''''' ===
'''Presented by''' '''Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====
Publishing articles involves submitting scholarly manuscripts to peer-reviewed journals. A key component of publishing manuscripts involves receiving commentary about your work from peers in your field, and satisfactorily responding to such commentary. Yet, researchers rarely receive formal training on responding to peer review commentary. The first thing to know when submitting an article for peer review is that it can take several months to get feedback and the feedback among reviewers hardly converges. However, the good news is, typically if you receive a revise and resubmit, revise accordingly and resubmit the paper, the finished product will likely be of higher quality than when you first submitted for publication. When submitting a paper for publication, you should consider which journal you believe will give you a fair review and you should submit 2-4 reviewers in your cover letter that you think have the expertise to review your work. Once you get your decision from peer-review, you should wait a few days before working on revisions and/or responding. Next you should itemize the decision letter creating a to-do list of the revisions (this will become a template for the cover letter you send in response). If there are suggested revisions you cannot do, you need a solid reason as to why you are unable to and you should cite this if possible. If you need help, it is okay to reach out to the editor and/or colleagues for support and advice. If reviewers give conflicting advice, you should give a reason as to why you went with one reviewer's suggestion over the others. Lastly, you can sometimes get a reviewer who is particularly mean, in that case you should let the editor know about the review and if you consistently have a hard time with a reviewer, you can let the editor know in your cover letter that you wish for the person to not review your paper.

Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=cJCIRuzD5sY here].

=== '''''Strategies for Developing a Research Program''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====
Research isn’t all elegant study designs, accurate data collection, and sophisticated equations. Researchers must also communicate their ideas and findings with scholarly audiences, and do so effectively. These audiences are no different from those found at your local theater: They understand each paper you write or talk you deliver insofar as it tells a compelling story. Yet, your storytelling doesn’t stop with a single paper or talk. Scholarly records span years and multiple pieces of work. Successful researchers learn to synthesize their records to tell a larger story: a research program. Finding your &quot;burning question&quot; will help when developing a research program. This can then be used to build the theoretical framework to which you begin to answer your &quot;burning question&quot;. The goal of developing a research program should be similar to that of making a film, you want to elicit positive emotion in your audience and make them think. The first three papers you write in your lab should have a role in building your research program and connect back to your &quot;burning question&quot;. Each paper should propel your next paper in this &quot;trilogy&quot; and the third paper should bring you back to the beginning one. Tailored to the lives of early career researchers, these tools reveal keen insights into nailing the job talk that launches your career.

Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=HRRvMJiI7zs here].

=== '''''New Models of Collaboration and Dissemination''''' ===
'''Presented by Dr. Eric Youngstrom, Ph.D.'''

===== Description =====
Wikipedia and Wikiversity offer powerful tools for disseminating knowledge to diverse audiences, including scientists and other key stakeholders (e.g., parents and policy makers). These tools greatly increase in utility if scientists receive training on how to leverage these tools for disseminating knowledge. In this workshop, Dr. Eric Youngstrom provides attendees with the know-how for using Wikipedia and Wikiversity, with a focus on how these tools help advance the mission of the open science movement. He begins by outlining copyright for open source information including a description of creative commons (CC) licensing that Wiki platforms use. There are several ways to use a CC license one of which is more high tech and involves registering your work on the CC website and the other which involves you placing CC BY 4.0 NAME on your work. Next Dr. Youngstrom describes how to use OSF to promote open access science. Housing materials on OSF allows others to easily access the work you are doing without having to go through paywalls. Lastly, Dr. Youngstrom describes the non-profit, Helping Give Away Psychological Science (HGAPS) and the work that HGAPS is doing to promote open science. This includes utilizing free platforms such as OSF and Wiki platforms to disseminate information. He concludes with a discussion on the HGAPS Assessment Center and the free assessment resources housed there. 

Watch the YouTube recording of the video [https://www.youtube.com/watch?v=9hIrVr1RaOc here]. 

=== '''''Tools For Lab Building: Training Undergraduate Research Assistants''''' ===
'''Presented by Dr. Sarah Racz, Ph.D. and Dr. Yo Jackson, Ph.D.'''

===== Description =====
For many research teams, undergraduate research assistants (RA) form a core component of their personnel. A key challenge involves not only the varying motivations of these personnel and their ultimate career goals, but also their relative inexperience with research generally. Often, we found ourselves immersing these students in their first research experiences. In this workshop, Dr. Racz and Jackson discuss concrete strategies for providing standardized research training experiences for undergraduates, with a focus on developing personnel to assist in accurate data collection and creating a hospitable work environment for students, post-doctoral fellows, staff, and faculty. There are many benefits of having undergraduate research assistants in your lab such as showing you are invested in training students, building your lab, and bringing in new ideas to the lab. When recruiting undergrad RAs, your university may have systems in place to assign these students to your lab or you can have open recruitment usually by advertisement or word of mouth. You should also have an application for students to apply, screening criteria for applicants, and an interview process. Once RAs have been hired, you should set clear expectations from the beginning and they should be outlined in a contract that the RA signs. You should have documents outlining the lab processes that RAs can go through and you can also utilize advanced RAs to help train new ones. It is good to set a hours per week expectation for RAs and to have a policy for when they miss scheduled lab time. Give specific tasks to RAs and have regular meetings to discuss tasks and lab duties. It is good to provide a range of skills and experiences to RAs and to have a benchmark so they have something to work towards. You may want to require a specific commitment to the lab such as 2 semesters to ensure retention in the lab as well as reward good performance with more responsibilities. You should also have clear guidelines for publishing with undergrad RAs and these opportunities should be reserved for RAs who have been in the lab for a while and have shown good performance. 

Watch the YouTube Recording [https://www.youtube.com/watch?v=pN1Jpg3_5do here]. 

=== '''''Building and Maintaining Research Partnerships with Schools''''' ===
'''Presented by Dr. Tim Cavell, Ph.D. and Dr. Elizabeth Talbott, Ph.D.'''

===== Description =====
A key component of research embedded in primary and secondary schools involves building long-term partnerships with key stakeholders in the school system. These stakeholders include administrators, teachers, classroom aids, school staff, and parents. In this workshop, Drs. Cavell and Talbott provide concrete advice on how to build lasting partnerships with school systems in an effort to conduct research with meaningful impacts on these systems. Schools are important and useful settings to conduct research in since they serve all children from pre-k to 21 years old. When thinking about partnering with a school for research, it is important to consider the culture and history of the school, the current events in the district, the geography of the schools you would like to partner with, and the district's needs. Before trying to partner with a school, you should familiarize yourself with the relationship the school may already have with researchers and whether your project would be better served in another community such as schools that primarily serve underrepresented students. You should also determine who has the authority to greenlight or stop your project and should contact that individual. Sometimes it is helpful to get a referral from someone who has a good relationship with that school or district. It is usually best to start with a phone call when trying to make contact. You should have an initial ask prepared and be ready to play the long game. It can be helpful to have the school identify a point of contact for you so you know who to communicate with. When thinking about conducting research in schools you should consider some key aspects of what will be required of schools. Your research should try to require little if any work from school staff, have minimal disruption to instructional time, be valuable to the teachers and staff, require minimal space in the school for your research team to conduct their work, among others. Obtaining consent is a very important aspect to conducting research in schools and you should think about how to best obtain it. Think about how you want to collect the data in schools considering both personnel needed and the measures you will use. Lastly, prepare your graduate students to conduct research in schools using your knowledge from working with schools.

Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=7LxkQ6X0aqQ here]. 

=== '''''Getting Your First Grant''''' ===
'''Presented by Dr. Deborah Drabick, Ph.D. and Dr. Katie Ehrlich, Ph.D.'''

===== Description =====
Submitting your first grant as a Ph.D. can appear on the surface to be a daunting task, with many expectations, requirements, and complicated forms. In this workshop, Drs. Drabick and Ehrlich leverage years of experience with extramural funding to explain the grant submission process, and provide attendees with concrete tools for submitting successful applications via multiple post-Ph.D. mechanisms, including project grants and K Series applications. First, they cover why it is valuable to apply for a training grant. These reasons include gaining opportunities that may not otherwise be available, getting additional mentorship, have protected research time, and career development. There are several types of funding grants to choose from including NIH T series, K series, F series, and R series, the National Science Foundation training grant, and grants from private agencies. Depending on which grant type you choose, there are different requirements you need to consider. When telling your story in a grant application, you should build off of your graduate school focus. You should also discuss what is missing/unresolved in the field, discuss how you will learn from this research, and how the goal of the grant is relevant for the field. Consider the big trends in the field and how your work could fill the gaps. In order to make your application successful, you will need to show some preliminary data usually in the form of a pilot project. You will typically need a &quot;track record&quot; i.e., few publications in your area of interest in order for your application to be successful. Next, consider whether or not your need to bring in more senior individuals to the team and what other personnel you will need for the grant. It is important to be clear up front in what you are asking your team members to do. The next step is to formulate the budget. Think about staffing costs, including hiring full time personnel, graduate students, and Co-PIs. Once your application is under review, there are typically 3 reviewers who will rate your application in 5 areas on a scale from 1 to 9. These scores are used to create an overall impact score which are then used to determine who to fund. It is ideal to have multiple grants under review at a time. Once you have written one grant, it is easier to use that as a template and modify for the requirements of other funding agencies. 

Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=KcYjR9IznGo here]. 

=== '''''Demystifying Academic Job Interviewing''''' ===
'''Presented by Dr. Kathryn Humphreys and Dr. Jessica Schleider'''

===== Description =====
The academic job interview factors prominently into faculty hiring decisions. It represents a public sample of your program of research and your style of teaching, as well as your critical thinking, responsiveness to feedback, and a whole range of non-specific variables, like your &quot;accessibility,&quot; &quot;collegiality,&quot; or &quot;likeability.&quot; Typically the academic job market &quot;opens up&quot; in June and July with applications being due September-December. The next stages involve screening interviews and then on-campus interviews with job offers, negotiations, and decisions happening December through April. In your application package, you will usually have a research statement, teaching statement, diversity statement, CV, cover letter, and references. To prepare for a interviews and job talks, you should attend as many of these as possible during graduate school, you should also research the institution and department in which you are applying to, and practice responses to questions they may ask. The screening interview is typically brief and standardized across candidates. When in interviews, remember to keep your answers concise as time is usually short. Be prepared to answer questions about your research, don't be afraid to ask clarifying questions, and remember to show enthusiasm for the job. Always send a thank you after interviews. If invited fro a job talk, make sure to practice with people in and outside of your field. Familiarize yourself with the faculty you will be meeting and their research. Make sure to ask questions that show you are interested, but be careful not to come across as arrogant. Think about how your work might connect with other faculty in the department. Be prepared to answer personal questions and know what you are/aren't willing to disclose and how you will handle those types of questions.  

Watch the YouTube Recording of the Workshop [https://www.youtube.com/watch?v=8iZLmNOOIDQ here]. 

=== '''''Preparing a Training Grant: Overview''''' ===
'''Presented by Dr. Stephen Becker, Ph.D. and Dr. Meghan Miller, Ph.D.'''

===== Description =====
Submitting a training grant involves considering multiple factors that focus on not only a proposed study but also a concrete plan for developing the skills needed to execute this study. By construction, these applications carry many expectations, requirements, and complicated forms. In this workshop, Dr. Becker and Miller leverage their years of experience with extramural funding to clarify the process of submitting a training grant, and provide attendees with concrete tools for submitting successful training grant applications. There are several reasons to apply to training grants including grant-writing experience, practice applying to a federal funding agency, opportunities gained, additional mentorship and/or consultation, protected time for independent research, and career advancement. The NIH has F and K grants available for training. The National Science Foundation and Institute for Education Services among others also offer training grant opportunities. Make sure to check the requirements for each grant. The presenters go into detail about the anatomy of each grant type in this presentation. When telling your story make sure to build on your graduate school focus, extend to a new area or skill set, discuss the gaps and how your research will allow you to learn, and how the goal/topic is relevant and important to the field. 3 reviewers rate your application on a scale from 1-9 and those scores are used to create an average impact score which is then used for the final decision. Each grant has specific areas in which they rate your application, so make sure to know those areas and address each one in your application. It is best to start on your application early and build a timeline for getting it done. Getting an example of a funded grant can also be helpful. 

Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=7qbPoHiAmFk here]. 

== '''Ceremony for the ''Future Directions Launch Award''''' ==
=== Jessie Greenlee ===
* Award Winner in the area of Autism Spectrum Disorder
* Postdoctoral Fellow at the University of Wisonsin-Madson

==== About the award recipient ====
Jessie is a recipient of the 2020 Future Directions Launch Award in Autism. Jessie completed a Postdoctoral Fellow at the Waisman Center at the University of Wisconsin-Madison. She received her Ph.D. in Developmental Psychology from Virginia Commonwealth University in 2019. Jessie’s research investigates the mechanisms through which individual and contextual factors are associated with mental and physical health disparities in vulnerable populations. She is particularly interested in understanding how families promote healthy social and emotional development in individuals with special healthcare needs. Jessie is currently working on several projects aimed at understanding how different sub-system family processes (e.g., marital conflict, co-parenting, parent-child relationship quality) impact outcomes for children and youth with autism spectrum disorder. Jessie is currently hold a position as an Assistant Professor of Psychology at Lafayette College in PA. Learn more about Jessie's work here: [https://www.researchgate.net/profile/Jessica&amp;#x20;Greenlee www.researchgate.net/profile/Jessica_Greenlee]

Watch the YouTube recording of the remarks [https://www.youtube.com/watch?v=BRggTSAhIW0 here].

=== Tyler McFayden ===
* Award Winner in the area of Autism Spectrum Disorder and Language Development
* Received Ph.D. at Virginia Tech

==== About the award recipient ====
Tyler received the 2020 Future Directions Launch Award in Autism. She is a current predoctoral clinical intern at the University of North Carolina- Chapel Hill’s (UNC-CH) Carolina Institute for Developmental Disabilities. Tyler plans to remain at UNC-CH for her postdoctoral training to participate in the NIMH-funded T32 Postdoctoral Research Training Program focusing on intellectual and developmental disabilities. Tyler attended Virginia Tech’s Clinical and Developmental Psychological doctoral program under the mentorship of Dr. Thomas Ollendick, where she worked in a typically developing infant lab, studying early language development, and an autism lab investigating endophenotypes of social communication. Tyler is particularly interested in how language develops in early infancy and in groups without spoken language (e.g., Deaf/Hard of Hearing and minimally-verbal/partially speaking autistic youth) to best inform social communication interventions. Learn more about Tyler’s work here: [https://www.researchgate.net/profile/Tyler&amp;#x20;Mcfayden www.researchgate.net/profile/Tyler_Mcfayden] 

Watch the YouTube recording of the remarks [https://www.youtube.com/watch?v=L_ebFjTEJXk here].

== References ==</text>
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      <text bytes="29423" xml:space="preserve">== '''Addresses''' ==

=== '''''Future Directions Address 1: Father Inclusion, Engagement, Retention, and Positive Outcomes in Child and Adolescent Research''''' ===
'''Presented by Greg Fabiano, Ph.D.'''

====Description====
In this address, Dr. Greg Fabiano outlines future directions in the next generation of father-focused studies in the child and adolescent psychology literature, with an emphasis on improving the study of the parameters of inclusion, engagement, retention, and measurement of outcomes. Research shows that parents may play a key role in child resilience. It is important to focus on fathers in research because father involvement lags behind mother involvement and father involvement in childhood has been a long-standing national priority. Research has found father involvement to contribute to language development and early literacy skills as well as the development of appropriate socialization skills. However, few studies have looked at father involvement in child treatment. The format most research studies are in, may actually discourage fathers to participate. The COACHES program works to address this gap in research by focusing on father involvement in the intervention. Results show that fathers are satisfied with the outcomes of the program (100%) and satisfied with the treatment process (80%). This research shows initial promise for interventions that include fathers. &lt;ref&gt;{{Cite journal|last=Fabiano|first=Gregory A.|last2=Caserta|first2=Abigail|date=2018-09-03|title=Future Directions in Father Inclusion, Engagement, Retention, and Positive Outcomes in Child and Adolescent Research|url=https://doi.org/10.1080/15374416.2018.1485106|journal=Journal of Clinical Child &amp; Adolescent Psychology|volume=47|issue=5|pages=847–862|doi=10.1080/15374416.2018.1485106|issn=1537-4416|pmid=30110185}}&lt;/ref&gt;

Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=EogVEgi3n80&amp;feature=youtu.be here]

=== '''''Future Directions Address 2: Research and Intervention with Youths in Poverty''''' ===
'''Presented by Martha Wadsworth, Ph.D.'''

====Description====
In this address, Dr. Martha Wadsworth integrates theory and empirical findings about understanding and fostering the process of resilience and adaptation in children and families who live in poverty. She starts off by defining what &quot;normal&quot; is and how it is culturally defined. The stress response system gathers information from the environment which it then uses to guide the development of the individual in a way that matches the environment. There are many ways in which our bodies develop and respond to the environment. Because development is extremely dependent on environment, children that grow up in dangerous and unpredictable environments, develop in a way suited to handle those environments and usually do not develop behaviors that are well suited to a more enriched environment. We develop abilities that match our environment rather than abilities that may be useful in other environments. Dr. Wadsworth proposes that focusing on skills that are strengths may be more beneficial than focusing on deficits. Cultural identity should also play a large role in interventions and can be used as a &quot;coping resource&quot; for youth in poverty. By building a strong identity, children can gain coping skills. Interventions for youth in poverty should begin by focusing on stress and coping. Next, the intervention can focus on cultural identity and finish with collective empowerment. Overall, a multifaceted approach is necessary to treatment youth in poverty. &lt;ref&gt;{{Cite journal|last=Wadsworth|first=Martha E.|last2=Ahlkvist|first2=Jarl A.|last3=McDonald|first3=Ashley|last4=Tilghman-Osborne|first4=Emile M.|date=2018-11-02|title=Future Directions in Research and Intervention with Youths in Poverty|url=https://doi.org/10.1080/15374416.2018.1485108|journal=Journal of Clinical Child &amp; Adolescent Psychology|volume=47|issue=6|pages=1023–1038|doi=10.1080/15374416.2018.1485108|issn=1537-4416|pmc=PMC6348127|pmid=30052089}}&lt;/ref&gt; 

Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=Mj3bgzgXI6g here]

=== '''''Future Directions Address 3: Examination of Brain Networks in Neurodevelopmental Disorders''''' ===
'''Presented by Lucina Uddin, Ph.D.'''

====Description====
In this address, Dr. Lucina Uddin discusses future directions for neuroscience researchers examining brain networks in neurodevelopmental disorders, highlighting gaps in the current literature. Cognitive flexibility is important in individuals with autism spectrum disorder (ASD). Research has shown atypical patterns of brain dynamics often seen in those with neurodevelopmental disorders which is characterized by cognitive inflexibility. Dr. Uddin and her lab have conducted several studies looking at brain networks in those with neurodevelopmental disorders. They concluded that ASD and ADHD must be considered in the context of phenotypic heterogeneity. There are also associated difficulties in parsing heterogeneity in neurodevelopmental disorders. She suggests that researchers use a novel data-driven approach to inform diagnostic nosology. There are new studies looking at whether bilingualism can help with executive functioning in individuals with ASD. &lt;ref&gt;{{Cite journal|last=Uddin|first=Lucina Q.|last2=Karlsgodt|first2=Katherine H.|date=2018-05-04|title=Future Directions for Examination of Brain Networks in Neurodevelopmental Disorders|url=https://doi.org/10.1080/15374416.2018.1443461|journal=Journal of Clinical Child &amp; Adolescent Psychology|volume=47|issue=3|pages=483–497|doi=10.1080/15374416.2018.1443461|issn=1537-4416|pmc=PMC6842321|pmid=29634380}}&lt;/ref&gt;  

Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=CoSM8C2ucRE here]

=== '''''Future Directions Address 4: the Treatment of Youth Mental Health''''' ===
'''Presented by Bruce Chorpita, Ph.D.'''

====Description====
In this address, Dr. Bruce Chorpita discusses mental health care systems and presents ideas and examples of methods that may preserve the strengths of the two major paradigms in children’s mental health, evidence-based treatments, and individualized care models, but that also have the potential to extend their applicability and impact. While over 20% of the world experiences mental health concerns at some point, most don't receive services and those who do are typically not receiving evidence-based services. There is also an insufficient workforce to meet society's needs. The future of research needs to make sure that the evidence-base treatments match diverse populations, are continuing in development, and are dynamic. We need to &quot;chunk and filter&quot; treatments per individual who is receiving them. When modular design, feedback, and expert process managing were combined for treatment of youth, they performed better than evidence-based treatments and usual care. The MATCH program performed well in multiple research clinical trials. In the future, treatment engagement, handling a patient with a crisis, and ethnic and cultural diversity need to be addressed in treatments. Chunking, filtering, and discovering are the best ways to empower individuals to act on best ideas first and self correct. &lt;ref&gt;{{Cite journal|last=Chorpita|first=Bruce F.|last2=Daleiden|first2=Eric L.|date=2014-03-01|title=Structuring the Collaboration of Science and Service in Pursuit of a Shared Vision|url=https://doi.org/10.1080/15374416.2013.828297|journal=Journal of Clinical Child &amp; Adolescent Psychology|volume=43|issue=2|pages=323–338|doi=10.1080/15374416.2013.828297|issn=1537-4416|pmid=23981145}}&lt;/ref&gt;

Watch the YouTube video recording of the address enter link [https://www.youtube.com/watch?v=bnKkc2iyeLo here].

== '''Workshops''' ==

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====
Applying to doctoral programs marks an important life milestone for you and other undergraduate majors and post-baccalaureate trainees. Importantly, some of the considerations for choosing where to receive undergraduate training (e.g., faculty-to-student ratio, quality of institution) take a &quot;back seat&quot; to the key factor in doctoral training that most impacts your career: Identifying the person who will serve as your mentor. Undergraduate programs rarely offer formal instruction in choosing doctoral mentors, and some of the factors you might consider could vary from year-to-year and by mentor. Overall, when selecting a mentor you should consider who fits your needs and learning style. Because the quality of your doctoral training is mostly impacted by your mentor, you should think about who is going to fit best with your goals. When searching for mentors to apply to, you should first identify researchers who study topics similar to your interests. You can do this by reading research articles and looking for ones that align with your &quot;burning question&quot;. Other factors you should consider when selecting mentors include: the size of their lab, the time they devote to their mentees, the area in which their research encompasses, and the resources they have available. To determine what to look for in these factors, it is important to know your values and learning style. 

Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=OXpO0FqAY9E here].

=== '''''Responding to Peer Review Commentary''''' ===
'''Presented by''' '''Andres De Los Reyes, Ph.D.'''

===== Description =====
Publishing articles involves submitting scholarly manuscripts to peer-reviewed journals. A key component of publishing manuscripts involves receiving commentary about your work from peers in your field, and satisfactorily responding to such commentary. Yet, researchers rarely receive formal training on responding to peer review commentary. The first thing to know when submitting an article for peer review is that it can take several months to get feedback and the feedback among reviewers hardly converges. However, the good news is, typically if you receive a revise and resubmit, revise accordingly and resubmit the paper, the finished product will likely be of higher quality than when you first submitted for publication. When submitting a paper for publication, you should consider which journal you believe will give you a fair review and you should submit 2-4 reviewers in your cover letter that you think have the expertise to review your work. Once you get your decision from peer-review, you should wait a few days before working on revisions and/or responding. Next you should itemize the decision letter creating a to-do list of the revisions (this will become a template for the cover letter you send in response). If there are suggested revisions you cannot do, you need a solid reason as to why you are unable to and you should cite this if possible. If you need help, it is okay to reach out to the editor and/or colleagues for support and advice. If reviewers give conflicting advice, you should give a reason as to why you went with one reviewer's suggestion over the others. Lastly, you can sometimes get a reviewer who is particularly mean, in that case you should let the editor know about the review and if you consistently have a hard time with a reviewer, you can let the editor know in your cover letter that you wish for the person to not review your paper.

Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=cJCIRuzD5sY here].

=== '''''Strategies for Developing a Research Program''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====
Research isn’t all elegant study designs, accurate data collection, and sophisticated equations. Researchers must also communicate their ideas and findings with scholarly audiences, and do so effectively. These audiences are no different from those found at your local theater: They understand each paper you write or talk you deliver insofar as it tells a compelling story. Yet, your storytelling doesn’t stop with a single paper or talk. Scholarly records span years and multiple pieces of work. Successful researchers learn to synthesize their records to tell a larger story: a research program. Finding your &quot;burning question&quot; will help when developing a research program. This can then be used to build the theoretical framework to which you begin to answer your &quot;burning question&quot;. The goal of developing a research program should be similar to that of making a film, you want to elicit positive emotion in your audience and make them think. The first three papers you write in your lab should have a role in building your research program and connect back to your &quot;burning question&quot;. Each paper should propel your next paper in this &quot;trilogy&quot; and the third paper should bring you back to the beginning one. Tailored to the lives of early career researchers, these tools reveal keen insights into nailing the job talk that launches your career.

Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=HRRvMJiI7zs here].

=== '''''New Models of Collaboration and Dissemination''''' ===
'''Presented by Eric Youngstrom, Ph.D.'''

===== Description =====
Wikipedia and Wikiversity offer powerful tools for disseminating knowledge to diverse audiences, including scientists and other key stakeholders (e.g., parents and policy makers). These tools greatly increase in utility if scientists receive training on how to leverage these tools for disseminating knowledge. In this workshop, Dr. Eric Youngstrom provides attendees with the know-how for using Wikipedia and Wikiversity, with a focus on how these tools help advance the mission of the open science movement. He begins by outlining copyright for open source information including a description of creative commons (CC) licensing that Wiki platforms use. There are several ways to use a CC license one of which is more high tech and involves registering your work on the CC website and the other which involves you placing CC BY 4.0 NAME on your work. Next Dr. Youngstrom describes how to use OSF to promote open access science. Housing materials on OSF allows others to easily access the work you are doing without having to go through paywalls. Lastly, Dr. Youngstrom describes the non-profit, Helping Give Away Psychological Science (HGAPS) and the work that HGAPS is doing to promote open science. This includes utilizing free platforms such as OSF and Wiki platforms to disseminate information. He concludes with a discussion on the HGAPS Assessment Center and the free assessment resources housed there. 

Watch the YouTube recording of the video [https://www.youtube.com/watch?v=9hIrVr1RaOc here]. 

=== '''''Tools For Lab Building: Training Undergraduate Research Assistants''''' ===
'''Presented by Sarah Racz, Ph.D. and Yo Jackson, Ph.D.'''

===== Description =====
For many research teams, undergraduate research assistants (RA) form a core component of their personnel. A key challenge involves not only the varying motivations of these personnel and their ultimate career goals, but also their relative inexperience with research generally. Often, we found ourselves immersing these students in their first research experiences. In this workshop, Dr. Racz and Jackson discuss concrete strategies for providing standardized research training experiences for undergraduates, with a focus on developing personnel to assist in accurate data collection and creating a hospitable work environment for students, post-doctoral fellows, staff, and faculty. There are many benefits of having undergraduate research assistants in your lab such as showing you are invested in training students, building your lab, and bringing in new ideas to the lab. When recruiting undergrad RAs, your university may have systems in place to assign these students to your lab or you can have open recruitment usually by advertisement or word of mouth. You should also have an application for students to apply, screening criteria for applicants, and an interview process. Once RAs have been hired, you should set clear expectations from the beginning and they should be outlined in a contract that the RA signs. You should have documents outlining the lab processes that RAs can go through and you can also utilize advanced RAs to help train new ones. It is good to set a hours per week expectation for RAs and to have a policy for when they miss scheduled lab time. Give specific tasks to RAs and have regular meetings to discuss tasks and lab duties. It is good to provide a range of skills and experiences to RAs and to have a benchmark so they have something to work towards. You may want to require a specific commitment to the lab such as 2 semesters to ensure retention in the lab as well as reward good performance with more responsibilities. You should also have clear guidelines for publishing with undergrad RAs and these opportunities should be reserved for RAs who have been in the lab for a while and have shown good performance. 

Watch the YouTube Recording [https://www.youtube.com/watch?v=pN1Jpg3_5do here]. 

=== '''''Building and Maintaining Research Partnerships with Schools''''' ===
'''Presented by Tim Cavell, Ph.D. and Elizabeth Talbott, Ph.D.'''

===== Description =====
A key component of research embedded in primary and secondary schools involves building long-term partnerships with key stakeholders in the school system. These stakeholders include administrators, teachers, classroom aids, school staff, and parents. In this workshop, Drs. Cavell and Talbott provide concrete advice on how to build lasting partnerships with school systems in an effort to conduct research with meaningful impacts on these systems. Schools are important and useful settings to conduct research in since they serve all children from pre-k to 21 years old. When thinking about partnering with a school for research, it is important to consider the culture and history of the school, the current events in the district, the geography of the schools you would like to partner with, and the district's needs. Before trying to partner with a school, you should familiarize yourself with the relationship the school may already have with researchers and whether your project would be better served in another community such as schools that primarily serve underrepresented students. You should also determine who has the authority to greenlight or stop your project and should contact that individual. Sometimes it is helpful to get a referral from someone who has a good relationship with that school or district. It is usually best to start with a phone call when trying to make contact. You should have an initial ask prepared and be ready to play the long game. It can be helpful to have the school identify a point of contact for you so you know who to communicate with. When thinking about conducting research in schools you should consider some key aspects of what will be required of schools. Your research should try to require little if any work from school staff, have minimal disruption to instructional time, be valuable to the teachers and staff, require minimal space in the school for your research team to conduct their work, among others. Obtaining consent is a very important aspect to conducting research in schools and you should think about how to best obtain it. Think about how you want to collect the data in schools considering both personnel needed and the measures you will use. Lastly, prepare your graduate students to conduct research in schools using your knowledge from working with schools.

Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=7LxkQ6X0aqQ here]. 

=== '''''Getting Your First Grant''''' ===
'''Presented by Deborah Drabick, Ph.D. and Katie Ehrlich, Ph.D.'''

===== Description =====
Submitting your first grant as a Ph.D. can appear on the surface to be a daunting task, with many expectations, requirements, and complicated forms. In this workshop, Drs. Drabick and Ehrlich leverage years of experience with extramural funding to explain the grant submission process, and provide attendees with concrete tools for submitting successful applications via multiple post-Ph.D. mechanisms, including project grants and K Series applications. First, they cover why it is valuable to apply for a training grant. These reasons include gaining opportunities that may not otherwise be available, getting additional mentorship, have protected research time, and career development. There are several types of funding grants to choose from including NIH T series, K series, F series, and R series, the National Science Foundation training grant, and grants from private agencies. Depending on which grant type you choose, there are different requirements you need to consider. When telling your story in a grant application, you should build off of your graduate school focus. You should also discuss what is missing/unresolved in the field, discuss how you will learn from this research, and how the goal of the grant is relevant for the field. Consider the big trends in the field and how your work could fill the gaps. In order to make your application successful, you will need to show some preliminary data usually in the form of a pilot project. You will typically need a &quot;track record&quot; i.e., few publications in your area of interest in order for your application to be successful. Next, consider whether or not your need to bring in more senior individuals to the team and what other personnel you will need for the grant. It is important to be clear up front in what you are asking your team members to do. The next step is to formulate the budget. Think about staffing costs, including hiring full time personnel, graduate students, and Co-PIs. Once your application is under review, there are typically 3 reviewers who will rate your application in 5 areas on a scale from 1 to 9. These scores are used to create an overall impact score which are then used to determine who to fund. It is ideal to have multiple grants under review at a time. Once you have written one grant, it is easier to use that as a template and modify for the requirements of other funding agencies. 

Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=KcYjR9IznGo here]. 

=== '''''Demystifying Academic Job Interviewing''''' ===
'''Presented by Kathryn Humphreys and Jessica Schleider'''

===== Description =====
The academic job interview factors prominently into faculty hiring decisions. It represents a public sample of your program of research and your style of teaching, as well as your critical thinking, responsiveness to feedback, and a whole range of non-specific variables, like your &quot;accessibility,&quot; &quot;collegiality,&quot; or &quot;likeability.&quot; Typically the academic job market &quot;opens up&quot; in June and July with applications being due September-December. The next stages involve screening interviews and then on-campus interviews with job offers, negotiations, and decisions happening December through April. In your application package, you will usually have a research statement, teaching statement, diversity statement, CV, cover letter, and references. To prepare for a interviews and job talks, you should attend as many of these as possible during graduate school, you should also research the institution and department in which you are applying to, and practice responses to questions they may ask. The screening interview is typically brief and standardized across candidates. When in interviews, remember to keep your answers concise as time is usually short. Be prepared to answer questions about your research, don't be afraid to ask clarifying questions, and remember to show enthusiasm for the job. Always send a thank you after interviews. If invited fro a job talk, make sure to practice with people in and outside of your field. Familiarize yourself with the faculty you will be meeting and their research. Make sure to ask questions that show you are interested, but be careful not to come across as arrogant. Think about how your work might connect with other faculty in the department. Be prepared to answer personal questions and know what you are/aren't willing to disclose and how you will handle those types of questions.  

Watch the YouTube Recording of the Workshop [https://www.youtube.com/watch?v=8iZLmNOOIDQ here]. 

=== '''''Preparing a Training Grant: Overview''''' ===
'''Presented by Stephen Becker, Ph.D. and Meghan Miller, Ph.D.'''

===== Description =====
Submitting a training grant involves considering multiple factors that focus on not only a proposed study but also a concrete plan for developing the skills needed to execute this study. By construction, these applications carry many expectations, requirements, and complicated forms. In this workshop, Dr. Becker and Miller leverage their years of experience with extramural funding to clarify the process of submitting a training grant, and provide attendees with concrete tools for submitting successful training grant applications. There are several reasons to apply to training grants including grant-writing experience, practice applying to a federal funding agency, opportunities gained, additional mentorship and/or consultation, protected time for independent research, and career advancement. The NIH has F and K grants available for training. The National Science Foundation and Institute for Education Services among others also offer training grant opportunities. Make sure to check the requirements for each grant. The presenters go into detail about the anatomy of each grant type in this presentation. When telling your story make sure to build on your graduate school focus, extend to a new area or skill set, discuss the gaps and how your research will allow you to learn, and how the goal/topic is relevant and important to the field. 3 reviewers rate your application on a scale from 1-9 and those scores are used to create an average impact score which is then used for the final decision. Each grant has specific areas in which they rate your application, so make sure to know those areas and address each one in your application. It is best to start on your application early and build a timeline for getting it done. Getting an example of a funded grant can also be helpful. 

Watch the YouTube recording of the workshop [https://www.youtube.com/watch?v=7qbPoHiAmFk here]. 

== '''Ceremony for the ''Future Directions Launch Award''''' ==
=== Jessie Greenlee ===
* Award Winner in the area of Autism Spectrum Disorder
* Postdoctoral Fellow at the University of Wisonsin-Madson

==== About the award recipient ====
Jessie is a recipient of the 2020 Future Directions Launch Award in Autism. Jessie completed a Postdoctoral Fellow at the Waisman Center at the University of Wisconsin-Madison. She received her Ph.D. in Developmental Psychology from Virginia Commonwealth University in 2019. Jessie’s research investigates the mechanisms through which individual and contextual factors are associated with mental and physical health disparities in vulnerable populations. She is particularly interested in understanding how families promote healthy social and emotional development in individuals with special healthcare needs. Jessie is currently working on several projects aimed at understanding how different sub-system family processes (e.g., marital conflict, co-parenting, parent-child relationship quality) impact outcomes for children and youth with autism spectrum disorder. Jessie is currently hold a position as an Assistant Professor of Psychology at Lafayette College in PA. Learn more about Jessie's work here: [https://www.researchgate.net/profile/Jessica&amp;#x20;Greenlee www.researchgate.net/profile/Jessica_Greenlee]

Watch the YouTube recording of the remarks [https://www.youtube.com/watch?v=BRggTSAhIW0 here].

=== Tyler McFayden ===
* Award Winner in the area of Autism Spectrum Disorder and Language Development
* Received Ph.D. at Virginia Tech

==== About the award recipient ====
Tyler received the 2020 Future Directions Launch Award in Autism. She is a current predoctoral clinical intern at the University of North Carolina- Chapel Hill’s (UNC-CH) Carolina Institute for Developmental Disabilities. Tyler plans to remain at UNC-CH for her postdoctoral training to participate in the NIMH-funded T32 Postdoctoral Research Training Program focusing on intellectual and developmental disabilities. Tyler attended Virginia Tech’s Clinical and Developmental Psychological doctoral program under the mentorship of Dr. Thomas Ollendick, where she worked in a typically developing infant lab, studying early language development, and an autism lab investigating endophenotypes of social communication. Tyler is particularly interested in how language develops in early infancy and in groups without spoken language (e.g., Deaf/Hard of Hearing and minimally-verbal/partially speaking autistic youth) to best inform social communication interventions. Learn more about Tyler’s work here: [https://www.researchgate.net/profile/Tyler&amp;#x20;Mcfayden www.researchgate.net/profile/Tyler_Mcfayden] 

Watch the YouTube recording of the remarks [https://www.youtube.com/watch?v=L_ebFjTEJXk here].

== References ==</text>
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    <title>Talk:WikiJournal Preprints/Melioidosis</title>
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== Plagiarism check ==

{{Pass}} [https://tools.wmflabs.org/copyvios/?lang=en&amp;project=wikiversity&amp;action=search&amp;turnitin=1&amp;title=WikiJournal_Preprints/Melioidosis Report from WMF copyvios tool]: Overlaps were found with other publicationps, but similarities were trivial or common phrases (e.g. &quot;contributes to an increased risk of contracting melioidosis&quot; or &quot;direct immunofluorescent antibody test (IFAT) and latex agglutination&quot;). [[User:Evolution and evolvability|T.Shafee(Evo&amp;#65120;Evo)]]&lt;sup&gt;[[User talk:Evolution and evolvability|talk]]&lt;/sup&gt; 10:03, 17 May 2020 (UTC)

== Editorial comment: Problems with illustrations ==
{{editorial comment
|name=Candace Makeda Moore
|date = 07:04, 18 May 2020 (UTC) 
|text = The illustration &quot;[[:File:Signs of melioidosis.svg|Signs of Meliodosis]]&quot; has several problems. These include:different fonts for different labels, the label for neck abscess is spelled with an extra S in subscript, the picture has been made as semi-transparent, and therefore will show a checkered background at certain sizes. I would also note the muscles seem to crowd the image, and the pathologies listed are not actually illustrated in any way other than showing where in the body they would be. 
The radiological images are not incorrect, however they do now show something pathognomonic or very typical for the disease. We should ask the authors if these were from actual patients with the disease or from cases of other disease and just used for illustrative purposes.

{{response|
* The picture of &quot;Signs of melioidosis&quot; removed. All the radiological images come from patients with melioidosis in case reports. I have clarified in the image captions that the radiological findings are from patients with melioidosis. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 07:41, 1 June 2020 (UTC)
}}
}}

== Peer review 1 ==

{{review
|reviewer = Chirag Dhar
|affiliation = University of California, San Diego
|credentials = 
|link = https://profiles.ucsd.edu/chirag.dhar
|version = https://en.wikiversity.org/w/index.php?title=WikiJournal_Preprints/Melioidosis&amp;oldid=2167776
|date = 24 July 2020
|text =
Based on my expertise, I have reviewed the section on the diagnosis of melioidosis:

Firstly, thank you for your comprehensive review of the topic. Few minor points:

#Detection from CSF fluid is generally hard. Maybe this can be mentioned.
#It would be good if the low sensitivity of API 20NE/20E is mentioned with relevant references.
#The readability might improve by breaking the section into sub-sections such as: culture, microscopy, imaging, etc.
#Are there reports about the reactivation of latent melioidosis in patients receiving immune checkpoint inhibitors? This is something we see in tuberculosis as I covered in a recent correspondence with the Indian Journal of Cancer (https://escholarship.org/content/qt2cj454g2/qt2cj454g2.pdf).

{{response|text=
#Added the reference &quot;Gassiep I, Armstrong M, Norton R (11 March 2020) that only 29% of the CSF cultures are positive.
#However from Gassiep 2020 reference, it cited a 2007 study which found that API 20NE or 20E biochemical kit is 99% sensitive in identifying B, pseudomallei and they are not sure why previous studies found significantly lower sensitivity than their study.
#Done
#Found two references [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5562302/ Persistent infection due to a small-colony variant of Burkholderia pseudomallei leads to PD-1 upregulation on circulating immune cells and mononuclear infiltration in viscera of experimental BALB/c mice] and [https://www.microbiologyresearch.org/content/journal/acmi/10.1099/acmi.0.000110?crawler=true Experimental exposure of Burkholderia pseudomallei crude culture filtrate upregulates PD-1 on T lymphocytes] which stated that ''B. pseudomallei'' are able to upregulate PD-1, which leads to poor immunity response against the bacteria. I found no articles regarding the effect of immune checkpoint inhibitors on the reactivation of latent melioidosis. However, according to the logic presented in the above two articles, immune checkpoint inhibitors may actually inhibit the action of ''B. pseudomallei''. 
[[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 14:58, 9 August 2020 (UTC)
}}
}}

==Peer review 2==
{{review
|reviewer = David Dance
|affiliation = MORU/LOMWRU/University of Oxford/London School of Hygiene and Tropical Medicine
|credentials = 
|link = https://www.tropicalmedicine.ox.ac.uk/team/david-dance
|version = https://en.wikiversity.org/w/index.php?title=WikiJournal_Preprints/Melioidosis&amp;oldid=2167776
|date = 26 August 2020
|pdf=Melioidosis_-_David_Dance.pdf
|text = Comments attached as annotated PDF

{{response|text=
Thanks for your comprehensive review, sir! I have learned a lot from your comments. It is a humbling experience to have you as my peer reviewer because you have extensive experiences and expertise in this field. Since this article is intended for Wikipedia integration, it should be written so that a knowledgable generalist can understand them. Besides,  &quot;medical statements must be supported by secondary sources, ideally meta-analyses, clinical guidelines, standard textbooks or position statements from national or international expert bodies (not only primary literature)&quot; as mentioned in this publishing guideline https://en.wikiversity.org/wiki/WikiJournal_of_Medicine/Publishing. This is to ensure that verified and replicable information can reach more people rather than confined to professional readers. All the replies are attached in the annotated pdf. Thank you and have a nice day.
|pdf=Melioidosis - David Dance (replied comments).pdf
}}
}}
==Editorial note==
Reviewer #2 will provide some up to date references for the authors to add to the pre-print [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 10:13, 21 June 2022 (UTC)
:Dear [[User:Rwatson1955|Rwatson1955]], all the concerns has been addressed. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 01:23, 16 July 2022 (UTC)

==Additional comments from Reviewer 2 - David Dance - 24 June 2022==
This version of the review is improved, but I do have a few suggestions for further improvement and update.

1.	It would be worth including mention of the recent cases in the USA linked to an imported aromatherapy spray – see Gee JE, Bower WA, Kunkel A, Petras J, Gettings J, Bye M, et al. Multistate Outbreak of Melioidosis Associated with Imported Aromatherapy Spray. The New England journal of medicine. 2022;386(9):861-8. - Done. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 05:41, 26 June 2022 (UTC)

2.	In relation to prevention, it would also be worth citing Suntornsut P, Teparrukkul P, Wongsuvan G, Chaowagul W, Michie S, Day NPJ, et al. Effectiveness of a multifaceted prevention programme for melioidosis in diabetics (PREMEL): A stepped-wedge cluster-randomised controlled trial. PLoS neglected tropical diseases. 2021;15(6):e0009060. - Done. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 14:19, 26 June 2022 (UTC)

3.	The use of cotrimoxazole prophylaxis in dialysis patients has not been universally  successful – see Chau KWT, Smith S, Kang K, Dheda S, Hanson J. Antibiotic Prophylaxis for Melioidosis in Patients Receiving Hemodialysis in the Tropics? One Size Does Not Fit All. The American journal of tropical medicine and hygiene. 2018;99(3):597-600. - Done. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 05:43, 27 June 2022 (UTC)

4.	The sentences ‘Most people exposed to B. pseudomallei experience no symptoms’ and ‘Approximately 85% of the people experience symptomatic infection’ are mutually contradictory. - Removed the later sentence. Added new citation supporting the earlier sentence. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 05:41, 26 June 2022 (UTC)
 
5.	It would be worth citing Currie BJ, Mayo M, Ward LM, Kaestli M, Meumann EM, Webb JR, et al. The Darwin Prospective Melioidosis Study: a 30-year prospective, observational investigation. The Lancet Infectious diseases. 2021;21(12):1737-46, which is the largest published series of melioidosis cases. - Done. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 08:46, 28 June 2022 (UTC)

6.	In relation to CNS infection, the role of bimABm warrants mention – see Gora H, Hasan T, Smith S, Wilson I, Mayo M, Woerle C, et al. Melioidosis of the central nervous system; impact of the bimABm allele on patient presentation and outcome. Clinical infectious diseases : an official publication of the Infectious Diseases Society of America. 2022. - Done. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 08:24, 29 June 2022 (UTC)

7.	It is 16SrDNA sequencing not ‘16S rDNA probes’ that may be used for identification. - Done - [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 07:58, 30 June 2022 (UTC)

8.	It would be worth mentioning both antigen detection and NAATs on clinical samples as potential diagnostic methods. There are numerous references that could be used for this. - Done.[[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 07:58, 30 June 2022 (UTC)

9.	If co-amoxiclav is used to treat melioidosis, very high doses must be used and treatment failure is more frequent than with ceftazidime (see Cheng AC, Chierakul W, Chaowagul W, Chetchotisakd P, Limmathurotsakul D, Dance DAB, et al. Consensus guidelines for dosing of amoxicillin-clavulanate in melioidosis. American Journal of Tropical Medicine and Hygiene. 2008;78(2):208-9 and Suputtamongkol Y, Rajchanuwong A, Chaowagul W, Dance DAB, Smith MD, Wuthiekanun V, et al. Ceftazidime vs. amoxicillin/clavulanate in the treatment of severe melioidosis. Clinical infectious diseases : an official publication of the Infectious Diseases Society of America. 1994;19(5):846-53). - Done. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 06:24, 1 July 2022 (UTC)

10.	The paper Anunnatsiri S, Chaowagul W, Teparrukkul P, Chetchotisakd P, Tanwisaid K, Khemla S, et al. A Comparison Between 12 Versus 20 Weeks of Trimethoprim-sulfamethoxazole as Oral Eradication Treatment for Melioidosis: An Open-label, Pragmatic, Multicenter, Non-inferiority, Randomized Controlled Trial. Clinical infectious diseases : an official publication of the Infectious Diseases Society of America. 2021;73(11):e3627-e33 also warrants mention. - Done. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 07:35, 3 July 2022 (UTC)

11.	It would be better to use the 2020 Darwin guidelines than the 2014 version throughout – see Sullivan RP, Marshall CS, Anstey NM, Ward L, Currie BJ. 2020 Review and revision of the 2015 Darwin melioidosis treatment guideline; paradigm drift not shift. PLoS neglected tropical diseases. 2020;14(9):e0008659. - Done. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 07:35, 3 July 2022 (UTC)

12.	Resistance to cotrimoxazole is also difficult to detect reliably – see for example Saiprom N, Amornchai P, Wuthiekanun V, Day NP, Limmathurotsakul D, Peacock SJ, et al. Trimethoprim/sulfamethoxazole resistance in clinical isolates of Burkholderia pseudomallei from Thailand. International journal of antimicrobial agents. 2015;45(5):557-9. Disc diffusion methods frequently overcall resistance. - Done. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 03:13, 13 July 2022 (UTC)

13.	The statement ‘Recurrent melioidosis is rare since 2014 due to improved antibiotic therapy and prolongation of the intensive phase of therapy’ relates specifically to Darwin. - Done. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 13:28, 13 July 2022 (UTC)

14.	I would suggest saying ‘Recrudescence rates may be reduced by ensuring adherence to a full course of eradication therapy e.g. by reducing self-discharge…’ - Done. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 13:28, 13 July 2022 (UTC)

15.	The sentence ‘One of the melioidosis is encephalomyelitis’ does not make sense. - Done. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 13:28, 13 July 2022 (UTC)

16.	It may be worth mentioning that EUCAST have recently published interpretative criteria for disc diffusion susceptibility testing - see Dance DAB, Wuthiekanun V, Baird RW, Norton R, Limmathurotsakul D, Currie BJ. Interpreting Burkholderia pseudomallei disc diffusion susceptibility test results by the EUCAST method. Clinical microbiology and infection : the official publication of the European Society of Clinical Microbiology and Infectious Diseases. 2021;27(6):827-9. -Done.[[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 07:05, 14 July 2022 (UTC)

17.	When discussing surveillance it may be worth mentioning the fact that until very recently the official notification system in Thailand significantly underestimated the incidence of culture-positive melioidosis and mortality (Ref 25), and also that even in the UK, where reporting by laboratories is mandatory, 41.3% of cases imported since 2010 were not notified (O'Connor C, Kenna D, Walsh A, Zamarreño DV, Dance D. Imported melioidosis in the United Kingdom: Increasing incidence but continued under-reporting. Clinical Infection in Practice. 2020:100051). - Done.[[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 07:05, 14 July 2022 (UTC)

18.	An alternative reference regarding the Neglected Tropical Disease issue is Savelkoel J, Dance DAB, Currie BJ, Limmathurotsakul D, Wiersinga WJ. A call to action: time to recognise melioidosis as a neglected tropical disease. The Lancet Infectious Diseases. 2022;22(6):e176-e82. -Done.[[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 14:21, 14 July 2022 (UTC)

19.	The first author for reference 63 should be Cossaboom not Marinova-Petkova. - Done.[[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 14:36, 14 July 2022 (UTC)

20.	Another recent imported US case worthy of mention is Dawson P, Duwell MM, Elrod MG, Thompson RJ, Crum DA, Jacobs RM, et al. Human Melioidosis Caused by Novel Transmission of Burkholderia pseudomallei from Freshwater Home Aquarium, United States(1). Emerging infectious diseases. 2021;27(12):3030-5. -Done. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 13:22, 15 July 2022 (UTC)

21.	The Sherlock Holmes story is more usually called ‘The Adventure of the Dying Detective’. -Done.[[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 13:22, 15 July 2022 (UTC)

22.	Although his first name was Ambrose, Sir AT Stanton was usually known by his second name, Thomas. -Done.[[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 13:22, 15 July 2022 (UTC)

23.	Although it is essentially derived from B. pseudomallei, as a result of gene loss B. mallei actually has a much smaller genome than B. pseudomallei – see Nierman WC, DeShazer D, Kim HS, Tettelin H, Nelson KE, Feldblyum T, et al. Structural flexibility in the Burkholderia mallei genome. Proceedings of the National Academy of Sciences of the United States of America. 2004;101(39):14246-51. -Done.[[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]])

24.	Far from spreading from SE Asia to Australia, the genomic evidence suggests that B. pseudomallei originated in Australia and spread from there to SE Asia and onward to other parts of the tropics (Ref 14), so the sentence ‘The discovery of melioidosis in Australia had led to a debate on when and how the disease spread from Southeast Asia to a new distant environment’ is misleading. -Done. Moved the last sentence of the &quot;History&quot; section to just behind this sentence for clarification.[[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]])

25.	The suggestion that B. pseudomallei ‘is cost-effective to produce’ as a weapon is not supported by any evidence as far as I am aware. - Removed this statement as source from Foong YC cited only single source. There is no other sources that can collaborate with this statement. [[User:Cerevisae|Cerevisae]] ([[User talk:Cerevisae|discuss]] • [[Special:Contributions/Cerevisae|contribs]]) 06:26, 27 June 2022 (UTC)
[[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 15:23, 24 June 2022 (UTC)

==Editorial note==
The authors have addressed the points raised above [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 07:03, 5 August 2022 (UTC)</text>
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== [[WikiJournal Preprints/A Survey on Internet Protocol version 4 (IPv4)]] ==

Hello Emanuele. We are trying to tackle the backlog of submissions. I noticed that you signed up to arrange for the search of peer reviewers. I was wondering what is the status of this article? The author made some changes to this submission back in March so I don't think they will be abandoning/withdrawing the submissions. If you are no longer able to coordinator for the search of peer reviewers, please let me know so that I can reassign it to someone else. Thanks. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 20:15, 10 July 2022 (UTC)

:Hello Andrew, I informed Thomas and Kate by email that I would not be able to work for some time. I can resume my task on the paper at the end of July in case you can't find another editor. [[User:Natematic|Natematic]] ([[User talk:Natematic|discuss]] • [[Special:Contributions/Natematic|contribs]]) 10:03, 11 July 2022 (UTC)
::Thanks for the prompt reply. We can wait for a bit. If you need additional help, I can assign a few [[WikiJournal User Group/Technical editors|technical editors]] who can assist with your search of peer reviewers for this article. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 19:42, 11 July 2022 (UTC)
:::@[[User:OhanaUnited|OhanaUnited]], this is just a notice that I have already found a reviewer (granting two months of time to him), and I'm looking for a couple more. [[User:Natematic|Natematic]] ([[User talk:Natematic|discuss]] • [[Special:Contributions/Natematic|contribs]]) 12:17, 3 August 2022 (UTC)
::::Terrific. Would you like me to assign one to two technical editors to you to assist with the peer reviewer search? [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 18:04, 3 August 2022 (UTC)
:::::I am first checking for potential reviewers among colleagues, if I don't get any further reviewer in a few days I'll get back to you. [[User:Natematic|Natematic]] ([[User talk:Natematic|discuss]] • [[Special:Contributions/Natematic|contribs]]) 09:41, 5 August 2022 (UTC)</text>
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      <contributor>
        <username>D.H</username>
        <id>52339</id>
      </contributor>
      <comment>/* {{anchor|Bour}} Bour (1856) – Homogeneous coordinates */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="88165" xml:space="preserve">{{../Lorentz transformation (header)}}
==Most general Lorentz transformations==
===General quadratic form===
The general [[w:quadratic form]] ''q(x)'' with coefficients of a [[w:symmetric matrix]] '''A''', the associated [[w:bilinear form]] ''b(x,y)'', and the [[w:linear transformation]]s of ''q(x)'' and ''b(x,y)'' into ''q(x′)'' and ''b(x′,y′)'' using the [[w:transformation matrix]] '''g''', can be written as&lt;ref&gt;Bôcher (1907), chapter X&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\begin{align}\begin{align}q=\sum_{0}^{n}A_{ij}x_{i}x_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}\end{align}
 &amp; =q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{x}'\\
b=\sum_{0}^{n}A_{ij}x_{i}y_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{y} &amp; =b'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{y}'
\end{align}
\quad\left(A_{ij}=A_{ji}\right)\\
\hline \left.\begin{align}x_{i}^{\prime} &amp; =\sum_{j=0}^{n}g_{ij}x_{j}=\mathbf{g}\cdot\mathbf{x}\\
x_{i} &amp; =\sum_{j=0}^{n}g_{ij}^{(-1)}x_{j}^{\prime}=\mathbf{g}^{-1}\cdot\mathbf{x}'
\end{align}
\right|\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g}=\mathbf{A}'
\end{matrix}&lt;/math&gt;|{{equationRef|Q1}}}}

The case ''n=1'' is the [[w:binary quadratic form]] introduced by [[#Lagrange|Lagrange (1773)]] and [[#Gauss|Gauss (1798/1801)]], ''n=2'' is the ternary quadratic form introduced by [[#Gauss2|Gauss (1798/1801)]], ''n=3'' is the quaternary quadratic form etc.

===Most general Lorentz transformation===
The general Lorentz transformation follows from ({{equationNote|Q1}}) by setting '''A'''='''A′'''=diag(-1,1,...,1) and det '''g'''=±1. It forms an [[w:indefinite orthogonal group]] called the [[w:Lorentz group]] O(1,n), while the case det '''g'''=+1 forms the restricted [[w:Lorentz group]] SO(1,n). The quadratic form ''q(x)'' becomes the [[w:Lorentz interval]] in terms of an [[w:indefinite quadratic form]] of [[w:Minkowski space]] (being a special case of [[w:pseudo-Euclidean space]]), and the associated bilinear form ''b(x)'' becomes the [[w:Minkowski inner product]]:&lt;ref name=ratcliffe&gt;Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1&lt;/ref&gt;&lt;ref&gt;Naimark (1964), 2 in four dimensions&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\begin{align}-x_{0}^{2}+\cdots+x_{n}^{2} &amp; =-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}\\
-x_{0}y_{0}+\cdots+x_{n}y_{n} &amp; =-x_{0}^{\prime}y_{0}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime}
\end{align}
\\
\hline \left.\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\
\downarrow\\
\begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}+\dots+x_{n}g_{0n}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}+\dots+x_{n}g_{1n}\\
 &amp; \dots\\
x_{n}^{\prime} &amp; =x_{0}g_{n0}+x_{1}g_{n1}+\dots+x_{n}g_{nn}
\end{align}
\\
\\
\mathbf{x}=\mathbf{g}^{-1}\cdot\mathbf{x}'\\
\downarrow\\
\begin{align}x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}-\dots-x_{n}^{\prime}g_{n0}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}+\dots+x_{n}^{\prime}g_{n1}\\
 &amp; \dots\\
x_{n} &amp; =-x_{0}^{\prime}g_{0n}+x_{1}^{\prime}g_{1n}+\dots+x_{n}^{\prime}g_{nn}
\end{align}
\end{matrix}\right|\begin{matrix}\begin{align}\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}}\cdot\mathbf{A} &amp; =\mathbf{g}^{-1}\\
\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g} &amp; =\mathbf{A}\\
\mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} &amp; =\mathbf{A}\\
\\
\end{align}
\\
\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} &amp; =\left\{ \begin{align}-1\quad &amp; (j=k=0)\\
1\quad &amp; (j=k&gt;0)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} &amp; =\left\{ \begin{align}-1\quad &amp; (i=k=0)\\
1\quad &amp; (i=k&gt;0)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
\end{matrix}
\end{matrix}&lt;/math&gt;|{{equationRef|1a}}}}

Such general Lorentz transformations ({{equationNote|1a}}) for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]] in order to simplify computations of [[w:elliptic function]]s and integrals.&lt;ref&gt;Musen (1970) pointed out the intimate connection of Hill's scalar development and Minkowski's pseudo-Euclidean 3D space.&lt;/ref&gt;&lt;ref&gt;Touma et al. (2009) showed the analogy between Gauss and Hill's equations and Lorentz transformations, see eq. 22-29.&lt;/ref&gt; They were also used by [[#Chasles|Chasles (1829)]] and [[#Weddle|Weddle (1847)]] to describe relations on hyperboloids, as well as by [[#Poincare|Poincaré (1881)]], [[#Cox|Cox (1881/82)]], [[#Picard|Picard (1882, 1884)]], [[#Killing|Killing (1885, 1893)]], [[#Gerard|Gérard (1892)]], [[#Hausdorff|Hausdorff (1899)]], [[#Woods2|Woods (1901, 1903)]], [[#Liebmann|Liebmann (1904/05)]] to describe [[w:hyperbolic motion]]s (i.e. rigid motions in the [[w:hyperbolic plane]] or [[w:hyperbolic space]]), which were expressed in terms of Weierstrass coordinates of the [[w:hyperboloid model]] satisfying the relation &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=-1&lt;/math&gt; or in terms of the [[w:Cayley–Klein metric]] of [[w:projective geometry]] using the &quot;absolute&quot; form &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=0&lt;/math&gt;.&lt;ref group=M&gt;Killing (1885), p. 71&lt;/ref&gt;&lt;ref&gt;Müller (1910), p. 661, in particular footnote 247.&lt;/ref&gt;&lt;ref&gt;Sommerville (1911), p. 286, section K6.&lt;/ref&gt; In addition, [[w:infinitesimal transformation]]s related to the [[w:Lie algebra]] of the group of hyperbolic motions were given in terms of Weierstrass coordinates &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=-1&lt;/math&gt; by [[#Killing3|Killing (1888-1897)]].

===Most general Lorentz transformation of velocity===

If &lt;math&gt;x_{i},\ x_{i}^{\prime}&lt;/math&gt; in ({{equationNote|1a}}) are interpreted as [[w:homogeneous coordinates]], then the corresponding inhomogenous coordinates &lt;math&gt;u_{s},\ u_{s}^{\prime}&lt;/math&gt; follow by

:&lt;math&gt;\left[\frac{x_{0}}{x_{0}},\ \frac{x_{s}}{x_{0}}\right]=\left[1,\ u_{s}\right],\ \left[\frac{x_{0}^{\prime}}{x_{0}^{\prime}},\ \frac{x_{s}^{\prime}}{x_{0}^{\prime}}\right]=\left[1,\ u_{s}^{\prime}\right],\ (s=1,2\dots n)&lt;/math&gt;

so that the Lorentz transformation becomes a [[w:homography]] leaving invariant the equation of the [[w:unit sphere]], which [[w:John Lighton Synge]] called &quot;the most general formula for the composition of velocities&quot; in terms of special relativity (the transformation matrix '''g''' stays the same as in ({{equationNote|1a}})):&lt;ref&gt;Synge (1955), p. 129 for ''n''=3&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\begin{matrix}-x_{0}^{2}+\cdots+x_{n}^{2}=-x_{0}^{\prime2}+\dots+x_{n}^{\prime2} &amp; \rightarrow &amp; \begin{align}-1+u_{1}^{2}+\cdots+u_{n}^{2} &amp; ={\scriptstyle \frac{-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}}{\left(g_{00}+g_{01}u_{1}^{\prime}+\dots+g_{0n}u_{n}^{\prime}\right)^{2}}}\\
{\scriptstyle \frac{-1+u_{1}^{2}+\cdots+u_{n}^{2}}{\left(g_{00}-g_{10}u_{1}-\dots-g_{n0}u_{n}\right)^{2}}} &amp; =-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}
\end{align}
\\
\hline -x_{0}^{2}+\cdots+x_{n}^{2}=-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}=0 &amp; \rightarrow &amp; -1+u_{1}^{2}+\cdots+u_{n}^{2}=-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0
\end{matrix}\\
\hline \begin{align}u_{s}^{\prime} &amp; =\frac{g_{s0}+g_{s1}u_{1}+\dots+g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots+g_{0n}u_{n}}\\
\\
u_{s} &amp; =\frac{-g_{0s}+g_{1s}u_{1}^{\prime}+\dots+g_{ns}u_{n}^{\prime}}{g_{00}-g_{10}u_{1}^{\prime}-\dots-g_{n0}u_{n}^{\prime}}
\end{align}
\left|\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} &amp; =\left\{ \begin{align}-1\quad &amp; (j=k=0)\\
1\quad &amp; (j=k&gt;0)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} &amp; =\left\{ \begin{align}-1\quad &amp; (i=k=0)\\
1\quad &amp; (i=k&gt;0)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
\right.
\end{matrix}&lt;/math&gt;|{{equationRef|1b}}}}

Such Lorentz transformations for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827–1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]], [[#Callandreau|Callandreau (1885)]] in order to simplify computations of elliptic functions and integrals, by [[#Picard|Picard (1882-1884)]] in relation to [[w:Hermitian form|Hermitian quadratic form]]s, or by [[#Woods2|Woods (1901, 1903)]] in terms of the [[w:Beltrami–Klein model]] of hyperbolic geometry. In addition, infinitesimal transformations in terms of the [[w:Lie algebra]] of the group of hyperbolic motions leaving invariant the unit sphere &lt;math&gt;-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0&lt;/math&gt; were given by [[#Lie3|Lie (1885-1893) and Werner (1889)]] and [[#Killing3|Killing (1888-1897)]].

==Historical notation==
===Lagrange (1773) – Binary quadratic forms {{anchor|Lagrange}}===
After the invariance of the sum of squares under linear substitutions was discussed by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]], the general expressions of a [[w:binary quadratic form]] and its transformation was formulated by [[w:Joseph-Louis Lagrange]] (1773/75) as follows&lt;ref group=M&gt;Lagrange (1773/75), section 22&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\
\hline \begin{align}y &amp; =Ms+Nx\\
z &amp; =ms+nx
\end{align}
\left|\begin{matrix}\begin{align}P &amp; =pM^{2}+2qMm+rm^{2}\\
Q &amp; =pMN+q(Mn+Nm)+rmn\\
R &amp; =pN^{2}+2qNn+rn^{2}
\end{align}
\\
\downarrow\\
PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to ({{equationNote|Q1}}) ''(n=1)''. The Lorentz interval &lt;math&gt;-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(p,q,r)=(P,Q,R)=(1,0,-1)''.&lt;/p&gt;

==={{anchor|Gauss}} Gauss (1798–1818)===

===={{anchor|Gauss1}} Binary quadratic forms====

The theory of binary quadratic forms was considerably expanded by [[w:Carl Friedrich Gauss]] (1798, published 1801) in his [[w:Disquisitiones Arithmeticae]]. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:&lt;ref group=M&gt;Gauss (1798/1801), articles 157–158;&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\
F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\
\hline \begin{align}x &amp; =\alpha x'+\beta y'\\
y &amp; =\gamma x'+\delta y'\\
\\
x' &amp; =\delta x-\beta y\\
y' &amp; =-\gamma x+\alpha y
\end{align}
\left|\begin{matrix}\begin{align}a' &amp; =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\
b' &amp; =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\
c' &amp; =a\beta^{2}+2b\beta\delta+c\delta^{2}
\end{align}
\\
\downarrow\\
b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

which is equivalent to ({{equationNote|Q1}}) ''(n=1)''. As pointed out by Gauss, ''F'' and ''F′'' are called &quot;proper equivalent&quot; if αδ-βγ=1, so that ''F'' is contained in ''F′'' as well as ''F′'' is contained in ''F''. In addition, if another form ''F″'' is contained by the same procedure in ''F′'' it is also contained in ''F'' and so forth.&lt;ref group=M&gt;Gauss (1798/1801), section 159&lt;/ref&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The Lorentz interval &lt;math&gt;-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(a,b,c)=(a',b',c')=(1,0,-1)''.&lt;/p&gt;

===={{anchor|Gauss2}} Ternary quadratic forms====

Gauss (1798/1801)&lt;ref group=M&gt;Gauss (1798/1801), articles 266–285&lt;/ref&gt; also discussed ternary quadratic forms with the general expression

:&lt;math&gt;\begin{matrix}f=ax^{2}+a'x^{\prime2}+a''x^{\prime\prime2}+2bx'x''+2b'xx''+2b''xx'=\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)\\
g=my^{2}+m'y^{\prime2}+m''y^{\prime\prime2}+2ny'y''+2n'yy''+2n''yy'=\left(\begin{matrix}m, &amp; m', &amp; m''\\
n, &amp; n', &amp; n''
\end{matrix}\right)\\
\hline \begin{align}x &amp; =\alpha y+\beta y'+\gamma y''\\
x' &amp; =\alpha'y+\beta'y'+\gamma'y''\\
x'' &amp; =\alpha''y+\beta''y'+\gamma''y''
\end{align}
\end{matrix}&lt;/math&gt;

which is equivalent to ({{equationNote|Q1}}) ''(n=2)''. Gauss called these forms definite when they have the same sign such as ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;+z&lt;sup&gt;2&lt;/sup&gt;'', or indefinite in the case of different signs such as ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;''. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:&lt;ref group=M&gt;Gauss (1798/1801), article 277&lt;/ref&gt;

:&lt;math&gt;\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)\Rightarrow\begin{matrix}\left(\begin{matrix}1, &amp; -1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; 1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\\
\left(\begin{matrix}1, &amp; -1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; -1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval &lt;math&gt;x^{2}+x^{\prime2}-x^{\prime\prime2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' is an indefinite ternary quadratic form, which follows from the general ternary form by setting:
&lt;br&gt;&lt;math&gt;\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)=\left(\begin{matrix}m, &amp; m', &amp; m''\\
n, &amp; n', &amp; n''
\end{matrix}\right)=\left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)&lt;/math&gt;&lt;/p&gt;

===={{anchor|Gauss4}} Homogeneous coordinates====

Gauss (1818) discussed planetary motions together with formulating [[w:elliptic function]]s. In order to simplify the integration, he transformed the expression

:&lt;math&gt;(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E&lt;/math&gt;

into

:&lt;math&gt;G+G'\cos T^{2}+G''\sin T^{2}&lt;/math&gt;

in which the [[w:eccentric anomaly]] ''E'' is connected to the new variable ''T'' by the following transformation including an arbitrary constant ''k'', which Gauss then rewrote by setting ''k''=1:&lt;ref group=M&gt;Gauss (1818), pp. 5–10&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}{\scriptstyle \left(\alpha+\alpha'\cos T+\alpha''\sin T\right)^{2}+\left(\beta+\beta'\cos T+\beta''\sin T\right)^{2}-\left(\gamma+\gamma'\cos T+\gamma''\sin T\right)^{2}}=0\\
k\left(\cos^{2}T+\sin^{2}T-1\right)=0\\
\hline \begin{align}\cos E &amp; =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\
\sin E &amp; =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =k &amp; \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' &amp; =-k\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-k &amp; \beta\beta-\beta'\beta'-\beta''\beta'' &amp; =-k\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-k &amp; \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' &amp; =+k\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0 &amp; \beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0 &amp; \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0 &amp; \alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
}\right.\\
\hline k=1\\
\begin{align}t\cos E &amp; =\alpha+\alpha'\cos T+\alpha''\sin T\\
t\sin E &amp; =\beta+\beta'\cos T+\beta''\sin T\\
t &amp; =\gamma+\gamma'\cos T+\gamma''\sin T
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =1\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-1\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-1\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The coefficients α,β,γ,... of Gauss' case ''k''=1 are equivalent to the coefficient system in Lorentz transformations ({{equationNote|1a}}) and ({{equationNote|1b}})  ''(n=2)''.

Further setting &lt;math&gt;[\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]&lt;/math&gt;, Gauss' transformation becomes Lorentz transformation ({{equationNote|1b}})  ''(n=2)''.&lt;/p&gt;

Subsequently, he showed that these relations can be reformulated using three variables ''x,y,z'' and ''u,u′,u″'', so that

:&lt;math&gt;aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz&lt;/math&gt;

can be transformed into

:&lt;math&gt;Guu+G'u'u'+G''u''u''&lt;/math&gt;,

in which ''x,y,z'' and ''u,u′,u″'' are related by the transformation:&lt;ref group=M&gt;Gauss (1818), pp. 9–10&lt;/ref&gt;

:&lt;math&gt;\begin{align}x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
y &amp; =\beta u+\beta'u'+\beta''u''\\
z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
\\
u &amp; =-\alpha x-\beta y+\gamma z\\
u' &amp; =\alpha'x+\beta'y-\gamma'z\\
u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =1\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-1\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-1\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0
\end{align}
}\right.&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}})  ''(n=2)'' satisfying &lt;math&gt;x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}&lt;/math&gt;, and can be related to Gauss' previous equations in terms of homogeneous coordinates &lt;math&gt;\left[\cos T,\sin T,\cos E,\sin E\right]=\left[\tfrac{x}{z},\ \tfrac{y}{z},\ \tfrac{u'}{u},\ \tfrac{u''}{u}\right]&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Jacobi}} Jacobi (1827, 1833/34) – Homogeneous coordinates===

Following [[#Gauss4|Gauss (1818)]], [[w:Carl Gustav Jacob Jacobi]] extended Gauss' transformation to 3 dimensions in 1827:&lt;ref group=M&gt;Jacobi (1827), p. 235, 239–240&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}\cos P^{2}+\sin P^{2}\cos\vartheta^{2}+\sin P^{2}\sin\vartheta^{2}=1\\
k\left(\cos\psi^{2}+\sin\psi^{2}\cos\varphi^{2}+\sin\psi^{2}\sin\varphi^{2}-1\right)=0\\
\hline {\left.\begin{matrix}\mathbf{(1)}\begin{align}\cos P &amp; =\frac{\alpha+\alpha'\cos\psi+\alpha''\sin\psi\cos\varphi+\alpha'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\sin P\cos\vartheta &amp; =\frac{\beta+\beta'\cos\psi+\beta''\sin\psi\cos\varphi+\beta'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\sin P\sin\vartheta &amp; =\frac{\gamma+\beta'\cos\psi+\gamma''\sin\psi\cos\varphi+\gamma'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\\
\cos\psi &amp; =\frac{-\delta'+\alpha'\cos P+\beta'\sin P\cos\vartheta+\gamma'\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\
\sin\psi\cos\varphi &amp; =\frac{-\delta''+\alpha''\cos P+\beta''\sin P\cos\vartheta+\gamma''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\
\sin\psi\sin\varphi &amp; =\frac{-\delta'''+\alpha'''\cos P+\beta'''\sin P\cos\vartheta+\gamma'''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}
\end{align}
\\
\\
\hline \mathbf{(2)}\begin{align}\alpha\mu+\beta x+\gamma y+\delta z &amp; =m\\
\alpha'\mu+\beta'x+\gamma'y+\delta'z &amp; =m'\\
\alpha''\mu+\beta''x+\gamma''y+\delta''z &amp; =m''\\
\alpha'''\mu+\beta'''x+\gamma'''y+\delta'''z &amp; =m'''\\
\\
Am+A'm'+A''m''+A'''m''' &amp; =\mu\\
Bm+B'm'+B''m''+B'''m''' &amp; =x\\
Cm+C'm'+C''m''+C'''m''' &amp; =y\\
Dm+D'm'+D''m''+D'''m''' &amp; =z\\
\\
\end{align}
\\
\begin{align}\alpha &amp; =-kA, &amp; \beta &amp; =-kB, &amp; \gamma &amp; =-kC, &amp; \delta &amp; =kD,\\
\alpha' &amp; =kA', &amp; \beta' &amp; =kB', &amp; \gamma' &amp; =kC', &amp; \delta' &amp; =-kD',\\
\alpha'' &amp; =kA'', &amp; \beta'' &amp; =kB'', &amp; \gamma'' &amp; =kC'', &amp; \delta'' &amp; =-kD'',\\
\alpha''' &amp; =kA''', &amp; \beta''' &amp; =kB''', &amp; \gamma''' &amp; =kC''', &amp; \delta''' &amp; =-kD''',
\end{align}
\end{matrix}\right|\begin{matrix}\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta &amp; =-k\\
\alpha'\alpha'+\beta'\beta'+\gamma'\gamma'-\delta'\delta' &amp; =k\\
\alpha''\alpha''+\beta''\beta''+\gamma''\gamma''-\delta''\delta'' &amp; =k\\
\alpha'''\alpha'''+\beta'''\beta'''+\gamma'''\gamma'''-\delta'''\delta''' &amp; =k\\
\alpha\alpha'+\beta\beta'+\gamma\gamma'-\delta\delta' &amp; =0\\
\alpha\alpha''+\beta\beta''+\gamma\gamma''-\delta\delta'' &amp; =0\\
\alpha\alpha'''+\beta\beta'''+\gamma\gamma'''-\delta\delta''' &amp; =0\\
\alpha''\alpha'''+\beta''\beta'''+\gamma''\gamma'''-\delta''\delta''' &amp; =0\\
\alpha'''\alpha'+\beta'''\beta'+\gamma'''\gamma'-\delta'''\delta' &amp; =0\\
\alpha'\alpha''+\beta'\beta''+\gamma'\gamma''-\delta'\delta'' &amp; =0\\
\\
-\alpha\alpha+\alpha'\alpha'+\alpha''\alpha''+\alpha'''\alpha''' &amp; =k\\
-\beta\beta+\beta'\beta'+\beta''\beta''+\beta'''\beta''' &amp; =k\\
-\gamma\gamma+\gamma'\gamma'+\gamma''\gamma''+\gamma'''\gamma''' &amp; =k\\
-\delta\delta+\delta'\delta'+\delta''\delta''+\delta'''\delta''' &amp; =-k\\
-\alpha\beta+\alpha'\beta'+\alpha''\beta''+\alpha'''\beta''' &amp; =0\\
-\alpha\gamma+\alpha'\gamma'+\alpha''\gamma''+\alpha'''\gamma''' &amp; =0\\
-\alpha\delta+\alpha'\delta'+\alpha''\delta''+\alpha'''\delta''' &amp; =0\\
-\beta\gamma+\beta'\gamma'+\beta''\gamma''+\beta'''\gamma''' &amp; =0\\
-\gamma\delta+\gamma'\delta'+\gamma''\delta''+\gamma'''\delta''' &amp; =0\\
-\delta\beta+\delta'\beta'+\delta''\beta''+\delta'''\beta''' &amp; =0
\end{align}
\end{matrix}}
\end{matrix}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;By setting &lt;math&gt;{\scriptstyle \begin{align}\left[\cos P,\ \sin P\cos\varphi,\ \sin P\sin\varphi\right] &amp; =\left[u_{1},\ u_{2},\ u_{3}\right]\\{}
[\cos\psi,\ \sin\psi\cos\vartheta,\ \sin\psi\sin\vartheta] &amp; =\left[u_{1}^{\prime},\ u_{2}^{\prime},\ u_{3}^{\prime}\right]
\end{align}
}&lt;/math&gt; and ''k''=1 in the (1827) formulas, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'', and by setting ''k''=1 in transformation system (2) it becomes equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' producing &lt;math&gt;m^{2}+m^{\prime2}+m^{\prime\prime2}-m^{\prime\prime\prime2}=\mu^{2}+x^{2}+y^{2}-z^{2}&lt;/math&gt;.&lt;/p&gt;

Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change) in the case of 2 dimensions:&lt;ref group=M&gt;The orthogonal substitution and the imaginary transformation was defined in Jacobi (1832a), pp. 257, 265–267; Transformation system (2) and (3) and coefficients in Jacobi (1832b), pp. 321-325.&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}xx+yy+zz=ss+s's'+s''s''=0\\
\mathbf{(1)}\begin{align}x &amp; =\alpha s+\alpha's'+\alpha''s''\\
y &amp; =\beta s+\beta's'+\beta''s''\\
z &amp; =\gamma s+\gamma's'+\gamma''s''\\
\\
s &amp; =\alpha x+\beta y+\gamma z\\
s' &amp; =\alpha'x+\beta'y+\gamma'z\\
s'' &amp; =\alpha''x+\beta''y+\gamma''z
\end{align}
\left|\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma &amp; =1 &amp; \alpha\alpha+\alpha'\alpha'+\alpha''\alpha'' &amp; =1\\
\alpha'\alpha'+\beta'\beta'+\gamma'\gamma' &amp; =1 &amp; \beta\beta+\beta'\beta'+\beta''\beta'' &amp; =1\\
\alpha''\alpha''+\beta''\beta''+\gamma''\gamma'' &amp; =1 &amp; \gamma\gamma+\gamma'\gamma'+\gamma''\gamma'' &amp; =1\\
\alpha'\alpha''+\beta'\beta''+\gamma'\gamma'' &amp; =0 &amp; \beta\gamma+\beta'\gamma'+\beta''\gamma'' &amp; =0\\
\alpha''\alpha+\beta''\beta+\gamma''\gamma &amp; =0 &amp; \gamma\alpha+\gamma'\alpha'+\gamma''\alpha'' &amp; =0\\
\alpha\alpha'+\beta\beta'+\gamma\gamma' &amp; =0 &amp; \alpha\beta+\alpha'\beta'+\alpha''\beta'' &amp; =0
\end{align}
\right.\\
\hline \left[\frac{y}{x},\ \frac{z}{x},\ \frac{s'}{s},\ \frac{s''}{s}\right]=\left[-i\cos\varphi,\ -i\sin\varphi,\ i\cos\eta,\ i\sin\eta\right]\\
\left[\alpha',\ \alpha'',\ \beta,\ \gamma\right]=\left[i\alpha',\ i\alpha'',\ -i\beta,\ -i\gamma\right]\\
\hline \begin{matrix}\mathbf{(2)}\begin{matrix}\left(\alpha-\alpha'\cos\eta-\alpha''\sin\eta\right)^{2}=\left(\beta-\beta'\cos\eta-\beta''\sin\eta\right)^{2}+\left(\gamma-\gamma'\cos\eta-\gamma''\sin\eta\right)^{2}\\
\left(\alpha-\beta\cos\phi-\gamma\sin\phi\right)^{2}=\left(\alpha'-\beta'\cos\phi-\gamma'\sin\phi\right)^{2}+\left(\alpha''-\beta''\cos\phi-\gamma''\sin\phi\right)^{2}\\
\hline \begin{align}\cos\phi &amp; =\frac{\beta-\beta'\cos\eta-\beta''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, &amp; \cos\eta &amp; =\frac{\alpha'-\beta'\cos\phi-\gamma'\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}\\
\sin\phi &amp; =\frac{\gamma-\gamma'\cos\eta-\gamma''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, &amp; \sin\eta &amp; =\frac{\alpha''-\beta''\cos\phi-\gamma''\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}
\end{align}
\end{matrix}\\
\hline \\
\mathbf{(3)}\begin{matrix}1-zz-yy=\frac{1-s's'-s''s''}{\left(\alpha-\alpha's'-\alpha''s''\right)^{2}}\\
\hline \begin{align}y &amp; =\frac{\beta-\beta's'-\beta''s''}{\alpha-\alpha's'-\alpha''s''}, &amp; s' &amp; =\frac{\alpha'-\beta'y-\gamma'z}{\alpha-\beta y-\gamma z},\\
z &amp; =\frac{\gamma-\gamma's'-\gamma''s''}{\alpha-\alpha's'-\alpha''s'''}, &amp; s'' &amp; =\frac{\alpha''-\beta''y-\gamma''z}{\alpha-\beta y-\gamma z},
\end{align}
\end{matrix}
\end{matrix}\left|\begin{align}\alpha\alpha-\beta\beta-\gamma\gamma &amp; =1\\
\alpha'\alpha'-\beta'\beta'-\gamma'\gamma' &amp; =-1\\
\alpha''\alpha''-\beta''\beta''-\gamma''\gamma'' &amp; =-1\\
\alpha'\alpha''-\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\alpha''\alpha-\beta''\beta-\gamma''\gamma &amp; =0\\
\alpha\alpha'-\beta\beta'-\gamma\gamma' &amp; =0\\
\\
\alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' &amp; =1\\
\beta\beta-\beta'\beta'-\beta''\beta'' &amp; =-1\\
\gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' &amp; =-1\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
\gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' &amp; =0\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
\right.
\end{matrix}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;By setting &lt;math&gt;[\cos\phi,\ \sin\phi,\ \cos\eta,\ \sin\eta]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]&lt;/math&gt;, transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Also transformation system (3) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' up to a sign change.&lt;/p&gt;

Extending his previous result, Jacobi (1833) started with [[#Cauchy|Cauchy's (1829)]] orthogonal transformation for ''n'' dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of ''n'' dimensions:&lt;ref group =M&gt;Jacobi (1833/34), pp. 7–8, 34–35, 41; Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 229–230.&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots+x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots+y_{n}y_{n}\\
\hline \mathbf{(1)\ }\begin{align}y_{\varkappa} &amp; =\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}\\
x_{\varkappa} &amp; =\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}\\
\\
\frac{y_{\varkappa}}{y_{n}} &amp; =\frac{\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}\\
\frac{x_{\varkappa}}{x_{n}} &amp; =\frac{\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}
\end{align}
\left|\begin{align}\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\lambda}^{(n)} &amp; =0\\
\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\varkappa}^{(n)} &amp; =1\\
\\
\alpha_{1}^{(\varkappa)}\alpha_{1}^{(\lambda)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\lambda)} &amp; =0\\
\alpha_{1}^{(\varkappa)}\alpha_{1}^{(\varkappa)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\varkappa)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\varkappa)} &amp; =1
\end{align}
\right.\\
\hline \frac{x_{\varkappa}}{x_{n}}=-i\xi_{\varkappa},\ \frac{y_{\varkappa}}{y_{n}}=i\nu_{\varkappa}\\
1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{y_{n}y_{n}}{x_{n}x_{n}}\left(1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}\right)\\
\alpha_{n}^{(\varkappa)}=i\alpha^{(\varkappa)},\ \alpha_{\varkappa}^{(n)}=-i\alpha_{\varkappa},\ \alpha_{n}^{(n)}=\alpha\\
1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}}{\left[\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}\right]^{2}}\\
\hline \mathbf{(2)\ }\begin{align}\nu_{\varkappa} &amp; =\frac{\alpha^{(\varkappa)}-\alpha_{1}^{(\varkappa)}\xi_{1}-\alpha_{2}^{(\varkappa)}\xi_{2}\dots-\alpha_{n-1}^{(\varkappa)}\xi_{n-1}}{\alpha-\alpha_{1}\xi_{1}-\alpha_{2}\xi_{2}\dots-\alpha_{n-1}\xi_{n-1}}\\
\\
\xi_{\varkappa} &amp; =\frac{\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\nu_{1}-\alpha_{2}^{\prime\prime}\nu_{2}\dots-\alpha_{\varkappa}^{(n-1)}\nu_{n-1}}{\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}}
\end{align}
\\
\hline \xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=1\ \Rightarrow\ \nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}=1
\end{matrix}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) up to a sign change.&lt;/p&gt;

He also stated the following transformation leaving invariant the Lorentz interval:&lt;ref group=M&gt;Jacobi (1833/34), p. 37. Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 232–233.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots-u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots-w_{n-1}w_{n-1}\\
\hline {\scriptstyle \begin{align}u &amp; =\alpha w-\alpha'w_{1}-\alpha''w_{2}-\dots-\alpha^{(n-1)}w_{n-1}\\
u_{1} &amp; =\alpha_{1}w-\alpha_{1}^{\prime}w_{1}-\alpha_{1}^{\prime\prime}w_{2}-\dots-\alpha_{1}^{(n-1)}w_{n-1}\\
 &amp; \dots\\
u_{n-1} &amp; =\alpha_{n-1}w-\alpha_{n-1}^{\prime}w_{1}-\alpha_{n-1}^{\prime\prime}w_{2}-\dots-\alpha_{n-1}^{(n-1)}w_{n-1}\\
\\
w &amp; =\alpha u-\alpha_{1}u_{1}-\alpha_{2}^{\prime\prime}u_{2}-\dots-\alpha_{n-1}u_{n-1}\\
w_{1} &amp; =\alpha'u-\alpha_{1}^{\prime}u_{1}-\alpha_{2}^{\prime}u_{2}-\dots-\alpha_{n-1}^{\prime}u_{n-1}\\
 &amp; \dots\\
w_{n-1} &amp; =\alpha^{(n-1)}u-\alpha_{1}^{(n-1)}u_{1}-\alpha_{2}^{(n-1)}u_{2}-\dots-\alpha_{n-1}^{(n-1)}u_{n-1}
\end{align}
\left|\begin{align}\alpha\alpha-\alpha'\alpha'-\alpha''\alpha''\dots-\alpha^{(n-1)}\alpha^{(n-1)} &amp; =+1\\
\alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\varkappa}^{(n-1)} &amp; =-1\\
\alpha\alpha_{\varkappa}-\alpha^{\prime}\alpha_{\varkappa}^{\prime}-\alpha^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha^{(n-1)}\alpha_{\varkappa}^{(n-1)} &amp; =0\\
\alpha_{\varkappa}\alpha_{\lambda}-\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\lambda}^{(n-1)} &amp; =0\\
\\
\alpha\alpha-\alpha_{1}\alpha_{1}-\alpha_{2}\alpha_{2}\dots-\alpha_{n-1}\alpha_{n-1} &amp; =+1\\
\alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{1}^{\varkappa}\alpha_{1}^{\varkappa}-\alpha_{2}^{\prime\prime}\alpha_{2}^{\prime\prime}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\varkappa)} &amp; =-1\\
\alpha\alpha^{(\varkappa)}-\alpha_{1}\alpha_{1}^{(\varkappa)}-\alpha_{2}\alpha_{2}^{(\varkappa)}\dots-\alpha_{n-1}\alpha_{n-1}^{(\varkappa)} &amp; =0\\
\alpha^{(\varkappa)}\alpha^{(\lambda)}-\alpha_{1}^{(\varkappa)}\alpha_{1}^{\lambda l)}-\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\lambda)} &amp; =0
\end{align}
\text{ }\right.}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}}) up to a sign change.&lt;/p&gt;

==={{anchor|Chasles}} Chasles (1829) – Conjugate hyperboloids ===

[[w:Michel Chasles]] (1829) independently introduced the same equation systems as [[#Gauss4|Gauss (1818)]] and [[#Jacobi|Jacobi (1827)]], albeit in the different context of conjugate hyperboloids. He started with two equation systems (a) and (b) from which he derived systems (c), (d) and others:&lt;ref group=M&gt;Chasles (1829), p. 139&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =1\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =1\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =-1
\end{align}
\right\}  &amp; \dots(a)\\ \\
\left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma' &amp; =0
\end{align}
\right\}  &amp; \dots(b)\\ \\
\left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =1\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =1\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-1
\end{align}
\right\}  &amp; \dots(c)\\ \\
\left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right\}  &amp; \dots(d)
\end{matrix}&lt;/math&gt;

He noted that those quantities become the “frequently employed” formulas of Lagrange [i.e. the coefficients of the Euclidean orthogonal transformation first given by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]]] by setting:&lt;ref group=M&gt;Chasles (1829), p. 141&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\gamma\quad\Rightarrow\quad-\gamma\sqrt{-1}\\
\gamma'\quad\Rightarrow\quad-\gamma'\sqrt{-1}\\
\alpha''\quad\Rightarrow\quad\alpha''\sqrt{-1}\\
\beta''\quad\Rightarrow\quad\beta''\sqrt{-1}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Equations (a,b,c,d) are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).&lt;/p&gt;

Chasles now showed that equation systems (a,b,c,d) are of importance when discussing the relations between conjugate diameters of hyperboloids. He used the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid having the same principal axes (x,y,z), thus sharing the same conjugate axes, and having the common asymptotic cone &lt;math&gt;\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0&lt;/math&gt;. He then transformed those two hyperboloids to new axes (x',y',z') sharing the property of conjugacy:&lt;ref group=M&gt;Chasles (1829), pp. 143-144&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\
\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\
\hline \begin{align}x &amp; =lx'+l'y'+l''z'\\
y &amp; =mx'+m'y'+m''z'\\
z &amp; =nx'+n'y'+n''z'
\end{align}
\\
\left\{ \begin{align}\frac{ll'}{a{{}^2}}+\frac{mm'}{b^{2}}-\frac{nn'}{c^{2}} &amp; =0\\
\frac{ll''}{a{{}^2}}+\frac{mm''}{b^{2}}-\frac{nn''}{c^{2}} &amp; =0\\
\frac{l'l''}{a{{}^2}}+\frac{m'm''}{b^{2}}-\frac{n'n''}{c^{2}} &amp; =0
\end{align}
\right\} \\
\hline \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=1\\
\left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=-1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Chasles defined the conditional equations of ''l,m,n'' in the same way as those of &lt;math&gt;\alpha,\beta,\gamma&lt;/math&gt; in equation system (b) above, so his transformation of x,y,z into x',y',z' represents Lorentz transformation ({{equationNote|1a}}, n=2) by applying equation system (a) as well.&lt;/p&gt;

He went on to use two semi-diameters of the one-sheet hyperboloid and one semi-diameter of the two-sheet hyperboloid in order to define equation system (A), and went on to suggest that the other equations related to this system can be obtained using the above transformation from oblique coordinates to other oblique ones, but he deemed it more simple to use a geometric argument to obtain system (B), which together with (A) then allowed him to algebraically determine systems (C), (D) and additional ones, leading Chasles to announce that “''from these formulas one can very easily conclude the various properties of conjugated diameters of hyperboloids''”:&lt;ref group=M&gt;Chasles (1829), pp. 145-146&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =a^{2}\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =b^{2}\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =-c^{2}
\end{align}
\right\}  &amp; \dots(A)\\
\left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma' &amp; =0
\end{align}
\right\}  &amp; \dots(B)\\
\left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =a^{2}\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =b^{2}\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-c^{2}
\end{align}
\right\}  &amp; \dots(C)\\
\left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right\}  &amp; \dots(D)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Equation systems (A,B,C,D), being equivalent to systems (a,b,c,d) above, are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.&lt;/p&gt;

==={{anchor|Lebesgue}} Lebesgue (1837) – Homogeneous coordinates===

[[w:Victor-Amédée Lebesgue]] (1837) summarized the previous work of [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Cauchy|Cauchy (1829)]]. He started with the orthogonal transformation&lt;ref group=M&gt;Lebesgue (1837), pp. 338-341&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}\ (9)\\
\hline {\scriptstyle \begin{align}x_{1} &amp; =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\
x_{2} &amp; =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\
\dots\\
x_{n} &amp; =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\
\\
y_{1} &amp; =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n,1}x_{n}\\
y_{2} &amp; =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n,2}x_{n}\ (12)\ \\
\dots\\
y_{n} &amp; =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n,n}x_{n}
\end{align}
\left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n,\alpha}^{2} &amp; =1 &amp; (10)\\
a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n,\alpha}a_{n,\beta} &amp; =0 &amp; (11)\\
a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n}^{2} &amp; =1 &amp; (13)\\
a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n}a_{\beta,n} &amp; =0 &amp; (14)
\end{align}
\right.}
\end{matrix}&lt;/math&gt;

In order to achieve the invariance of the Lorentz interval&lt;ref group=M&gt;Lebesgue (1837), pp. 353–354&lt;/ref&gt;

:&lt;math&gt;x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}&lt;/math&gt;

he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first ''n-1'' equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=''n'' change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the ''n''-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the ''n''-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These instructions give Lorentz transformation ({{equationNote|1a}}) in the form:

&lt;br&gt;&lt;math&gt;{\scriptstyle \begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\
\hline \begin{align}x_{1} &amp; =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\
x_{2} &amp; =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\
\dots\\
x_{n} &amp; =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\
\\
y_{1} &amp; =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\
y_{2} &amp; =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\
\dots\\
-y_{n} &amp; =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n-1,n}x_{n-1}-a_{n,n}x_{n}
\end{align}
\left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n-1,\alpha}^{2}-a_{n,\alpha}^{2} &amp; =1\\
a_{1,n}^{2}+a_{2,n}^{2}+\dots+a_{n-1,n}^{2}-a_{n,n}^{2} &amp; =-1\\
a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n-1,\alpha}a_{n-1,\beta}-a_{n,\alpha}a_{n,\beta} &amp; =0\\
a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n-1}^{2}-a_{\alpha,n}^{2} &amp; =1\\
a_{n,1}^{2}+a_{n,2}^{2}+\dots+a_{n,n-1}^{2}-a_{n,n}^{2} &amp; =-1\\
a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n-1}a_{\beta,n-1}-a_{\alpha,n}a_{\beta,n} &amp; =0
\end{align}
\right.
\end{matrix}}&lt;/math&gt;&lt;/p&gt;

He went on to redefine the variables of the Lorentz interval and its transformation:&lt;ref group=M&gt;Lebesgue (1837), pp. 353–355&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\
\downarrow\\
\begin{align}x_{1} &amp; =x_{n}\cos\theta_{1}, &amp; x_{2} &amp; =x_{n}\cos\theta_{2},\dots &amp; x_{n-1} &amp; =x_{n}\cos\theta_{n-1}\\
y_{1} &amp; =y_{n}\cos\phi_{1}, &amp; y_{2} &amp; =y_{n}\cos\phi_{2},\dots &amp; y_{n-1} &amp; =y_{n}\cos\phi_{n-1}
\end{align}
\\
\downarrow\\
\cos^{2}\theta_{1}+\cos^{2}\theta_{2}+\dots+\cos^{2}\theta_{n-1}=1\\
\cos^{2}\phi_{1}+\cos^{2}\phi_{2}+\dots+\cos^{2}\phi_{n-1}=1\\
\hline \\
\cos\theta_{i}=\frac{a_{i,1}\cos\phi_{1}+a_{i,2}\cos\phi_{2}+\dots+a_{i,n-1}\cos\phi_{n-1}+a_{i,n}}{a_{n,1}\cos\phi_{1}+a_{n,2}\cos\phi_{2}+\dots+a_{n,n-1}\cos\phi_{n-1}+a_{n,n}}\\
(i=1,2,3\dots n)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Setting &lt;math&gt;[\cos\theta_{i},\ \cos\phi_{i}]=\left[u_{s},\ u_{s}^{\prime}\right]&lt;/math&gt; it is equivalent to Lorentz transformation ({{equationNote|1b}}).&lt;/p&gt;

==={{anchor|Weddle}} Weddle (1847) – Conjugate hyperboloids===
Very similar to [[#Chasles|Chasles (1829)]], though without reference to him, [[w:Thomas Weddle]] discussed conjugate hyperboloids using the following equation system (&amp;alpha;), from which he derived equations (&amp;beta;) and others:&lt;ref group=M&gt;Weddle (1847), p. 274&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} &amp; =1, &amp; l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} &amp; =0\\
l_{2}^{2}+m_{2}^{2}-n_{2}^{2} &amp; =1, &amp; l_{1}l_{3}+m_{1}m_{3}-n_{1}n_{3} &amp; =0\\
l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =-1, &amp; l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} &amp; =0
\end{align}
\right\}  &amp; \dots(\alpha)\\ \\
\left.\begin{align}l_{1}^{2}+l_{2}^{2}-l_{3}^{2} &amp; =1, &amp; l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} &amp; =0\\
m_{1}^{2}+m_{2}^{2}-m_{3}^{2} &amp; =1, &amp; l_{1}n_{1}+l_{2}n_{2}-l_{3}n_{3} &amp; =0\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =-1, &amp; m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} &amp; =0
\end{align}
\right\}  &amp; \dots(\beta)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).&lt;/p&gt;

Using the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid sharing the same conjugate axes, and having the common asymptotic cone &lt;math&gt;\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0&lt;/math&gt;, he defined three conjugate points &lt;math&gt;(x_{1}\dots,y_{1}\dots,z_{1}\dots)&lt;/math&gt; on those two conjugate hyperboloids, related to each other in the same way as equations (&amp;alpha;, &amp;beta;) stated above:&lt;ref group=M&gt;Weddle (1847), pp. 275-276&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\
\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\
\hline \begin{align}\frac{x_{1}x_{2}}{a{{}^2}}+\frac{y_{1}y_{2}}{b^{2}}-\frac{z_{1}z_{2}}{c^{2}} &amp; =0\\
\frac{x_{1}x_{3}}{a{{}^2}}+\frac{y_{1}y_{3}}{b^{2}}-\frac{z_{1}z_{3}}{c^{2}} &amp; =0\\
\frac{x_{2}x_{3}}{a{{}^2}}+\frac{y_{2}y_{3}}{b^{2}}-\frac{z_{2}z_{3}}{c^{2}} &amp; =0
\end{align}
\quad\begin{align}\frac{x_{1}^{2}}{a{{}^2}}+\frac{y_{1}^{2}}{b^{2}}-\frac{z_{1}^{2}}{c^{2}} &amp; =1\\
\frac{x_{2}^{2}}{a{{}^2}}+\frac{y_{2}^{2}}{b^{2}}-\frac{z_{2}^{2}}{c^{2}} &amp; =1\\
\frac{x_{3}^{2}}{a{{}^2}}+\frac{y_{3}^{2}}{b^{2}}-\frac{z_{3}^{2}}{c^{2}} &amp; =-1
\end{align}
\\
\begin{align}x_{1}^{2}+x_{2}^{2}-x_{3}^{2} &amp; =a^{2}\\
y_{1}^{2}+y_{2}^{2}-y_{3}^{2} &amp; =b^{2}\\
z_{1}^{2}+z_{2}^{2}-z_{3}^{2} &amp; =-c^{2}
\end{align}
\quad\begin{align}x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3} &amp; =0\\
x_{1}z_{1}+x_{2}z_{2}-x_{3}z_{3} &amp; =0\\
y_{1}z_{1}+y_{2}z_{2}-y_{3}z_{3} &amp; =0
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt; These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.&lt;/p&gt;

==={{anchor|Bour}} Bour (1856) – Homogeneous coordinates===

Following [[#Gauss4|Gauss (1818)]], [[w:Edmond Bour]] (1856) wrote the transformations:&lt;ref group=M&gt;Bour (1856), pp. 61; 64–65&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\cos^{2}E+\sin^{2}E-1=k\left(\cos^{2}T+\sin^{2}T-1\right)\\
\hline \left.\begin{matrix}\mathbf{(1)}\ \begin{align}\cos E &amp; =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\
\sin E &amp; =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}
\end{align}
\\
\hline \\
k=+1\\
t=\gamma+\gamma'\cos T+\gamma''\sin T,\\
1=u,\ \cos T=u',\ \sin T=u',\\
t=z,\ t\cos E=x,\ t\sin E=y\\
\downarrow\\
\mathbf{(2)}\begin{align}x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
y &amp; =\beta u+\beta'u'+\beta''u''\\
z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
\\
u &amp; =\gamma z-\alpha x-\beta y\\
u' &amp; =\alpha'x+\beta'y'-\gamma'z\\
u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\end{matrix}\right|{\scriptstyle \begin{align}-\alpha^{2}-\beta^{2}+\gamma^{2} &amp; =k\\
-\alpha^{\prime2}-\beta^{\prime2}+\gamma^{\prime2} &amp; =-k\\
-\alpha^{\prime\prime2}-\beta^{\prime\prime2}+\gamma^{\prime\prime2} &amp; =-k\\
\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =-k\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =-k\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =k\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
\alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'', implying &lt;math&gt;x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}&lt;/math&gt;. Furthermore, setting &lt;math&gt;[k,\cos T,\sin T,\cos E,\sin E]=\left[1,u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt; in transformation system (1) produces Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.&lt;/p&gt;

=== {{anchor|Somov}} Somov (1863) – Homogeneous coordinates ===

Following [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], and [[#Bour|Bour (1856)]], [[w:Osip Ivanovich Somov]] (1863) wrote the transformation systems:&lt;ref group=M&gt;Somov (1863), pp. 12–14; p. 18 for differentials.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\cos\phi &amp; =\frac{m\cos\psi+n\sin\psi+s}{m''\cos\psi+n''\sin\psi+s''}\\
\sin\phi &amp; =\frac{m'\cos\psi+n'\sin\psi+s'}{m''\cos\psi+n''\sin\psi+s''}
\end{align}
\right|\begin{matrix}\cos^{2}\phi+\cos^{2}\phi=1\\
\cos^{2}\psi+\cos^{2}\psi=1
\end{matrix}\\
\hline \mathbf{(1)}\ \begin{align}\cos\phi &amp; =x, &amp; \cos\psi &amp; =x'\\
\sin\phi &amp; =y, &amp; \sin\psi &amp; =y'
\end{align}
\ \left|\begin{align}x &amp; =\frac{mx'+ny'+s}{m''x'+n''y'+s''}\\
y &amp; =\frac{m'x'+n'y'+s'}{m''x'+n''y'+s''}
\end{align}
\right|\ \begin{matrix}x^{2}+y^{2}=1\\
x^{\prime2}+y^{\prime2}=1
\end{matrix}\\
\hline \begin{align}\cos\phi &amp; =\frac{x}{z}, &amp; \cos\psi &amp; =\frac{x'}{z'}\\
\sin\phi &amp; =\frac{y}{z}, &amp; \sin\psi &amp; =\frac{y'}{z'}
\end{align}
\ \left|\begin{align}\frac{x}{z} &amp; =\frac{mx'+ny'+sz'}{m''x'+n''y'+s''z'}\\
\frac{y}{z} &amp; =\frac{m'x'+n'y'+s'z'}{m''x'+n''y'+s''z'}
\end{align}
\right|\ \begin{matrix}x^{2}+y^{2}=z^{2}\\
x^{\prime2}+y^{\prime2}=z^{\prime2}
\end{matrix}\\
\hline \mathbf{(2)}\ \left.\begin{align}x &amp; =mx'+ny'+sz'\\
y &amp; =m'x'+n'y'+s'z'\\
z &amp; =m''x'+n''y'+s''z'\\
\\
x' &amp; =mx+m'y-m''z\\
y' &amp; =nx+n'y-n''z\\
z' &amp; =-sx-s'y+s''z\\
\\
dx &amp; =mdx'+ndy'+sdz'\\
dy &amp; =m'dx'+n'dy'+s'dz'\\
dz &amp; =m''dx'+n''dy'+s''dz'
\end{align}
\right|{\scriptstyle \begin{align}m^{2}+m^{\prime2}-m^{\prime\prime2} &amp; =1\\
n^{2}+n^{\prime2}-n^{\prime\prime2} &amp; =1\\
-s^{2}-s^{\prime2}+s^{\prime\prime2} &amp; =1\\
ns+n's'-n''s'' &amp; =0\\
sm+s'm'-s''m'' &amp; =0\\
mn+m'n'-m''n'' &amp; =0\\
\\
m^{2}+n^{2}-s^{2} &amp; =1\\
m^{\prime2}+n^{\prime2}-s^{\prime2} &amp; =1\\
-m^{\prime\prime2}-n^{\prime\prime2}+s^{\prime\prime2} &amp; =1\\
-m'm''-n'n''+s's'' &amp; =0\\
-m''m-n''n+s''s &amp; =0\\
mm'+nn'-ss' &amp; =0
\end{align}
}\\
dx^{2}+dy^{2}-dz^{2}=dx^{\prime2}+dy^{\prime2}-dz^{\prime2}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.

Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.&lt;/p&gt;

==={{anchor|Killing}} Killing (1878–1893)===

===={{anchor|Killing1}} Weierstrass coordinates====

[[w:Wilhelm Killing]] (1878–1880) described non-Euclidean geometry by using [[w:hyperboloid model|Weierstrass coordinates]] (named after [[w:Karl Weierstrass]] who described them in lectures in 1872 which Killing attended) obeying the form

:&lt;math&gt;k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}&lt;/math&gt;&lt;ref group=M&gt;Killing (1877/78), p. 74; Killing (1880), p. 279&lt;/ref&gt; with &lt;math&gt;ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}&lt;/math&gt;&lt;ref group=M&gt;Killing (1880), eq. 25 on p. 283&lt;/ref&gt;

or&lt;ref group=M&gt;Killing (1880), p. 283&lt;/ref&gt;

:&lt;math&gt;k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}&lt;/math&gt;

where ''k'' is the reciprocal measure of curvature, &lt;math&gt;k^{2}=\infty&lt;/math&gt; denotes [[w:Euclidean geometry]], &lt;math&gt;k^{2}&gt;0&lt;/math&gt; [[w:elliptic geometry]], and &lt;math&gt;k^{2}&lt;0&lt;/math&gt; hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form.&lt;ref group=M&gt;Killing (1877/78), eq. 25 on p. 283&lt;/ref&gt; In (1879/80) he tried to formulate the corresponding transformations by plugging &lt;math&gt;k^{2}&lt;/math&gt; into a [[w:Rotation matrix#Rotation matrix from axis and angle|general rotation matrix]]:&lt;ref group=M&gt;Killing (1879/80), p. 274&lt;/ref&gt;

&lt;math&gt;\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\
\hline \begin{matrix}\cos\eta\tau+\lambda^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \nu\frac{\sin\eta\tau}{\eta}+\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; -\mu\frac{\sin\eta\tau}{\eta}+\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}\\
-k^{2}\nu\frac{\sin\eta\tau}{\eta}+k^{2}\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \cos\eta\tau+\mu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}\\
k^{2}\mu\frac{\sin\eta\tau}{\eta}+k^{2}\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; -\lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \cos\eta\tau+\nu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}
\end{matrix}\\
\left(\lambda^{2}+k^{2}\mu^{2}+k^{2}\nu^{2}=\eta^{2}\right)
\end{matrix}&lt;/math&gt;

In (1885) he wrote the Weierstrass coordinates and their transformation as follows:&lt;ref group=M&gt;Killing (1885), pp. 18, 28–30, 53&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\
k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime2}+x^{\prime2}+y^{\prime2}\\
ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\
\hline \begin{align}k^{2}p' &amp; =k^{2}wp+w'x+w''y\\
x' &amp; =ap+a'x+a''y\\
y' &amp; =bp+b'x+b''y\\
\\
k^{2}p &amp; =k^{2}wp'+ax'+by'\\
x &amp; =w'p'+a'x+b'y'\\
y &amp; =w''p'+a''x'+b''y'
\end{align}
\left|{\scriptstyle \begin{align}k^{2}w^{2}+w^{\prime2}+w^{\prime\prime2} &amp; =k^{2}\\
\frac{a^{2}}{k^{2}}+a^{\prime2}+a^{\prime\prime2} &amp; =1\\
\frac{b^{2}}{k^{2}}+b^{\prime2}+b^{\prime\prime2} &amp; =1\\
aw+a'w'+a''w'' &amp; =0\\
bw+b'w'+b''w'' &amp; =0\\
\frac{ab}{k^{2}}+a'b'+a''b'' &amp; =0\\
\\
k^{2}w^{2}+a^{2}+b^{2} &amp; =k^{2}\\
\frac{w^{\prime2}}{k^{2}}+a^{\prime2}+b^{\prime2} &amp; =1\\
\frac{w^{\prime\prime2}}{k^{2}}+a^{\prime\prime2}+b^{\prime\prime2} &amp; =1\\
ww'+aa'+bb' &amp; =0\\
ww''+aa''+bb'' &amp; =0\\
\frac{w'w''}{k^{2}}+a'a''+b'b'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

In (1885) he also gave the transformation for ''n'' dimensions:&lt;ref group=M&gt;Killing (1884/85), pp. 42–43; Killing (1885), pp. 73–74, 222&lt;/ref&gt;&lt;ref&gt;Ratcliffe (1994), § 3.6&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots+dx_{n}^{2}\\
\hline \left.\begin{align}k^{2}\xi_{0} &amp; =k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{0}\\
\xi_{\varkappa} &amp; =a_{\varkappa0}x_{0}+a_{\varkappa1}x_{1}+\dots+a_{\varkappa n}x_{n}\\
\\
k^{2}x_{0} &amp; =a_{00}k^{2}\xi_{0}+a_{10}\xi_{1}+\dots+a_{n0}\xi_{n}\\
x_{\varkappa} &amp; =a_{0\varkappa}\xi_{0}+a_{1\varkappa}\xi_{1}+\dots+a_{n\varkappa}\xi_{n}
\end{align}
\right|{\scriptstyle \begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} &amp; =k^{2}\\
a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} &amp; =0\\
\frac{a_{0\iota}a_{0\varkappa}}{k^{2}}+a_{0\iota}a_{1\varkappa}+\dots+a_{n\iota}a_{n\varkappa}=\delta_{\iota\kappa} &amp; =1\ (\iota=\kappa)\ \text{or}\ 0\ (\iota\ne\kappa)
\end{align}
}
\end{matrix}&lt;/math&gt;

In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force.&lt;ref group=M&gt;Killing (1884/85), pp. 4–5&lt;/ref&gt; Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting &lt;math&gt;k^{2}=-1&lt;/math&gt; and using ''p,x,y'' as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid &lt;math&gt;p^{2}-x^{2}-y^{2}=1&lt;/math&gt; (known as [[w:hyperboloid model]]),&lt;ref group=M&gt;Killing (1885), Note 9 on p. 260&lt;/ref&gt;&lt;ref name=rey /&gt; by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;All of Killing's transformations between 1879 and 1885 don't work when &lt;math&gt;k^{2}&lt;/math&gt; is negative, thus they fail to produce Lorentz transformation ({{equationNote|1a}}) with &lt;math&gt;k^{2}=-1&lt;/math&gt;.&lt;/p&gt;

Finally, in (1893) he wrote:&lt;ref group=M&gt;Killing (1893), see pp. 144, 327–328&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\
\hline \begin{align}t' &amp; =at+bu+cv\\
u' &amp; =a't+b'u+c'v\\
v' &amp; =a''t+b''u+c''v
\end{align}
\left|\begin{align}k^{2}a^{2}+a^{\prime2}+a^{\prime\prime2} &amp; =k^{2}\\
k^{2}b^{2}+b^{\prime2}+b^{\prime\prime2} &amp; =1\\
k^{2}c^{2}+b^{\prime2}+c^{\prime\prime2} &amp; =1\\
k^{2}ab+a'b'+a''b'' &amp; =0\\
k^{2}ac+a'c'+a''c'' &amp; =0\\
k^{2}bc+b'c'+b''c'' &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

and in ''n'' dimensions&lt;ref group=M&gt;Killing (1893), pp. 314–316, 216–217&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
k^{2}y_{0}y_{0}^{\prime}+y_{1}y_{1}^{\prime}+\cdots+y_{n}y_{n}^{\prime}=k^{2}x_{0}x_{0}^{\prime}+x_{1}x_{1}^{\prime}+\cdots+x_{n}x_{n}^{\prime}\\
ds^{2}=k^{2}dx_{0}^{2}+\dots+dx_{n}^{2}\\
\hline \begin{align}y_{0} &amp; =a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{n}\\
y_{1} &amp; =a_{10}x_{0}+a_{11}x_{1}+\dots+a_{1n}x_{n}\\
 &amp; \,\,\,\vdots\\
y_{n} &amp; =a_{n0}x_{0}+a_{n1}x_{1}+\dots+a_{nn}x_{n}
\end{align}
\left|\begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} &amp; =k^{2}\\
k^{2}a_{0\varkappa}^{2}+a_{1\varkappa}^{2}+\dots+a_{n\varkappa}^{2} &amp; =1\\
k^{2}a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} &amp; =0\\
k^{2}a_{0\varkappa}a_{0\lambda}+a_{1\varkappa}a_{1\lambda}+\dots+a_{n\varkappa}a_{n\lambda} &amp; =0\\
(\varkappa,\lambda=1,\dots, n,\ \lambda\lessgtr\varkappa)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}}) with &lt;math&gt;k^{2}=-1&lt;/math&gt;.&lt;/p&gt;

===={{anchor|Killing3}} Infinitesimal transformations and Lie group====

After [[#Lie3|Lie (1885/86)]] identified the projective group of a general surface of second degree &lt;math&gt;\sum f_{ik}x_{i}'x_{k}'=0&lt;/math&gt; with the group of non-Euclidean motions, Killing (1887/88)&lt;ref group=M&gt;Killing (1887/88a), pp. 274–275&lt;/ref&gt; defined the infinitesimal projective transformations (Lie algebra) in relation to the unit hypersphere:

:&lt;math&gt;\begin{matrix}x_{1}^{2}+\dots+x_{m+1}^{2}=1\\
\hline X_{\iota\varkappa}f=x_{i}\frac{\partial f}{\partial x_{\varkappa}}-x_{\varkappa}\frac{\partial f}{\partial x_{\iota}}\\
\text{where}\\
\left(X_{\iota\varkappa},X_{\iota\lambda}\right)=X_{\varkappa\lambda};\ \left(X_{\iota\varkappa},X_{\lambda\mu}\right)=0;\\
\left[\iota\ne\varkappa\ne\lambda\ne\mu\right]
\end{matrix}&lt;/math&gt;

and in (1892) he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:&lt;ref group=M&gt;Killing (1892), p. 177&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
\hline X_{\iota\varkappa}=x_{\iota}p_{\varkappa}-x_{\varkappa}p_{\iota},\quad X_{\iota}=x_{0}p_{\iota}-\frac{x_{\iota}p_{0}}{k^{2}}\\
\text{where}\\
\left(X_{\iota}X_{\iota\varkappa}\right)=X_{\varkappa}f;\ \left(X_{\iota}X_{\varkappa\lambda}\right)=0;\ \left(X_{\iota}X_{\varkappa}\right)=-\frac{1}{k^{2}}X_{\iota\varkappa}f;
\end{matrix}&lt;/math&gt;

In (1897/98) he showed the relation between Weierstrass coordinates &lt;math&gt;k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}&lt;/math&gt; and corrdinates &lt;math&gt;k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0&lt;/math&gt; used by himself in (1887/88) and by [[#Lie3|Werner (1889), Lie (1890)]]:&lt;ref group=M&gt;Killing (1897/98), pp. 255–256&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2} &amp; (a)\\
k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2} &amp; (b)
\end{matrix}\\
\hline V_{\varkappa}=k^{2}x_{0}p_{\varkappa}-x_{\varkappa}p_{0},\quad U_{\iota\varkappa}=p_{\iota}x_{\varkappa}-p_{\varkappa}x_{\iota}\\
\text{where}\\
\left(V_{\iota},V_{\varkappa}\right)=k^{2}U_{\iota\varkappa},\ \left(V_{\iota},U_{\iota\varkappa}\right)=-V_{\varkappa},\ \left(V_{\iota},U_{\varkappa\lambda}\right)=0,\\
\left(U_{\iota\varkappa},U_{\iota\lambda}\right)=U_{\varkappa\lambda},\ \left(U_{\iota\varkappa},U_{\lambda\mu}\right)=0\\
\left[\iota,\varkappa,\lambda,\mu=1,2,\dots n\right]\\
\hline \begin{matrix}y_{1}=\frac{x_{1}}{x_{0}},\ y_{2}=\frac{x_{2}}{x_{0}},\dots y_{n}=\frac{x_{n}}{x_{0}}\\
\downarrow\\
k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0\\
\hline q_{\varkappa}+\frac{y_{\varkappa}}{k^{2}}\sum_{\varrho}y_{y}q_{\varrho},\quad q_{\iota}y_{\varkappa}-q_{\varkappa}y_{\iota}
\end{matrix}
\end{matrix}&lt;/math&gt;

He pointed out that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to quadratic form (a) and [[w:Group action (mathematics)|transitive]] when related to quadratic form (b).

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Setting &lt;math&gt;k^{2}=-1&lt;/math&gt; denotes the group of hyperbolic motions and thus the Lorentz group.&lt;/p&gt;

=== {{anchor|Poincare}} Poincaré (1881) – Weierstrass coordinates ===
[[w:Henri Poincaré]] (1881) connected the work of [[../Lorentz transformation (Cayley-Hermite)#Hermite|E:Hermite (1853)]] and [[../Lorentz transformation (Möbius)#Selling|E:Selling (1873)]] on indefinite quadratic forms with non-Euclidean geometry (Poincaré already discussed such relations in an unpublished manuscript in 1880).&lt;ref&gt;Gray (1997)&lt;/ref&gt; He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates (without using that expression) connected by a transformation with integer coefficients:&lt;ref group=M name=p1&gt;Poincaré (1881a), pp. 133–134&lt;/ref&gt;&lt;ref&gt;Dickson (1923), pp. 220–221&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}F &amp; =(ax+by+cz)^{2}+(a'x+b'y+c'z)^{2}-(a''x+b''y+c''z)^{2}\\
 &amp; =\xi^{2}+\eta^{2}-\zeta^{2}=-1\\
F &amp; =(ax'+by'+cz')^{2}+(a'x'+b'y'+c'z')^{2}-(a''x'+b''y'+c''z')^{2}\\
 &amp; =\xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1
\end{align}
\\
\hline \begin{align}\xi' &amp; =\alpha\xi+\beta\eta+\gamma\zeta\\
\eta' &amp; =\alpha'\xi+\beta'\eta+\gamma'\zeta\\
\zeta' &amp; =\alpha''\xi+\beta''\eta+\gamma''\zeta
\end{align}
\left|\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =1\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =1\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-1\\
\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

He went on to describe the properties of &quot;hyperbolic coordinates&quot;.&lt;ref group=M name=poinc&gt;Poincaré (1881b), p. 333&lt;/ref&gt;&lt;ref name=rey&gt;Reynolds (1993)&lt;/ref&gt; Poincaré mentioned the hyperboloid model also in (1887).&lt;ref group=M&gt;Poincaré (1887), p. 206&lt;/ref&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.&lt;/p&gt;

=== {{anchor|Cox}} Cox (1881–1883) – Weierstrass coordinates ===
[[w:Homersham Cox (mathematician)|Homersham Cox]] (1881/82) – referring to similar rectangular coordinates used by [[w:Christoph Gudermann|Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt; and [[w:George Salmon]] (1862)&lt;ref group=M&gt;Salmon (1862), section 212, p. 165&lt;/ref&gt; on a sphere, and to [[#Escherich|Escherich (1874)]] as reported by [[w:Johannes Frischauf]] (1876)&lt;ref group=M&gt;Frischauf (1876), pp. 86–87&lt;/ref&gt; in the hyperbolic plane – defined the Weierstrass coordinates (without using that expression) and their transformation:&lt;ref group=M&gt;Cox (1881), p. 186 for Weierstrass coordinates; (1881/82), pp. 193–194 for Lorentz transformation. On p. 193, the misprinted expression &lt;math&gt;x^{2}-y^{2}-z^{2}&lt;/math&gt; should read &lt;math&gt;z^{2}-y^{2}-x^{2}&lt;/math&gt;&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}z^{2}-x^{2}-y^{2}=1\\
z^{2}-y^{2}-x^{2}=Z^{2}-Y^{2}-X^{2}\\
\hline \begin{align}x &amp; =l_{1}X+l_{2}Y+l_{3}Z\\
y &amp; =m_{1}X+m_{2}Y+m_{3}Z\\
z &amp; =n_{1}X+n_{2}Y+n_{3}Z\\
\\
X &amp; =l_{1}x+m_{1}y-n_{1}z\\
Y &amp; =l_{2}x+m_{2}y-n_{2}z\\
Z &amp; =l_{3}x+m_{3}y-n_{3}z
\end{align}
\left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} &amp; =1\\
l_{2}^{2}+m_{2}^{2}-n_{2}^{2} &amp; =1\\
l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =1\\
l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} &amp; =0\\
l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} &amp; =0\\
l_{3}l_{1}+m_{3}m_{1}-n_{3}n_{1} &amp; =0\\
\\
l_{1}^{2}+l_{2}^{2}-l_{3}^{2} &amp; =1\\
m_{1}^{2}+m_{2}^{2}-m_{3}^{2} &amp; =1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =1\\
l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} &amp; =0\\
m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} &amp; =0\\
n_{1}l_{1}+n_{2}l_{2}-n_{3}l_{3} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Replacing &lt;math&gt;{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =1
\end{align}
}&lt;/math&gt; with &lt;math&gt;{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =-1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =-1
\end{align}
}&lt;/math&gt;, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' up to a sign change in the inverse transformation.&lt;/p&gt;

Cox also gave the Weierstrass coordinates and their transformation in hyperbolic space:&lt;ref group=M&gt;Cox (1881), pp. 199, 206–207&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\
w^{2}-x^{2}-y^{2}-z^{2}=w^{\prime2}-x^{\prime2}-y^{\prime2}-z^{\prime2}\\
\hline \begin{align}x &amp; =l_{1}x'+l_{2}y'+l_{3}z'-l_{4}w'\\
y &amp; =m_{1}x'+m_{2}y'+m_{3}z'-m_{4}w'\\
z &amp; =n_{1}x'+n_{2}y'+n_{3}z'-n_{4}w'\\
w &amp; =r_{1}x'+r_{2}y'+r_{3}z'-r_{4}w'\\
\\
x' &amp; =l_{1}x+m_{1}y+n_{1}z-r_{1}w\\
y' &amp; =l_{2}x+m_{2}y+n_{2}z-r_{2}w\\
z' &amp; =l_{3}x+m_{3}y+n_{3}z-r_{3}w\\
w' &amp; =l_{4}x+m_{4}y+n_{4}z-r_{4}w
\end{align}
\left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}+n_{1}^{2}-r_{1}^{2} &amp; =1\\
l_{2}^{2}+m_{2}^{2}+n_{2}^{2}-r_{2}^{2} &amp; =1\\
l_{3}^{2}+m_{3}^{2}+n_{3}^{2}-r_{3}^{2} &amp; =1\\
l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2} &amp; =1\\
l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3}-r_{2}r_{3} &amp; =0\\
l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}-r_{3}r_{1} &amp; =0\\
l_{1}l_{4}+m_{1}m_{4}+n_{1}n_{4}-r_{1}r_{4} &amp; =0\\
l_{2}l_{4}+m_{2}m_{4}+n_{2}n_{4}-r_{2}r_{4} &amp; =0\\
l_{3}l_{4}+m_{3}m_{4}+n_{3}n_{4}-r_{3}r_{4} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Replacing &lt;math&gt;{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=1}&lt;/math&gt; with &lt;math&gt;{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=-1}&lt;/math&gt;, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' up to a sign change in both the first as well as inverse transformation.&lt;/p&gt;

=== {{anchor|Hill}} Hill (1882) – Homogeneous coordinates ===

Following [[#Gauss4|Gauss (1818)]], [[w:George William Hill]] (1882) formulated the equations&lt;ref group=M&gt;Hill (1882), pp. 323–325&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)\\
k\left(\sin^{2}E+\cos^{2}E-1\right)\\
\hline \begin{align} &amp;  &amp; \cos E' &amp; =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
 &amp; \mathbf{(1)} &amp; \sin E' &amp; =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
\hline \\
 &amp;  &amp; x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
 &amp;  &amp; y &amp; =\beta u+\beta'u'+\beta''u''\\
 &amp;  &amp; z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
 &amp; \mathbf{(2)}\\
 &amp;  &amp; u &amp; =-\alpha x-\beta y+\gamma z\\
 &amp;  &amp; u' &amp; =\alpha'x+\beta'y'-\gamma'z\\
 &amp;  &amp; u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\left|{\scriptstyle \begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =-1\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =1\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =1\\
\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
(k=-1)\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =k\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =k\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-k\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with &lt;math&gt;[\cos T,\sin T,\cos E',\sin E']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt;.

Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' .&lt;/p&gt;

=== {{anchor|Picard}} Picard (1882-1884) – Quadratic forms ===

[[w:Émile Picard]] (1882) analyzed the invariance of indefinite ternary [[w:Hermitian form|Hermitian quadratic forms]] with integer coefficients and their relation to [[w:Group action (mathematics)|discontinuous groups]], extending Poincaré's Fuchsian functions of one complex variable related to a circle, to &quot;hyperfuchsian&quot; functions of two complex variables related to a [[w:hypersphere]]. He formulated the following special case of an Hermitian form:&lt;ref group=M&gt;Picard (1882), pp. 307–308 first transformation system; pp. 315-317 second transformation system&lt;/ref&gt;&lt;ref&gt;Dickson (1923), pp. 280-281&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{matrix}xx_{0}+yy_{0}-zz_{0}\\
\\
\mathbf{(1)}\ \begin{align}x &amp; =M_{1}X+P_{1}Y+R_{1}Z\\
y &amp; =M_{2}X+P_{2}Y+R_{2}Z\\
z &amp; =M_{3}X+P_{3}Y+R_{3}Z
\end{align}
\\
\\
\left[\begin{align}[][x,y,z]=\text{complex}\\
\left[x_{0},y_{0},z_{0}\right]=\text{conjugate}
\end{align}
\right]\\
\\
\hline \\
x^{\prime2}+x^{\prime\prime2}+y^{\prime2}+y^{\prime\prime2}=1\\
x=x'+ix'',\quad y=y'+iy''\\
\\
\mathbf{(2)}\ \begin{align}X &amp; =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\
Y &amp; =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}}
\end{align}
\end{matrix}\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3} &amp; =1\\
P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} &amp; =1\\
R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} &amp; =-1\\
P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3} &amp; =0\\
M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} &amp; =0\\
P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} &amp; =0\\
\\
M_{1}\mu_{1}+P_{1}\pi_{1}-R_{1}\rho_{1} &amp; =1\\
M_{2}\mu_{2}+P_{2}\pi_{2}-R_{2}\rho_{2} &amp; =1\\
M_{3}\mu_{3}+P_{3}\pi_{3}-R_{3}\rho_{3} &amp; =-1\\
\mu_{2}M_{1}+\pi_{2}P_{1}-R_{1}\rho_{2} &amp; =0\\
\mu_{2}M_{3}+\pi_{2}P_{3}-R_{3}\rho_{2} &amp; =0\\
\mu_{3}M_{1}+\pi_{3}P_{1}-R_{1}\rho_{3} &amp; =0\\
\\
\left[\begin{align}[][M,P,R\dots]=\text{complex}\\
\left[\mu,\pi,\rho\dots\right]=\text{conjugate}
\end{align}
\right]
\end{align}
}\right.\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;-Z&lt;sup&gt;2&lt;/sup&gt;'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;=1''.&lt;/p&gt;

Or in (1884a) in relation to indefinite binary Hermitian quadratic forms:&lt;ref group=M&gt;Picard (1884a), p. 13&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}UU_{0}-VV_{0}=uu_{0}-vv_{0}\\
\hline \begin{align}U &amp; =\mathcal{A}u+\mathcal{B}v\\
V &amp; =\mathcal{C}u+\mathcal{D}v
\end{align}
\left|\begin{align}\mathcal{A}\mathcal{A}_{0}-\mathcal{C}\mathcal{C}_{0} &amp; =1\\
\mathcal{A}\mathcal{B}_{0}-\mathcal{C}\mathcal{D}_{0} &amp; =0\\
\mathcal{B}\mathcal{B}_{0}-\mathcal{D}\mathcal{D}_{0} &amp; =-1\\
\mathcal{D}\mathcal{D}_{0}-\mathcal{C}\mathcal{C}_{0} &amp; =1
\end{align}
\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' producing ''U&lt;sup&gt;2&lt;/sup&gt;-V&lt;sup&gt;2&lt;/sup&gt;=u&lt;sup&gt;2&lt;/sup&gt;-v&lt;sup&gt;2&lt;/sup&gt;''.&lt;/p&gt;

Or in (1884b):&lt;ref group=M&gt;Picard (1884b), p. 416&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}xx_{0}+yy_{0}-1=0\\
\hline \begin{align}X &amp; =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\
Y &amp; =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}}
\end{align}
\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3}=P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} &amp; =1\\
R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} &amp; =-1\\
P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3}=M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3}=P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} &amp; =0\\
M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;=1''.&lt;/p&gt;

Or in (1884c):&lt;ref group=M&gt;Picard (1884c), pp. 123–124; 163&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}UU_{0}+VV_{0}-WW_{0}=uu_{0}+vv_{0}-ww_{0}\\
\hline \mathbf{(1)}\ \begin{align}U &amp; =Mu+Pv+Rw\\
V &amp; =M'u+P'v+R'w\\
W &amp; =M''u+P''v+R''w\\
\\
u &amp; =M_{0}U+M_{0}^{\prime}V-M_{0}^{\prime\prime}W\\
v &amp; =P_{0}U+P_{0}^{\prime}V-P_{0}^{\prime\prime}W\\
w &amp; =-R_{0}U-R_{0}^{\prime}V+R_{0}^{\prime\prime}W
\end{align}
\left|{\scriptstyle \begin{align}MM_{0}+M'M_{0}^{\prime}-M''M_{0}^{\prime\prime} &amp; =1\\
PP_{0}+P'P_{0}^{\prime}-P''P_{0}^{\prime\prime} &amp; =1\\
RR_{0}+R'R_{0}^{\prime}-R''R_{0}^{\prime\prime} &amp; =-1\\
MP_{0}+M'P_{0}^{\prime}-M''P_{0}^{\prime\prime} &amp; =0\\
MR_{0}+M'R_{0}^{\prime}-M''R_{0}^{\prime\prime} &amp; =0\\
PR_{0}+P'R_{0}^{\prime}-P''R_{0}^{\prime\prime} &amp; =0\\
\\
MM_{0}+PP_{0}-RR_{0} &amp; =1\\
M'M_{0}^{\prime}+P'P_{0}^{\prime}-R'R_{0}^{\prime} &amp; =1\\
M''M_{0}^{\prime\prime}+P''P_{0}^{\prime\prime}-R''R_{0}^{\prime\prime} &amp; =-1\\
M_{0}M'+P_{0}P'-R_{0}R' &amp; =0\\
M_{0}M''+P_{0}P''-R_{0}R'' &amp; =0\\
M_{0}^{\prime}M''+P_{0}^{\prime}P''-R_{0}^{\prime}R'' &amp; =0
\end{align}
}\right.\\
\hline \text{Invariance of unit hypersphere:}\\
\mathbf{(2)}\ \begin{align}\xi' &amp; =\frac{A\xi+A'\eta+A''}{C\xi+C'\eta+C''}\\
\eta' &amp; =\frac{B\xi+B'\eta+B''}{C\xi+C'\eta+C''}
\end{align}
\left|{\scriptstyle \begin{align}AA_{0}+A'A_{0}^{\prime}-A''A_{0}^{\prime\prime} &amp; =1\\
BB_{0}+B'B_{0}^{\prime}-B''B_{0}^{\prime\prime} &amp; =1\\
CC_{0}+C'C_{0}^{\prime}-C''C_{0}^{\prime\prime} &amp; =-1\\
AB_{0}+A'B_{0}^{\prime}-A''B_{0}^{\prime\prime} &amp; =0\\
AC_{0}+A'C_{0}^{\prime}-A''C_{0}^{\prime\prime} &amp; =0\\
BC_{0}+B'C_{0}^{\prime}-B''C_{0}^{\prime\prime} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''U&lt;sup&gt;2&lt;/sup&gt;+V&lt;sup&gt;2&lt;/sup&gt;-W&lt;sup&gt;2&lt;/sup&gt;=u&lt;sup&gt;2&lt;/sup&gt;+v&lt;sup&gt;2&lt;/sup&gt;-w&lt;sup&gt;2&lt;/sup&gt;'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing &lt;math&gt;\xi^{\prime2}+\eta^{\prime2}=\xi^{2}+\eta^{2}=1&lt;/math&gt;.&lt;/p&gt;

=== {{anchor|Callandreau}} Callandreau (1885) – Homography ===

Following [[#Gauss4|Gauss (1818)]] and [[#Hill|Hill (1882)]], [[w:Octave Callandreau]] (1885) formulated the equations&lt;ref group=M&gt;Callandreau (1885), pp. A.7; A.12&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)=\\
{\scriptstyle (\alpha+\alpha'\sin T+\alpha''\cos T)^{2}+(\beta+\beta'\sin T+\beta''\cos T)^{2}-(\gamma+\gamma'\sin T+\gamma''\cos T)^{2}}\\
\hline \begin{align}\cos\varepsilon' &amp; =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
\sin\varepsilon' &amp; =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}
\end{align}
\left|{\scriptstyle \begin{align} &amp; \left(k=1\right)\\
\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =-k &amp; \alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =+k &amp; \alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =+k &amp; \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =-1 &amp; \alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =-1 &amp; \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =+1 &amp; \beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The transformation system is equivalent to Lorentz transformation ({{equationNote|1b}})  ''(n=2)'' with &lt;math&gt;[\cos T,\sin T,\cos\varepsilon',\sin\varepsilon']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Lie3}} Lie (1885-1890) – Lie group, hyperbolic motions, and infinitesimal transformations===

In (1885/86), [[w:Sophus Lie]] identified the projective group of a general surface of second degree &lt;math&gt;\sum f_{ik}x_{i}'x_{k}'=0&lt;/math&gt; with the group of non-Euclidean motions.&lt;ref group=M&gt;Lie (1885/86), p. 411&lt;/ref&gt; In a thesis guided by Lie, [[w:Hermann Werner]] (1889) discussed this projective group by using the equation of a unit hypersphere as the surface of second degree (which was already given before by [[#Killing3|Killing (1887)]]), and also gave the corresponding infinitesimal projective transformations (Lie algebra):&lt;ref group=M&gt;Werner (1889), pp. 4, 28&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=1\\
\hline x_{i}p_{\varkappa}-x_{\varkappa}p_{i},\quad p_{i}-x_{i}\sum_{1}^{n}{\scriptstyle j}\ x_{j}p_{j}\quad(i,\varkappa=1,\dots, n)\\
\text{where}\\
\left(Q_{i},Q_{\varkappa}\right)=R_{i,\varkappa};\ \left(Q_{i},Q_{j,\varkappa}\right)=\varepsilon_{i,j}Q_{\varkappa}-\varepsilon_{i,\varkappa}Q_{j};\\
\left(R_{i,\varkappa},R_{\mu,\nu}\right)=\varepsilon_{\varkappa,\mu}R_{i,\nu}-\varepsilon_{\varkappa,\nu}R_{i,\mu}-\varepsilon_{,\mu}R_{\varkappa,\nu}+\varepsilon_{i,\nu}R_{\varkappa,\mu}\\
\left[\varepsilon_{i,\varkappa}\equiv0\ \text{for}\ i\ne\varkappa;\ \varepsilon_{i,i}=1\right]
\end{matrix}&lt;/math&gt;

More generally, Lie (1890)&lt;ref group=M&gt;Lie (1890a), p. 295;&lt;/ref&gt; defined non-Euclidean motions in terms of two forms &lt;math&gt;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\pm1=0&lt;/math&gt; in which the imaginary form with &lt;math&gt;+1&lt;/math&gt; denotes the group of elliptic motions (in Klein's terminology), the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:&lt;ref group=M&gt;Lie (1890a), p. 311&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+\dots+x_{n}^{2}-1=0\\
\hline p_{k}-x_{k}\sum j_{1}^{0}x_{j}p_{j},\quad x_{i}p_{k}-x_{k}p_{i}\quad(i,k=1\dots n)
\end{matrix}&lt;/math&gt;

Summarizing, Lie (1893) discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form:

:&lt;math&gt;x^{2}+y^{2}-1=0&lt;/math&gt;&lt;ref group=M&gt;Lie (1893), p. 474&lt;/ref&gt; or &lt;math&gt;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1=0&lt;/math&gt;&lt;ref group=M&gt;Lie (1893), p. 479&lt;/ref&gt; or &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-1=0&lt;/math&gt;.&lt;ref group=M&gt;Lie (1893), p. 481&lt;/ref&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The group of hyperbolic motions is isomorphic to the Lorentz group. The interval &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-1=0&lt;/math&gt; becomes the Lorentz interval &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-x_{0}^{2}=0&lt;/math&gt; by setting
&lt;br&gt;&lt;math&gt;(x_{1},\dots,\ x_{n},\ 1)=\left(\frac{x_{1}}{x_{0}},\dots,\ \frac{x_{n}}{x_{0}},\ \frac{x_{0}}{x_{0}}\right)&lt;/math&gt;&lt;/p&gt;

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane using the following invariant and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)'':&lt;ref group=M&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}-Z^{2}=1\\
X^{2}+Y^{2}-Z^{2}=X^{\prime2}+Y^{\prime2}-Z^{\prime2}\\
\hline \begin{align}X &amp; =aX'+a'Y'+a''Z'\\
Y &amp; =bX'+b'Y'+b''Z'\\
Z &amp; =cX'+c'Y'+c''Z'\\
\\
X' &amp; =aX+bY-cZ\\
Y' &amp; =a'X+b'Y-c'Z\\
Z' &amp; =-a''X-b''Y+c''Z
\end{align}
\left|\begin{align}a^{2}+b^{2}-c^{2} &amp; =1\\
a^{\prime2}+b^{\prime2}-c^{\prime2} &amp; =1\\
a^{\prime\prime2}+b^{\prime\prime2}-c^{\prime\prime2} &amp; =-1\\
aa'+bb'-cc' &amp; =0\\
a'a''+b'b''-c'c'' &amp; =0\\
a''a+b''b-c''c &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.&lt;/p&gt;

He gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Hausdorff}} Hausdorff (1899) – Weierstrass coordinates===

[[w:Felix Hausdorff]] (1899) – citing Killing (1885) – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:&lt;ref group=M&gt;Hausdorff (1899), p. 165, pp. 181-182&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{2}-x^{2}-y^{2}=1\\
\hline \begin{align}x &amp; =a_{1}x'+a_{2}y'+x_{0}p'\\
y &amp; =b_{1}x'+b{}_{2}y'+y_{0}p'\\
p &amp; =e_{1}x'+e_{2}y'+p_{0}p'\\
\\
x' &amp; =a_{1}x+b_{1}y-e_{1}p\\
y' &amp; =a_{2}x+b_{2}y-e_{2}p\\
-p' &amp; =x_{0}x+y_{0}y-p_{0}p
\end{align}
\left|{\scriptstyle \begin{align}a_{1}^{2}+b_{1}^{2}-e_{1}^{2} &amp; =1\\
a_{2}^{2}+b_{2}^{2}-e_{2}^{2} &amp; =1\\
-x_{0}^{2}-y_{0}^{2}+p_{0}^{2} &amp; =1\\
a_{2}x_{0}+b_{2}y_{0}-e_{2}p_{0} &amp; =0\\
a_{1}x_{0}+b_{1}y_{0}-e_{1}p_{0} &amp; =0\\
a_{1}a_{2}+b_{1}b_{2}-e_{1}e_{2} &amp; =0\\
\\
a_{1}^{2}+a_{2}^{2}-x_{0}^{2} &amp; =1\\
b_{1}^{2}+b_{2}^{2}-y_{0}^{2} &amp; =1\\
-e_{1}^{2}-e_{2}^{2}+p_{0}^{2} &amp; =1\\
b_{1}e_{1}+b_{2}e_{2}-y_{0}p_{0} &amp; =0\\
a_{1}e_{1}+a_{2}e_{2}-x_{0}p_{0} &amp; =0\\
a_{1}b_{1}+a_{2}b_{2}-x_{0}y_{0} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.&lt;/p&gt;

==={{anchor|Woods2}} Woods (1901-05) – Beltrami and Weierstrass coordinates ===
In (1901/02) [[w:Frederick S. Woods]] defined the following invariant quadratic form and its [[w:projective transformation]] in terms of Beltrami coordinates (he pointed out that this can be connected to hyperbolic geometry by setting &lt;math&gt;k=\sqrt{-1}R&lt;/math&gt; with ''R'' as real quantity):&lt;ref group=M&gt;Woods (1901/02), p. 98, 104&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}\left(u^{2}+v^{2}+w^{2}\right)+1=0\\
\hline \begin{align}u' &amp; =\frac{\alpha_{1}u+\alpha_{2}v+\alpha_{3}w+\alpha_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\
v' &amp; =\frac{\beta_{1}u+\beta_{2}v+\beta_{3}w+\beta_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\
w' &amp; =\frac{\gamma_{1}u+\gamma_{2}v+\gamma_{3}w+\gamma_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}
\end{align}
\left|\begin{align}k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right)+\delta_{i}^{2} &amp; =k^{2}\\
(i=1,2,3)\\
k^{2}\left(\alpha_{4}^{2}+\beta_{4}^{2}+\gamma_{4}^{2}\right)+\delta_{4}^{2} &amp; =1\\
\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}+\delta_{i}\delta_{h} &amp; =0\\
(i,h=1,2,3,4;\ i\ne h)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'' with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.&lt;/p&gt;

Alternatively, Woods (1903, published 1905) – citing Killing (1885) – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation (with &lt;math&gt;k=\sqrt{-1}k&lt;/math&gt; for hyperbolic space):&lt;ref group=M&gt;Woods (1903/05), pp. 45–46; p. 48)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\\
ds^{2}=\frac{1}{k^{2}}dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\\
\hline \begin{align}x_{1}^{\prime} &amp; =\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+\alpha_{0}x_{0}\\
x_{2}^{\prime} &amp; =\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{0}x_{0}\\
x_{3}^{\prime} &amp; =\gamma_{1}x_{1}+\gamma_{2}x_{2}+\gamma_{3}x_{3}+\gamma_{0}x_{0}\\
x_{0}^{\prime} &amp; =\delta_{1}x_{1}+\delta_{2}x_{2}+\delta_{3}x_{3}+\delta_{0}x_{0}
\end{align}
\left|\begin{align}\delta_{0}^{2}+k^{2}\left(\alpha_{0}^{2}+\beta_{0}^{2}+\gamma_{0}^{2}\right) &amp; =1\\
\delta_{i}^{2}+k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right) &amp; =k^{2}\\
(i=1,2,3)\\
\delta_{i}\delta_{h}+k^{2}\left(\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}\right) &amp; =0\\
(i,h=0,1,2,3;\ i\ne h)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.&lt;/p&gt;

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – used the invariant quadratic form and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)''&lt;ref group=M&gt;Liebmann (1904/05), p. 168; pp. 175–176&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{\prime2}-x^{\prime2}-y^{\prime2}=1\\
\hline \begin{align}x_{1} &amp; =\alpha_{11}x+\alpha_{12}y+\alpha_{13}p\\
y_{1} &amp; =\alpha_{21}x+\alpha_{22}y+\alpha_{23}p\\
x_{1} &amp; =\alpha_{31}x+\alpha_{32}y+\alpha_{33}p\\
\\
x &amp; =\alpha_{11}x_{1}+\alpha_{21}y_{1}-\alpha_{31}p_{1}\\
y &amp; =\alpha_{12}x_{1}+\alpha_{22}y_{1}-\alpha_{32}p_{1}\\
p &amp; =-\alpha_{13}x_{1}-\alpha_{23}y_{1}+\alpha_{33}p_{1}
\end{align}
\left|\begin{align}\alpha_{33}^{2}-\alpha_{13}^{2}-\alpha_{23}^{2} &amp; =1\\
-\alpha_{31}^{2}+\alpha_{11}^{2}+\alpha_{21}^{2} &amp; =1\\
-\alpha_{32}^{2}+\alpha_{12}^{2}+\alpha_{22}^{2} &amp; =1\\
\alpha_{31}\alpha_{32}-\alpha_{11}\alpha_{12}-\alpha_{21}\alpha_{22} &amp; =0\\
\alpha_{32}\alpha_{33}-\alpha_{12}\alpha_{13}-\alpha_{22}\alpha_{23} &amp; =0\\
\alpha_{33}\alpha_{31}-\alpha_{23}\alpha_{11}-\alpha_{23}\alpha_{21} &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.&lt;/p&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bour56att}} 
*{{#section:History of Topics in Special Relativity/mathsource|chal82sec}} 
*{{#section:History of Topics in Special Relativity/mathsource|chas29}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|fris76}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau98}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau18}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|haus99}} 
*{{#section:History of Topics in Special Relativity/mathsource|hill82}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac27}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac32a}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac32b}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac33}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil77}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil79}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil84}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil85}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag73}} 
*{{#section:History of Topics in Special Relativity/mathsource|leb37}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie85}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie90}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie93}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic82}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84a}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84b}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84c}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81a}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81b}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin87}} 
*{{#section:History of Topics in Special Relativity/mathsource|sal62}} 
*{{#section:History of Topics in Special Relativity/mathsource|som63}} 
*{{#section:History of Topics in Special Relativity/mathsource|wedd47}} 
*{{#section:History of Topics in Special Relativity/mathsource|wern89}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L1}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
      <sha1>dggw5ykur4zf6nahn9dphi50esmny2h</sha1>
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      <comment>/* Most general Lorentz transformation of velocity */</comment>
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      <text bytes="87351" xml:space="preserve">{{../Lorentz transformation (header)}}
==Most general Lorentz transformations==
===General quadratic form===
The general [[w:quadratic form]] ''q(x)'' with coefficients of a [[w:symmetric matrix]] '''A''', the associated [[w:bilinear form]] ''b(x,y)'', and the [[w:linear transformation]]s of ''q(x)'' and ''b(x,y)'' into ''q(x′)'' and ''b(x′,y′)'' using the [[w:transformation matrix]] '''g''', can be written as&lt;ref&gt;Bôcher (1907), chapter X&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\begin{align}\begin{align}q=\sum_{0}^{n}A_{ij}x_{i}x_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}\end{align}
 &amp; =q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{x}'\\
b=\sum_{0}^{n}A_{ij}x_{i}y_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{y} &amp; =b'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{y}'
\end{align}
\quad\left(A_{ij}=A_{ji}\right)\\
\hline \left.\begin{align}x_{i}^{\prime} &amp; =\sum_{j=0}^{n}g_{ij}x_{j}=\mathbf{g}\cdot\mathbf{x}\\
x_{i} &amp; =\sum_{j=0}^{n}g_{ij}^{(-1)}x_{j}^{\prime}=\mathbf{g}^{-1}\cdot\mathbf{x}'
\end{align}
\right|\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g}=\mathbf{A}'
\end{matrix}&lt;/math&gt;|{{equationRef|Q1}}}}

The case ''n=1'' is the [[w:binary quadratic form]] introduced by [[#Lagrange|Lagrange (1773)]] and [[#Gauss|Gauss (1798/1801)]], ''n=2'' is the ternary quadratic form introduced by [[#Gauss2|Gauss (1798/1801)]], ''n=3'' is the quaternary quadratic form etc.

===Most general Lorentz transformation===
The general Lorentz transformation follows from ({{equationNote|Q1}}) by setting '''A'''='''A′'''=diag(-1,1,...,1) and det '''g'''=±1. It forms an [[w:indefinite orthogonal group]] called the [[w:Lorentz group]] O(1,n), while the case det '''g'''=+1 forms the restricted [[w:Lorentz group]] SO(1,n). The quadratic form ''q(x)'' becomes the [[w:Lorentz interval]] in terms of an [[w:indefinite quadratic form]] of [[w:Minkowski space]] (being a special case of [[w:pseudo-Euclidean space]]), and the associated bilinear form ''b(x)'' becomes the [[w:Minkowski inner product]]:&lt;ref name=ratcliffe&gt;Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1&lt;/ref&gt;&lt;ref&gt;Naimark (1964), 2 in four dimensions&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\begin{align}-x_{0}^{2}+\cdots+x_{n}^{2} &amp; =-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}\\
-x_{0}y_{0}+\cdots+x_{n}y_{n} &amp; =-x_{0}^{\prime}y_{0}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime}
\end{align}
\\
\hline \left.\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\
\downarrow\\
\begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}+\dots+x_{n}g_{0n}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}+\dots+x_{n}g_{1n}\\
 &amp; \dots\\
x_{n}^{\prime} &amp; =x_{0}g_{n0}+x_{1}g_{n1}+\dots+x_{n}g_{nn}
\end{align}
\\
\\
\mathbf{x}=\mathbf{g}^{-1}\cdot\mathbf{x}'\\
\downarrow\\
\begin{align}x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}-\dots-x_{n}^{\prime}g_{n0}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}+\dots+x_{n}^{\prime}g_{n1}\\
 &amp; \dots\\
x_{n} &amp; =-x_{0}^{\prime}g_{0n}+x_{1}^{\prime}g_{1n}+\dots+x_{n}^{\prime}g_{nn}
\end{align}
\end{matrix}\right|\begin{matrix}\begin{align}\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}}\cdot\mathbf{A} &amp; =\mathbf{g}^{-1}\\
\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g} &amp; =\mathbf{A}\\
\mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} &amp; =\mathbf{A}\\
\\
\end{align}
\\
\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} &amp; =\left\{ \begin{align}-1\quad &amp; (j=k=0)\\
1\quad &amp; (j=k&gt;0)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} &amp; =\left\{ \begin{align}-1\quad &amp; (i=k=0)\\
1\quad &amp; (i=k&gt;0)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
\end{matrix}
\end{matrix}&lt;/math&gt;|{{equationRef|1a}}}}

Such general Lorentz transformations ({{equationNote|1a}}) for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]] in order to simplify computations of [[w:elliptic function]]s and integrals.&lt;ref&gt;Musen (1970) pointed out the intimate connection of Hill's scalar development and Minkowski's pseudo-Euclidean 3D space.&lt;/ref&gt;&lt;ref&gt;Touma et al. (2009) showed the analogy between Gauss and Hill's equations and Lorentz transformations, see eq. 22-29.&lt;/ref&gt; They were also used by [[#Chasles|Chasles (1829)]] and [[#Weddle|Weddle (1847)]] to describe relations on hyperboloids, as well as by [[#Poincare|Poincaré (1881)]], [[#Cox|Cox (1881/82)]], [[#Picard|Picard (1882, 1884)]], [[#Killing|Killing (1885, 1893)]], [[#Gerard|Gérard (1892)]], [[#Hausdorff|Hausdorff (1899)]], [[#Woods2|Woods (1901, 1903)]], [[#Liebmann|Liebmann (1904/05)]] to describe [[w:hyperbolic motion]]s (i.e. rigid motions in the [[w:hyperbolic plane]] or [[w:hyperbolic space]]), which were expressed in terms of Weierstrass coordinates of the [[w:hyperboloid model]] satisfying the relation &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=-1&lt;/math&gt; or in terms of the [[w:Cayley–Klein metric]] of [[w:projective geometry]] using the &quot;absolute&quot; form &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=0&lt;/math&gt;.&lt;ref group=M&gt;Killing (1885), p. 71&lt;/ref&gt;&lt;ref&gt;Müller (1910), p. 661, in particular footnote 247.&lt;/ref&gt;&lt;ref&gt;Sommerville (1911), p. 286, section K6.&lt;/ref&gt; In addition, [[w:infinitesimal transformation]]s related to the [[w:Lie algebra]] of the group of hyperbolic motions were given in terms of Weierstrass coordinates &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=-1&lt;/math&gt; by [[#Killing3|Killing (1888-1897)]].

===Most general Lorentz transformation of velocity===

If &lt;math&gt;x_{i},\ x_{i}^{\prime}&lt;/math&gt; in ({{equationNote|1a}}) are interpreted as [[w:homogeneous coordinates]], then the corresponding inhomogenous coordinates &lt;math&gt;u_{s},\ u_{s}^{\prime}&lt;/math&gt; follow by

:&lt;math&gt;\frac{x_{s}}{x_{0}}=u_{s},\ \frac{x_{s}^{\prime}}{x_{0}^{\prime}}=u_{s}^{\prime}\ (s=1,2\dots n)&lt;/math&gt;

so that the Lorentz transformation becomes a [[w:homography]] inside the [[w:unit sphere]], which [[w:John Lighton Synge]] called &quot;the most general formula for the composition of velocities&quot; in terms of special relativity (the transformation matrix '''g''' stays the same as in ({{equationNote|1a}})):&lt;ref&gt;Synge (1955), p. 129 for ''n''=3&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{align}u_{s}^{\prime} &amp; =\frac{g_{s0}+g_{s1}u_{1}+\dots+g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots+g_{0n}u_{n}}\\
\\
u_{s} &amp; =\frac{-g_{0s}+g_{1s}u_{1}^{\prime}+\dots+g_{ns}u_{n}^{\prime}}{g_{00}-g_{10}u_{1}^{\prime}-\dots-g_{n0}u_{n}^{\prime}}
\end{align}
\left|\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} &amp; =\left\{ \begin{align}-1\quad &amp; (j=k=0)\\
1\quad &amp; (j=k&gt;0)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} &amp; =\left\{ \begin{align}-1\quad &amp; (i=k=0)\\
1\quad &amp; (i=k&gt;0)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
\right.&lt;/math&gt;|{{equationRef|1b}}}}

Such Lorentz transformations for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827–1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]], [[#Callandreau|Callandreau (1885)]] in order to simplify computations of elliptic functions and integrals, by [[#Picard|Picard (1882-1884)]] in relation to [[w:Hermitian form|Hermitian quadratic form]]s, or by [[#Woods2|Woods (1901, 1903)]] in terms of the [[w:Beltrami–Klein model]] of hyperbolic geometry. In addition, infinitesimal transformations in terms of the [[w:Lie algebra]] of the group of hyperbolic motions leaving invariant the unit sphere &lt;math&gt;-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0&lt;/math&gt; were given by [[#Lie3|Lie (1885-1893) and Werner (1889)]] and [[#Killing3|Killing (1888-1897)]].

==Historical notation==
===Lagrange (1773) – Binary quadratic forms {{anchor|Lagrange}}===
After the invariance of the sum of squares under linear substitutions was discussed by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]], the general expressions of a [[w:binary quadratic form]] and its transformation was formulated by [[w:Joseph-Louis Lagrange]] (1773/75) as follows&lt;ref group=M&gt;Lagrange (1773/75), section 22&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\
\hline \begin{align}y &amp; =Ms+Nx\\
z &amp; =ms+nx
\end{align}
\left|\begin{matrix}\begin{align}P &amp; =pM^{2}+2qMm+rm^{2}\\
Q &amp; =pMN+q(Mn+Nm)+rmn\\
R &amp; =pN^{2}+2qNn+rn^{2}
\end{align}
\\
\downarrow\\
PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to ({{equationNote|Q1}}) ''(n=1)''. The Lorentz interval &lt;math&gt;-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(p,q,r)=(P,Q,R)=(1,0,-1)''.&lt;/p&gt;

==={{anchor|Gauss}} Gauss (1798–1818)===

===={{anchor|Gauss1}} Binary quadratic forms====

The theory of binary quadratic forms was considerably expanded by [[w:Carl Friedrich Gauss]] (1798, published 1801) in his [[w:Disquisitiones Arithmeticae]]. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:&lt;ref group=M&gt;Gauss (1798/1801), articles 157–158;&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\
F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\
\hline \begin{align}x &amp; =\alpha x'+\beta y'\\
y &amp; =\gamma x'+\delta y'\\
\\
x' &amp; =\delta x-\beta y\\
y' &amp; =-\gamma x+\alpha y
\end{align}
\left|\begin{matrix}\begin{align}a' &amp; =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\
b' &amp; =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\
c' &amp; =a\beta^{2}+2b\beta\delta+c\delta^{2}
\end{align}
\\
\downarrow\\
b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

which is equivalent to ({{equationNote|Q1}}) ''(n=1)''. As pointed out by Gauss, ''F'' and ''F′'' are called &quot;proper equivalent&quot; if αδ-βγ=1, so that ''F'' is contained in ''F′'' as well as ''F′'' is contained in ''F''. In addition, if another form ''F″'' is contained by the same procedure in ''F′'' it is also contained in ''F'' and so forth.&lt;ref group=M&gt;Gauss (1798/1801), section 159&lt;/ref&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The Lorentz interval &lt;math&gt;-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(a,b,c)=(a',b',c')=(1,0,-1)''.&lt;/p&gt;

===={{anchor|Gauss2}} Ternary quadratic forms====

Gauss (1798/1801)&lt;ref group=M&gt;Gauss (1798/1801), articles 266–285&lt;/ref&gt; also discussed ternary quadratic forms with the general expression

:&lt;math&gt;\begin{matrix}f=ax^{2}+a'x^{\prime2}+a''x^{\prime\prime2}+2bx'x''+2b'xx''+2b''xx'=\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)\\
g=my^{2}+m'y^{\prime2}+m''y^{\prime\prime2}+2ny'y''+2n'yy''+2n''yy'=\left(\begin{matrix}m, &amp; m', &amp; m''\\
n, &amp; n', &amp; n''
\end{matrix}\right)\\
\hline \begin{align}x &amp; =\alpha y+\beta y'+\gamma y''\\
x' &amp; =\alpha'y+\beta'y'+\gamma'y''\\
x'' &amp; =\alpha''y+\beta''y'+\gamma''y''
\end{align}
\end{matrix}&lt;/math&gt;

which is equivalent to ({{equationNote|Q1}}) ''(n=2)''. Gauss called these forms definite when they have the same sign such as ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;+z&lt;sup&gt;2&lt;/sup&gt;'', or indefinite in the case of different signs such as ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;''. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:&lt;ref group=M&gt;Gauss (1798/1801), article 277&lt;/ref&gt;

:&lt;math&gt;\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)\Rightarrow\begin{matrix}\left(\begin{matrix}1, &amp; -1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; 1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\\
\left(\begin{matrix}1, &amp; -1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; -1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval &lt;math&gt;x^{2}+x^{\prime2}-x^{\prime\prime2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' is an indefinite ternary quadratic form, which follows from the general ternary form by setting:
&lt;br&gt;&lt;math&gt;\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)=\left(\begin{matrix}m, &amp; m', &amp; m''\\
n, &amp; n', &amp; n''
\end{matrix}\right)=\left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)&lt;/math&gt;&lt;/p&gt;

===={{anchor|Gauss4}} Homogeneous coordinates====

Gauss (1818) discussed planetary motions together with formulating [[w:elliptic function]]s. In order to simplify the integration, he transformed the expression

:&lt;math&gt;(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E&lt;/math&gt;

into

:&lt;math&gt;G+G'\cos T^{2}+G''\sin T^{2}&lt;/math&gt;

in which the [[w:eccentric anomaly]] ''E'' is connected to the new variable ''T'' by the following transformation including an arbitrary constant ''k'', which Gauss then rewrote by setting ''k''=1:&lt;ref group=M&gt;Gauss (1818), pp. 5–10&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}{\scriptstyle \left(\alpha+\alpha'\cos T+\alpha''\sin T\right)^{2}+\left(\beta+\beta'\cos T+\beta''\sin T\right)^{2}-\left(\gamma+\gamma'\cos T+\gamma''\sin T\right)^{2}}=0\\
k\left(\cos^{2}T+\sin^{2}T-1\right)=0\\
\hline \begin{align}\cos E &amp; =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\
\sin E &amp; =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =k &amp; \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' &amp; =-k\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-k &amp; \beta\beta-\beta'\beta'-\beta''\beta'' &amp; =-k\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-k &amp; \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' &amp; =+k\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0 &amp; \beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0 &amp; \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0 &amp; \alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
}\right.\\
\hline k=1\\
\begin{align}t\cos E &amp; =\alpha+\alpha'\cos T+\alpha''\sin T\\
t\sin E &amp; =\beta+\beta'\cos T+\beta''\sin T\\
t &amp; =\gamma+\gamma'\cos T+\gamma''\sin T
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =1\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-1\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-1\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The coefficients α,β,γ,... of Gauss' case ''k''=1 are equivalent to the coefficient system in Lorentz transformations ({{equationNote|1a}}) and ({{equationNote|1b}})  ''(n=2)''.

Further setting &lt;math&gt;[\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]&lt;/math&gt;, Gauss' transformation becomes Lorentz transformation ({{equationNote|1b}})  ''(n=2)''.&lt;/p&gt;

Subsequently, he showed that these relations can be reformulated using three variables ''x,y,z'' and ''u,u′,u″'', so that

:&lt;math&gt;aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz&lt;/math&gt;

can be transformed into

:&lt;math&gt;Guu+G'u'u'+G''u''u''&lt;/math&gt;,

in which ''x,y,z'' and ''u,u′,u″'' are related by the transformation:&lt;ref group=M&gt;Gauss (1818), pp. 9–10&lt;/ref&gt;

:&lt;math&gt;\begin{align}x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
y &amp; =\beta u+\beta'u'+\beta''u''\\
z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
\\
u &amp; =-\alpha x-\beta y+\gamma z\\
u' &amp; =\alpha'x+\beta'y-\gamma'z\\
u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =1\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-1\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-1\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0
\end{align}
}\right.&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}})  ''(n=2)'' satisfying &lt;math&gt;x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}&lt;/math&gt;, and can be related to Gauss' previous equations in terms of homogeneous coordinates &lt;math&gt;\left[\cos T,\sin T,\cos E,\sin E\right]=\left[\tfrac{x}{z},\ \tfrac{y}{z},\ \tfrac{u'}{u},\ \tfrac{u''}{u}\right]&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Jacobi}} Jacobi (1827, 1833/34) – Homogeneous coordinates===

Following [[#Gauss4|Gauss (1818)]], [[w:Carl Gustav Jacob Jacobi]] extended Gauss' transformation to 3 dimensions in 1827:&lt;ref group=M&gt;Jacobi (1827), p. 235, 239–240&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}\cos P^{2}+\sin P^{2}\cos\vartheta^{2}+\sin P^{2}\sin\vartheta^{2}=1\\
k\left(\cos\psi^{2}+\sin\psi^{2}\cos\varphi^{2}+\sin\psi^{2}\sin\varphi^{2}-1\right)=0\\
\hline {\left.\begin{matrix}\mathbf{(1)}\begin{align}\cos P &amp; =\frac{\alpha+\alpha'\cos\psi+\alpha''\sin\psi\cos\varphi+\alpha'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\sin P\cos\vartheta &amp; =\frac{\beta+\beta'\cos\psi+\beta''\sin\psi\cos\varphi+\beta'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\sin P\sin\vartheta &amp; =\frac{\gamma+\beta'\cos\psi+\gamma''\sin\psi\cos\varphi+\gamma'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\\
\cos\psi &amp; =\frac{-\delta'+\alpha'\cos P+\beta'\sin P\cos\vartheta+\gamma'\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\
\sin\psi\cos\varphi &amp; =\frac{-\delta''+\alpha''\cos P+\beta''\sin P\cos\vartheta+\gamma''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\
\sin\psi\sin\varphi &amp; =\frac{-\delta'''+\alpha'''\cos P+\beta'''\sin P\cos\vartheta+\gamma'''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}
\end{align}
\\
\\
\hline \mathbf{(2)}\begin{align}\alpha\mu+\beta x+\gamma y+\delta z &amp; =m\\
\alpha'\mu+\beta'x+\gamma'y+\delta'z &amp; =m'\\
\alpha''\mu+\beta''x+\gamma''y+\delta''z &amp; =m''\\
\alpha'''\mu+\beta'''x+\gamma'''y+\delta'''z &amp; =m'''\\
\\
Am+A'm'+A''m''+A'''m''' &amp; =\mu\\
Bm+B'm'+B''m''+B'''m''' &amp; =x\\
Cm+C'm'+C''m''+C'''m''' &amp; =y\\
Dm+D'm'+D''m''+D'''m''' &amp; =z\\
\\
\end{align}
\\
\begin{align}\alpha &amp; =-kA, &amp; \beta &amp; =-kB, &amp; \gamma &amp; =-kC, &amp; \delta &amp; =kD,\\
\alpha' &amp; =kA', &amp; \beta' &amp; =kB', &amp; \gamma' &amp; =kC', &amp; \delta' &amp; =-kD',\\
\alpha'' &amp; =kA'', &amp; \beta'' &amp; =kB'', &amp; \gamma'' &amp; =kC'', &amp; \delta'' &amp; =-kD'',\\
\alpha''' &amp; =kA''', &amp; \beta''' &amp; =kB''', &amp; \gamma''' &amp; =kC''', &amp; \delta''' &amp; =-kD''',
\end{align}
\end{matrix}\right|\begin{matrix}\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta &amp; =-k\\
\alpha'\alpha'+\beta'\beta'+\gamma'\gamma'-\delta'\delta' &amp; =k\\
\alpha''\alpha''+\beta''\beta''+\gamma''\gamma''-\delta''\delta'' &amp; =k\\
\alpha'''\alpha'''+\beta'''\beta'''+\gamma'''\gamma'''-\delta'''\delta''' &amp; =k\\
\alpha\alpha'+\beta\beta'+\gamma\gamma'-\delta\delta' &amp; =0\\
\alpha\alpha''+\beta\beta''+\gamma\gamma''-\delta\delta'' &amp; =0\\
\alpha\alpha'''+\beta\beta'''+\gamma\gamma'''-\delta\delta''' &amp; =0\\
\alpha''\alpha'''+\beta''\beta'''+\gamma''\gamma'''-\delta''\delta''' &amp; =0\\
\alpha'''\alpha'+\beta'''\beta'+\gamma'''\gamma'-\delta'''\delta' &amp; =0\\
\alpha'\alpha''+\beta'\beta''+\gamma'\gamma''-\delta'\delta'' &amp; =0\\
\\
-\alpha\alpha+\alpha'\alpha'+\alpha''\alpha''+\alpha'''\alpha''' &amp; =k\\
-\beta\beta+\beta'\beta'+\beta''\beta''+\beta'''\beta''' &amp; =k\\
-\gamma\gamma+\gamma'\gamma'+\gamma''\gamma''+\gamma'''\gamma''' &amp; =k\\
-\delta\delta+\delta'\delta'+\delta''\delta''+\delta'''\delta''' &amp; =-k\\
-\alpha\beta+\alpha'\beta'+\alpha''\beta''+\alpha'''\beta''' &amp; =0\\
-\alpha\gamma+\alpha'\gamma'+\alpha''\gamma''+\alpha'''\gamma''' &amp; =0\\
-\alpha\delta+\alpha'\delta'+\alpha''\delta''+\alpha'''\delta''' &amp; =0\\
-\beta\gamma+\beta'\gamma'+\beta''\gamma''+\beta'''\gamma''' &amp; =0\\
-\gamma\delta+\gamma'\delta'+\gamma''\delta''+\gamma'''\delta''' &amp; =0\\
-\delta\beta+\delta'\beta'+\delta''\beta''+\delta'''\beta''' &amp; =0
\end{align}
\end{matrix}}
\end{matrix}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;By setting &lt;math&gt;{\scriptstyle \begin{align}\left[\cos P,\ \sin P\cos\varphi,\ \sin P\sin\varphi\right] &amp; =\left[u_{1},\ u_{2},\ u_{3}\right]\\{}
[\cos\psi,\ \sin\psi\cos\vartheta,\ \sin\psi\sin\vartheta] &amp; =\left[u_{1}^{\prime},\ u_{2}^{\prime},\ u_{3}^{\prime}\right]
\end{align}
}&lt;/math&gt; and ''k''=1 in the (1827) formulas, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'', and by setting ''k''=1 in transformation system (2) it becomes equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' producing &lt;math&gt;m^{2}+m^{\prime2}+m^{\prime\prime2}-m^{\prime\prime\prime2}=\mu^{2}+x^{2}+y^{2}-z^{2}&lt;/math&gt;.&lt;/p&gt;

Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change) in the case of 2 dimensions:&lt;ref group=M&gt;The orthogonal substitution and the imaginary transformation was defined in Jacobi (1832a), pp. 257, 265–267; Transformation system (2) and (3) and coefficients in Jacobi (1832b), pp. 321-325.&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}xx+yy+zz=ss+s's'+s''s''=0\\
\mathbf{(1)}\begin{align}x &amp; =\alpha s+\alpha's'+\alpha''s''\\
y &amp; =\beta s+\beta's'+\beta''s''\\
z &amp; =\gamma s+\gamma's'+\gamma''s''\\
\\
s &amp; =\alpha x+\beta y+\gamma z\\
s' &amp; =\alpha'x+\beta'y+\gamma'z\\
s'' &amp; =\alpha''x+\beta''y+\gamma''z
\end{align}
\left|\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma &amp; =1 &amp; \alpha\alpha+\alpha'\alpha'+\alpha''\alpha'' &amp; =1\\
\alpha'\alpha'+\beta'\beta'+\gamma'\gamma' &amp; =1 &amp; \beta\beta+\beta'\beta'+\beta''\beta'' &amp; =1\\
\alpha''\alpha''+\beta''\beta''+\gamma''\gamma'' &amp; =1 &amp; \gamma\gamma+\gamma'\gamma'+\gamma''\gamma'' &amp; =1\\
\alpha'\alpha''+\beta'\beta''+\gamma'\gamma'' &amp; =0 &amp; \beta\gamma+\beta'\gamma'+\beta''\gamma'' &amp; =0\\
\alpha''\alpha+\beta''\beta+\gamma''\gamma &amp; =0 &amp; \gamma\alpha+\gamma'\alpha'+\gamma''\alpha'' &amp; =0\\
\alpha\alpha'+\beta\beta'+\gamma\gamma' &amp; =0 &amp; \alpha\beta+\alpha'\beta'+\alpha''\beta'' &amp; =0
\end{align}
\right.\\
\hline \left[\frac{y}{x},\ \frac{z}{x},\ \frac{s'}{s},\ \frac{s''}{s}\right]=\left[-i\cos\varphi,\ -i\sin\varphi,\ i\cos\eta,\ i\sin\eta\right]\\
\left[\alpha',\ \alpha'',\ \beta,\ \gamma\right]=\left[i\alpha',\ i\alpha'',\ -i\beta,\ -i\gamma\right]\\
\hline \begin{matrix}\mathbf{(2)}\begin{matrix}\left(\alpha-\alpha'\cos\eta-\alpha''\sin\eta\right)^{2}=\left(\beta-\beta'\cos\eta-\beta''\sin\eta\right)^{2}+\left(\gamma-\gamma'\cos\eta-\gamma''\sin\eta\right)^{2}\\
\left(\alpha-\beta\cos\phi-\gamma\sin\phi\right)^{2}=\left(\alpha'-\beta'\cos\phi-\gamma'\sin\phi\right)^{2}+\left(\alpha''-\beta''\cos\phi-\gamma''\sin\phi\right)^{2}\\
\hline \begin{align}\cos\phi &amp; =\frac{\beta-\beta'\cos\eta-\beta''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, &amp; \cos\eta &amp; =\frac{\alpha'-\beta'\cos\phi-\gamma'\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}\\
\sin\phi &amp; =\frac{\gamma-\gamma'\cos\eta-\gamma''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, &amp; \sin\eta &amp; =\frac{\alpha''-\beta''\cos\phi-\gamma''\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}
\end{align}
\end{matrix}\\
\hline \\
\mathbf{(3)}\begin{matrix}1-zz-yy=\frac{1-s's'-s''s''}{\left(\alpha-\alpha's'-\alpha''s''\right)^{2}}\\
\hline \begin{align}y &amp; =\frac{\beta-\beta's'-\beta''s''}{\alpha-\alpha's'-\alpha''s''}, &amp; s' &amp; =\frac{\alpha'-\beta'y-\gamma'z}{\alpha-\beta y-\gamma z},\\
z &amp; =\frac{\gamma-\gamma's'-\gamma''s''}{\alpha-\alpha's'-\alpha''s'''}, &amp; s'' &amp; =\frac{\alpha''-\beta''y-\gamma''z}{\alpha-\beta y-\gamma z},
\end{align}
\end{matrix}
\end{matrix}\left|\begin{align}\alpha\alpha-\beta\beta-\gamma\gamma &amp; =1\\
\alpha'\alpha'-\beta'\beta'-\gamma'\gamma' &amp; =-1\\
\alpha''\alpha''-\beta''\beta''-\gamma''\gamma'' &amp; =-1\\
\alpha'\alpha''-\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\alpha''\alpha-\beta''\beta-\gamma''\gamma &amp; =0\\
\alpha\alpha'-\beta\beta'-\gamma\gamma' &amp; =0\\
\\
\alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' &amp; =1\\
\beta\beta-\beta'\beta'-\beta''\beta'' &amp; =-1\\
\gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' &amp; =-1\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
\gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' &amp; =0\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
\right.
\end{matrix}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;By setting &lt;math&gt;[\cos\phi,\ \sin\phi,\ \cos\eta,\ \sin\eta]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]&lt;/math&gt;, transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Also transformation system (3) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' up to a sign change.&lt;/p&gt;

Extending his previous result, Jacobi (1833) started with [[#Cauchy|Cauchy's (1829)]] orthogonal transformation for ''n'' dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of ''n'' dimensions:&lt;ref group =M&gt;Jacobi (1833/34), pp. 7–8, 34–35, 41; Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 229–230.&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots+x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots+y_{n}y_{n}\\
\hline \mathbf{(1)\ }\begin{align}y_{\varkappa} &amp; =\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}\\
x_{\varkappa} &amp; =\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}\\
\\
\frac{y_{\varkappa}}{y_{n}} &amp; =\frac{\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}\\
\frac{x_{\varkappa}}{x_{n}} &amp; =\frac{\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}
\end{align}
\left|\begin{align}\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\lambda}^{(n)} &amp; =0\\
\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\varkappa}^{(n)} &amp; =1\\
\\
\alpha_{1}^{(\varkappa)}\alpha_{1}^{(\lambda)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\lambda)} &amp; =0\\
\alpha_{1}^{(\varkappa)}\alpha_{1}^{(\varkappa)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\varkappa)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\varkappa)} &amp; =1
\end{align}
\right.\\
\hline \frac{x_{\varkappa}}{x_{n}}=-i\xi_{\varkappa},\ \frac{y_{\varkappa}}{y_{n}}=i\nu_{\varkappa}\\
1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{y_{n}y_{n}}{x_{n}x_{n}}\left(1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}\right)\\
\alpha_{n}^{(\varkappa)}=i\alpha^{(\varkappa)},\ \alpha_{\varkappa}^{(n)}=-i\alpha_{\varkappa},\ \alpha_{n}^{(n)}=\alpha\\
1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}}{\left[\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}\right]^{2}}\\
\hline \mathbf{(2)\ }\begin{align}\nu_{\varkappa} &amp; =\frac{\alpha^{(\varkappa)}-\alpha_{1}^{(\varkappa)}\xi_{1}-\alpha_{2}^{(\varkappa)}\xi_{2}\dots-\alpha_{n-1}^{(\varkappa)}\xi_{n-1}}{\alpha-\alpha_{1}\xi_{1}-\alpha_{2}\xi_{2}\dots-\alpha_{n-1}\xi_{n-1}}\\
\\
\xi_{\varkappa} &amp; =\frac{\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\nu_{1}-\alpha_{2}^{\prime\prime}\nu_{2}\dots-\alpha_{\varkappa}^{(n-1)}\nu_{n-1}}{\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}}
\end{align}
\\
\hline \xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=1\ \Rightarrow\ \nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}=1
\end{matrix}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) up to a sign change.&lt;/p&gt;

He also stated the following transformation leaving invariant the Lorentz interval:&lt;ref group=M&gt;Jacobi (1833/34), p. 37. Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 232–233.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots-u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots-w_{n-1}w_{n-1}\\
\hline {\scriptstyle \begin{align}u &amp; =\alpha w-\alpha'w_{1}-\alpha''w_{2}-\dots-\alpha^{(n-1)}w_{n-1}\\
u_{1} &amp; =\alpha_{1}w-\alpha_{1}^{\prime}w_{1}-\alpha_{1}^{\prime\prime}w_{2}-\dots-\alpha_{1}^{(n-1)}w_{n-1}\\
 &amp; \dots\\
u_{n-1} &amp; =\alpha_{n-1}w-\alpha_{n-1}^{\prime}w_{1}-\alpha_{n-1}^{\prime\prime}w_{2}-\dots-\alpha_{n-1}^{(n-1)}w_{n-1}\\
\\
w &amp; =\alpha u-\alpha_{1}u_{1}-\alpha_{2}^{\prime\prime}u_{2}-\dots-\alpha_{n-1}u_{n-1}\\
w_{1} &amp; =\alpha'u-\alpha_{1}^{\prime}u_{1}-\alpha_{2}^{\prime}u_{2}-\dots-\alpha_{n-1}^{\prime}u_{n-1}\\
 &amp; \dots\\
w_{n-1} &amp; =\alpha^{(n-1)}u-\alpha_{1}^{(n-1)}u_{1}-\alpha_{2}^{(n-1)}u_{2}-\dots-\alpha_{n-1}^{(n-1)}u_{n-1}
\end{align}
\left|\begin{align}\alpha\alpha-\alpha'\alpha'-\alpha''\alpha''\dots-\alpha^{(n-1)}\alpha^{(n-1)} &amp; =+1\\
\alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\varkappa}^{(n-1)} &amp; =-1\\
\alpha\alpha_{\varkappa}-\alpha^{\prime}\alpha_{\varkappa}^{\prime}-\alpha^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha^{(n-1)}\alpha_{\varkappa}^{(n-1)} &amp; =0\\
\alpha_{\varkappa}\alpha_{\lambda}-\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\lambda}^{(n-1)} &amp; =0\\
\\
\alpha\alpha-\alpha_{1}\alpha_{1}-\alpha_{2}\alpha_{2}\dots-\alpha_{n-1}\alpha_{n-1} &amp; =+1\\
\alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{1}^{\varkappa}\alpha_{1}^{\varkappa}-\alpha_{2}^{\prime\prime}\alpha_{2}^{\prime\prime}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\varkappa)} &amp; =-1\\
\alpha\alpha^{(\varkappa)}-\alpha_{1}\alpha_{1}^{(\varkappa)}-\alpha_{2}\alpha_{2}^{(\varkappa)}\dots-\alpha_{n-1}\alpha_{n-1}^{(\varkappa)} &amp; =0\\
\alpha^{(\varkappa)}\alpha^{(\lambda)}-\alpha_{1}^{(\varkappa)}\alpha_{1}^{\lambda l)}-\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\lambda)} &amp; =0
\end{align}
\text{ }\right.}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}}) up to a sign change.&lt;/p&gt;

==={{anchor|Chasles}} Chasles (1829) – Conjugate hyperboloids ===

[[w:Michel Chasles]] (1829) independently introduced the same equation systems as [[#Gauss4|Gauss (1818)]] and [[#Jacobi|Jacobi (1827)]], albeit in the different context of conjugate hyperboloids. He started with two equation systems (a) and (b) from which he derived systems (c), (d) and others:&lt;ref group=M&gt;Chasles (1829), p. 139&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =1\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =1\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =-1
\end{align}
\right\}  &amp; \dots(a)\\ \\
\left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma' &amp; =0
\end{align}
\right\}  &amp; \dots(b)\\ \\
\left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =1\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =1\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-1
\end{align}
\right\}  &amp; \dots(c)\\ \\
\left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right\}  &amp; \dots(d)
\end{matrix}&lt;/math&gt;

He noted that those quantities become the “frequently employed” formulas of Lagrange [i.e. the coefficients of the Euclidean orthogonal transformation first given by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]]] by setting:&lt;ref group=M&gt;Chasles (1829), p. 141&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\gamma\quad\Rightarrow\quad-\gamma\sqrt{-1}\\
\gamma'\quad\Rightarrow\quad-\gamma'\sqrt{-1}\\
\alpha''\quad\Rightarrow\quad\alpha''\sqrt{-1}\\
\beta''\quad\Rightarrow\quad\beta''\sqrt{-1}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Equations (a,b,c,d) are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).&lt;/p&gt;

Chasles now showed that equation systems (a,b,c,d) are of importance when discussing the relations between conjugate diameters of hyperboloids. He used the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid having the same principal axes (x,y,z), thus sharing the same conjugate axes, and having the common asymptotic cone &lt;math&gt;\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0&lt;/math&gt;. He then transformed those two hyperboloids to new axes (x',y',z') sharing the property of conjugacy:&lt;ref group=M&gt;Chasles (1829), pp. 143-144&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\
\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\
\hline \begin{align}x &amp; =lx'+l'y'+l''z'\\
y &amp; =mx'+m'y'+m''z'\\
z &amp; =nx'+n'y'+n''z'
\end{align}
\\
\left\{ \begin{align}\frac{ll'}{a{{}^2}}+\frac{mm'}{b^{2}}-\frac{nn'}{c^{2}} &amp; =0\\
\frac{ll''}{a{{}^2}}+\frac{mm''}{b^{2}}-\frac{nn''}{c^{2}} &amp; =0\\
\frac{l'l''}{a{{}^2}}+\frac{m'm''}{b^{2}}-\frac{n'n''}{c^{2}} &amp; =0
\end{align}
\right\} \\
\hline \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=1\\
\left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=-1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Chasles defined the conditional equations of ''l,m,n'' in the same way as those of &lt;math&gt;\alpha,\beta,\gamma&lt;/math&gt; in equation system (b) above, so his transformation of x,y,z into x',y',z' represents Lorentz transformation ({{equationNote|1a}}, n=2) by applying equation system (a) as well.&lt;/p&gt;

He went on to use two semi-diameters of the one-sheet hyperboloid and one semi-diameter of the two-sheet hyperboloid in order to define equation system (A), and went on to suggest that the other equations related to this system can be obtained using the above transformation from oblique coordinates to other oblique ones, but he deemed it more simple to use a geometric argument to obtain system (B), which together with (A) then allowed him to algebraically determine systems (C), (D) and additional ones, leading Chasles to announce that “''from these formulas one can very easily conclude the various properties of conjugated diameters of hyperboloids''”:&lt;ref group=M&gt;Chasles (1829), pp. 145-146&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =a^{2}\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =b^{2}\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =-c^{2}
\end{align}
\right\}  &amp; \dots(A)\\
\left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma' &amp; =0
\end{align}
\right\}  &amp; \dots(B)\\
\left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =a^{2}\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =b^{2}\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-c^{2}
\end{align}
\right\}  &amp; \dots(C)\\
\left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right\}  &amp; \dots(D)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Equation systems (A,B,C,D), being equivalent to systems (a,b,c,d) above, are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.&lt;/p&gt;

==={{anchor|Lebesgue}} Lebesgue (1837) – Homogeneous coordinates===

[[w:Victor-Amédée Lebesgue]] (1837) summarized the previous work of [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Cauchy|Cauchy (1829)]]. He started with the orthogonal transformation&lt;ref group=M&gt;Lebesgue (1837), pp. 338-341&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}\ (9)\\
\hline {\scriptstyle \begin{align}x_{1} &amp; =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\
x_{2} &amp; =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\
\dots\\
x_{n} &amp; =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\
\\
y_{1} &amp; =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n,1}x_{n}\\
y_{2} &amp; =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n,2}x_{n}\ (12)\ \\
\dots\\
y_{n} &amp; =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n,n}x_{n}
\end{align}
\left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n,\alpha}^{2} &amp; =1 &amp; (10)\\
a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n,\alpha}a_{n,\beta} &amp; =0 &amp; (11)\\
a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n}^{2} &amp; =1 &amp; (13)\\
a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n}a_{\beta,n} &amp; =0 &amp; (14)
\end{align}
\right.}
\end{matrix}&lt;/math&gt;

In order to achieve the invariance of the Lorentz interval&lt;ref group=M&gt;Lebesgue (1837), pp. 353–354&lt;/ref&gt;

:&lt;math&gt;x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}&lt;/math&gt;

he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first ''n-1'' equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=''n'' change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the ''n''-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the ''n''-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These instructions give Lorentz transformation ({{equationNote|1a}}) in the form:

&lt;br&gt;&lt;math&gt;{\scriptstyle \begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\
\hline \begin{align}x_{1} &amp; =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\
x_{2} &amp; =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\
\dots\\
x_{n} &amp; =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\
\\
y_{1} &amp; =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\
y_{2} &amp; =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\
\dots\\
-y_{n} &amp; =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n-1,n}x_{n-1}-a_{n,n}x_{n}
\end{align}
\left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n-1,\alpha}^{2}-a_{n,\alpha}^{2} &amp; =1\\
a_{1,n}^{2}+a_{2,n}^{2}+\dots+a_{n-1,n}^{2}-a_{n,n}^{2} &amp; =-1\\
a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n-1,\alpha}a_{n-1,\beta}-a_{n,\alpha}a_{n,\beta} &amp; =0\\
a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n-1}^{2}-a_{\alpha,n}^{2} &amp; =1\\
a_{n,1}^{2}+a_{n,2}^{2}+\dots+a_{n,n-1}^{2}-a_{n,n}^{2} &amp; =-1\\
a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n-1}a_{\beta,n-1}-a_{\alpha,n}a_{\beta,n} &amp; =0
\end{align}
\right.
\end{matrix}}&lt;/math&gt;&lt;/p&gt;

He went on to redefine the variables of the Lorentz interval and its transformation:&lt;ref group=M&gt;Lebesgue (1837), pp. 353–355&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\
\downarrow\\
\begin{align}x_{1} &amp; =x_{n}\cos\theta_{1}, &amp; x_{2} &amp; =x_{n}\cos\theta_{2},\dots &amp; x_{n-1} &amp; =x_{n}\cos\theta_{n-1}\\
y_{1} &amp; =y_{n}\cos\phi_{1}, &amp; y_{2} &amp; =y_{n}\cos\phi_{2},\dots &amp; y_{n-1} &amp; =y_{n}\cos\phi_{n-1}
\end{align}
\\
\downarrow\\
\cos^{2}\theta_{1}+\cos^{2}\theta_{2}+\dots+\cos^{2}\theta_{n-1}=1\\
\cos^{2}\phi_{1}+\cos^{2}\phi_{2}+\dots+\cos^{2}\phi_{n-1}=1\\
\hline \\
\cos\theta_{i}=\frac{a_{i,1}\cos\phi_{1}+a_{i,2}\cos\phi_{2}+\dots+a_{i,n-1}\cos\phi_{n-1}+a_{i,n}}{a_{n,1}\cos\phi_{1}+a_{n,2}\cos\phi_{2}+\dots+a_{n,n-1}\cos\phi_{n-1}+a_{n,n}}\\
(i=1,2,3\dots n)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Setting &lt;math&gt;[\cos\theta_{i},\ \cos\phi_{i}]=\left[u_{s},\ u_{s}^{\prime}\right]&lt;/math&gt; it is equivalent to Lorentz transformation ({{equationNote|1b}}).&lt;/p&gt;

==={{anchor|Weddle}} Weddle (1847) – Conjugate hyperboloids===
Very similar to [[#Chasles|Chasles (1829)]], though without reference to him, [[w:Thomas Weddle]] discussed conjugate hyperboloids using the following equation system (&amp;alpha;), from which he derived equations (&amp;beta;) and others:&lt;ref group=M&gt;Weddle (1847), p. 274&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} &amp; =1, &amp; l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} &amp; =0\\
l_{2}^{2}+m_{2}^{2}-n_{2}^{2} &amp; =1, &amp; l_{1}l_{3}+m_{1}m_{3}-n_{1}n_{3} &amp; =0\\
l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =-1, &amp; l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} &amp; =0
\end{align}
\right\}  &amp; \dots(\alpha)\\ \\
\left.\begin{align}l_{1}^{2}+l_{2}^{2}-l_{3}^{2} &amp; =1, &amp; l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} &amp; =0\\
m_{1}^{2}+m_{2}^{2}-m_{3}^{2} &amp; =1, &amp; l_{1}n_{1}+l_{2}n_{2}-l_{3}n_{3} &amp; =0\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =-1, &amp; m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} &amp; =0
\end{align}
\right\}  &amp; \dots(\beta)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).&lt;/p&gt;

Using the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid sharing the same conjugate axes, and having the common asymptotic cone &lt;math&gt;\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0&lt;/math&gt;, he defined three conjugate points &lt;math&gt;(x_{1}\dots,y_{1}\dots,z_{1}\dots)&lt;/math&gt; on those two conjugate hyperboloids, related to each other in the same way as equations (&amp;alpha;, &amp;beta;) stated above:&lt;ref group=M&gt;Weddle (1847), pp. 275-276&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\
\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\
\hline \begin{align}\frac{x_{1}x_{2}}{a{{}^2}}+\frac{y_{1}y_{2}}{b^{2}}-\frac{z_{1}z_{2}}{c^{2}} &amp; =0\\
\frac{x_{1}x_{3}}{a{{}^2}}+\frac{y_{1}y_{3}}{b^{2}}-\frac{z_{1}z_{3}}{c^{2}} &amp; =0\\
\frac{x_{2}x_{3}}{a{{}^2}}+\frac{y_{2}y_{3}}{b^{2}}-\frac{z_{2}z_{3}}{c^{2}} &amp; =0
\end{align}
\quad\begin{align}\frac{x_{1}^{2}}{a{{}^2}}+\frac{y_{1}^{2}}{b^{2}}-\frac{z_{1}^{2}}{c^{2}} &amp; =1\\
\frac{x_{2}^{2}}{a{{}^2}}+\frac{y_{2}^{2}}{b^{2}}-\frac{z_{2}^{2}}{c^{2}} &amp; =1\\
\frac{x_{3}^{2}}{a{{}^2}}+\frac{y_{3}^{2}}{b^{2}}-\frac{z_{3}^{2}}{c^{2}} &amp; =-1
\end{align}
\\
\begin{align}x_{1}^{2}+x_{2}^{2}-x_{3}^{2} &amp; =a^{2}\\
y_{1}^{2}+y_{2}^{2}-y_{3}^{2} &amp; =b^{2}\\
z_{1}^{2}+z_{2}^{2}-z_{3}^{2} &amp; =-c^{2}
\end{align}
\quad\begin{align}x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3} &amp; =0\\
x_{1}z_{1}+x_{2}z_{2}-x_{3}z_{3} &amp; =0\\
y_{1}z_{1}+y_{2}z_{2}-y_{3}z_{3} &amp; =0
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt; These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.&lt;/p&gt;

==={{anchor|Bour}} Bour (1856) – Homogeneous coordinates===

Following [[#Gauss4|Gauss (1818)]], [[w:Edmond Bour]] (1856) wrote the transformations:&lt;ref group=M&gt;Bour (1856), pp. 61; 64–65&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\cos^{2}E+\sin^{2}E-1=k\left(\cos^{2}T+\sin^{2}T-1\right)\\
\hline \left.\begin{matrix}\mathbf{(1)}\ \begin{align}\cos E &amp; =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\
\sin E &amp; =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}
\end{align}
\\
\hline \\
k=+1\\
t=\gamma+\gamma'\cos T+\gamma''\sin T,\\
1=u,\ \cos T=u',\ \sin T=u',\\
t=z,\ t\cos E=x,\ t\sin E=y\\
\downarrow\\
\mathbf{(2)}\begin{align}x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
y &amp; =\beta u+\beta'u'+\beta''u''\\
z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
\\
u &amp; =\gamma z-\alpha x-\beta y\\
u' &amp; =\alpha'x+\beta'y'-\gamma'z\\
u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\end{matrix}\right|{\scriptstyle \begin{align}-\alpha^{2}-\beta^{2}+\gamma^{2} &amp; =k\\
-\alpha^{\prime2}-\beta^{\prime2}+\gamma^{\prime2} &amp; =-k\\
-\alpha^{\prime\prime2}-\beta^{\prime\prime2}+\gamma^{\prime\prime2} &amp; =-k\\
\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =-k\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =-k\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =k\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
\alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'', implying &lt;math&gt;x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}&lt;/math&gt;. Furthermore, setting &lt;math&gt;[k,\cos T,\sin T,\cos E,\sin E]=\left[1,u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt; in transformation system (1) produces Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.&lt;/p&gt;

=== {{anchor|Somov}} Somov (1863) – Homogeneous coordinates ===

Following [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], and [[#Bour|Bour (1856)]], [[w:Osip Ivanovich Somov]] (1863) wrote the transformation systems:&lt;ref group=M&gt;Somov (1863), pp. 12–14; p. 18 for differentials.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\cos\phi &amp; =\frac{m\cos\psi+n\sin\psi+s}{m''\cos\psi+n''\sin\psi+s''}\\
\sin\phi &amp; =\frac{m'\cos\psi+n'\sin\psi+s'}{m''\cos\psi+n''\sin\psi+s''}
\end{align}
\right|\begin{matrix}\cos^{2}\phi+\cos^{2}\phi=1\\
\cos^{2}\psi+\cos^{2}\psi=1
\end{matrix}\\
\hline \mathbf{(1)}\ \begin{align}\cos\phi &amp; =x, &amp; \cos\psi &amp; =x'\\
\sin\phi &amp; =y, &amp; \sin\psi &amp; =y'
\end{align}
\ \left|\begin{align}x &amp; =\frac{mx'+ny'+s}{m''x'+n''y'+s''}\\
y &amp; =\frac{m'x'+n'y'+s'}{m''x'+n''y'+s''}
\end{align}
\right|\ \begin{matrix}x^{2}+y^{2}=1\\
x^{\prime2}+y^{\prime2}=1
\end{matrix}\\
\hline \begin{align}\cos\phi &amp; =\frac{x}{z}, &amp; \cos\psi &amp; =\frac{x'}{z'}\\
\sin\phi &amp; =\frac{y}{z}, &amp; \sin\psi &amp; =\frac{y'}{z'}
\end{align}
\ \left|\begin{align}\frac{x}{z} &amp; =\frac{mx'+ny'+sz'}{m''x'+n''y'+s''z'}\\
\frac{y}{z} &amp; =\frac{m'x'+n'y'+s'z'}{m''x'+n''y'+s''z'}
\end{align}
\right|\ \begin{matrix}x^{2}+y^{2}=z^{2}\\
x^{\prime2}+y^{\prime2}=z^{\prime2}
\end{matrix}\\
\hline \mathbf{(2)}\ \left.\begin{align}x &amp; =mx'+ny'+sz'\\
y &amp; =m'x'+n'y'+s'z'\\
z &amp; =m''x'+n''y'+s''z'\\
\\
x' &amp; =mx+m'y-m''z\\
y' &amp; =nx+n'y-n''z\\
z' &amp; =-sx-s'y+s''z\\
\\
dx &amp; =mdx'+ndy'+sdz'\\
dy &amp; =m'dx'+n'dy'+s'dz'\\
dz &amp; =m''dx'+n''dy'+s''dz'
\end{align}
\right|{\scriptstyle \begin{align}m^{2}+m^{\prime2}-m^{\prime\prime2} &amp; =1\\
n^{2}+n^{\prime2}-n^{\prime\prime2} &amp; =1\\
-s^{2}-s^{\prime2}+s^{\prime\prime2} &amp; =1\\
ns+n's'-n''s'' &amp; =0\\
sm+s'm'-s''m'' &amp; =0\\
mn+m'n'-m''n'' &amp; =0\\
\\
m^{2}+n^{2}-s^{2} &amp; =1\\
m^{\prime2}+n^{\prime2}-s^{\prime2} &amp; =1\\
-m^{\prime\prime2}-n^{\prime\prime2}+s^{\prime\prime2} &amp; =1\\
-m'm''-n'n''+s's'' &amp; =0\\
-m''m-n''n+s''s &amp; =0\\
mm'+nn'-ss' &amp; =0
\end{align}
}\\
dx^{2}+dy^{2}-dz^{2}=dx^{\prime2}+dy^{\prime2}-dz^{\prime2}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.

Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.&lt;/p&gt;

==={{anchor|Killing}} Killing (1878–1893)===

===={{anchor|Killing1}} Weierstrass coordinates====

[[w:Wilhelm Killing]] (1878–1880) described non-Euclidean geometry by using [[w:hyperboloid model|Weierstrass coordinates]] (named after [[w:Karl Weierstrass]] who described them in lectures in 1872 which Killing attended) obeying the form

:&lt;math&gt;k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}&lt;/math&gt;&lt;ref group=M&gt;Killing (1877/78), p. 74; Killing (1880), p. 279&lt;/ref&gt; with &lt;math&gt;ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}&lt;/math&gt;&lt;ref group=M&gt;Killing (1880), eq. 25 on p. 283&lt;/ref&gt;

or&lt;ref group=M&gt;Killing (1880), p. 283&lt;/ref&gt;

:&lt;math&gt;k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}&lt;/math&gt;

where ''k'' is the reciprocal measure of curvature, &lt;math&gt;k^{2}=\infty&lt;/math&gt; denotes [[w:Euclidean geometry]], &lt;math&gt;k^{2}&gt;0&lt;/math&gt; [[w:elliptic geometry]], and &lt;math&gt;k^{2}&lt;0&lt;/math&gt; hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form.&lt;ref group=M&gt;Killing (1877/78), eq. 25 on p. 283&lt;/ref&gt; In (1879/80) he tried to formulate the corresponding transformations by plugging &lt;math&gt;k^{2}&lt;/math&gt; into a [[w:Rotation matrix#Rotation matrix from axis and angle|general rotation matrix]]:&lt;ref group=M&gt;Killing (1879/80), p. 274&lt;/ref&gt;

&lt;math&gt;\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\
\hline \begin{matrix}\cos\eta\tau+\lambda^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \nu\frac{\sin\eta\tau}{\eta}+\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; -\mu\frac{\sin\eta\tau}{\eta}+\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}\\
-k^{2}\nu\frac{\sin\eta\tau}{\eta}+k^{2}\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \cos\eta\tau+\mu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}\\
k^{2}\mu\frac{\sin\eta\tau}{\eta}+k^{2}\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; -\lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \cos\eta\tau+\nu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}
\end{matrix}\\
\left(\lambda^{2}+k^{2}\mu^{2}+k^{2}\nu^{2}=\eta^{2}\right)
\end{matrix}&lt;/math&gt;

In (1885) he wrote the Weierstrass coordinates and their transformation as follows:&lt;ref group=M&gt;Killing (1885), pp. 18, 28–30, 53&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\
k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime2}+x^{\prime2}+y^{\prime2}\\
ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\
\hline \begin{align}k^{2}p' &amp; =k^{2}wp+w'x+w''y\\
x' &amp; =ap+a'x+a''y\\
y' &amp; =bp+b'x+b''y\\
\\
k^{2}p &amp; =k^{2}wp'+ax'+by'\\
x &amp; =w'p'+a'x+b'y'\\
y &amp; =w''p'+a''x'+b''y'
\end{align}
\left|{\scriptstyle \begin{align}k^{2}w^{2}+w^{\prime2}+w^{\prime\prime2} &amp; =k^{2}\\
\frac{a^{2}}{k^{2}}+a^{\prime2}+a^{\prime\prime2} &amp; =1\\
\frac{b^{2}}{k^{2}}+b^{\prime2}+b^{\prime\prime2} &amp; =1\\
aw+a'w'+a''w'' &amp; =0\\
bw+b'w'+b''w'' &amp; =0\\
\frac{ab}{k^{2}}+a'b'+a''b'' &amp; =0\\
\\
k^{2}w^{2}+a^{2}+b^{2} &amp; =k^{2}\\
\frac{w^{\prime2}}{k^{2}}+a^{\prime2}+b^{\prime2} &amp; =1\\
\frac{w^{\prime\prime2}}{k^{2}}+a^{\prime\prime2}+b^{\prime\prime2} &amp; =1\\
ww'+aa'+bb' &amp; =0\\
ww''+aa''+bb'' &amp; =0\\
\frac{w'w''}{k^{2}}+a'a''+b'b'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

In (1885) he also gave the transformation for ''n'' dimensions:&lt;ref group=M&gt;Killing (1884/85), pp. 42–43; Killing (1885), pp. 73–74, 222&lt;/ref&gt;&lt;ref&gt;Ratcliffe (1994), § 3.6&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots+dx_{n}^{2}\\
\hline \left.\begin{align}k^{2}\xi_{0} &amp; =k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{0}\\
\xi_{\varkappa} &amp; =a_{\varkappa0}x_{0}+a_{\varkappa1}x_{1}+\dots+a_{\varkappa n}x_{n}\\
\\
k^{2}x_{0} &amp; =a_{00}k^{2}\xi_{0}+a_{10}\xi_{1}+\dots+a_{n0}\xi_{n}\\
x_{\varkappa} &amp; =a_{0\varkappa}\xi_{0}+a_{1\varkappa}\xi_{1}+\dots+a_{n\varkappa}\xi_{n}
\end{align}
\right|{\scriptstyle \begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} &amp; =k^{2}\\
a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} &amp; =0\\
\frac{a_{0\iota}a_{0\varkappa}}{k^{2}}+a_{0\iota}a_{1\varkappa}+\dots+a_{n\iota}a_{n\varkappa}=\delta_{\iota\kappa} &amp; =1\ (\iota=\kappa)\ \text{or}\ 0\ (\iota\ne\kappa)
\end{align}
}
\end{matrix}&lt;/math&gt;

In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force.&lt;ref group=M&gt;Killing (1884/85), pp. 4–5&lt;/ref&gt; Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting &lt;math&gt;k^{2}=-1&lt;/math&gt; and using ''p,x,y'' as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid &lt;math&gt;p^{2}-x^{2}-y^{2}=1&lt;/math&gt; (known as [[w:hyperboloid model]]),&lt;ref group=M&gt;Killing (1885), Note 9 on p. 260&lt;/ref&gt;&lt;ref name=rey /&gt; by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;All of Killing's transformations between 1879 and 1885 don't work when &lt;math&gt;k^{2}&lt;/math&gt; is negative, thus they fail to produce Lorentz transformation ({{equationNote|1a}}) with &lt;math&gt;k^{2}=-1&lt;/math&gt;.&lt;/p&gt;

Finally, in (1893) he wrote:&lt;ref group=M&gt;Killing (1893), see pp. 144, 327–328&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\
\hline \begin{align}t' &amp; =at+bu+cv\\
u' &amp; =a't+b'u+c'v\\
v' &amp; =a''t+b''u+c''v
\end{align}
\left|\begin{align}k^{2}a^{2}+a^{\prime2}+a^{\prime\prime2} &amp; =k^{2}\\
k^{2}b^{2}+b^{\prime2}+b^{\prime\prime2} &amp; =1\\
k^{2}c^{2}+b^{\prime2}+c^{\prime\prime2} &amp; =1\\
k^{2}ab+a'b'+a''b'' &amp; =0\\
k^{2}ac+a'c'+a''c'' &amp; =0\\
k^{2}bc+b'c'+b''c'' &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

and in ''n'' dimensions&lt;ref group=M&gt;Killing (1893), pp. 314–316, 216–217&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
k^{2}y_{0}y_{0}^{\prime}+y_{1}y_{1}^{\prime}+\cdots+y_{n}y_{n}^{\prime}=k^{2}x_{0}x_{0}^{\prime}+x_{1}x_{1}^{\prime}+\cdots+x_{n}x_{n}^{\prime}\\
ds^{2}=k^{2}dx_{0}^{2}+\dots+dx_{n}^{2}\\
\hline \begin{align}y_{0} &amp; =a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{n}\\
y_{1} &amp; =a_{10}x_{0}+a_{11}x_{1}+\dots+a_{1n}x_{n}\\
 &amp; \,\,\,\vdots\\
y_{n} &amp; =a_{n0}x_{0}+a_{n1}x_{1}+\dots+a_{nn}x_{n}
\end{align}
\left|\begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} &amp; =k^{2}\\
k^{2}a_{0\varkappa}^{2}+a_{1\varkappa}^{2}+\dots+a_{n\varkappa}^{2} &amp; =1\\
k^{2}a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} &amp; =0\\
k^{2}a_{0\varkappa}a_{0\lambda}+a_{1\varkappa}a_{1\lambda}+\dots+a_{n\varkappa}a_{n\lambda} &amp; =0\\
(\varkappa,\lambda=1,\dots, n,\ \lambda\lessgtr\varkappa)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}}) with &lt;math&gt;k^{2}=-1&lt;/math&gt;.&lt;/p&gt;

===={{anchor|Killing3}} Infinitesimal transformations and Lie group====

After [[#Lie3|Lie (1885/86)]] identified the projective group of a general surface of second degree &lt;math&gt;\sum f_{ik}x_{i}'x_{k}'=0&lt;/math&gt; with the group of non-Euclidean motions, Killing (1887/88)&lt;ref group=M&gt;Killing (1887/88a), pp. 274–275&lt;/ref&gt; defined the infinitesimal projective transformations (Lie algebra) in relation to the unit hypersphere:

:&lt;math&gt;\begin{matrix}x_{1}^{2}+\dots+x_{m+1}^{2}=1\\
\hline X_{\iota\varkappa}f=x_{i}\frac{\partial f}{\partial x_{\varkappa}}-x_{\varkappa}\frac{\partial f}{\partial x_{\iota}}\\
\text{where}\\
\left(X_{\iota\varkappa},X_{\iota\lambda}\right)=X_{\varkappa\lambda};\ \left(X_{\iota\varkappa},X_{\lambda\mu}\right)=0;\\
\left[\iota\ne\varkappa\ne\lambda\ne\mu\right]
\end{matrix}&lt;/math&gt;

and in (1892) he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:&lt;ref group=M&gt;Killing (1892), p. 177&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
\hline X_{\iota\varkappa}=x_{\iota}p_{\varkappa}-x_{\varkappa}p_{\iota},\quad X_{\iota}=x_{0}p_{\iota}-\frac{x_{\iota}p_{0}}{k^{2}}\\
\text{where}\\
\left(X_{\iota}X_{\iota\varkappa}\right)=X_{\varkappa}f;\ \left(X_{\iota}X_{\varkappa\lambda}\right)=0;\ \left(X_{\iota}X_{\varkappa}\right)=-\frac{1}{k^{2}}X_{\iota\varkappa}f;
\end{matrix}&lt;/math&gt;

In (1897/98) he showed the relation between Weierstrass coordinates &lt;math&gt;k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}&lt;/math&gt; and corrdinates &lt;math&gt;k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0&lt;/math&gt; used by himself in (1887/88) and by [[#Lie3|Werner (1889), Lie (1890)]]:&lt;ref group=M&gt;Killing (1897/98), pp. 255–256&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2} &amp; (a)\\
k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2} &amp; (b)
\end{matrix}\\
\hline V_{\varkappa}=k^{2}x_{0}p_{\varkappa}-x_{\varkappa}p_{0},\quad U_{\iota\varkappa}=p_{\iota}x_{\varkappa}-p_{\varkappa}x_{\iota}\\
\text{where}\\
\left(V_{\iota},V_{\varkappa}\right)=k^{2}U_{\iota\varkappa},\ \left(V_{\iota},U_{\iota\varkappa}\right)=-V_{\varkappa},\ \left(V_{\iota},U_{\varkappa\lambda}\right)=0,\\
\left(U_{\iota\varkappa},U_{\iota\lambda}\right)=U_{\varkappa\lambda},\ \left(U_{\iota\varkappa},U_{\lambda\mu}\right)=0\\
\left[\iota,\varkappa,\lambda,\mu=1,2,\dots n\right]\\
\hline \begin{matrix}y_{1}=\frac{x_{1}}{x_{0}},\ y_{2}=\frac{x_{2}}{x_{0}},\dots y_{n}=\frac{x_{n}}{x_{0}}\\
\downarrow\\
k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0\\
\hline q_{\varkappa}+\frac{y_{\varkappa}}{k^{2}}\sum_{\varrho}y_{y}q_{\varrho},\quad q_{\iota}y_{\varkappa}-q_{\varkappa}y_{\iota}
\end{matrix}
\end{matrix}&lt;/math&gt;

He pointed out that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to quadratic form (a) and [[w:Group action (mathematics)|transitive]] when related to quadratic form (b).

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Setting &lt;math&gt;k^{2}=-1&lt;/math&gt; denotes the group of hyperbolic motions and thus the Lorentz group.&lt;/p&gt;

=== {{anchor|Poincare}} Poincaré (1881) – Weierstrass coordinates ===
[[w:Henri Poincaré]] (1881) connected the work of [[../Lorentz transformation (Cayley-Hermite)#Hermite|E:Hermite (1853)]] and [[../Lorentz transformation (Möbius)#Selling|E:Selling (1873)]] on indefinite quadratic forms with non-Euclidean geometry (Poincaré already discussed such relations in an unpublished manuscript in 1880).&lt;ref&gt;Gray (1997)&lt;/ref&gt; He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates (without using that expression) connected by a transformation with integer coefficients:&lt;ref group=M name=p1&gt;Poincaré (1881a), pp. 133–134&lt;/ref&gt;&lt;ref&gt;Dickson (1923), pp. 220–221&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}F &amp; =(ax+by+cz)^{2}+(a'x+b'y+c'z)^{2}-(a''x+b''y+c''z)^{2}\\
 &amp; =\xi^{2}+\eta^{2}-\zeta^{2}=-1\\
F &amp; =(ax'+by'+cz')^{2}+(a'x'+b'y'+c'z')^{2}-(a''x'+b''y'+c''z')^{2}\\
 &amp; =\xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1
\end{align}
\\
\hline \begin{align}\xi' &amp; =\alpha\xi+\beta\eta+\gamma\zeta\\
\eta' &amp; =\alpha'\xi+\beta'\eta+\gamma'\zeta\\
\zeta' &amp; =\alpha''\xi+\beta''\eta+\gamma''\zeta
\end{align}
\left|\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =1\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =1\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-1\\
\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

He went on to describe the properties of &quot;hyperbolic coordinates&quot;.&lt;ref group=M name=poinc&gt;Poincaré (1881b), p. 333&lt;/ref&gt;&lt;ref name=rey&gt;Reynolds (1993)&lt;/ref&gt; Poincaré mentioned the hyperboloid model also in (1887).&lt;ref group=M&gt;Poincaré (1887), p. 206&lt;/ref&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.&lt;/p&gt;

=== {{anchor|Cox}} Cox (1881–1883) – Weierstrass coordinates ===
[[w:Homersham Cox (mathematician)|Homersham Cox]] (1881/82) – referring to similar rectangular coordinates used by [[w:Christoph Gudermann|Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt; and [[w:George Salmon]] (1862)&lt;ref group=M&gt;Salmon (1862), section 212, p. 165&lt;/ref&gt; on a sphere, and to [[#Escherich|Escherich (1874)]] as reported by [[w:Johannes Frischauf]] (1876)&lt;ref group=M&gt;Frischauf (1876), pp. 86–87&lt;/ref&gt; in the hyperbolic plane – defined the Weierstrass coordinates (without using that expression) and their transformation:&lt;ref group=M&gt;Cox (1881), p. 186 for Weierstrass coordinates; (1881/82), pp. 193–194 for Lorentz transformation. On p. 193, the misprinted expression &lt;math&gt;x^{2}-y^{2}-z^{2}&lt;/math&gt; should read &lt;math&gt;z^{2}-y^{2}-x^{2}&lt;/math&gt;&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}z^{2}-x^{2}-y^{2}=1\\
z^{2}-y^{2}-x^{2}=Z^{2}-Y^{2}-X^{2}\\
\hline \begin{align}x &amp; =l_{1}X+l_{2}Y+l_{3}Z\\
y &amp; =m_{1}X+m_{2}Y+m_{3}Z\\
z &amp; =n_{1}X+n_{2}Y+n_{3}Z\\
\\
X &amp; =l_{1}x+m_{1}y-n_{1}z\\
Y &amp; =l_{2}x+m_{2}y-n_{2}z\\
Z &amp; =l_{3}x+m_{3}y-n_{3}z
\end{align}
\left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} &amp; =1\\
l_{2}^{2}+m_{2}^{2}-n_{2}^{2} &amp; =1\\
l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =1\\
l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} &amp; =0\\
l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} &amp; =0\\
l_{3}l_{1}+m_{3}m_{1}-n_{3}n_{1} &amp; =0\\
\\
l_{1}^{2}+l_{2}^{2}-l_{3}^{2} &amp; =1\\
m_{1}^{2}+m_{2}^{2}-m_{3}^{2} &amp; =1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =1\\
l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} &amp; =0\\
m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} &amp; =0\\
n_{1}l_{1}+n_{2}l_{2}-n_{3}l_{3} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Replacing &lt;math&gt;{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =1
\end{align}
}&lt;/math&gt; with &lt;math&gt;{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =-1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =-1
\end{align}
}&lt;/math&gt;, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' up to a sign change in the inverse transformation.&lt;/p&gt;

Cox also gave the Weierstrass coordinates and their transformation in hyperbolic space:&lt;ref group=M&gt;Cox (1881), pp. 199, 206–207&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\
w^{2}-x^{2}-y^{2}-z^{2}=w^{\prime2}-x^{\prime2}-y^{\prime2}-z^{\prime2}\\
\hline \begin{align}x &amp; =l_{1}x'+l_{2}y'+l_{3}z'-l_{4}w'\\
y &amp; =m_{1}x'+m_{2}y'+m_{3}z'-m_{4}w'\\
z &amp; =n_{1}x'+n_{2}y'+n_{3}z'-n_{4}w'\\
w &amp; =r_{1}x'+r_{2}y'+r_{3}z'-r_{4}w'\\
\\
x' &amp; =l_{1}x+m_{1}y+n_{1}z-r_{1}w\\
y' &amp; =l_{2}x+m_{2}y+n_{2}z-r_{2}w\\
z' &amp; =l_{3}x+m_{3}y+n_{3}z-r_{3}w\\
w' &amp; =l_{4}x+m_{4}y+n_{4}z-r_{4}w
\end{align}
\left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}+n_{1}^{2}-r_{1}^{2} &amp; =1\\
l_{2}^{2}+m_{2}^{2}+n_{2}^{2}-r_{2}^{2} &amp; =1\\
l_{3}^{2}+m_{3}^{2}+n_{3}^{2}-r_{3}^{2} &amp; =1\\
l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2} &amp; =1\\
l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3}-r_{2}r_{3} &amp; =0\\
l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}-r_{3}r_{1} &amp; =0\\
l_{1}l_{4}+m_{1}m_{4}+n_{1}n_{4}-r_{1}r_{4} &amp; =0\\
l_{2}l_{4}+m_{2}m_{4}+n_{2}n_{4}-r_{2}r_{4} &amp; =0\\
l_{3}l_{4}+m_{3}m_{4}+n_{3}n_{4}-r_{3}r_{4} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Replacing &lt;math&gt;{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=1}&lt;/math&gt; with &lt;math&gt;{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=-1}&lt;/math&gt;, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' up to a sign change in both the first as well as inverse transformation.&lt;/p&gt;

=== {{anchor|Hill}} Hill (1882) – Homogeneous coordinates ===

Following [[#Gauss4|Gauss (1818)]], [[w:George William Hill]] (1882) formulated the equations&lt;ref group=M&gt;Hill (1882), pp. 323–325&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)\\
k\left(\sin^{2}E+\cos^{2}E-1\right)\\
\hline \begin{align} &amp;  &amp; \cos E' &amp; =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
 &amp; \mathbf{(1)} &amp; \sin E' &amp; =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
\hline \\
 &amp;  &amp; x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
 &amp;  &amp; y &amp; =\beta u+\beta'u'+\beta''u''\\
 &amp;  &amp; z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
 &amp; \mathbf{(2)}\\
 &amp;  &amp; u &amp; =-\alpha x-\beta y+\gamma z\\
 &amp;  &amp; u' &amp; =\alpha'x+\beta'y'-\gamma'z\\
 &amp;  &amp; u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\left|{\scriptstyle \begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =-1\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =1\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =1\\
\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
(k=-1)\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =k\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =k\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-k\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with &lt;math&gt;[\cos T,\sin T,\cos E',\sin E']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt;.

Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' .&lt;/p&gt;

=== {{anchor|Picard}} Picard (1882-1884) – Quadratic forms ===

[[w:Émile Picard]] (1882) analyzed the invariance of indefinite ternary [[w:Hermitian form|Hermitian quadratic forms]] with integer coefficients and their relation to [[w:Group action (mathematics)|discontinuous groups]], extending Poincaré's Fuchsian functions of one complex variable related to a circle, to &quot;hyperfuchsian&quot; functions of two complex variables related to a [[w:hypersphere]]. He formulated the following special case of an Hermitian form:&lt;ref group=M&gt;Picard (1882), pp. 307–308 first transformation system; pp. 315-317 second transformation system&lt;/ref&gt;&lt;ref&gt;Dickson (1923), pp. 280-281&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{matrix}xx_{0}+yy_{0}-zz_{0}\\
\\
\mathbf{(1)}\ \begin{align}x &amp; =M_{1}X+P_{1}Y+R_{1}Z\\
y &amp; =M_{2}X+P_{2}Y+R_{2}Z\\
z &amp; =M_{3}X+P_{3}Y+R_{3}Z
\end{align}
\\
\\
\left[\begin{align}[][x,y,z]=\text{complex}\\
\left[x_{0},y_{0},z_{0}\right]=\text{conjugate}
\end{align}
\right]\\
\\
\hline \\
x^{\prime2}+x^{\prime\prime2}+y^{\prime2}+y^{\prime\prime2}=1\\
x=x'+ix'',\quad y=y'+iy''\\
\\
\mathbf{(2)}\ \begin{align}X &amp; =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\
Y &amp; =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}}
\end{align}
\end{matrix}\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3} &amp; =1\\
P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} &amp; =1\\
R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} &amp; =-1\\
P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3} &amp; =0\\
M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} &amp; =0\\
P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} &amp; =0\\
\\
M_{1}\mu_{1}+P_{1}\pi_{1}-R_{1}\rho_{1} &amp; =1\\
M_{2}\mu_{2}+P_{2}\pi_{2}-R_{2}\rho_{2} &amp; =1\\
M_{3}\mu_{3}+P_{3}\pi_{3}-R_{3}\rho_{3} &amp; =-1\\
\mu_{2}M_{1}+\pi_{2}P_{1}-R_{1}\rho_{2} &amp; =0\\
\mu_{2}M_{3}+\pi_{2}P_{3}-R_{3}\rho_{2} &amp; =0\\
\mu_{3}M_{1}+\pi_{3}P_{1}-R_{1}\rho_{3} &amp; =0\\
\\
\left[\begin{align}[][M,P,R\dots]=\text{complex}\\
\left[\mu,\pi,\rho\dots\right]=\text{conjugate}
\end{align}
\right]
\end{align}
}\right.\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;-Z&lt;sup&gt;2&lt;/sup&gt;'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;=1''.&lt;/p&gt;

Or in (1884a) in relation to indefinite binary Hermitian quadratic forms:&lt;ref group=M&gt;Picard (1884a), p. 13&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}UU_{0}-VV_{0}=uu_{0}-vv_{0}\\
\hline \begin{align}U &amp; =\mathcal{A}u+\mathcal{B}v\\
V &amp; =\mathcal{C}u+\mathcal{D}v
\end{align}
\left|\begin{align}\mathcal{A}\mathcal{A}_{0}-\mathcal{C}\mathcal{C}_{0} &amp; =1\\
\mathcal{A}\mathcal{B}_{0}-\mathcal{C}\mathcal{D}_{0} &amp; =0\\
\mathcal{B}\mathcal{B}_{0}-\mathcal{D}\mathcal{D}_{0} &amp; =-1\\
\mathcal{D}\mathcal{D}_{0}-\mathcal{C}\mathcal{C}_{0} &amp; =1
\end{align}
\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' producing ''U&lt;sup&gt;2&lt;/sup&gt;-V&lt;sup&gt;2&lt;/sup&gt;=u&lt;sup&gt;2&lt;/sup&gt;-v&lt;sup&gt;2&lt;/sup&gt;''.&lt;/p&gt;

Or in (1884b):&lt;ref group=M&gt;Picard (1884b), p. 416&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}xx_{0}+yy_{0}-1=0\\
\hline \begin{align}X &amp; =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\
Y &amp; =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}}
\end{align}
\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3}=P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} &amp; =1\\
R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} &amp; =-1\\
P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3}=M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3}=P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} &amp; =0\\
M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;=1''.&lt;/p&gt;

Or in (1884c):&lt;ref group=M&gt;Picard (1884c), pp. 123–124; 163&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}UU_{0}+VV_{0}-WW_{0}=uu_{0}+vv_{0}-ww_{0}\\
\hline \mathbf{(1)}\ \begin{align}U &amp; =Mu+Pv+Rw\\
V &amp; =M'u+P'v+R'w\\
W &amp; =M''u+P''v+R''w\\
\\
u &amp; =M_{0}U+M_{0}^{\prime}V-M_{0}^{\prime\prime}W\\
v &amp; =P_{0}U+P_{0}^{\prime}V-P_{0}^{\prime\prime}W\\
w &amp; =-R_{0}U-R_{0}^{\prime}V+R_{0}^{\prime\prime}W
\end{align}
\left|{\scriptstyle \begin{align}MM_{0}+M'M_{0}^{\prime}-M''M_{0}^{\prime\prime} &amp; =1\\
PP_{0}+P'P_{0}^{\prime}-P''P_{0}^{\prime\prime} &amp; =1\\
RR_{0}+R'R_{0}^{\prime}-R''R_{0}^{\prime\prime} &amp; =-1\\
MP_{0}+M'P_{0}^{\prime}-M''P_{0}^{\prime\prime} &amp; =0\\
MR_{0}+M'R_{0}^{\prime}-M''R_{0}^{\prime\prime} &amp; =0\\
PR_{0}+P'R_{0}^{\prime}-P''R_{0}^{\prime\prime} &amp; =0\\
\\
MM_{0}+PP_{0}-RR_{0} &amp; =1\\
M'M_{0}^{\prime}+P'P_{0}^{\prime}-R'R_{0}^{\prime} &amp; =1\\
M''M_{0}^{\prime\prime}+P''P_{0}^{\prime\prime}-R''R_{0}^{\prime\prime} &amp; =-1\\
M_{0}M'+P_{0}P'-R_{0}R' &amp; =0\\
M_{0}M''+P_{0}P''-R_{0}R'' &amp; =0\\
M_{0}^{\prime}M''+P_{0}^{\prime}P''-R_{0}^{\prime}R'' &amp; =0
\end{align}
}\right.\\
\hline \text{Invariance of unit hypersphere:}\\
\mathbf{(2)}\ \begin{align}\xi' &amp; =\frac{A\xi+A'\eta+A''}{C\xi+C'\eta+C''}\\
\eta' &amp; =\frac{B\xi+B'\eta+B''}{C\xi+C'\eta+C''}
\end{align}
\left|{\scriptstyle \begin{align}AA_{0}+A'A_{0}^{\prime}-A''A_{0}^{\prime\prime} &amp; =1\\
BB_{0}+B'B_{0}^{\prime}-B''B_{0}^{\prime\prime} &amp; =1\\
CC_{0}+C'C_{0}^{\prime}-C''C_{0}^{\prime\prime} &amp; =-1\\
AB_{0}+A'B_{0}^{\prime}-A''B_{0}^{\prime\prime} &amp; =0\\
AC_{0}+A'C_{0}^{\prime}-A''C_{0}^{\prime\prime} &amp; =0\\
BC_{0}+B'C_{0}^{\prime}-B''C_{0}^{\prime\prime} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''U&lt;sup&gt;2&lt;/sup&gt;+V&lt;sup&gt;2&lt;/sup&gt;-W&lt;sup&gt;2&lt;/sup&gt;=u&lt;sup&gt;2&lt;/sup&gt;+v&lt;sup&gt;2&lt;/sup&gt;-w&lt;sup&gt;2&lt;/sup&gt;'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing &lt;math&gt;\xi^{\prime2}+\eta^{\prime2}=\xi^{2}+\eta^{2}=1&lt;/math&gt;.&lt;/p&gt;

=== {{anchor|Callandreau}} Callandreau (1885) – Homography ===

Following [[#Gauss4|Gauss (1818)]] and [[#Hill|Hill (1882)]], [[w:Octave Callandreau]] (1885) formulated the equations&lt;ref group=M&gt;Callandreau (1885), pp. A.7; A.12&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)=\\
{\scriptstyle (\alpha+\alpha'\sin T+\alpha''\cos T)^{2}+(\beta+\beta'\sin T+\beta''\cos T)^{2}-(\gamma+\gamma'\sin T+\gamma''\cos T)^{2}}\\
\hline \begin{align}\cos\varepsilon' &amp; =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
\sin\varepsilon' &amp; =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}
\end{align}
\left|{\scriptstyle \begin{align} &amp; \left(k=1\right)\\
\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =-k &amp; \alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =+k &amp; \alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =+k &amp; \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =-1 &amp; \alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =-1 &amp; \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =+1 &amp; \beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The transformation system is equivalent to Lorentz transformation ({{equationNote|1b}})  ''(n=2)'' with &lt;math&gt;[\cos T,\sin T,\cos\varepsilon',\sin\varepsilon']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Lie3}} Lie (1885-1890) – Lie group, hyperbolic motions, and infinitesimal transformations===

In (1885/86), [[w:Sophus Lie]] identified the projective group of a general surface of second degree &lt;math&gt;\sum f_{ik}x_{i}'x_{k}'=0&lt;/math&gt; with the group of non-Euclidean motions.&lt;ref group=M&gt;Lie (1885/86), p. 411&lt;/ref&gt; In a thesis guided by Lie, [[w:Hermann Werner]] (1889) discussed this projective group by using the equation of a unit hypersphere as the surface of second degree (which was already given before by [[#Killing3|Killing (1887)]]), and also gave the corresponding infinitesimal projective transformations (Lie algebra):&lt;ref group=M&gt;Werner (1889), pp. 4, 28&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=1\\
\hline x_{i}p_{\varkappa}-x_{\varkappa}p_{i},\quad p_{i}-x_{i}\sum_{1}^{n}{\scriptstyle j}\ x_{j}p_{j}\quad(i,\varkappa=1,\dots, n)\\
\text{where}\\
\left(Q_{i},Q_{\varkappa}\right)=R_{i,\varkappa};\ \left(Q_{i},Q_{j,\varkappa}\right)=\varepsilon_{i,j}Q_{\varkappa}-\varepsilon_{i,\varkappa}Q_{j};\\
\left(R_{i,\varkappa},R_{\mu,\nu}\right)=\varepsilon_{\varkappa,\mu}R_{i,\nu}-\varepsilon_{\varkappa,\nu}R_{i,\mu}-\varepsilon_{,\mu}R_{\varkappa,\nu}+\varepsilon_{i,\nu}R_{\varkappa,\mu}\\
\left[\varepsilon_{i,\varkappa}\equiv0\ \text{for}\ i\ne\varkappa;\ \varepsilon_{i,i}=1\right]
\end{matrix}&lt;/math&gt;

More generally, Lie (1890)&lt;ref group=M&gt;Lie (1890a), p. 295;&lt;/ref&gt; defined non-Euclidean motions in terms of two forms &lt;math&gt;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\pm1=0&lt;/math&gt; in which the imaginary form with &lt;math&gt;+1&lt;/math&gt; denotes the group of elliptic motions (in Klein's terminology), the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:&lt;ref group=M&gt;Lie (1890a), p. 311&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+\dots+x_{n}^{2}-1=0\\
\hline p_{k}-x_{k}\sum j_{1}^{0}x_{j}p_{j},\quad x_{i}p_{k}-x_{k}p_{i}\quad(i,k=1\dots n)
\end{matrix}&lt;/math&gt;

Summarizing, Lie (1893) discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form:

:&lt;math&gt;x^{2}+y^{2}-1=0&lt;/math&gt;&lt;ref group=M&gt;Lie (1893), p. 474&lt;/ref&gt; or &lt;math&gt;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1=0&lt;/math&gt;&lt;ref group=M&gt;Lie (1893), p. 479&lt;/ref&gt; or &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-1=0&lt;/math&gt;.&lt;ref group=M&gt;Lie (1893), p. 481&lt;/ref&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The group of hyperbolic motions is isomorphic to the Lorentz group. The interval &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-1=0&lt;/math&gt; becomes the Lorentz interval &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-x_{0}^{2}=0&lt;/math&gt; by setting
&lt;br&gt;&lt;math&gt;(x_{1},\dots,\ x_{n},\ 1)=\left(\frac{x_{1}}{x_{0}},\dots,\ \frac{x_{n}}{x_{0}},\ \frac{x_{0}}{x_{0}}\right)&lt;/math&gt;&lt;/p&gt;

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane using the following invariant and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)'':&lt;ref group=M&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}-Z^{2}=1\\
X^{2}+Y^{2}-Z^{2}=X^{\prime2}+Y^{\prime2}-Z^{\prime2}\\
\hline \begin{align}X &amp; =aX'+a'Y'+a''Z'\\
Y &amp; =bX'+b'Y'+b''Z'\\
Z &amp; =cX'+c'Y'+c''Z'\\
\\
X' &amp; =aX+bY-cZ\\
Y' &amp; =a'X+b'Y-c'Z\\
Z' &amp; =-a''X-b''Y+c''Z
\end{align}
\left|\begin{align}a^{2}+b^{2}-c^{2} &amp; =1\\
a^{\prime2}+b^{\prime2}-c^{\prime2} &amp; =1\\
a^{\prime\prime2}+b^{\prime\prime2}-c^{\prime\prime2} &amp; =-1\\
aa'+bb'-cc' &amp; =0\\
a'a''+b'b''-c'c'' &amp; =0\\
a''a+b''b-c''c &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.&lt;/p&gt;

He gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Hausdorff}} Hausdorff (1899) – Weierstrass coordinates===

[[w:Felix Hausdorff]] (1899) – citing Killing (1885) – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:&lt;ref group=M&gt;Hausdorff (1899), p. 165, pp. 181-182&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{2}-x^{2}-y^{2}=1\\
\hline \begin{align}x &amp; =a_{1}x'+a_{2}y'+x_{0}p'\\
y &amp; =b_{1}x'+b{}_{2}y'+y_{0}p'\\
p &amp; =e_{1}x'+e_{2}y'+p_{0}p'\\
\\
x' &amp; =a_{1}x+b_{1}y-e_{1}p\\
y' &amp; =a_{2}x+b_{2}y-e_{2}p\\
-p' &amp; =x_{0}x+y_{0}y-p_{0}p
\end{align}
\left|{\scriptstyle \begin{align}a_{1}^{2}+b_{1}^{2}-e_{1}^{2} &amp; =1\\
a_{2}^{2}+b_{2}^{2}-e_{2}^{2} &amp; =1\\
-x_{0}^{2}-y_{0}^{2}+p_{0}^{2} &amp; =1\\
a_{2}x_{0}+b_{2}y_{0}-e_{2}p_{0} &amp; =0\\
a_{1}x_{0}+b_{1}y_{0}-e_{1}p_{0} &amp; =0\\
a_{1}a_{2}+b_{1}b_{2}-e_{1}e_{2} &amp; =0\\
\\
a_{1}^{2}+a_{2}^{2}-x_{0}^{2} &amp; =1\\
b_{1}^{2}+b_{2}^{2}-y_{0}^{2} &amp; =1\\
-e_{1}^{2}-e_{2}^{2}+p_{0}^{2} &amp; =1\\
b_{1}e_{1}+b_{2}e_{2}-y_{0}p_{0} &amp; =0\\
a_{1}e_{1}+a_{2}e_{2}-x_{0}p_{0} &amp; =0\\
a_{1}b_{1}+a_{2}b_{2}-x_{0}y_{0} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.&lt;/p&gt;

==={{anchor|Woods2}} Woods (1901-05) – Beltrami and Weierstrass coordinates ===
In (1901/02) [[w:Frederick S. Woods]] defined the following invariant quadratic form and its [[w:projective transformation]] in terms of Beltrami coordinates (he pointed out that this can be connected to hyperbolic geometry by setting &lt;math&gt;k=\sqrt{-1}R&lt;/math&gt; with ''R'' as real quantity):&lt;ref group=M&gt;Woods (1901/02), p. 98, 104&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}\left(u^{2}+v^{2}+w^{2}\right)+1=0\\
\hline \begin{align}u' &amp; =\frac{\alpha_{1}u+\alpha_{2}v+\alpha_{3}w+\alpha_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\
v' &amp; =\frac{\beta_{1}u+\beta_{2}v+\beta_{3}w+\beta_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\
w' &amp; =\frac{\gamma_{1}u+\gamma_{2}v+\gamma_{3}w+\gamma_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}
\end{align}
\left|\begin{align}k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right)+\delta_{i}^{2} &amp; =k^{2}\\
(i=1,2,3)\\
k^{2}\left(\alpha_{4}^{2}+\beta_{4}^{2}+\gamma_{4}^{2}\right)+\delta_{4}^{2} &amp; =1\\
\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}+\delta_{i}\delta_{h} &amp; =0\\
(i,h=1,2,3,4;\ i\ne h)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'' with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.&lt;/p&gt;

Alternatively, Woods (1903, published 1905) – citing Killing (1885) – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation (with &lt;math&gt;k=\sqrt{-1}k&lt;/math&gt; for hyperbolic space):&lt;ref group=M&gt;Woods (1903/05), pp. 45–46; p. 48)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\\
ds^{2}=\frac{1}{k^{2}}dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\\
\hline \begin{align}x_{1}^{\prime} &amp; =\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+\alpha_{0}x_{0}\\
x_{2}^{\prime} &amp; =\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{0}x_{0}\\
x_{3}^{\prime} &amp; =\gamma_{1}x_{1}+\gamma_{2}x_{2}+\gamma_{3}x_{3}+\gamma_{0}x_{0}\\
x_{0}^{\prime} &amp; =\delta_{1}x_{1}+\delta_{2}x_{2}+\delta_{3}x_{3}+\delta_{0}x_{0}
\end{align}
\left|\begin{align}\delta_{0}^{2}+k^{2}\left(\alpha_{0}^{2}+\beta_{0}^{2}+\gamma_{0}^{2}\right) &amp; =1\\
\delta_{i}^{2}+k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right) &amp; =k^{2}\\
(i=1,2,3)\\
\delta_{i}\delta_{h}+k^{2}\left(\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}\right) &amp; =0\\
(i,h=0,1,2,3;\ i\ne h)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.&lt;/p&gt;

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – used the invariant quadratic form and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)''&lt;ref group=M&gt;Liebmann (1904/05), p. 168; pp. 175–176&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{\prime2}-x^{\prime2}-y^{\prime2}=1\\
\hline \begin{align}x_{1} &amp; =\alpha_{11}x+\alpha_{12}y+\alpha_{13}p\\
y_{1} &amp; =\alpha_{21}x+\alpha_{22}y+\alpha_{23}p\\
x_{1} &amp; =\alpha_{31}x+\alpha_{32}y+\alpha_{33}p\\
\\
x &amp; =\alpha_{11}x_{1}+\alpha_{21}y_{1}-\alpha_{31}p_{1}\\
y &amp; =\alpha_{12}x_{1}+\alpha_{22}y_{1}-\alpha_{32}p_{1}\\
p &amp; =-\alpha_{13}x_{1}-\alpha_{23}y_{1}+\alpha_{33}p_{1}
\end{align}
\left|\begin{align}\alpha_{33}^{2}-\alpha_{13}^{2}-\alpha_{23}^{2} &amp; =1\\
-\alpha_{31}^{2}+\alpha_{11}^{2}+\alpha_{21}^{2} &amp; =1\\
-\alpha_{32}^{2}+\alpha_{12}^{2}+\alpha_{22}^{2} &amp; =1\\
\alpha_{31}\alpha_{32}-\alpha_{11}\alpha_{12}-\alpha_{21}\alpha_{22} &amp; =0\\
\alpha_{32}\alpha_{33}-\alpha_{12}\alpha_{13}-\alpha_{22}\alpha_{23} &amp; =0\\
\alpha_{33}\alpha_{31}-\alpha_{23}\alpha_{11}-\alpha_{23}\alpha_{21} &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.&lt;/p&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bour56att}} 
*{{#section:History of Topics in Special Relativity/mathsource|chal82sec}} 
*{{#section:History of Topics in Special Relativity/mathsource|chas29}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|fris76}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau98}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau18}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|haus99}} 
*{{#section:History of Topics in Special Relativity/mathsource|hill82}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac27}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac32a}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac32b}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac33}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil77}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil79}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil84}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil85}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag73}} 
*{{#section:History of Topics in Special Relativity/mathsource|leb37}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie85}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie90}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie93}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic82}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84a}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84b}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84c}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81a}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81b}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin87}} 
*{{#section:History of Topics in Special Relativity/mathsource|sal62}} 
*{{#section:History of Topics in Special Relativity/mathsource|som63}} 
*{{#section:History of Topics in Special Relativity/mathsource|wedd47}} 
*{{#section:History of Topics in Special Relativity/mathsource|wern89}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L1}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
      <sha1>juy2da2ehwybnp86e966kql2xidxojo</sha1>
    </revision>
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    <title>History of Topics in Special Relativity/Lorentz transformation (hyperbolic)</title>
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      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="47082" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via hyperbolic functions==
===Translation in the hyperbolic plane===
The case of a Lorentz transformation without spatial rotation is called a [[w:Lorentz boost]]. The simplest case can be given, for instance, by setting ''n=1'' in the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}\\
\\
x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}
\end{align}
\left|\begin{align}g_{01}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{10}^{2} &amp; =1\\
g_{01}g_{11}-g_{00}g_{10} &amp; =0\\
g_{10}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{01}^{2} &amp; =1\\
g_{10}g_{11}-g_{00}g_{01} &amp; =0
\end{align}
\rightarrow\begin{align}g_{00}^{2} &amp; =g_{11}^{2}\\
g_{01}^{2} &amp; =g_{10}^{2}
\end{align}
\right.
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}g_{00} &amp; -g_{10}\\
-g_{01} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\right|\det\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;|{{equationRef|3a}}}}

which resembles precisely the relations of [[w:hyperbolic function]]s in terms of [[w:hyperbolic angle]] &lt;math&gt;\eta&lt;/math&gt;. Thus a Lorentz boost or [[w:hyperbolic rotation]] (being the same as a rotation around an imaginary angle &lt;math&gt;i\eta=\phi&lt;/math&gt; in [[../Lorentz transformation (imaginary)#math_2b|E:'''(2b)''']] or a [[w:Translation (geometry)|translation]] in the hyperbolic plane in terms of the hyperboloid model) is given by

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline g_{00}=g_{11}=\cosh\eta,\ g_{01}=g_{10}=-\sinh\eta\\
\hline \left.\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta &amp;  &amp; =\frac{x_{0}-x_{1}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}-x_{1}v}{\sqrt{1-v^{2}}}\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta &amp;  &amp; =\frac{x_{1}-x_{0}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}-x_{0}v}{\sqrt{1-v^{2}}}\\
x_{2}^{\prime} &amp; =x_{2} &amp;  &amp; =x_{2} &amp;  &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta &amp;  &amp; =\frac{x_{0}^{\prime}-x_{1}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}^{\prime}-x_{1}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta &amp;  &amp; =\frac{x_{1}^{\prime}-x_{0}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}^{\prime}-x_{0}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{2} &amp; =x_{2}^{\prime} &amp;  &amp; =x_{2}^{\prime} &amp;  &amp; =x_{2}^{\prime}
\end{align}
\right|{\scriptstyle \begin{align}\sinh^{2}\eta-\cosh^{2}\eta &amp; =-1 &amp; (a)\\
\cosh^{2}\eta-\sinh^{2}\eta &amp; =1 &amp; (b)\\
\frac{\sinh\eta}{\cosh\eta} &amp; =\tanh\eta=v &amp; (c)\\
\frac{1}{\sqrt{1-\tanh^{2}\eta}} &amp; =\cosh\eta &amp; (d)\\
\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\sinh\eta &amp; (e)\\
\frac{\tanh q\pm\tanh\eta}{1\pm\tanh q\tanh\eta} &amp; =\tanh\left(q\pm\eta\right) &amp; (f)
\end{align}
}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\right|\det\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;|{{equationRef|3b}}}}

Hyperbolic identities (a,b) on the right of ({{equationNote|3b}}) were given by [[#Riccati|Riccati (1757)]], all identities (a,b,c,d,e,f) by [[#Lambert|Lambert (1768–1770)]]. Lorentz transformations ({{equationNote|3b}}-A) were given by [[#Laisant|Laisant (1874)]], [[#Cox|Cox (1882)]], [[#Lindemann|Lindemann (1890/91)]], [[#Gerard|Gérard (1892)]], [[#Killing2|Killing (1893, 1897/98)]], [[#Whitehead|Whitehead (1897/98)]], [[#Woods2|Woods (1903/05)]], [[#Elliott|Elliott (1903)]] and [[#Liebmann|Liebmann (1904/05)]] in terms of Weierstrass coordinates of the [[w:hyperboloid model]], while transformations similar to ({{equationNote|3b}}-C) have been used by [[#Lipschitz1|Lipschitz (1885/86)]].

Rapidity can be composed of arbitrary many rapidities &lt;math&gt;\eta_{1},\eta_{2}\dots&lt;/math&gt; as per the [[w:Hyperbolic functions#Sums of arguments|w:angle sum laws of hyperbolic sines and cosines]], so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between [[w:List of trigonometric identities#Angle sum and difference identities|w:angle sum laws of circular trigonometry]] and spatial rotations. Alternatively, the hyperbolic angle sum laws ''themselves'' can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the [[w:unit hyperbola]]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \left[\eta=\eta_{2}-\eta_{1}\right]\\
\begin{align}x_{0}^{\prime} &amp; =\sinh\eta_{1} &amp;  &amp; =\sinh\left(\eta_{2}-\eta\right) &amp;  &amp; =\sinh\eta_{2}\cosh\eta-\cosh\eta_{2}\sinh\eta &amp;  &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =\cosh\eta_{1} &amp;  &amp; =\cosh\left(\eta_{2}-\eta\right) &amp;  &amp; =-\sinh\eta_{2}\sinh\eta+\cosh\eta_{2}\cosh\eta &amp;  &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta\\
\\
x_{0} &amp; =\sinh\eta_{2} &amp;  &amp; =\sinh\left(\eta_{1}+\eta\right) &amp;  &amp; =\sinh\eta_{1}\cosh\eta+\cosh\eta_{1}\sinh\eta &amp;  &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =\cosh\eta_{2} &amp;  &amp; =\cosh\left(\eta_{1}+\eta\right) &amp;  &amp; =\sinh\eta_{1}\sinh\eta+\cosh\eta_{1}\cosh\eta &amp;  &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \begin{align} &amp; \begin{bmatrix}x_{0}^{\prime}\\
x_{1}^{\prime}
\end{bmatrix}=\begin{bmatrix}\sinh\eta_{1}\\
\cosh\eta_{1}
\end{bmatrix}=\begin{bmatrix}\sinh\left(\eta_{2}-\eta\right)\\
\cosh\left(\eta_{2}-\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}\sinh\eta_{2}\\
\cosh\eta_{2}
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}x_{0}\\
x_{1}
\end{bmatrix}\\
 &amp; \begin{bmatrix}x_{0}\\
x_{1}
\end{bmatrix}=\begin{bmatrix}\sinh\eta_{2}\\
\cosh\eta_{2}
\end{bmatrix}=\begin{bmatrix}\sinh\left(\eta_{1}+\eta\right)\\
\cosh\left(\eta_{1}+\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}\sinh\eta_{1}\\
\cosh\eta_{1}
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}x_{0}^{\prime}\\
x_{1}^{\prime}
\end{bmatrix}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|3c}}}}

Hyperbolic angle sum laws equivalent to Lorentz boost ({{equationNote|3c}}) were given by [[#Riccati|Riccati (1757)]] and [[#Lambert|Lambert (1768–1770)]], while the matrix representation was given by [[#Glaisher|Glaisher (1878)]] and [[#Gunther2|Günther (1880/81)]]. In special relativity, hyperbolic functions were used by [[#Frank|Frank (1909)]] and [[#Varicak|Varićak (1910)]].

Finally, because of &lt;math&gt;\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;, Lorentz boost ({{equationNote|3b}}) assumes a simple form by using [[w:squeeze mapping]]s in analogy to Euler's formula in [[../Lorentz transformation (imaginary)#math_2c|E:'''(2c)''']]:&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \begin{matrix}x_{1}^{\prime}-x_{0}^{\prime}=k\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime}=\frac{1}{k}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime}=x_{2}
\end{matrix}\\
\hline \begin{align}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}-x_{0} &amp; =e^{-\eta}\left(x_{1}^{\prime}-x_{0}^{\prime}\right)\\
x_{1}+x_{0} &amp; =e^{\eta}\left(x_{1}^{\prime}+x_{0}^{\prime}\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\\
\hline k=e^{\eta}=\cosh\eta+\sinh\eta=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}=\sqrt{\frac{1+v}{1-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|3d}}}} 

Lorentz transformations ({{equationNote|3d}}) for arbitrary ''k'' were given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]), while the above exponential form was explicitly used by [[#Lindemann|Lindemann (1890/91)]], [[#Elliott|Elliott (1903)]], [[#Herglotz1|Herglotz (1909)]]. 

===Hyperbolic law of cosines===
In line with equation [[../Lorentz transformation (general)#math_1b|E:'''(1b)''']] one can use coordinates &lt;math&gt;[u_{1},\ u_{2}]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}}\right]&lt;/math&gt; inside the [[w:unit circle]] &lt;math&gt;u_{1}^{2}+u_{2}^{2}=1&lt;/math&gt;, thus the corresponding Lorentz transformations ({{equationNote|3b}}) obtain the form:

{{NumBlk|:|&lt;math&gt;\begin{matrix}u_{x}^{2}+u_{y}^{2}=u_{x}^{\prime2}+u_{y}^{\prime2}=1\\
\hline \begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
u_{1}^{\prime} &amp; =\frac{-\sinh\eta+u_{1}\cosh\eta}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{1}-\tanh\eta}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{1}-v}{1-u_{1}v}\\
u_{2}^{\prime} &amp; =\frac{u_{2}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-v^{2}}}{1-u_{1}v}\\
\\
u_{1} &amp; =\frac{\sinh\eta+u_{1}^{\prime}\cosh\eta}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+\tanh\eta}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+v}{1+u_{1}^{\prime}v}\\
u_{2} &amp; =\frac{u_{2}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|3e}}}}

Transformations (A) were given by [[#Escherich|Escherich (1874)]] and [[#Killing2|Killing (1898)]], and transformations (B) by [[#Beltrami|Beltrami (1868)]] and [[#Schur|Schur (1885/86, 1900/02)]] in terms of [[w:Beltrami–Klein model|Beltrami coordinates]]&lt;ref&gt;Rosenfeld (1988), p. 231&lt;/ref&gt; of hyperbolic geometry. By using the scalar product of &lt;math&gt;\left[u_{1},u_{2}\right]&lt;/math&gt;, the resulting Lorentz transformation can be seen as equivalent to the [[w:hyperbolic law of cosines]]:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{matrix}u^{2}=u_{1}^{2}+u_{2}^{2}\\
u'^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}
\end{matrix}\left|\begin{matrix}u_{1}=u\cos\alpha\\
u_{2}=u\sin\alpha\\
\\
u_{1}^{\prime}=u'\cos\alpha'\\
u_{2}^{\prime}=u'\sin\alpha'
\end{matrix}\right|\begin{align}u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+vu'\cos\alpha'}, &amp; u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-vu\cos\alpha}\\
u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{1+vu'\cos\alpha'}, &amp; u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{1-vu\cos\alpha}\\
\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{u'\cos\alpha'+v}, &amp; \tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{u\cos\alpha-v}
\end{align}
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(vu'\sin\alpha'\right){}^{2}}}{1+vu'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(vu\sin\alpha\right){}^{2}}}{1-vu\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-u^{\prime2}}}=\frac{1}{\sqrt{1-v^{2}}}\frac{1}{\sqrt{1-u^{2}}}-\frac{v}{\sqrt{1-v^{2}}}\frac{u}{\sqrt{1-u^{2}}}\cos\alpha &amp; (b)\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha &amp; (a)
\end{matrix}&lt;/math&gt;|{{equationRef|3f}}}}

The hyperbolic law of cosines (a) was given by [[#Taurinus|Taurinus (1826) and Lobachevsky (1829/30)]] and others, while variant (b) was given by [[#Schur|Schur (1900/02)]]. By further setting ''u=u′'' it follows:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\cos\alpha=\frac{\cos\alpha'+v}{1+v\cos\alpha'},\ \sin\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{1+v\cos\alpha'},\ \tan\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{\cos\alpha'+v},\ \tan\frac{\alpha}{2}=\sqrt{\frac{1-v}{1+v}}\tan\frac{\alpha'}{2}\\
\cos\alpha'=\frac{\cos\alpha-v}{1-v\cos\alpha},\ \sin\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{1-v\cos\alpha},\ \tan\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{\cos\alpha-v},\ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+v}{1-v}}\tan\frac{\alpha}{2}
\end{matrix}&lt;/math&gt;|{{equationRef|3g}}}}

Formulas (3b) resemble the equations of an [[w:ellipse]] of [[w:Orbital eccentricity|eccentricity]] ''v/c'', [[w:eccentric anomaly]] α' and [[w:true anomaly]] α, first geometrically formulated by [[#Euler|Kepler (1609)]] and explicitly written down by [[#Euler|Euler (1735, 1748), Lagrange (1770)]] and many others in relation to planetary motions. They were also used by [[../Lorentz transformation (conformal)#Darboux|E:Darboux (1873)]] as a sphere transformation. In special relativity, these formulas describes the aberration of light, see [[../Lorentz transformation (velocity)#Velocity addition and aberration|E:velocity addition and aberration]].

==Historical notation==
==={{anchor|Euler}} Euler (1735) – True and eccentric anomaly===

[[w:Johannes Kepler]] (1609) geometrically formulated [[w:Kepler's equation]] and the relations between the [[w:mean anomaly]], [[w:true anomaly]], and [[w:eccentric anomaly]].&lt;ref group=M&gt;Kepler (1609), chapter 60. The editors of Kepler's collected papers remark (p. 482), that Kepler's relations correspond to &lt;math&gt;{\scriptstyle \alpha=\beta+e\sin\beta}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\nu=\frac{e+\cos\beta}{1+e\cos\beta}}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\beta=\frac{\cos\nu-e}{1-e\cos\nu}}&lt;/math&gt;&lt;/ref&gt;&lt;ref&gt;Volk (1976), p. 366&lt;/ref&gt; The relation between the true anomaly ''z'' and the eccentric anomaly ''P'' was algebraically expressed by [[w:Leonhard Euler]] (1735/40) as follows:&lt;ref group=M&gt;Euler (1735/40), § 19&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{\cos P+v}{1+v\cos P},\ \cos P=\frac{\cos z-v}{1-v\cos z},\ \int P=\frac{\int z\sqrt{1-v^{2}}}{1-v\cos z}&lt;/math&gt;

and in 1748:&lt;ref group=M&gt;Euler (1748a), section VIII&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{n+\cos y}{1+n\cos y},\ \sin z=\frac{\sin y\sqrt{1-n^{2}}}{1+n\cos y},\ \tan z=\frac{\sin y\sqrt{1-n^{2}}}{n+\cos y}&lt;/math&gt;

while [[w:Joseph-Louis Lagrange]] (1770/71) expressed them as follows&lt;ref group=M&gt;Lagrange (1770/71), section I&lt;/ref&gt;

:&lt;math&gt;\sin u=\frac{m\sin x}{1+n\cos x},\ \cos u=\frac{n+\cos x}{1+n\cos x},\ \operatorname{tang}\frac{1}{2}u=\frac{m}{1+n}\operatorname{tang}\frac{1}{2}x,\ \left(m^{2}=1-n^{2}\right)&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt; These relations resemble formulas ({{equationNote|3g}}), while ({{equationNote|3f}}) follows by setting &lt;math&gt;[\cos z,\sin z]=\left[u_{x},u_{y}\right]&lt;/math&gt; in Euler's formulas or &lt;math&gt;[\cos u,\sin u]=\left[u_{x},u_{y}\right]&lt;/math&gt; in Lagrange's formulas.&lt;/p&gt;

==={{anchor|Riccati}} Riccati (1757) – hyperbolic addition===

[[w:Vincenzo Riccati]] introduced hyperbolic functions in 1757,&lt;ref group=M&gt;Riccati (1757), p. 71&lt;/ref&gt;&lt;ref group=M&gt;Günther (1880/81), pp. 7–13&lt;/ref&gt; in particular he formulated the angle sum laws for hyperbolic sine and cosine:

:&lt;math&gt;\begin{matrix}\mathrm{Ch}(\varphi+\pi)=\frac{\mathrm{Ch}\varphi\mathrm{Ch}\pi+\mathrm{Sh}\varphi\mathrm{Sh}\pi}{r}\\
\mathrm{Sh}(\varphi+\pi)=\frac{\mathrm{Ch}\varphi\mathrm{Sh}\pi+\mathrm{Ch}\pi\mathrm{Sh}\varphi}{r}\\
\left[\mathrm{Ch}^{2}-\mathrm{Sh}^{2}=rr\right]
\end{matrix}&lt;/math&gt;

He furthermore showed that &lt;math&gt;\mathrm{Ch}(\varphi-\pi)&lt;/math&gt; and &lt;math&gt;\mathrm{Sh}(\varphi-\pi)&lt;/math&gt; follow by setting &lt;math&gt;\mathrm{Ch}(\pi)\Rightarrow\mathrm{Ch}(-\pi)&lt;/math&gt; and &lt;math&gt;\mathrm{Sh}(\pi)\Rightarrow\mathrm{Sh}(-\pi)&lt;/math&gt; in the above formulas.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3c}}). (In modern publications, Riccati's additional factor ''r'' is set to unity.)&lt;/p&gt;

==={{anchor|Lambert}} Lambert (1768–1770) – hyperbolic addition===

While [[#Riccati|Riccati (1757)]] discussed the hyperbolic sine and cosine, [[w:Johann Heinrich Lambert]] (read 1767, published 1768) introduced the expression ''tang φ'' or abbreviated ''tφ'' as the [[w:tangens hyperbolicus]] &lt;math&gt;{\scriptstyle \frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}}&lt;/math&gt; of a variable ''u'', or in modern notation ''tφ=tanh(u)'':&lt;ref group=M&gt;Lambert (1761/68), pp. 309–318&lt;/ref&gt;&lt;ref&gt;Barnett (2004), pp. 22–23&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}\xi\xi-1 &amp; =\eta\eta &amp; (a)\\
1+\eta\eta &amp; =\xi\xi &amp; (b)\\
\frac{\eta}{\xi} &amp; =tang\ \phi=t\phi &amp; (c)\\
\xi &amp; =\frac{1}{\sqrt{1-t\phi^{2}}} &amp; (d)\\
\eta &amp; =\frac{t\phi}{\sqrt{1-t\phi^{2}}} &amp; (e)\\
t\phi'' &amp; =\frac{t\phi+t\phi'}{1+t\phi\cdot t\phi'} &amp; (f)\\
t\phi' &amp; =\frac{t\phi''-t\phi}{1-t\phi\cdot t\phi''} &amp; (g)
\end{align}
\right|\begin{align}2u &amp; =\log\frac{1+t\phi}{1-t\phi}\\
\xi &amp; =\frac{e^{u}+e^{-u}}{2}\\
\eta &amp; =\frac{e^{u}-e^{-u}}{2}\\
t\phi &amp; =\frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}\\
e^{u} &amp; =\xi+\eta\\
e^{-u} &amp; =\xi-\eta
\end{align}&lt;/math&gt;

In (1770) he rewrote the addition law for the hyperbolic tangens (f) or (g) as:&lt;ref group=M&gt;Lambert (1770), p. 335&lt;/ref&gt;

:&lt;math&gt;\begin{align}t(y+z) &amp; =(ty+tz):(1+ty\cdot tz) &amp; (f)\\
t(y-z) &amp; =(ty-tz):(1-ty\cdot tz) &amp; (g)
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The hyperbolic relations (a,b,c,d,e,f) are equivalent to the hyperbolic relations on the right of ({{equationNote|3b}}). Relations (f,g) can also be found in ({{equationNote|3e}}). By setting ''tφ=v/c'', formula (c) becomes the relative velocity between two frames, (d) the [[w:Lorentz factor]], (e) the [[w:proper velocity]], (f) or (g) becomes the Lorentz transformation of velocity (or relativistic [[w:velocity addition formula]]) for collinear velocities in [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] and [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']].&lt;/p&gt;

Lambert also formulated the addition laws for the hyperbolic cosine and sine (Lambert's &quot;cos&quot; and &quot;sin&quot; actually mean &quot;cosh&quot; and &quot;sinh&quot;):

:&lt;math&gt;\begin{align}\sin(y+z) &amp; =\sin y\cos z+\cos y\sin z\\
\sin(y-z) &amp; =\sin y\cos z-\cos y\sin z\\
\cos(y+z) &amp; =\cos y\cos z+\sin y\sin z\\
\cos(y-z) &amp; =\cos y\cos z-\sin y\sin z
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{Anchor|Taurinus}} Taurinus (1826) – Hyperbolic law of cosines===

After the addition theorem for the tangens hyperbolicus was given by [[#Lambert|Lambert (1768)]], [[w:hyperbolic geometry]] was used by [[w:Franz Taurinus]] (1826), and later by [[w:Nikolai Lobachevsky]] (1829/30) and others, to formulate the [[w:hyperbolic law of cosines]]:&lt;ref group=M&gt;Taurinus (1826), p. 66; see also p. 272 in the translation by Engel and Stäckel (1899)&lt;/ref&gt;&lt;ref&gt;Bonola (1912), p. 79&lt;/ref&gt;&lt;ref&gt;Gray (1979), p. 242&lt;/ref&gt;

:&lt;math&gt;A=\operatorname{arccos}\frac{\cos\left(\alpha\sqrt{-1}\right)-\cos\left(\beta\sqrt{-1}\right)\cos\left(\gamma\sqrt{-1}\right)}{\sin\left(\beta\sqrt{-1}\right)\sin\left(\gamma\sqrt{-1}\right)}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;When solved for &lt;math&gt;\cos\left(\alpha\sqrt{-1}\right)&lt;/math&gt; it corresponds to the Lorentz transformation in Beltrami coordinates ({{equationNote|3f}}), and by defining the rapidities &lt;math&gt;{\scriptstyle \left(\left[\frac{U}{c},\ \frac{v}{c},\ \frac{u}{c}\right]=\left[\tanh\alpha,\ \tanh\beta,\ \tanh\gamma\right]\right)}&lt;/math&gt; it corresponds to the relativistic velocity addition formula [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']].&lt;/p&gt;

==={{anchor|Beltrami}} Beltrami (1868) – Beltrami coordinates===

[[w:Eugenio Beltrami]] (1868a) introduced coordinates of the [[w:Beltrami–Klein model]] of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:&lt;ref group=M&gt;Beltrami (1868a), pp. 287-288; Note I; Note II&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds^{2}=R^{2}\frac{\left(a^{2}+v^{2}\right)du^{2}-2uv\,du\,dv+\left(a^{2}+v^{2}\right)dv^{2}}{\left(a^{2}+u^{2}+v^{2}\right)^{2}}\\
u^{2}+v^{2}=a^{2}\\
\hline u''=\frac{aa_{0}\left(u'-r_{0}\right)}{a^{2}-r_{0}u'},\ v''=\frac{a_{0}w_{0}v'}{a^{2}-r_{0}u'},\\
\left(r_{0}=\sqrt{u_{0}^{2}+v_{0}^{2}},\ w_{0}=\sqrt{a^{2}-r_{0}^{2}}\right)\\
\hline ds^{2}=R^{2}\frac{\left(a^{2}-v^{2}\right)du^{2}+2uv\,du\,dv+\left(a^{2}-v^{2}\right)dv^{2}}{\left(a^{2}-u^{2}-v^{2}\right)^{2}}\\
(R=R\sqrt{-1},\ a=a\sqrt{-1})
\end{matrix}&lt;/math&gt;

(where the disk radius ''a'' and the [[w:radius of curvature]] ''R'' are real in spherical geometry, in hyperbolic geometry they are imaginary), and for arbitrary dimensions in (1868b)&lt;ref group=M&gt;Beltrami (1868b), pp. 232, 240–241, 253–254&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds=R\frac{\sqrt{dx^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}}}{x}\\
x^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=a^{2}\\
\hline y_{1}=\frac{ab\left(x_{1}-a_{1}\right)}{a^{2}-a_{1}x_{1}}\ \text{or}\ x_{1}=\frac{a\left(ay_{1}+a_{1}b\right)}{ab+a_{1}y_{1}},\ x_{r}=\pm\frac{ay_{r}\sqrt{a^{2}-a_{1}^{2}}}{ab+a_{1}y_{1}}\ (r=2,3,\dots,n)\\
\hline ds=R\frac{\sqrt{dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}-dx^{2}}}{x}\\
x^{2}=a^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\\
\left(R=R\sqrt{-1},\ x=x\sqrt{-1},\ a=a\sqrt{-1}\right)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Setting ''a=a&lt;sub&gt;0&lt;/sub&gt;'' Beltrami's (1868a) formulas become formulas ({{equationNote|3e}}), or in his (1868b) formulas one sets ''a=b'' for arbitrary dimensions.&lt;/p&gt;

=== {{anchor|Laisant2}} Laisant (1874) – Equipollences===
In his French translation of [[w:Giusto Bellavitis]]' principal work on [[w:Equipollence (geometry)|w:equipollences]], [[w:Charles-Ange Laisant]] (1874) added a chapter related to hyperbolas. The equipollence OM and its tangent MT of a hyperbola is defined by Laisant as&lt;ref group=M&gt;Laisant (1874b), pp. 134–135&lt;/ref&gt;

:(1) &lt;math&gt;\begin{matrix} &amp; \mathrm{OM}\bumpeq x\mathrm{OA}+y\mathrm{OB}\\
 &amp; \mathrm{MT}\bumpeq y\mathrm{OA}+x\mathrm{OB}\\
 &amp; \left[x^{2}-y^{2}=1;\ x=\cosh t,\ y=\sinh t\right]\\
\Rightarrow &amp; \mathrm{OM}\bumpeq\cosh t\cdot\mathrm{OA}+\sinh t\cdot\mathrm{OB}
\end{matrix}&lt;/math&gt;

Here, OA and OB are [[w:Conjugate diameters|conjugate semi-diameters]] of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation:

:&lt;math&gt;\begin{matrix}\begin{align}\mathrm{OC} &amp; \bumpeq c\mathrm{OA}+d\mathrm{OB} &amp; \qquad &amp;  &amp; \mathrm{OA} &amp; \bumpeq c\mathrm{OC}-d\mathrm{OD}\\
\mathrm{OD} &amp; \bumpeq d\mathrm{OA}+c\mathrm{OB} &amp;  &amp;  &amp; \mathrm{OB} &amp; \bumpeq-d\mathrm{OC}+c\mathrm{OD}
\end{align}
\\
\left[c^{2}-d^{2}=1\right]
\end{matrix}&lt;/math&gt;

producing the invariant relation

:&lt;math&gt;(\mathrm{OC})^{2}-(\mathrm{OD})^{2}\bumpeq(\mathrm{OA})^{2}-(\mathrm{OB})^{2}&lt;/math&gt;.

Substituting into (1), he showed that OM retains its form

:&lt;math&gt;\begin{matrix}\mathrm{OM}\bumpeq(cx-dy)\mathrm{OC}+(cy-dx)\mathrm{OD}\\
\left[(cx-dy)^{2}-(cy-dx)^{2}=1\right]
\end{matrix}&lt;/math&gt;

He also defined velocity and acceleration by differentiation of (1).

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These relations are equivalent to several Lorentz boosts or hyperbolic rotations producing the invariant Lorentz interval in line with ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Escherich}} Escherich (1874) – Beltrami coordinates===

[[w:Gustav von Escherich]] (1874) discussed the plane of constant negative curvature&lt;ref&gt;Sommerville (1911), p. 297&lt;/ref&gt; based on the [[w:Beltrami–Klein model]] of hyperbolic geometry by [[#Beltrami|Beltrami (1868)]]. Similar to [[w:Christoph Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt;  who introduced axial coordinates ''x''=tan(a) and ''y''=tan(b) in sphere geometry in order to perform coordinate transformations in the case of rotation and translation, Escherich used hyperbolic functions ''x''=tanh(a/k) and ''y''=tanh(b/k)&lt;ref group=M&gt;Escherich (1874), p. 508&lt;/ref&gt; in order to give the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation have the form:&lt;ref group=M name=escher&gt;Escherich (1874), p. 510&lt;/ref&gt;

:&lt;math&gt;x=\frac{\sinh\frac{a}{k}+x'\cosh\frac{a}{k}}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt; and &lt;math&gt;y=\frac{y'}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}), also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting &lt;math&gt;\tfrac{a}{k}=\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt; and multiplying ''[x,y,x′,y′]'' by 1/''c'', and equivalent to Lorentz boost ({{equationNote|3b}}) by setting &lt;math&gt;\scriptstyle (x,\ y,\ x',\ y')=\left(\frac{x_{1}}{x_{0}},\ \frac{x_{2}}{x_{0}},\ \frac{x_{1}^{\prime}}{x_{0}^{\prime}},\ \frac{x_{2}^{\prime}}{x_{0}^{\prime}}\right)&lt;/math&gt;. This is the relation between the [[w:Beltrami–Klein model|Beltrami coordinates]] in terms of Gudermann-Escherich coordinates, and the Weierstrass coordinates of the [[w:hyperboloid model]] introduced by [[../Lorentz transformation (general)#Killing1|E:Killing (1878–1893)]], [[../Lorentz transformation (general)#Poincare|E:Poincaré (1881)]], and [[../Lorentz transformation (general)#Cox|E:Cox (1881)]]. Both coordinate systems were compared by Cox (1881).&lt;ref group=M&gt;Cox (1881), p. 186&lt;/ref&gt;&lt;/p&gt;

==={{anchor|Glaisher}} Glaisher (1878) – hyperbolic addition===

It was shown by [[w:James Whitbread Lee Glaisher]] (1878) that the hyperbolic addition laws can be written as matrix multiplication&lt;ref group=M&gt;Glaisher (1878), p. 30&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\cosh x, &amp; \sinh x\\
\sinh x, &amp; \cosh x
\end{vmatrix}=1,\ \begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y
\end{vmatrix}=1\\
\left[c_{1},c_{2},c_{3},c_{4}\right]=\left[\cosh x,\cosh y,\sinh x,\sinh y\right]\\
\Rightarrow\begin{vmatrix}c_{1}c_{2}+s_{1}s_{2}, &amp; s_{1}c_{2}+c_{1}s_{2}\\
c_{1}s_{2}+s_{1}c_{2}, &amp; s_{1}s_{2}+c_{1}c_{2}
\end{vmatrix}=1\ \Rightarrow\begin{vmatrix}\cosh(x+y), &amp; \sinh(x+y)\\
\sinh(x+y), &amp; \cosh(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{anchor|Gunther1}} Günther (1880/81) – hyperbolic addition ===

Following [[#Glaisher|Glaisher (1878)]], [[w:Siegmund Günther]] (1880/81) formulated the hyperbolic addition laws in matrix form as&lt;ref group=M&gt;Günther (1880/81), p. 405&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\mathfrak{Cos}x, &amp; \mathfrak{Sin}x\\
\mathfrak{Sin}x, &amp; \mathfrak{Cos}x
\end{vmatrix}\cdot\begin{vmatrix}\mathfrak{Cos}y, &amp; \mathfrak{Sin}y\\
\mathfrak{Sin}y, &amp; \mathfrak{Cos}y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}x\mathfrak{Cos}y+\mathfrak{Sin}x\mathfrak{Sin}y, &amp; \mathfrak{Cos}x\mathfrak{Sin}y+\mathfrak{Sin}x\mathfrak{Cos}y\\
\mathfrak{Sin}x\mathfrak{Cos}y+\mathfrak{Cos}x\mathfrak{Sin}y, &amp; \mathfrak{Sin}x\mathfrak{Sin}y+\mathfrak{Cos}x\mathfrak{Cos}y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}(x+y), &amp; \mathfrak{Sin}(x+y)\\
\mathfrak{Sin}(x+y), &amp; \mathfrak{Cos}(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{anchor|Lipschitz1}} Lipschitz (1885/86) ===
[[w:Rudolf Lipschitz]] (1885/86) formulated transformations leaving invariant the sum of squares &lt;math&gt;x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}&lt;/math&gt;, which he rewrote as &lt;math&gt;x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots+x_{n}^{2}-y_{n}^{2}=0&lt;/math&gt;. This led to the problem of finding transformations leaving invariant the pairs &lt;math&gt;x_{a}^{2}-y_{a}^{2}&lt;/math&gt; (''a''=1...n) for which he gave the following solution:&lt;ref group=M&gt;Lipschitz (1886), pp. 90–92&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{a}^{2}-y_{a}^{2}=\mathfrak{x}_{a}^{2}-\mathfrak{y}_{a}^{2}\\
\hline \begin{align}x_{a}-y_{a} &amp; =\left(\mathfrak{x}_{a}-\mathfrak{y}_{a}\right)r_{a}\\
x_{a}+y_{a} &amp; =\left(\mathfrak{x}_{a}+\mathfrak{y}_{a}\right)\frac{1}{r_{a}}
\end{align}
\quad(a)\\
\hline \begin{matrix}\begin{align}2\mathfrak{x}_{a} &amp; =\left(r_{a}+\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}-\frac{1}{r_{a}}\right)y_{a}\\
2\mathfrak{y}_{a} &amp; =\left(r_{a}-\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}+\frac{1}{r_{a}}\right)y_{a}
\end{align}
\quad(b)\end{matrix}\\
\hline \left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}\Rightarrow\begin{align}\mathfrak{x}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}\\
\mathfrak{y}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}
\end{align}
\quad(c)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Equation (a) is identical to Lorentz boost ({{equationNote|3d}}), while (c) is similar, though not identical, to Lorentz boost ({{equationNote|3b}}-C). The difference stems from his definition&lt;br&gt;&lt;math&gt;\qquad\left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}&lt;/math&gt;&lt;br&gt;whereas in accordance to expression &lt;math&gt;\sqrt{\tfrac{1+v}{1-v}}&lt;/math&gt; in ({{equationNote|3d}}) he should have stated &lt;br&gt;&lt;math&gt;\qquad\left\{ \begin{matrix}r_{a}=\frac{\sqrt{1+s_{a}}}{\sqrt{1-s_{a}}}\\
s_{a}&lt;1
\end{matrix}\right\}&lt;/math&gt;.&lt;br&gt;Using the latter choice, equations (c) would assume a form equivalent to ({{equationNote|3b}}):&lt;br&gt;&lt;math&gt;\qquad\begin{align}\mathfrak{x}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{1-s_{a}}\sqrt{1+s_{a}}}\\
\mathfrak{y}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{1-s_{a}}\sqrt{1+s_{a}}}\end{align}&lt;/math&gt;&lt;/p&gt;

==={{Anchor|Schur}} Schur (1885/86, 1900/02) – Beltrami coordinates===

[[w:Friedrich Schur]] (1885/86) discussed spaces of constant Riemann curvature, and by following [[#Beltrami|Beltrami (1868)]] he used the transformation&lt;ref group=M&gt;Schur (1885/86), p. 167&lt;/ref&gt;

:&lt;math&gt;x_{1}=R^{2}\frac{y_{1}+a_{1}}{R^{2}+a_{1}y_{1}},\ x_{2}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{2}}{R^{2}+a_{1}y_{1}},\dots,\ x_{n}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{n}}{R^{2}+a_{1}y_{1}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in arbitrary dimensions by setting ''R=c'' as the speed of light and ''a&lt;sub&gt;1&lt;/sub&gt;=v'' as relative velocity.&lt;/p&gt;

In (1900/02) he derived basic formulas of non-Eucliden geometry, including the case of translation for which he obtained the transformation similar to his previous one:&lt;ref group=M&gt;Schur (1900/02), p. 290; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;x'=\frac{x-a}{1-\mathfrak{k}ax},\quad y'=\frac{y\sqrt{1-\mathfrak{k}a^{2}}}{1-\mathfrak{k}ax}&lt;/math&gt;

where &lt;math&gt;\mathfrak{k}&lt;/math&gt; can have values &gt;0, &lt;0 or ∞.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting ''a=v'' and &lt;math&gt;\mathfrak{k}=\tfrac{1}{c^{2}}&lt;/math&gt;.&lt;/p&gt;

He also defined the triangle&lt;ref group=M&gt;Schur (1900/02), p. 291; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;\frac{1}{\sqrt{1-\mathfrak{k}c^{2}}}=\frac{1}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{1}{\sqrt{1-\mathfrak{k}b^{2}}}-\frac{a}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{b}{\sqrt{1-\mathfrak{k}b^{2}}}\cos\gamma&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to the hyperbolic law of cosines and the relativistic velocity addition ({{equationNote|3f}}, b) or [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']] by setting &lt;math&gt;[\mathfrak{k},\ c,\ a,\ b]=\left[\tfrac{1}{c^{2}},\ \sqrt{u_{x}^{\prime2}+u_{y}^{\prime2}},\ v,\ \sqrt{u_{x}^{2}+u_{y}^{2}}\right]&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Lindemann}} Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute===

[[w:Ferdinand von Lindemann]] discussed hyperbolic geometry in terms of the [[w:Cayley–Klein metric]] in his (1890/91) edition of the lectures on geometry of [[w:Alfred Clebsch]]. Citing [[../Lorentz transformation (general)#Killing|E:Killing (1885)]] and [[../Lorentz transformation (general)#Poincare|Poincaré (1887)]] in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 477–478, 524&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\Omega_{xx}=x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}-4k^{2}dx_{3}^{2}\\
\Omega_{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}
\end{matrix}&lt;/math&gt;

and used the following transformation&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 361–362&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
X_{1}X_{4}+X_{2}X_{3}=\Xi_{1}\Xi_{4}+\Xi_{2}\Xi_{3}\\
\hline \begin{align}X_{1} &amp; =\left(\lambda+\lambda_{1}\right)U_{4} &amp; \Xi_{1} &amp; =\left(\lambda-\lambda_{1}\right)U_{4} &amp; X_{1} &amp; =\frac{\lambda+\lambda_{1}}{\lambda-\lambda_{1}}\Xi_{1}\\
X_{2} &amp; =\left(\lambda+\lambda_{3}\right)U_{4} &amp; \Xi_{2} &amp; =\left(\lambda-\lambda_{3}\right)U_{4} &amp; X_{2} &amp; =\frac{\lambda+\lambda_{3}}{\lambda-\lambda_{3}}\Xi_{2}\\
X_{3} &amp; =\left(\lambda-\lambda_{3}\right)U_{2} &amp; \Xi_{3} &amp; =\left(\lambda+\lambda_{3}\right)U_{2} &amp; X_{3} &amp; =\frac{\lambda-\lambda_{3}}{\lambda+\lambda_{3}}\Xi_{3}\\
X_{4} &amp; =\left(\lambda-\lambda_{1}\right)U_{1} &amp; \Xi_{4} &amp; =\left(\lambda+\lambda_{1}\right)U_{1} &amp; X_{4} &amp; =\frac{\lambda-\lambda_{1}}{\lambda+\lambda_{1}}\Xi_{4}
\end{align}
\end{matrix}&lt;/math&gt;

into which he put&lt;ref group=M name=linde&gt;Lindemann &amp; Clebsch (1890/91), p. 496&lt;/ref&gt;

:&lt;math&gt;\begin{align}X_{1} &amp; =x_{1}+2kx_{4}, &amp; X_{2} &amp; =x_{2}+ix_{3}, &amp; \lambda+\lambda_{1} &amp; =\left(\lambda-\lambda_{1}\right)e^{a},\\
X_{4} &amp; =x_{1}-2kx_{4}, &amp; X_{3} &amp; =x_{2}-ix_{3}, &amp; \lambda+\lambda_{3} &amp; =\left(\lambda-\lambda_{3}\right)e^{\alpha i},
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3d}}) with &lt;math&gt;e^{\alpha i}=1&lt;/math&gt; and ''2k=1'' .&lt;/p&gt;

From that, he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations (''a''=0) or translations (α=0):&lt;ref group=M  name=linde /&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=0\\
\hline \begin{align}x_{2} &amp; =\xi_{2}\cos\alpha+\xi_{3}\sin\alpha, &amp; x_{1} &amp; =\xi_{1}\cos\frac{a}{i}+2ki\xi_{4}\sin\frac{a}{i},\\
x_{3} &amp; =-\xi_{2}\sin\alpha+\xi_{3}\cos\alpha, &amp; 2kx_{4} &amp; =i\xi_{1}\sin\frac{a}{i}+2k\xi_{4}\cos\frac{a}{i}.
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0 and ''2k=1''.&lt;/p&gt;

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane and gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Killing2}} Killing (1893,97) – Weierstrass coordinates===

[[w:Wilhelm Killing]] (1878–1880) gave case of translation in the form&lt;ref group=M name=killtra&gt;Killing (1893), p. 331&lt;/ref&gt;

:&lt;math&gt;y_{0}=x_{0}\operatorname{Ch}a+x_{1}\operatorname{Sh}a,\quad y_{1}=x_{0}\operatorname{Sh}a+x_{1}\operatorname{Ch}a,\quad y_{2}=x_{2}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

In 1898, Killing wrote that relation in a form similar to [[#Escherich|Escherich (1874)]], and derived the corresponding Lorentz transformation for the two cases were ''v'' is unchanged or ''u'' is unchanged:&lt;ref group=M name=kill98&gt;Killing (1898), p. 133&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\xi'=\frac{\xi\operatorname{Ch}\frac{\mu}{l}+l\operatorname{Sh}\frac{\mu}{l}}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}},\ \eta'=\frac{\eta}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}}\\
\hline \frac{u}{p}=\xi,\ \frac{v}{p}=\eta\\
\hline p'=p\operatorname{Ch}\frac{\mu}{l}+\frac{u}{l}\operatorname{Sh}\frac{\mu}{l},\quad u'=pl\operatorname{Sh}\frac{\mu}{l}+u\operatorname{Ch}\frac{\mu}{l},\quad v'=v\\
\text{or}\\
p'=p\operatorname{Ch}\frac{\nu}{l}+\frac{v}{l}\operatorname{Sh}\frac{\nu}{l},\quad u'=u,\quad v'=pl\operatorname{Sh}\frac{\nu}{l}+v\operatorname{Ch}\frac{\nu}{l}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The upper transformation system is equivalent to Lorentz transformation ({{equationNote|3e}}) and the velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] with ''l=c'' and &lt;math&gt;\mu=c\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt;, the system below is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Whitehead}} Whitehead (1897/98) – Universal algebra===

[[w:Alfred North Whitehead]] (1898) discussed the kinematics of hyperbolic space as part of his study of [[w:universal algebra]], and obtained the following transformation:&lt;ref group=M name=white&gt;Whitehead (1898), pp. 459–460&lt;/ref&gt;

:&lt;math&gt;\begin{align}x' &amp; =\left(\eta\cosh\frac{\delta}{\gamma}+\eta_{1}\sinh\frac{\delta}{\gamma}\right)e+\left(\eta\sinh\frac{\delta}{\gamma}+\eta_{1}\cosh\frac{\delta}{\gamma}\right)e_{1}\\
 &amp; \qquad+\left(\eta_{2}\cos\alpha+\eta_{3}\sin\alpha\right)e_{2}+\left(\eta_{3}\cos\alpha-\eta_{2}\sin\alpha\right)e_{3}
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0.&lt;/p&gt;

==={{anchor|Elliott}} Elliott (1903) – Invariant theory ===
[[w:Edwin Bailey Elliott]] (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:&lt;ref group=M&gt;Elliott (1903), p. 109&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=X\cosh\phi+Y\sinh\phi,\quad y=X\sinh\phi+Y\cosh\phi\\
\hline X+Y=e^{-\phi}(x+y),\quad X-Y=e^{\phi}(x-y)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}). The mentioned subgroup corresponds to the one-parameter subgroup generated by Lorentz boosts.&lt;/p&gt;

==={{anchor|Woods2}} Woods (1903) – Weierstrass coordinates ===
[[w:Frederick S. Woods]] (1903, published 1905) gave the case of translation in hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 55&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x_{1}\cos kl+x_{0}\frac{\sin kl}{k},\quad x_{2}^{\prime}=x_{2},\quad x_{2}^{\prime}=x_{3},\quad x_{0}^{\prime}=-x_{1}k\sin kl+x_{0}\cos kl&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.&lt;/p&gt;

and the loxodromic substitution for hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 72&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}x_{1}^{\prime} &amp; =x_{1}\cosh\alpha-x_{0}\sinh\alpha\\
x_{2}^{\prime} &amp; =x_{2}\cos\beta-x_{3}\sin\beta\\
x_{3}^{\prime} &amp; =x_{2}\sin\beta+x_{3}\cos\beta\\
x_{0}^{\prime} &amp; =-x_{1}\sinh\alpha+x_{0}\cosh\alpha
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with β=0.&lt;/p&gt;

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – gave the case of translation in the hyperbolic plane:&lt;ref group=M name=lieb&gt;Liebmann (1904/05), p. 174&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x'\operatorname{ch}a+p'\operatorname{sh}a,\quad y_{1}^{\prime}=y',\quad p_{1}^{\prime}=x'\operatorname{sh}a+p'\operatorname{ch}a&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Frank}} Frank (1909) – Special relativity===
In special relativity, hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity:&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;
 	
=== {{anchor|Herglotz1}} Herglotz (1909/10) – Special relativity===

In special relativity, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic, with the hyperbolic case being:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;
	
:&lt;math&gt;\begin{matrix}Z=Z'e^{\vartheta}\\
	
\begin{aligned}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
	
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
	
\end{aligned}
	
\end{matrix}&lt;/math&gt;	
	
&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3d}}).&lt;/p&gt;

==={{anchor|Varicak}} Varićak (1910) – Special relativity===

In special relativity, hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

He showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bel68sag}} 
*{{#section:History of Topics in Special Relativity/mathsource|bel68fond}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay84non}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|eli03}} 
*{{#section:History of Topics in Special Relativity/mathsource|esch74}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|glai78}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|guen80}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|lais74b}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam67}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lind90}} 
*{{#section:History of Topics in Special Relativity/mathsource|ric57}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu85}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu00}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu09}} 
*{{#section:History of Topics in Special Relativity/mathsource|tau26}} 
*{{#section:History of Topics in Special Relativity/mathsource|whit98}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L3}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
      <sha1>8zjaklnt2qleuhitdphl7oqva3zp36k</sha1>
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    <revision>
      <id>2411815</id>
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      <timestamp>2022-08-04T12:07:22Z</timestamp>
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      <text bytes="47007" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via hyperbolic functions==
===Translation in the hyperbolic plane===
The case of a Lorentz transformation without spatial rotation is called a [[w:Lorentz boost]]. The simplest case can be given, for instance, by setting ''n=1'' in the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}\\
\\
x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}
\end{align}
\left|\begin{align}g_{01}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{10}^{2} &amp; =1\\
g_{01}g_{11}-g_{00}g_{10} &amp; =0\\
g_{10}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{01}^{2} &amp; =1\\
g_{10}g_{11}-g_{00}g_{01} &amp; =0
\end{align}
\rightarrow\begin{align}g_{00}^{2} &amp; =g_{11}^{2}\\
g_{01}^{2} &amp; =g_{10}^{2}
\end{align}
\right.
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}g_{00} &amp; -g_{10}\\
-g_{01} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\right|\det\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;|{{equationRef|3a}}}}

which resembles precisely the relations of [[w:hyperbolic function]]s in terms of [[w:hyperbolic angle]] &lt;math&gt;\eta&lt;/math&gt;. Thus a Lorentz boost or [[w:hyperbolic rotation]] (being the same as a rotation around an imaginary angle &lt;math&gt;i\eta=\phi&lt;/math&gt; in [[../Lorentz transformation (imaginary)#math_2b|E:'''(2b)''']] or a [[w:Translation (geometry)|translation]] in the hyperbolic plane in terms of the hyperboloid model) is given by

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline g_{00}=g_{11}=\cosh\eta,\ g_{01}=g_{10}=-\sinh\eta\\
\hline \left.\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta &amp;  &amp; =\frac{x_{0}-x_{1}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}-x_{1}v}{\sqrt{1-v^{2}}}\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta &amp;  &amp; =\frac{x_{1}-x_{0}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}-x_{0}v}{\sqrt{1-v^{2}}}\\
x_{2}^{\prime} &amp; =x_{2} &amp;  &amp; =x_{2} &amp;  &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta &amp;  &amp; =\frac{x_{0}^{\prime}-x_{1}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}^{\prime}-x_{1}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta &amp;  &amp; =\frac{x_{1}^{\prime}-x_{0}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}^{\prime}-x_{0}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{2} &amp; =x_{2}^{\prime} &amp;  &amp; =x_{2}^{\prime} &amp;  &amp; =x_{2}^{\prime}
\end{align}
\right|{\scriptstyle \begin{align}\sinh^{2}\eta-\cosh^{2}\eta &amp; =-1 &amp; (a)\\
\cosh^{2}\eta-\sinh^{2}\eta &amp; =1 &amp; (b)\\
\frac{\sinh\eta}{\cosh\eta} &amp; =\tanh\eta=v &amp; (c)\\
\frac{1}{\sqrt{1-\tanh^{2}\eta}} &amp; =\cosh\eta &amp; (d)\\
\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\sinh\eta &amp; (e)\\
\frac{\tanh q\pm\tanh\eta}{1\pm\tanh q\tanh\eta} &amp; =\tanh\left(q\pm\eta\right) &amp; (f)
\end{align}
}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\right|\det\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;|{{equationRef|3b}}}}

Hyperbolic identities (a,b) on the right of ({{equationNote|3b}}) were given by [[#Riccati|Riccati (1757)]], all identities (a,b,c,d,e,f) by [[#Lambert|Lambert (1768–1770)]]. Lorentz transformations ({{equationNote|3b}}-A) were given by [[#Laisant|Laisant (1874)]], [[#Cox|Cox (1882)]], [[#Lindemann|Lindemann (1890/91)]], [[#Gerard|Gérard (1892)]], [[#Killing2|Killing (1893, 1897/98)]], [[#Whitehead|Whitehead (1897/98)]], [[#Woods2|Woods (1903/05)]], [[#Elliott|Elliott (1903)]] and [[#Liebmann|Liebmann (1904/05)]] in terms of Weierstrass coordinates of the [[w:hyperboloid model]], while transformations similar to ({{equationNote|3b}}-C) have been used by [[#Lipschitz1|Lipschitz (1885/86)]].

Rapidity can be composed of arbitrary many rapidities &lt;math&gt;\eta_{1},\eta_{2}\dots&lt;/math&gt; as per the [[w:Hyperbolic functions#Sums of arguments|w:angle sum laws of hyperbolic sines and cosines]], so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between [[w:List of trigonometric identities#Angle sum and difference identities|w:angle sum laws of circular trigonometry]] and spatial rotations. Alternatively, the hyperbolic angle sum laws ''themselves'' can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the [[w:unit hyperbola]]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \left[\eta=\eta_{2}-\eta_{1}\right]\\
\begin{align}x_{0}^{\prime} &amp; =\sinh\eta_{1} &amp;  &amp; =\sinh\left(\eta_{2}-\eta\right) &amp;  &amp; =\sinh\eta_{2}\cosh\eta-\cosh\eta_{2}\sinh\eta &amp;  &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =\cosh\eta_{1} &amp;  &amp; =\cosh\left(\eta_{2}-\eta\right) &amp;  &amp; =-\sinh\eta_{2}\sinh\eta+\cosh\eta_{2}\cosh\eta &amp;  &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta\\
\\
x_{0} &amp; =\sinh\eta_{2} &amp;  &amp; =\sinh\left(\eta_{1}+\eta\right) &amp;  &amp; =\sinh\eta_{1}\cosh\eta+\cosh\eta_{1}\sinh\eta &amp;  &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =\cosh\eta_{2} &amp;  &amp; =\cosh\left(\eta_{1}+\eta\right) &amp;  &amp; =\sinh\eta_{1}\sinh\eta+\cosh\eta_{1}\cosh\eta &amp;  &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \begin{align} &amp; \begin{bmatrix}x_{0}^{\prime}\\
x_{1}^{\prime}
\end{bmatrix}=\begin{bmatrix}\sinh\eta_{1}\\
\cosh\eta_{1}
\end{bmatrix}=\begin{bmatrix}\sinh\left(\eta_{2}-\eta\right)\\
\cosh\left(\eta_{2}-\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}\sinh\eta_{2}\\
\cosh\eta_{2}
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}x_{0}\\
x_{1}
\end{bmatrix}\\
 &amp; \begin{bmatrix}x_{0}\\
x_{1}
\end{bmatrix}=\begin{bmatrix}\sinh\eta_{2}\\
\cosh\eta_{2}
\end{bmatrix}=\begin{bmatrix}\sinh\left(\eta_{1}+\eta\right)\\
\cosh\left(\eta_{1}+\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}\sinh\eta_{1}\\
\cosh\eta_{1}
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}x_{0}^{\prime}\\
x_{1}^{\prime}
\end{bmatrix}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|3c}}}}

Hyperbolic angle sum laws equivalent to Lorentz boost ({{equationNote|3c}}) were given by [[#Riccati|Riccati (1757)]] and [[#Lambert|Lambert (1768–1770)]], while the matrix representation was given by [[#Glaisher|Glaisher (1878)]] and [[#Gunther2|Günther (1880/81)]]. In special relativity, hyperbolic functions were used by [[#Frank|Frank (1909)]] and [[#Varicak|Varićak (1910)]].

Finally, because of &lt;math&gt;\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;, Lorentz boost ({{equationNote|3b}}) assumes a simple form by using [[w:squeeze mapping]]s in analogy to Euler's formula in [[../Lorentz transformation (imaginary)#math_2c|E:'''(2c)''']]:&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \begin{matrix}x_{1}^{\prime}-x_{0}^{\prime}=k\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime}=\frac{1}{k}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime}=x_{2}
\end{matrix}\\
\hline \begin{align}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}-x_{0} &amp; =e^{-\eta}\left(x_{1}^{\prime}-x_{0}^{\prime}\right)\\
x_{1}+x_{0} &amp; =e^{\eta}\left(x_{1}^{\prime}+x_{0}^{\prime}\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\\
\hline k=e^{\eta}=\cosh\eta+\sinh\eta=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}=\sqrt{\frac{1+v}{1-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|3d}}}} 

Lorentz transformations ({{equationNote|3d}}) for arbitrary ''k'' were given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]), while the above exponential form was explicitly used by [[#Lindemann|Lindemann (1890/91)]], [[#Elliott|Elliott (1903)]], [[#Herglotz1|Herglotz (1909)]]. 

===Hyperbolic law of cosines===
In line with equation [[../Lorentz transformation (general)#math_1b|E:'''(1b)''']] one can use coordinates &lt;math&gt;[u_{1},\ u_{2}]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}}\right]&lt;/math&gt; inside the [[w:unit circle]] &lt;math&gt;u_{1}^{2}+u_{2}^{2}=1&lt;/math&gt;, thus the corresponding Lorentz transformations ({{equationNote|3b}}) obtain the form:

{{NumBlk|:|&lt;math&gt;\begin{matrix}u_{x}^{2}+u_{y}^{2}=u_{x}^{\prime2}+u_{y}^{\prime2}=1\\
\hline \begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
u_{1}^{\prime} &amp; =\frac{-\sinh\eta+u_{1}\cosh\eta}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{1}-\tanh\eta}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{1}-v}{1-u_{1}v}\\
u_{2}^{\prime} &amp; =\frac{u_{2}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-v^{2}}}{1-u_{1}v}\\
\\
u_{1} &amp; =\frac{\sinh\eta+u_{1}^{\prime}\cosh\eta}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+\tanh\eta}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+v}{1+u_{1}^{\prime}v}\\
u_{2} &amp; =\frac{u_{2}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|3e}}}}

Transformations (A) were given by [[#Escherich|Escherich (1874)]] and [[#Killing2|Killing (1898)]], and transformations (B) by [[#Beltrami|Beltrami (1868)]] and [[#Schur|Schur (1885/86, 1900/02)]] in terms of [[w:Beltrami–Klein model|Beltrami coordinates]]&lt;ref&gt;Rosenfeld (1988), p. 231&lt;/ref&gt; of hyperbolic geometry. By using the scalar product of &lt;math&gt;\left[u_{1},u_{2}\right]&lt;/math&gt;, the resulting Lorentz transformation can be seen as equivalent to the [[w:hyperbolic law of cosines]]:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{matrix}u^{2}=u_{1}^{2}+u_{2}^{2}\\
u'^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}
\end{matrix}\left|\begin{matrix}u_{1}=u\cos\alpha\\
u_{2}=u\sin\alpha\\
\\
u_{1}^{\prime}=u'\cos\alpha'\\
u_{2}^{\prime}=u'\sin\alpha'
\end{matrix}\right|\begin{align}u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+vu'\cos\alpha'}, &amp; u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-vu\cos\alpha}\\
u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{1+vu'\cos\alpha'}, &amp; u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{1-vu\cos\alpha}\\
\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{u'\cos\alpha'+v}, &amp; \tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{u\cos\alpha-v}
\end{align}
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(vu'\sin\alpha'\right){}^{2}}}{1+vu'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(vu\sin\alpha\right){}^{2}}}{1-vu\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-u^{\prime2}}}=\frac{1}{\sqrt{1-v^{2}}}\frac{1}{\sqrt{1-u^{2}}}-\frac{v}{\sqrt{1-v^{2}}}\frac{u}{\sqrt{1-u^{2}}}\cos\alpha &amp; (b)\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha &amp; (a)
\end{matrix}&lt;/math&gt;|{{equationRef|3f}}}}

The hyperbolic law of cosines (a) was given by [[#Taurinus|Taurinus (1826) and Lobachevsky (1829/30)]] and others, while variant (b) was given by [[#Schur|Schur (1900/02)]]. By further setting ''u=u′'' it follows:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\cos\alpha=\frac{\cos\alpha'+v}{1+v\cos\alpha'},\ \sin\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{1+v\cos\alpha'},\ \tan\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{\cos\alpha'+v},\ \tan\frac{\alpha}{2}=\sqrt{\frac{1-v}{1+v}}\tan\frac{\alpha'}{2}\\
\cos\alpha'=\frac{\cos\alpha-v}{1-v\cos\alpha},\ \sin\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{1-v\cos\alpha},\ \tan\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{\cos\alpha-v},\ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+v}{1-v}}\tan\frac{\alpha}{2}
\end{matrix}&lt;/math&gt;|{{equationRef|3g}}}}

Formulas (3b) resemble the equations of an [[w:ellipse]] of [[w:Orbital eccentricity|eccentricity]] ''v/c'', [[w:eccentric anomaly]] α' and [[w:true anomaly]] α, first geometrically formulated by [[#Euler|Kepler (1609)]] and explicitly written down by [[#Euler|Euler (1735, 1748), Lagrange (1770)]] and many others in relation to planetary motions. They were also used by [[../Lorentz transformation (conformal)#Darboux|E:Darboux (1873)]] as a sphere transformation. In special relativity, these formulas describes the aberration of light, see [[../Lorentz transformation (velocity)#Velocity addition and aberration|E:velocity addition and aberration]].

==Historical notation==
==={{anchor|Euler}} Euler (1735) – True and eccentric anomaly===

[[w:Johannes Kepler]] (1609) geometrically formulated [[w:Kepler's equation]] and the relations between the [[w:mean anomaly]], [[w:true anomaly]], and [[w:eccentric anomaly]].&lt;ref group=M&gt;Kepler (1609), chapter 60. The editors of Kepler's collected papers remark (p. 482), that Kepler's relations correspond to &lt;math&gt;{\scriptstyle \alpha=\beta+e\sin\beta}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\nu=\frac{e+\cos\beta}{1+e\cos\beta}}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\beta=\frac{\cos\nu-e}{1-e\cos\nu}}&lt;/math&gt;&lt;/ref&gt;&lt;ref&gt;Volk (1976), p. 366&lt;/ref&gt; The relation between the true anomaly ''z'' and the eccentric anomaly ''P'' was algebraically expressed by [[w:Leonhard Euler]] (1735/40) as follows:&lt;ref group=M&gt;Euler (1735/40), § 19&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{\cos P+v}{1+v\cos P},\ \cos P=\frac{\cos z-v}{1-v\cos z},\ \int P=\frac{\int z\sqrt{1-v^{2}}}{1-v\cos z}&lt;/math&gt;

and in 1748:&lt;ref group=M&gt;Euler (1748a), section VIII&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{n+\cos y}{1+n\cos y},\ \sin z=\frac{\sin y\sqrt{1-n^{2}}}{1+n\cos y},\ \tan z=\frac{\sin y\sqrt{1-n^{2}}}{n+\cos y}&lt;/math&gt;

while [[w:Joseph-Louis Lagrange]] (1770/71) expressed them as follows&lt;ref group=M&gt;Lagrange (1770/71), section I&lt;/ref&gt;

:&lt;math&gt;\sin u=\frac{m\sin x}{1+n\cos x},\ \cos u=\frac{n+\cos x}{1+n\cos x},\ \operatorname{tang}\frac{1}{2}u=\frac{m}{1+n}\operatorname{tang}\frac{1}{2}x,\ \left(m^{2}=1-n^{2}\right)&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt; These relations resemble formulas ({{equationNote|3g}}), while ({{equationNote|3f}}) follows by setting &lt;math&gt;[\cos z,\sin z]=\left[u_{x},u_{y}\right]&lt;/math&gt; in Euler's formulas or &lt;math&gt;[\cos u,\sin u]=\left[u_{x},u_{y}\right]&lt;/math&gt; in Lagrange's formulas.&lt;/p&gt;

==={{anchor|Riccati}} Riccati (1757) – hyperbolic addition===

[[w:Vincenzo Riccati]] introduced hyperbolic functions in 1757,&lt;ref group=M&gt;Riccati (1757), p. 71&lt;/ref&gt;&lt;ref group=M&gt;Günther (1880/81), pp. 7–13&lt;/ref&gt; in particular he formulated the angle sum laws for hyperbolic sine and cosine:

:&lt;math&gt;\begin{matrix}\mathrm{Ch}(\varphi+\pi)=\frac{\mathrm{Ch}\varphi\mathrm{Ch}\pi+\mathrm{Sh}\varphi\mathrm{Sh}\pi}{r}\\
\mathrm{Sh}(\varphi+\pi)=\frac{\mathrm{Ch}\varphi\mathrm{Sh}\pi+\mathrm{Ch}\pi\mathrm{Sh}\varphi}{r}\\
\left[\mathrm{Ch}^{2}-\mathrm{Sh}^{2}=rr\right]
\end{matrix}&lt;/math&gt;

He furthermore showed that &lt;math&gt;\mathrm{Ch}(\varphi-\pi)&lt;/math&gt; and &lt;math&gt;\mathrm{Sh}(\varphi-\pi)&lt;/math&gt; follow by setting &lt;math&gt;\mathrm{Ch}(\pi)\Rightarrow\mathrm{Ch}(-\pi)&lt;/math&gt; and &lt;math&gt;\mathrm{Sh}(\pi)\Rightarrow\mathrm{Sh}(-\pi)&lt;/math&gt; in the above formulas.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3c}}). (In modern publications, Riccati's additional factor ''r'' is set to unity.)&lt;/p&gt;

==={{anchor|Lambert}} Lambert (1768–1770) – hyperbolic addition===

While [[#Riccati|Riccati (1757)]] discussed the hyperbolic sine and cosine, [[w:Johann Heinrich Lambert]] (read 1767, published 1768) introduced the expression ''tang φ'' or abbreviated ''tφ'' as the [[w:tangens hyperbolicus]] &lt;math&gt;{\scriptstyle \frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}}&lt;/math&gt; of a variable ''u'', or in modern notation ''tφ=tanh(u)'':&lt;ref group=M&gt;Lambert (1761/68), pp. 309–318&lt;/ref&gt;&lt;ref&gt;Barnett (2004), pp. 22–23&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}\xi\xi-1 &amp; =\eta\eta &amp; (a)\\
1+\eta\eta &amp; =\xi\xi &amp; (b)\\
\frac{\eta}{\xi} &amp; =tang\ \phi=t\phi &amp; (c)\\
\xi &amp; =\frac{1}{\sqrt{1-t\phi^{2}}} &amp; (d)\\
\eta &amp; =\frac{t\phi}{\sqrt{1-t\phi^{2}}} &amp; (e)\\
t\phi'' &amp; =\frac{t\phi+t\phi'}{1+t\phi\cdot t\phi'} &amp; (f)\\
t\phi' &amp; =\frac{t\phi''-t\phi}{1-t\phi\cdot t\phi''} &amp; (g)
\end{align}
\right|\begin{align}2u &amp; =\log\frac{1+t\phi}{1-t\phi}\\
\xi &amp; =\frac{e^{u}+e^{-u}}{2}\\
\eta &amp; =\frac{e^{u}-e^{-u}}{2}\\
t\phi &amp; =\frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}\\
e^{u} &amp; =\xi+\eta\\
e^{-u} &amp; =\xi-\eta
\end{align}&lt;/math&gt;

In (1770) he rewrote the addition law for the hyperbolic tangens (f) or (g) as:&lt;ref group=M&gt;Lambert (1770), p. 335&lt;/ref&gt;

:&lt;math&gt;\begin{align}t(y+z) &amp; =(ty+tz):(1+ty\cdot tz) &amp; (f)\\
t(y-z) &amp; =(ty-tz):(1-ty\cdot tz) &amp; (g)
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The hyperbolic relations (a,b,c,d,e,f) are equivalent to the hyperbolic relations on the right of ({{equationNote|3b}}). Relations (f,g) can also be found in ({{equationNote|3e}}). By setting ''tφ=v/c'', formula (c) becomes the relative velocity between two frames, (d) the [[w:Lorentz factor]], (e) the [[w:proper velocity]], (f) or (g) becomes the Lorentz transformation of velocity (or relativistic [[w:velocity addition formula]]) for collinear velocities in [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] and [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']].&lt;/p&gt;

Lambert also formulated the addition laws for the hyperbolic cosine and sine (Lambert's &quot;cos&quot; and &quot;sin&quot; actually mean &quot;cosh&quot; and &quot;sinh&quot;):

:&lt;math&gt;\begin{align}\sin(y+z) &amp; =\sin y\cos z+\cos y\sin z\\
\sin(y-z) &amp; =\sin y\cos z-\cos y\sin z\\
\cos(y+z) &amp; =\cos y\cos z+\sin y\sin z\\
\cos(y-z) &amp; =\cos y\cos z-\sin y\sin z
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{Anchor|Taurinus}} Taurinus (1826) – Hyperbolic law of cosines===

After the addition theorem for the tangens hyperbolicus was given by [[#Lambert|Lambert (1768)]], [[w:hyperbolic geometry]] was used by [[w:Franz Taurinus]] (1826), and later by [[w:Nikolai Lobachevsky]] (1829/30) and others, to formulate the [[w:hyperbolic law of cosines]]:&lt;ref group=M&gt;Taurinus (1826), p. 66; see also p. 272 in the translation by Engel and Stäckel (1899)&lt;/ref&gt;&lt;ref&gt;Bonola (1912), p. 79&lt;/ref&gt;&lt;ref&gt;Gray (1979), p. 242&lt;/ref&gt;

:&lt;math&gt;A=\operatorname{arccos}\frac{\cos\left(\alpha\sqrt{-1}\right)-\cos\left(\beta\sqrt{-1}\right)\cos\left(\gamma\sqrt{-1}\right)}{\sin\left(\beta\sqrt{-1}\right)\sin\left(\gamma\sqrt{-1}\right)}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;When solved for &lt;math&gt;\cos\left(\alpha\sqrt{-1}\right)&lt;/math&gt; it corresponds to the Lorentz transformation in Beltrami coordinates ({{equationNote|3f}}), and by defining the rapidities &lt;math&gt;{\scriptstyle \left(\left[\frac{U}{c},\ \frac{v}{c},\ \frac{u}{c}\right]=\left[\tanh\alpha,\ \tanh\beta,\ \tanh\gamma\right]\right)}&lt;/math&gt; it corresponds to the relativistic velocity addition formula [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']].&lt;/p&gt;

==={{anchor|Beltrami}} Beltrami (1868) – Beltrami coordinates===

[[w:Eugenio Beltrami]] (1868a) introduced coordinates of the [[w:Beltrami–Klein model]] of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:&lt;ref group=M&gt;Beltrami (1868a), pp. 287-288; Note I; Note II&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds^{2}=R^{2}\frac{\left(a^{2}+v^{2}\right)du^{2}-2uv\,du\,dv+\left(a^{2}+v^{2}\right)dv^{2}}{\left(a^{2}+u^{2}+v^{2}\right)^{2}}\\
u^{2}+v^{2}=a^{2}\\
\hline u''=\frac{aa_{0}\left(u'-r_{0}\right)}{a^{2}-r_{0}u'},\ v''=\frac{a_{0}w_{0}v'}{a^{2}-r_{0}u'},\\
\left(r_{0}=\sqrt{u_{0}^{2}+v_{0}^{2}},\ w_{0}=\sqrt{a^{2}-r_{0}^{2}}\right)\\
\hline ds^{2}=R^{2}\frac{\left(a^{2}-v^{2}\right)du^{2}+2uv\,du\,dv+\left(a^{2}-v^{2}\right)dv^{2}}{\left(a^{2}-u^{2}-v^{2}\right)^{2}}\\
(R=R\sqrt{-1},\ a=a\sqrt{-1})
\end{matrix}&lt;/math&gt;

(where the disk radius ''a'' and the [[w:radius of curvature]] ''R'' are real in spherical geometry, in hyperbolic geometry they are imaginary), and for arbitrary dimensions in (1868b)&lt;ref group=M&gt;Beltrami (1868b), pp. 232, 240–241, 253–254&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds=R\frac{\sqrt{dx^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}}}{x}\\
x^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=a^{2}\\
\hline y_{1}=\frac{ab\left(x_{1}-a_{1}\right)}{a^{2}-a_{1}x_{1}}\ \text{or}\ x_{1}=\frac{a\left(ay_{1}+a_{1}b\right)}{ab+a_{1}y_{1}},\ x_{r}=\pm\frac{ay_{r}\sqrt{a^{2}-a_{1}^{2}}}{ab+a_{1}y_{1}}\ (r=2,3,\dots,n)\\
\hline ds=R\frac{\sqrt{dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}-dx^{2}}}{x}\\
x^{2}=a^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\\
\left(R=R\sqrt{-1},\ x=x\sqrt{-1},\ a=a\sqrt{-1}\right)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Setting ''a=a&lt;sub&gt;0&lt;/sub&gt;'' Beltrami's (1868a) formulas become formulas ({{equationNote|3e}}), or in his (1868b) formulas one sets ''a=b'' for arbitrary dimensions.&lt;/p&gt;

=== {{anchor|Laisant2}} Laisant (1874) – Equipollences===
In his French translation of [[w:Giusto Bellavitis]]' principal work on [[w:Equipollence (geometry)|w:equipollences]], [[w:Charles-Ange Laisant]] (1874) added a chapter related to hyperbolas. The equipollence OM and its tangent MT of a hyperbola is defined by Laisant as&lt;ref group=M&gt;Laisant (1874b), pp. 134–135&lt;/ref&gt;

:(1) &lt;math&gt;\begin{matrix} &amp; \mathrm{OM}\bumpeq x\mathrm{OA}+y\mathrm{OB}\\
 &amp; \mathrm{MT}\bumpeq y\mathrm{OA}+x\mathrm{OB}\\
 &amp; \left[x^{2}-y^{2}=1;\ x=\cosh t,\ y=\sinh t\right]\\
\Rightarrow &amp; \mathrm{OM}\bumpeq\cosh t\cdot\mathrm{OA}+\sinh t\cdot\mathrm{OB}
\end{matrix}&lt;/math&gt;

Here, OA and OB are [[w:Conjugate diameters|conjugate semi-diameters]] of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation:

:&lt;math&gt;\begin{matrix}\begin{align}\mathrm{OC} &amp; \bumpeq c\mathrm{OA}+d\mathrm{OB} &amp; \qquad &amp;  &amp; \mathrm{OA} &amp; \bumpeq c\mathrm{OC}-d\mathrm{OD}\\
\mathrm{OD} &amp; \bumpeq d\mathrm{OA}+c\mathrm{OB} &amp;  &amp;  &amp; \mathrm{OB} &amp; \bumpeq-d\mathrm{OC}+c\mathrm{OD}
\end{align}
\\
\left[c^{2}-d^{2}=1\right]
\end{matrix}&lt;/math&gt;

producing the invariant relation

:&lt;math&gt;(\mathrm{OC})^{2}-(\mathrm{OD})^{2}\bumpeq(\mathrm{OA})^{2}-(\mathrm{OB})^{2}&lt;/math&gt;.

Substituting into (1), he showed that OM retains its form

:&lt;math&gt;\begin{matrix}\mathrm{OM}\bumpeq(cx-dy)\mathrm{OC}+(cy-dx)\mathrm{OD}\\
\left[(cx-dy)^{2}-(cy-dx)^{2}=1\right]
\end{matrix}&lt;/math&gt;

He also defined velocity and acceleration by differentiation of (1).

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These relations are equivalent to several Lorentz boosts or hyperbolic rotations producing the invariant Lorentz interval in line with ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Escherich}} Escherich (1874) – Beltrami coordinates===

[[w:Gustav von Escherich]] (1874) discussed the plane of constant negative curvature&lt;ref&gt;Sommerville (1911), p. 297&lt;/ref&gt; based on the [[w:Beltrami–Klein model]] of hyperbolic geometry by [[#Beltrami|Beltrami (1868)]]. Similar to [[w:Christoph Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt;  who introduced axial coordinates ''x''=tan(a) and ''y''=tan(b) in sphere geometry in order to perform coordinate transformations in the case of rotation and translation, Escherich used hyperbolic functions ''x''=tanh(a/k) and ''y''=tanh(b/k)&lt;ref group=M&gt;Escherich (1874), p. 508&lt;/ref&gt; in order to give the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation have the form:&lt;ref group=M name=escher&gt;Escherich (1874), p. 510&lt;/ref&gt;

:&lt;math&gt;x=\frac{\sinh\frac{a}{k}+x'\cosh\frac{a}{k}}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt; and &lt;math&gt;y=\frac{y'}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}), also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting &lt;math&gt;\tfrac{a}{k}=\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt; and multiplying ''[x,y,x′,y′]'' by 1/''c'', and equivalent to Lorentz boost ({{equationNote|3b}}) by setting &lt;math&gt;\scriptstyle (x,\ y,\ x',\ y')=\left(\frac{x_{1}}{x_{0}},\ \frac{x_{2}}{x_{0}},\ \frac{x_{1}^{\prime}}{x_{0}^{\prime}},\ \frac{x_{2}^{\prime}}{x_{0}^{\prime}}\right)&lt;/math&gt;. This is the relation between the [[w:Beltrami–Klein model|Beltrami coordinates]] in terms of Gudermann-Escherich coordinates, and the Weierstrass coordinates of the [[w:hyperboloid model]] introduced by [[../Lorentz transformation (general)#Killing1|E:Killing (1878–1893)]], [[../Lorentz transformation (general)#Poincare|E:Poincaré (1881)]], and [[../Lorentz transformation (general)#Cox|E:Cox (1881)]]. Both coordinate systems were compared by Cox (1881).&lt;ref group=M&gt;Cox (1881), p. 186&lt;/ref&gt;&lt;/p&gt;

==={{anchor|Glaisher}} Glaisher (1878) – hyperbolic addition===

It was shown by [[w:James Whitbread Lee Glaisher]] (1878) that the hyperbolic addition laws can be written as matrix multiplication&lt;ref group=M&gt;Glaisher (1878), p. 30&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\cosh x, &amp; \sinh x\\
\sinh x, &amp; \cosh x
\end{vmatrix}=1,\ \begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y
\end{vmatrix}=1\\
\left[c_{1},c_{2},c_{3},c_{4}\right]=\left[\cosh x,\cosh y,\sinh x,\sinh y\right]\\
\Rightarrow\begin{vmatrix}c_{1}c_{2}+s_{1}s_{2}, &amp; s_{1}c_{2}+c_{1}s_{2}\\
c_{1}s_{2}+s_{1}c_{2}, &amp; s_{1}s_{2}+c_{1}c_{2}
\end{vmatrix}=1\ \Rightarrow\begin{vmatrix}\cosh(x+y), &amp; \sinh(x+y)\\
\sinh(x+y), &amp; \cosh(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{anchor|Gunther1}} Günther (1880/81) – hyperbolic addition ===

Following [[#Glaisher|Glaisher (1878)]], [[w:Siegmund Günther]] (1880/81) formulated the hyperbolic addition laws in matrix form as&lt;ref group=M&gt;Günther (1880/81), p. 405&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\mathfrak{Cos}x, &amp; \mathfrak{Sin}x\\
\mathfrak{Sin}x, &amp; \mathfrak{Cos}x
\end{vmatrix}\cdot\begin{vmatrix}\mathfrak{Cos}y, &amp; \mathfrak{Sin}y\\
\mathfrak{Sin}y, &amp; \mathfrak{Cos}y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}x\mathfrak{Cos}y+\mathfrak{Sin}x\mathfrak{Sin}y, &amp; \mathfrak{Cos}x\mathfrak{Sin}y+\mathfrak{Sin}x\mathfrak{Cos}y\\
\mathfrak{Sin}x\mathfrak{Cos}y+\mathfrak{Cos}x\mathfrak{Sin}y, &amp; \mathfrak{Sin}x\mathfrak{Sin}y+\mathfrak{Cos}x\mathfrak{Cos}y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}(x+y), &amp; \mathfrak{Sin}(x+y)\\
\mathfrak{Sin}(x+y), &amp; \mathfrak{Cos}(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{anchor|Lipschitz1}} Lipschitz (1885/86) ===
[[w:Rudolf Lipschitz]] (1885/86) formulated transformations leaving invariant the sum of squares &lt;math&gt;x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}&lt;/math&gt;, which he rewrote as &lt;math&gt;x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots+x_{n}^{2}-y_{n}^{2}=0&lt;/math&gt;. This led to the problem of finding transformations leaving invariant the pairs &lt;math&gt;x_{a}^{2}-y_{a}^{2}&lt;/math&gt; (''a''=1...n) for which he gave the following solution:&lt;ref group=M&gt;Lipschitz (1886), pp. 90–92&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{a}^{2}-y_{a}^{2}=\mathfrak{x}_{a}^{2}-\mathfrak{y}_{a}^{2}\\
\hline \begin{align}x_{a}-y_{a} &amp; =\left(\mathfrak{x}_{a}-\mathfrak{y}_{a}\right)r_{a}\\
x_{a}+y_{a} &amp; =\left(\mathfrak{x}_{a}+\mathfrak{y}_{a}\right)\frac{1}{r_{a}}
\end{align}
\quad(a)\\
\hline \begin{matrix}\begin{align}2\mathfrak{x}_{a} &amp; =\left(r_{a}+\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}-\frac{1}{r_{a}}\right)y_{a}\\
2\mathfrak{y}_{a} &amp; =\left(r_{a}-\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}+\frac{1}{r_{a}}\right)y_{a}
\end{align}
\quad(b)\end{matrix}\\
\hline \left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}\Rightarrow\begin{align}\mathfrak{x}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}\\
\mathfrak{y}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}
\end{align}
\quad(c)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Equation (a) is identical to Lorentz boost ({{equationNote|3d}}), while (c) is similar, though not identical, to Lorentz boost ({{equationNote|3b}}-C). The difference stems from his definition&lt;br&gt;&lt;math&gt;\qquad\left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}&lt;/math&gt;&lt;br&gt;whereas in accordance to expression &lt;math&gt;\sqrt{\tfrac{1+v}{1-v}}&lt;/math&gt; in ({{equationNote|3d}}) he should have stated &lt;br&gt;&lt;math&gt;\qquad\left\{ \begin{matrix}r_{a}=\frac{\sqrt{1+s_{a}}}{\sqrt{1-s_{a}}}\\
s_{a}&lt;1
\end{matrix}\right\}&lt;/math&gt;.&lt;br&gt;Using the latter choice, equations (c) would assume a form equivalent to ({{equationNote|3b}}):&lt;br&gt;&lt;math&gt;\qquad\begin{align}\mathfrak{x}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{1-s_{a}}\sqrt{1+s_{a}}}\\
\mathfrak{y}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{1-s_{a}}\sqrt{1+s_{a}}}\end{align}&lt;/math&gt;&lt;/p&gt;

==={{Anchor|Schur}} Schur (1885/86, 1900/02) – Beltrami coordinates===

[[w:Friedrich Schur]] (1885/86) discussed spaces of constant Riemann curvature, and by following [[#Beltrami|Beltrami (1868)]] he used the transformation&lt;ref group=M&gt;Schur (1885/86), p. 167&lt;/ref&gt;

:&lt;math&gt;x_{1}=R^{2}\frac{y_{1}+a_{1}}{R^{2}+a_{1}y_{1}},\ x_{2}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{2}}{R^{2}+a_{1}y_{1}},\dots,\ x_{n}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{n}}{R^{2}+a_{1}y_{1}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in arbitrary dimensions by setting ''R=c'' as the speed of light and ''a&lt;sub&gt;1&lt;/sub&gt;=v'' as relative velocity.&lt;/p&gt;

In (1900/02) he derived basic formulas of non-Eucliden geometry, including the case of translation for which he obtained the transformation similar to his previous one:&lt;ref group=M&gt;Schur (1900/02), p. 290; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;x'=\frac{x-a}{1-\mathfrak{k}ax},\quad y'=\frac{y\sqrt{1-\mathfrak{k}a^{2}}}{1-\mathfrak{k}ax}&lt;/math&gt;

where &lt;math&gt;\mathfrak{k}&lt;/math&gt; can have values &gt;0, &lt;0 or ∞.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting ''a=v'' and &lt;math&gt;\mathfrak{k}=\tfrac{1}{c^{2}}&lt;/math&gt;.&lt;/p&gt;

He also defined the triangle&lt;ref group=M&gt;Schur (1900/02), p. 291; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;\frac{1}{\sqrt{1-\mathfrak{k}c^{2}}}=\frac{1}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{1}{\sqrt{1-\mathfrak{k}b^{2}}}-\frac{a}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{b}{\sqrt{1-\mathfrak{k}b^{2}}}\cos\gamma&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to the hyperbolic law of cosines and the relativistic velocity addition ({{equationNote|3f}}, b) or [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']] by setting &lt;math&gt;[\mathfrak{k},\ c,\ a,\ b]=\left[\tfrac{1}{c^{2}},\ \sqrt{u_{x}^{\prime2}+u_{y}^{\prime2}},\ v,\ \sqrt{u_{x}^{2}+u_{y}^{2}}\right]&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Lindemann}} Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute===

[[w:Ferdinand von Lindemann]] discussed hyperbolic geometry in terms of the [[w:Cayley–Klein metric]] in his (1890/91) edition of the lectures on geometry of [[w:Alfred Clebsch]]. Citing [[../Lorentz transformation (general)#Killing|E:Killing (1885)]] and [[../Lorentz transformation (general)#Poincare|Poincaré (1887)]] in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 477–478, 524&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\Omega_{xx}=x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}-4k^{2}dx_{3}^{2}\\
\Omega_{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}
\end{matrix}&lt;/math&gt;

and used the following transformation&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 361–362&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
X_{1}X_{4}+X_{2}X_{3}=\Xi_{1}\Xi_{4}+\Xi_{2}\Xi_{3}\\
\hline \begin{align}X_{1} &amp; =\left(\lambda+\lambda_{1}\right)U_{4} &amp; \Xi_{1} &amp; =\left(\lambda-\lambda_{1}\right)U_{4} &amp; X_{1} &amp; =\frac{\lambda+\lambda_{1}}{\lambda-\lambda_{1}}\Xi_{1}\\
X_{2} &amp; =\left(\lambda+\lambda_{3}\right)U_{4} &amp; \Xi_{2} &amp; =\left(\lambda-\lambda_{3}\right)U_{4} &amp; X_{2} &amp; =\frac{\lambda+\lambda_{3}}{\lambda-\lambda_{3}}\Xi_{2}\\
X_{3} &amp; =\left(\lambda-\lambda_{3}\right)U_{2} &amp; \Xi_{3} &amp; =\left(\lambda+\lambda_{3}\right)U_{2} &amp; X_{3} &amp; =\frac{\lambda-\lambda_{3}}{\lambda+\lambda_{3}}\Xi_{3}\\
X_{4} &amp; =\left(\lambda-\lambda_{1}\right)U_{1} &amp; \Xi_{4} &amp; =\left(\lambda+\lambda_{1}\right)U_{1} &amp; X_{4} &amp; =\frac{\lambda-\lambda_{1}}{\lambda+\lambda_{1}}\Xi_{4}
\end{align}
\end{matrix}&lt;/math&gt;

into which he put&lt;ref group=M name=linde&gt;Lindemann &amp; Clebsch (1890/91), p. 496&lt;/ref&gt;

:&lt;math&gt;\begin{align}X_{1} &amp; =x_{1}+2kx_{4}, &amp; X_{2} &amp; =x_{2}+ix_{3}, &amp; \lambda+\lambda_{1} &amp; =\left(\lambda-\lambda_{1}\right)e^{a},\\
X_{4} &amp; =x_{1}-2kx_{4}, &amp; X_{3} &amp; =x_{2}-ix_{3}, &amp; \lambda+\lambda_{3} &amp; =\left(\lambda-\lambda_{3}\right)e^{\alpha i},
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3d}}) with &lt;math&gt;e^{\alpha i}=1&lt;/math&gt; and ''2k=1'' .&lt;/p&gt;

From that, he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations (''a''=0) or translations (α=0):&lt;ref group=M  name=linde /&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=0\\
\hline \begin{align}x_{2} &amp; =\xi_{2}\cos\alpha+\xi_{3}\sin\alpha, &amp; x_{1} &amp; =\xi_{1}\cos\frac{a}{i}+2ki\xi_{4}\sin\frac{a}{i},\\
x_{3} &amp; =-\xi_{2}\sin\alpha+\xi_{3}\cos\alpha, &amp; 2kx_{4} &amp; =i\xi_{1}\sin\frac{a}{i}+2k\xi_{4}\cos\frac{a}{i}.
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0 and ''2k=1''.&lt;/p&gt;

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane and gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Killing2}} Killing (1893,97) – Weierstrass coordinates===

[[w:Wilhelm Killing]] (1878–1880) gave case of translation in the form&lt;ref group=M name=killtra&gt;Killing (1893), p. 331&lt;/ref&gt;

:&lt;math&gt;y_{0}=x_{0}\operatorname{Ch}a+x_{1}\operatorname{Sh}a,\quad y_{1}=x_{0}\operatorname{Sh}a+x_{1}\operatorname{Ch}a,\quad y_{2}=x_{2}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

In 1898, Killing wrote that relation in a form similar to [[#Escherich|Escherich (1874)]], and derived the corresponding Lorentz transformation for the two cases were ''v'' is unchanged or ''u'' is unchanged:&lt;ref group=M name=kill98&gt;Killing (1898), p. 133&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\xi'=\frac{\xi\operatorname{Ch}\frac{\mu}{l}+l\operatorname{Sh}\frac{\mu}{l}}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}},\ \eta'=\frac{\eta}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}}\\
\hline \frac{u}{p}=\xi,\ \frac{v}{p}=\eta\\
\hline p'=p\operatorname{Ch}\frac{\mu}{l}+\frac{u}{l}\operatorname{Sh}\frac{\mu}{l},\quad u'=pl\operatorname{Sh}\frac{\mu}{l}+u\operatorname{Ch}\frac{\mu}{l},\quad v'=v\\
\text{or}\\
p'=p\operatorname{Ch}\frac{\nu}{l}+\frac{v}{l}\operatorname{Sh}\frac{\nu}{l},\quad u'=u,\quad v'=pl\operatorname{Sh}\frac{\nu}{l}+v\operatorname{Ch}\frac{\nu}{l}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The upper transformation system is equivalent to Lorentz transformation ({{equationNote|3e}}) and the velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] with ''l=c'' and &lt;math&gt;\mu=c\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt;, the system below is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Whitehead}} Whitehead (1897/98) – Universal algebra===

[[w:Alfred North Whitehead]] (1898) discussed the kinematics of hyperbolic space as part of his study of [[w:universal algebra]], and obtained the following transformation:&lt;ref group=M name=white&gt;Whitehead (1898), pp. 459–460&lt;/ref&gt;

:&lt;math&gt;\begin{align}x' &amp; =\left(\eta\cosh\frac{\delta}{\gamma}+\eta_{1}\sinh\frac{\delta}{\gamma}\right)e+\left(\eta\sinh\frac{\delta}{\gamma}+\eta_{1}\cosh\frac{\delta}{\gamma}\right)e_{1}\\
 &amp; \qquad+\left(\eta_{2}\cos\alpha+\eta_{3}\sin\alpha\right)e_{2}+\left(\eta_{3}\cos\alpha-\eta_{2}\sin\alpha\right)e_{3}
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0.&lt;/p&gt;

==={{anchor|Elliott}} Elliott (1903) – Invariant theory ===
[[w:Edwin Bailey Elliott]] (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:&lt;ref group=M&gt;Elliott (1903), p. 109&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=X\cosh\phi+Y\sinh\phi,\quad y=X\sinh\phi+Y\cosh\phi\\
\hline X+Y=e^{-\phi}(x+y),\quad X-Y=e^{\phi}(x-y)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}). The mentioned subgroup corresponds to the one-parameter subgroup generated by Lorentz boosts.&lt;/p&gt;

==={{anchor|Woods2}} Woods (1903) – Weierstrass coordinates ===
[[w:Frederick S. Woods]] (1903, published 1905) gave the case of translation in hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 55&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x_{1}\cos kl+x_{0}\frac{\sin kl}{k},\quad x_{2}^{\prime}=x_{2},\quad x_{2}^{\prime}=x_{3},\quad x_{0}^{\prime}=-x_{1}k\sin kl+x_{0}\cos kl&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.&lt;/p&gt;

and the loxodromic substitution for hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 72&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}x_{1}^{\prime} &amp; =x_{1}\cosh\alpha-x_{0}\sinh\alpha\\
x_{2}^{\prime} &amp; =x_{2}\cos\beta-x_{3}\sin\beta\\
x_{3}^{\prime} &amp; =x_{2}\sin\beta+x_{3}\cos\beta\\
x_{0}^{\prime} &amp; =-x_{1}\sinh\alpha+x_{0}\cosh\alpha
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with β=0.&lt;/p&gt;

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – gave the case of translation in the hyperbolic plane:&lt;ref group=M name=lieb&gt;Liebmann (1904/05), p. 174&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x'\operatorname{ch}a+p'\operatorname{sh}a,\quad y_{1}^{\prime}=y',\quad p_{1}^{\prime}=x'\operatorname{sh}a+p'\operatorname{ch}a&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Frank}} Frank (1909) – Special relativity===
In special relativity, hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity:&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;
 	
=== {{anchor|Herglotz1}} Herglotz (1909/10) – Special relativity===

In special relativity, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic, with the hyperbolic case being:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;
	
:&lt;math&gt;\begin{matrix}Z=Z'e^{\vartheta}\\
	
\begin{aligned}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
	
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
	
\end{aligned}
	
\end{matrix}&lt;/math&gt;	
	
&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3d}}).&lt;/p&gt;

==={{anchor|Varicak}} Varićak (1910) – Special relativity===

In special relativity, hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

He showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bel68sag}} 
*{{#section:History of Topics in Special Relativity/mathsource|bel68fond}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|eli03}} 
*{{#section:History of Topics in Special Relativity/mathsource|esch74}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|glai78}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|guen80}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|lais74b}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam67}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lind90}} 
*{{#section:History of Topics in Special Relativity/mathsource|ric57}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu85}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu00}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu09}} 
*{{#section:History of Topics in Special Relativity/mathsource|tau26}} 
*{{#section:History of Topics in Special Relativity/mathsource|whit98}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L3}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
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      <text bytes="47296" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via hyperbolic functions==
===Translation in the hyperbolic plane===
The case of a Lorentz transformation without spatial rotation is called a [[w:Lorentz boost]]. The simplest case can be given, for instance, by setting ''n=1'' in the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}\\
\\
x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}
\end{align}
\left|\begin{align}g_{01}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{10}^{2} &amp; =1\\
g_{01}g_{11}-g_{00}g_{10} &amp; =0\\
g_{10}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{01}^{2} &amp; =1\\
g_{10}g_{11}-g_{00}g_{01} &amp; =0
\end{align}
\rightarrow\begin{align}g_{00}^{2} &amp; =g_{11}^{2}\\
g_{01}^{2} &amp; =g_{10}^{2}
\end{align}
\right.
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}g_{00} &amp; -g_{10}\\
-g_{01} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\right|\det\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;|{{equationRef|3a}}}}

which resembles precisely the relations of [[w:hyperbolic function]]s in terms of [[w:hyperbolic angle]] &lt;math&gt;\eta&lt;/math&gt;. Thus a Lorentz boost or [[w:hyperbolic rotation]] (being the same as a rotation around an imaginary angle &lt;math&gt;i\eta=\phi&lt;/math&gt; in [[../Lorentz transformation (imaginary)#math_2b|E:'''(2b)''']] or a [[w:Translation (geometry)|translation]] in the hyperbolic plane in terms of the hyperboloid model) is given by

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline g_{00}=g_{11}=\cosh\eta,\ g_{01}=g_{10}=-\sinh\eta\\
\hline \left.\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta &amp;  &amp; =\frac{x_{0}-x_{1}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}-x_{1}v}{\sqrt{1-v^{2}}}\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta &amp;  &amp; =\frac{x_{1}-x_{0}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}-x_{0}v}{\sqrt{1-v^{2}}}\\
x_{2}^{\prime} &amp; =x_{2} &amp;  &amp; =x_{2} &amp;  &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta &amp;  &amp; =\frac{x_{0}^{\prime}-x_{1}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}^{\prime}-x_{1}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta &amp;  &amp; =\frac{x_{1}^{\prime}-x_{0}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}^{\prime}-x_{0}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{2} &amp; =x_{2}^{\prime} &amp;  &amp; =x_{2}^{\prime} &amp;  &amp; =x_{2}^{\prime}
\end{align}
\right|{\scriptstyle \begin{align}\sinh^{2}\eta-\cosh^{2}\eta &amp; =-1 &amp; (a)\\
\cosh^{2}\eta-\sinh^{2}\eta &amp; =1 &amp; (b)\\
\frac{\sinh\eta}{\cosh\eta} &amp; =\tanh\eta=v &amp; (c)\\
\frac{1}{\sqrt{1-\tanh^{2}\eta}} &amp; =\cosh\eta &amp; (d)\\
\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\sinh\eta &amp; (e)\\
\frac{\tanh q\pm\tanh\eta}{1\pm\tanh q\tanh\eta} &amp; =\tanh\left(q\pm\eta\right) &amp; (f)
\end{align}
}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\right|\det\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;|{{equationRef|3b}}}}

Hyperbolic identities (a,b) on the right of ({{equationNote|3b}}) were given by [[#Riccati|Riccati (1757)]], all identities (a,b,c,d,e,f) by [[#Lambert|Lambert (1768–1770)]]. Lorentz transformations ({{equationNote|3b}}-A) were given by [[#Laisant|Laisant (1874)]], [[#Cox|Cox (1882)]], [[#Lindemann|Lindemann (1890/91)]], [[#Gerard|Gérard (1892)]], [[#Killing2|Killing (1893, 1897/98)]], [[#Whitehead|Whitehead (1897/98)]], [[#Woods2|Woods (1903/05)]], [[#Elliott|Elliott (1903)]] and [[#Liebmann|Liebmann (1904/05)]] in terms of Weierstrass coordinates of the [[w:hyperboloid model]], while transformations similar to ({{equationNote|3b}}-C) have been used by [[#Lipschitz1|Lipschitz (1885/86)]].

Rapidity can be composed of arbitrary many rapidities &lt;math&gt;\eta_{1},\eta_{2}\dots&lt;/math&gt; as per the [[w:Hyperbolic functions#Sums of arguments|w:angle sum laws of hyperbolic sines and cosines]], so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between [[w:List of trigonometric identities#Angle sum and difference identities|w:angle sum laws of circular trigonometry]] and spatial rotations. Alternatively, the hyperbolic angle sum laws ''themselves'' can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the [[w:unit hyperbola]]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \left[\eta=\eta_{2}-\eta_{1}\right]\\
\begin{align}x_{0}^{\prime} &amp; =\sinh\eta_{1} &amp;  &amp; =\sinh\left(\eta_{2}-\eta\right) &amp;  &amp; =\sinh\eta_{2}\cosh\eta-\cosh\eta_{2}\sinh\eta &amp;  &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =\cosh\eta_{1} &amp;  &amp; =\cosh\left(\eta_{2}-\eta\right) &amp;  &amp; =-\sinh\eta_{2}\sinh\eta+\cosh\eta_{2}\cosh\eta &amp;  &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta\\
\\
x_{0} &amp; =\sinh\eta_{2} &amp;  &amp; =\sinh\left(\eta_{1}+\eta\right) &amp;  &amp; =\sinh\eta_{1}\cosh\eta+\cosh\eta_{1}\sinh\eta &amp;  &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =\cosh\eta_{2} &amp;  &amp; =\cosh\left(\eta_{1}+\eta\right) &amp;  &amp; =\sinh\eta_{1}\sinh\eta+\cosh\eta_{1}\cosh\eta &amp;  &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \begin{align} &amp; \begin{bmatrix}x_{0}^{\prime}\\
x_{1}^{\prime}
\end{bmatrix}=\begin{bmatrix}\sinh\eta_{1}\\
\cosh\eta_{1}
\end{bmatrix}=\begin{bmatrix}\sinh\left(\eta_{2}-\eta\right)\\
\cosh\left(\eta_{2}-\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}\sinh\eta_{2}\\
\cosh\eta_{2}
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}x_{0}\\
x_{1}
\end{bmatrix}\\
 &amp; \begin{bmatrix}x_{0}\\
x_{1}
\end{bmatrix}=\begin{bmatrix}\sinh\eta_{2}\\
\cosh\eta_{2}
\end{bmatrix}=\begin{bmatrix}\sinh\left(\eta_{1}+\eta\right)\\
\cosh\left(\eta_{1}+\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}\sinh\eta_{1}\\
\cosh\eta_{1}
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}x_{0}^{\prime}\\
x_{1}^{\prime}
\end{bmatrix}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|3c}}}}

Hyperbolic angle sum laws equivalent to Lorentz boost ({{equationNote|3c}}) were given by [[#Riccati|Riccati (1757)]] and [[#Lambert|Lambert (1768–1770)]], while the matrix representation was given by [[#Glaisher|Glaisher (1878)]] and [[#Gunther2|Günther (1880/81)]]. In special relativity, hyperbolic functions were used by [[#Frank|Frank (1909)]] and [[#Varicak|Varićak (1910)]].

Finally, because of &lt;math&gt;\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;, Lorentz boost ({{equationNote|3b}}) assumes a simple form by using [[w:squeeze mapping]]s in analogy to Euler's formula in [[../Lorentz transformation (imaginary)#math_2c|E:'''(2c)''']]:&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \begin{matrix}x_{1}^{\prime}-x_{0}^{\prime}=k\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime}=\frac{1}{k}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime}=x_{2}
\end{matrix}\\
\hline \begin{align}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}-x_{0} &amp; =e^{-\eta}\left(x_{1}^{\prime}-x_{0}^{\prime}\right)\\
x_{1}+x_{0} &amp; =e^{\eta}\left(x_{1}^{\prime}+x_{0}^{\prime}\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\\
\hline k=e^{\eta}=\cosh\eta+\sinh\eta=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}=\sqrt{\frac{1+v}{1-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|3d}}}} 

Lorentz transformations ({{equationNote|3d}}) for arbitrary ''k'' were given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]), while the above exponential form was explicitly used by [[#Lindemann|Lindemann (1890/91)]], [[#Elliott|Elliott (1903)]], [[#Herglotz1|Herglotz (1909)]]. 

===Hyperbolic law of cosines===
In line with equation [[../Lorentz transformation (general)#math_1b|E:'''(1b)''']] one can use coordinates &lt;math&gt;[u_{1},\ u_{2}]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}}\right]&lt;/math&gt; inside the [[w:unit circle]] &lt;math&gt;u_{1}^{2}+u_{2}^{2}=1&lt;/math&gt;, thus the corresponding Lorentz transformations ({{equationNote|3b}}) obtain the form:

{{NumBlk|:|&lt;math&gt;\begin{matrix}u_{x}^{2}+u_{y}^{2}=u_{x}^{\prime2}+u_{y}^{\prime2}=1\\
\hline \begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
u_{1}^{\prime} &amp; =\frac{-\sinh\eta+u_{1}\cosh\eta}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{1}-\tanh\eta}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{1}-v}{1-u_{1}v}\\
u_{2}^{\prime} &amp; =\frac{u_{2}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-v^{2}}}{1-u_{1}v}\\
\\
u_{1} &amp; =\frac{\sinh\eta+u_{1}^{\prime}\cosh\eta}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+\tanh\eta}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+v}{1+u_{1}^{\prime}v}\\
u_{2} &amp; =\frac{u_{2}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|3e}}}}

Transformations (A) were given by [[#Escherich|Escherich (1874)]] and [[#Killing2|Killing (1898)]], and transformations (B) by [[#Beltrami|Beltrami (1868)]] and [[#Schur|Schur (1885/86, 1900/02)]] in terms of [[w:Beltrami–Klein model|Beltrami coordinates]]&lt;ref&gt;Rosenfeld (1988), p. 231&lt;/ref&gt; of hyperbolic geometry. By using the scalar product of &lt;math&gt;\left[u_{1},u_{2}\right]&lt;/math&gt;, the resulting Lorentz transformation can be seen as equivalent to the [[w:hyperbolic law of cosines]]:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{matrix}u^{2}=u_{1}^{2}+u_{2}^{2}\\
u'^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}
\end{matrix}\left|\begin{matrix}u_{1}=u\cos\alpha\\
u_{2}=u\sin\alpha\\
\\
u_{1}^{\prime}=u'\cos\alpha'\\
u_{2}^{\prime}=u'\sin\alpha'
\end{matrix}\right|\begin{align}u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+vu'\cos\alpha'}, &amp; u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-vu\cos\alpha}\\
u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{1+vu'\cos\alpha'}, &amp; u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{1-vu\cos\alpha}\\
\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{u'\cos\alpha'+v}, &amp; \tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{u\cos\alpha-v}
\end{align}
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(vu'\sin\alpha'\right){}^{2}}}{1+vu'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(vu\sin\alpha\right){}^{2}}}{1-vu\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-u^{\prime2}}}=\frac{1}{\sqrt{1-v^{2}}}\frac{1}{\sqrt{1-u^{2}}}-\frac{v}{\sqrt{1-v^{2}}}\frac{u}{\sqrt{1-u^{2}}}\cos\alpha &amp; (b)\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha &amp; (a)
\end{matrix}&lt;/math&gt;|{{equationRef|3f}}}}

The hyperbolic law of cosines (a) was given by [[#Taurinus|Taurinus (1826) and Lobachevsky (1829/30)]] and others, while variant (b) was given by [[#Schur|Schur (1900/02)]]. By further setting ''u=u′'' it follows:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\cos\alpha=\frac{\cos\alpha'+v}{1+v\cos\alpha'},\ \sin\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{1+v\cos\alpha'},\ \tan\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{\cos\alpha'+v},\ \tan\frac{\alpha}{2}=\sqrt{\frac{1-v}{1+v}}\tan\frac{\alpha'}{2}\\
\cos\alpha'=\frac{\cos\alpha-v}{1-v\cos\alpha},\ \sin\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{1-v\cos\alpha},\ \tan\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{\cos\alpha-v},\ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+v}{1-v}}\tan\frac{\alpha}{2}
\end{matrix}&lt;/math&gt;|{{equationRef|3g}}}}

Formulas (3b) resemble the equations of an [[w:ellipse]] of [[w:Orbital eccentricity|eccentricity]] ''v/c'', [[w:eccentric anomaly]] α' and [[w:true anomaly]] α, first geometrically formulated by [[#Euler|Kepler (1609)]] and explicitly written down by [[#Euler|Euler (1735, 1748), Lagrange (1770)]] and many others in relation to planetary motions. They were also used by [[../Lorentz transformation (conformal)#Darboux|E:Darboux (1873)]] as a sphere transformation. In special relativity, these formulas describes the aberration of light, see [[../Lorentz transformation (velocity)#Velocity addition and aberration|E:velocity addition and aberration]].

==Historical notation==
==={{anchor|Euler}} Euler (1735) – True and eccentric anomaly===

[[w:Johannes Kepler]] (1609) geometrically formulated [[w:Kepler's equation]] and the relations between the [[w:mean anomaly]], [[w:true anomaly]], and [[w:eccentric anomaly]].&lt;ref group=M&gt;Kepler (1609), chapter 60. The editors of Kepler's collected papers remark (p. 482), that Kepler's relations correspond to &lt;math&gt;{\scriptstyle \alpha=\beta+e\sin\beta}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\nu=\frac{e+\cos\beta}{1+e\cos\beta}}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\beta=\frac{\cos\nu-e}{1-e\cos\nu}}&lt;/math&gt;&lt;/ref&gt;&lt;ref&gt;Volk (1976), p. 366&lt;/ref&gt; The relation between the true anomaly ''z'' and the eccentric anomaly ''P'' was algebraically expressed by [[w:Leonhard Euler]] (1735/40) as follows:&lt;ref group=M&gt;Euler (1735/40), § 19&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{\cos P+v}{1+v\cos P},\ \cos P=\frac{\cos z-v}{1-v\cos z},\ \int P=\frac{\int z\sqrt{1-v^{2}}}{1-v\cos z}&lt;/math&gt;

and in 1748:&lt;ref group=M&gt;Euler (1748a), section VIII&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{n+\cos y}{1+n\cos y},\ \sin z=\frac{\sin y\sqrt{1-n^{2}}}{1+n\cos y},\ \tan z=\frac{\sin y\sqrt{1-n^{2}}}{n+\cos y}&lt;/math&gt;

while [[w:Joseph-Louis Lagrange]] (1770/71) expressed them as follows&lt;ref group=M&gt;Lagrange (1770/71), section I&lt;/ref&gt;

:&lt;math&gt;\sin u=\frac{m\sin x}{1+n\cos x},\ \cos u=\frac{n+\cos x}{1+n\cos x},\ \operatorname{tang}\frac{1}{2}u=\frac{m}{1+n}\operatorname{tang}\frac{1}{2}x,\ \left(m^{2}=1-n^{2}\right)&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt; These relations resemble formulas ({{equationNote|3g}}), while ({{equationNote|3f}}) follows by setting &lt;math&gt;[\cos z,\sin z]=\left[u_{x},u_{y}\right]&lt;/math&gt; in Euler's formulas or &lt;math&gt;[\cos u,\sin u]=\left[u_{x},u_{y}\right]&lt;/math&gt; in Lagrange's formulas.&lt;/p&gt;

==={{anchor|Riccati}} Riccati (1757) – hyperbolic addition===

[[w:Vincenzo Riccati]] introduced hyperbolic functions in 1757,&lt;ref group=M&gt;Riccati (1757), p. 71&lt;/ref&gt;&lt;ref group=M&gt;Günther (1880/81), pp. 7–13&lt;/ref&gt; in particular he formulated the angle sum laws for hyperbolic sine and cosine:

:&lt;math&gt;\begin{matrix}\mathrm{Ch}(\varphi+\pi)=\frac{\mathrm{Ch}\varphi\mathrm{Ch}\pi+\mathrm{Sh}\varphi\mathrm{Sh}\pi}{r}\\
\mathrm{Sh}(\varphi+\pi)=\frac{\mathrm{Ch}\varphi\mathrm{Sh}\pi+\mathrm{Ch}\pi\mathrm{Sh}\varphi}{r}\\
\left[\mathrm{Ch}^{2}-\mathrm{Sh}^{2}=rr\right]
\end{matrix}&lt;/math&gt;

He furthermore showed that &lt;math&gt;\mathrm{Ch}(\varphi-\pi)&lt;/math&gt; and &lt;math&gt;\mathrm{Sh}(\varphi-\pi)&lt;/math&gt; follow by setting &lt;math&gt;\mathrm{Ch}(\pi)\Rightarrow\mathrm{Ch}(-\pi)&lt;/math&gt; and &lt;math&gt;\mathrm{Sh}(\pi)\Rightarrow\mathrm{Sh}(-\pi)&lt;/math&gt; in the above formulas.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3c}}). (In modern publications, Riccati's additional factor ''r'' is set to unity.)&lt;/p&gt;

==={{anchor|Lambert}} Lambert (1768–1770) – hyperbolic addition===

While [[#Riccati|Riccati (1757)]] discussed the hyperbolic sine and cosine, [[w:Johann Heinrich Lambert]] (read 1767, published 1768) introduced the expression ''tang φ'' or abbreviated ''tφ'' as the [[w:tangens hyperbolicus]] &lt;math&gt;{\scriptstyle \frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}}&lt;/math&gt; of a variable ''u'', or in modern notation ''tφ=tanh(u)'':&lt;ref group=M&gt;Lambert (1761/68), pp. 309–318&lt;/ref&gt;&lt;ref&gt;Barnett (2004), pp. 22–23&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}\xi\xi-1 &amp; =\eta\eta &amp; (a)\\
1+\eta\eta &amp; =\xi\xi &amp; (b)\\
\frac{\eta}{\xi} &amp; =tang\ \phi=t\phi &amp; (c)\\
\xi &amp; =\frac{1}{\sqrt{1-t\phi^{2}}} &amp; (d)\\
\eta &amp; =\frac{t\phi}{\sqrt{1-t\phi^{2}}} &amp; (e)\\
t\phi'' &amp; =\frac{t\phi+t\phi'}{1+t\phi\cdot t\phi'} &amp; (f)\\
t\phi' &amp; =\frac{t\phi''-t\phi}{1-t\phi\cdot t\phi''} &amp; (g)
\end{align}
\right|\begin{align}2u &amp; =\log\frac{1+t\phi}{1-t\phi}\\
\xi &amp; =\frac{e^{u}+e^{-u}}{2}\\
\eta &amp; =\frac{e^{u}-e^{-u}}{2}\\
t\phi &amp; =\frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}\\
e^{u} &amp; =\xi+\eta\\
e^{-u} &amp; =\xi-\eta
\end{align}&lt;/math&gt;

In (1770) he rewrote the addition law for the hyperbolic tangens (f) or (g) as:&lt;ref group=M&gt;Lambert (1770), p. 335&lt;/ref&gt;

:&lt;math&gt;\begin{align}t(y+z) &amp; =(ty+tz):(1+ty\cdot tz) &amp; (f)\\
t(y-z) &amp; =(ty-tz):(1-ty\cdot tz) &amp; (g)
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The hyperbolic relations (a,b,c,d,e,f) are equivalent to the hyperbolic relations on the right of ({{equationNote|3b}}). Relations (f,g) can also be found in ({{equationNote|3e}}). By setting ''tφ=v/c'', formula (c) becomes the relative velocity between two frames, (d) the [[w:Lorentz factor]], (e) the [[w:proper velocity]], (f) or (g) becomes the Lorentz transformation of velocity (or relativistic [[w:velocity addition formula]]) for collinear velocities in [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] and [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']].&lt;/p&gt;

Lambert also formulated the addition laws for the hyperbolic cosine and sine (Lambert's &quot;cos&quot; and &quot;sin&quot; actually mean &quot;cosh&quot; and &quot;sinh&quot;):

:&lt;math&gt;\begin{align}\sin(y+z) &amp; =\sin y\cos z+\cos y\sin z\\
\sin(y-z) &amp; =\sin y\cos z-\cos y\sin z\\
\cos(y+z) &amp; =\cos y\cos z+\sin y\sin z\\
\cos(y-z) &amp; =\cos y\cos z-\sin y\sin z
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{Anchor|Taurinus}} Taurinus (1826) – Hyperbolic law of cosines===

After the addition theorem for the tangens hyperbolicus was given by [[#Lambert|Lambert (1768)]], [[w:hyperbolic geometry]] was used by [[w:Franz Taurinus]] (1826), and later by [[w:Nikolai Lobachevsky]] (1829/30) and others, to formulate the [[w:hyperbolic law of cosines]]:&lt;ref group=M&gt;Taurinus (1826), p. 66; see also p. 272 in the translation by Engel and Stäckel (1899)&lt;/ref&gt;&lt;ref&gt;Bonola (1912), p. 79&lt;/ref&gt;&lt;ref&gt;Gray (1979), p. 242&lt;/ref&gt;

:&lt;math&gt;A=\operatorname{arccos}\frac{\cos\left(\alpha\sqrt{-1}\right)-\cos\left(\beta\sqrt{-1}\right)\cos\left(\gamma\sqrt{-1}\right)}{\sin\left(\beta\sqrt{-1}\right)\sin\left(\gamma\sqrt{-1}\right)}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;When solved for &lt;math&gt;\cos\left(\alpha\sqrt{-1}\right)&lt;/math&gt; it corresponds to the Lorentz transformation in Beltrami coordinates ({{equationNote|3f}}), and by defining the rapidities &lt;math&gt;{\scriptstyle \left(\left[\frac{U}{c},\ \frac{v}{c},\ \frac{u}{c}\right]=\left[\tanh\alpha,\ \tanh\beta,\ \tanh\gamma\right]\right)}&lt;/math&gt; it corresponds to the relativistic velocity addition formula [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']].&lt;/p&gt;

==={{anchor|Beltrami}} Beltrami (1868) – Beltrami coordinates===

[[w:Eugenio Beltrami]] (1868a) introduced coordinates of the [[w:Beltrami–Klein model]] of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:&lt;ref group=M&gt;Beltrami (1868a), pp. 287-288; Note I; Note II&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds^{2}=R^{2}\frac{\left(a^{2}+v^{2}\right)du^{2}-2uv\,du\,dv+\left(a^{2}+v^{2}\right)dv^{2}}{\left(a^{2}+u^{2}+v^{2}\right)^{2}}\\
u^{2}+v^{2}=a^{2}\\
\hline u''=\frac{aa_{0}\left(u'-r_{0}\right)}{a^{2}-r_{0}u'},\ v''=\frac{a_{0}w_{0}v'}{a^{2}-r_{0}u'},\\
\left(r_{0}=\sqrt{u_{0}^{2}+v_{0}^{2}},\ w_{0}=\sqrt{a^{2}-r_{0}^{2}}\right)\\
\hline ds^{2}=R^{2}\frac{\left(a^{2}-v^{2}\right)du^{2}+2uv\,du\,dv+\left(a^{2}-v^{2}\right)dv^{2}}{\left(a^{2}-u^{2}-v^{2}\right)^{2}}\\
(R=R\sqrt{-1},\ a=a\sqrt{-1})
\end{matrix}&lt;/math&gt;

(where the disk radius ''a'' and the [[w:radius of curvature]] ''R'' are real in spherical geometry, in hyperbolic geometry they are imaginary), and for arbitrary dimensions in (1868b)&lt;ref group=M&gt;Beltrami (1868b), pp. 232, 240–241, 253–254&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds=R\frac{\sqrt{dx^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}}}{x}\\
x^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=a^{2}\\
\hline y_{1}=\frac{ab\left(x_{1}-a_{1}\right)}{a^{2}-a_{1}x_{1}}\ \text{or}\ x_{1}=\frac{a\left(ay_{1}+a_{1}b\right)}{ab+a_{1}y_{1}},\ x_{r}=\pm\frac{ay_{r}\sqrt{a^{2}-a_{1}^{2}}}{ab+a_{1}y_{1}}\ (r=2,3,\dots,n)\\
\hline ds=R\frac{\sqrt{dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}-dx^{2}}}{x}\\
x^{2}=a^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\\
\left(R=R\sqrt{-1},\ x=x\sqrt{-1},\ a=a\sqrt{-1}\right)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Setting ''a=a&lt;sub&gt;0&lt;/sub&gt;'' Beltrami's (1868a) formulas become formulas ({{equationNote|3e}}), or in his (1868b) formulas one sets ''a=b'' for arbitrary dimensions.&lt;/p&gt;

=== {{anchor|Laisant2}} Laisant (1874) – Equipollences===
In his French translation of [[w:Giusto Bellavitis]]' principal work on [[w:Equipollence (geometry)|w:equipollences]], [[w:Charles-Ange Laisant]] (1874) added a chapter related to hyperbolas. The equipollence OM and its tangent MT of a hyperbola is defined by Laisant as&lt;ref group=M&gt;Laisant (1874b), pp. 134–135&lt;/ref&gt;

:(1) &lt;math&gt;\begin{matrix} &amp; \mathrm{OM}\bumpeq x\mathrm{OA}+y\mathrm{OB}\\
 &amp; \mathrm{MT}\bumpeq y\mathrm{OA}+x\mathrm{OB}\\
 &amp; \left[x^{2}-y^{2}=1;\ x=\cosh t,\ y=\sinh t\right]\\
\Rightarrow &amp; \mathrm{OM}\bumpeq\cosh t\cdot\mathrm{OA}+\sinh t\cdot\mathrm{OB}
\end{matrix}&lt;/math&gt;

Here, OA and OB are [[w:Conjugate diameters|conjugate semi-diameters]] of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation:

:&lt;math&gt;\begin{matrix}\begin{align}\mathrm{OC} &amp; \bumpeq c\mathrm{OA}+d\mathrm{OB} &amp; \qquad &amp;  &amp; \mathrm{OA} &amp; \bumpeq c\mathrm{OC}-d\mathrm{OD}\\
\mathrm{OD} &amp; \bumpeq d\mathrm{OA}+c\mathrm{OB} &amp;  &amp;  &amp; \mathrm{OB} &amp; \bumpeq-d\mathrm{OC}+c\mathrm{OD}
\end{align}
\\
\left[c^{2}-d^{2}=1\right]
\end{matrix}&lt;/math&gt;

producing the invariant relation

:&lt;math&gt;(\mathrm{OC})^{2}-(\mathrm{OD})^{2}\bumpeq(\mathrm{OA})^{2}-(\mathrm{OB})^{2}&lt;/math&gt;.

Substituting into (1), he showed that OM retains its form

:&lt;math&gt;\begin{matrix}\mathrm{OM}\bumpeq(cx-dy)\mathrm{OC}+(cy-dx)\mathrm{OD}\\
\left[(cx-dy)^{2}-(cy-dx)^{2}=1\right]
\end{matrix}&lt;/math&gt;

He also defined velocity and acceleration by differentiation of (1).

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These relations are equivalent to several Lorentz boosts or hyperbolic rotations producing the invariant Lorentz interval in line with ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Escherich}} Escherich (1874) – Beltrami coordinates===

[[w:Gustav von Escherich]] (1874) discussed the plane of constant negative curvature&lt;ref&gt;Sommerville (1911), p. 297&lt;/ref&gt; based on the [[w:Beltrami–Klein model]] of hyperbolic geometry by [[#Beltrami|Beltrami (1868)]]. Similar to [[w:Christoph Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt;  who introduced axial coordinates ''x''=tan(a) and ''y''=tan(b) in sphere geometry in order to perform coordinate transformations in the case of rotation and translation, Escherich used hyperbolic functions ''x''=tanh(a/k) and ''y''=tanh(b/k)&lt;ref group=M&gt;Escherich (1874), p. 508&lt;/ref&gt; in order to give the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation have the form:&lt;ref group=M name=escher&gt;Escherich (1874), p. 510&lt;/ref&gt;

:&lt;math&gt;x=\frac{\sinh\frac{a}{k}+x'\cosh\frac{a}{k}}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt; and &lt;math&gt;y=\frac{y'}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}), also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting &lt;math&gt;\tfrac{a}{k}=\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt; and multiplying ''[x,y,x′,y′]'' by 1/''c'', and equivalent to Lorentz boost ({{equationNote|3b}}) by setting &lt;math&gt;\scriptstyle (x,\ y,\ x',\ y')=\left(\frac{x_{1}}{x_{0}},\ \frac{x_{2}}{x_{0}},\ \frac{x_{1}^{\prime}}{x_{0}^{\prime}},\ \frac{x_{2}^{\prime}}{x_{0}^{\prime}}\right)&lt;/math&gt;. This is the relation between the [[w:Beltrami–Klein model|Beltrami coordinates]] in terms of Gudermann-Escherich coordinates, and the Weierstrass coordinates of the [[w:hyperboloid model]] introduced by [[../Lorentz transformation (general)#Killing1|E:Killing (1878–1893)]], [[../Lorentz transformation (general)#Poincare|E:Poincaré (1881)]], and [[../Lorentz transformation (general)#Cox|E:Cox (1881)]]. Both coordinate systems were compared by Cox (1881).&lt;ref group=M&gt;Cox (1881), p. 186&lt;/ref&gt;&lt;/p&gt;

==={{anchor|Glaisher}} Glaisher (1878) – hyperbolic addition===

It was shown by [[w:James Whitbread Lee Glaisher]] (1878) that the hyperbolic addition laws can be written as matrix multiplication&lt;ref group=M&gt;Glaisher (1878), p. 30&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\cosh x, &amp; \sinh x\\
\sinh x, &amp; \cosh x
\end{vmatrix}=1,\ \begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y
\end{vmatrix}=1\\
\left[c_{1},c_{2},c_{3},c_{4}\right]=\left[\cosh x,\cosh y,\sinh x,\sinh y\right]\\
\Rightarrow\begin{vmatrix}c_{1}c_{2}+s_{1}s_{2}, &amp; s_{1}c_{2}+c_{1}s_{2}\\
c_{1}s_{2}+s_{1}c_{2}, &amp; s_{1}s_{2}+c_{1}c_{2}
\end{vmatrix}=1\ \Rightarrow\begin{vmatrix}\cosh(x+y), &amp; \sinh(x+y)\\
\sinh(x+y), &amp; \cosh(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{anchor|Gunther1}} Günther (1880/81) – hyperbolic addition ===

Following [[#Glaisher|Glaisher (1878)]], [[w:Siegmund Günther]] (1880/81) formulated the hyperbolic addition laws in matrix form as&lt;ref group=M&gt;Günther (1880/81), p. 405&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\mathfrak{Cos}x, &amp; \mathfrak{Sin}x\\
\mathfrak{Sin}x, &amp; \mathfrak{Cos}x
\end{vmatrix}\cdot\begin{vmatrix}\mathfrak{Cos}y, &amp; \mathfrak{Sin}y\\
\mathfrak{Sin}y, &amp; \mathfrak{Cos}y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}x\mathfrak{Cos}y+\mathfrak{Sin}x\mathfrak{Sin}y, &amp; \mathfrak{Cos}x\mathfrak{Sin}y+\mathfrak{Sin}x\mathfrak{Cos}y\\
\mathfrak{Sin}x\mathfrak{Cos}y+\mathfrak{Cos}x\mathfrak{Sin}y, &amp; \mathfrak{Sin}x\mathfrak{Sin}y+\mathfrak{Cos}x\mathfrak{Cos}y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}(x+y), &amp; \mathfrak{Sin}(x+y)\\
\mathfrak{Sin}(x+y), &amp; \mathfrak{Cos}(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{anchor|Lipschitz1}} Lipschitz (1885/86) ===
[[w:Rudolf Lipschitz]] (1885/86) formulated transformations leaving invariant the sum of squares &lt;math&gt;x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}&lt;/math&gt;, which he rewrote as &lt;math&gt;x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots+x_{n}^{2}-y_{n}^{2}=0&lt;/math&gt;. This led to the problem of finding transformations leaving invariant the pairs &lt;math&gt;x_{a}^{2}-y_{a}^{2}&lt;/math&gt; (''a''=1...n) for which he gave the following solution:&lt;ref group=M&gt;Lipschitz (1886), pp. 90–92&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{a}^{2}-y_{a}^{2}=\mathfrak{x}_{a}^{2}-\mathfrak{y}_{a}^{2}\\
\hline \begin{align}x_{a}-y_{a} &amp; =\left(\mathfrak{x}_{a}-\mathfrak{y}_{a}\right)r_{a}\\
x_{a}+y_{a} &amp; =\left(\mathfrak{x}_{a}+\mathfrak{y}_{a}\right)\frac{1}{r_{a}}
\end{align}
\quad(a)\\
\hline \begin{matrix}\begin{align}2\mathfrak{x}_{a} &amp; =\left(r_{a}+\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}-\frac{1}{r_{a}}\right)y_{a}\\
2\mathfrak{y}_{a} &amp; =\left(r_{a}-\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}+\frac{1}{r_{a}}\right)y_{a}
\end{align}
\quad(b)\end{matrix}\\
\hline \left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}\Rightarrow\begin{align}\mathfrak{x}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}\\
\mathfrak{y}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}
\end{align}
\quad(c)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Equation (a) is identical to Lorentz boost ({{equationNote|3d}}), while (c) is similar, though not identical, to Lorentz boost ({{equationNote|3b}}-C). The difference stems from his definition&lt;br&gt;&lt;math&gt;\qquad\left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}&lt;/math&gt;&lt;br&gt;whereas in accordance to expression &lt;math&gt;\sqrt{\tfrac{1+v}{1-v}}&lt;/math&gt; in ({{equationNote|3d}}) he should have stated &lt;br&gt;&lt;math&gt;\qquad\left\{ \begin{matrix}r_{a}=\frac{\sqrt{1+s_{a}}}{\sqrt{1-s_{a}}}\\
s_{a}&lt;1
\end{matrix}\right\}&lt;/math&gt;.&lt;br&gt;Using the latter choice, equations (c) would assume a form equivalent to ({{equationNote|3b}}):&lt;br&gt;&lt;math&gt;\qquad\begin{align}\mathfrak{x}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{1-s_{a}}\sqrt{1+s_{a}}}\\
\mathfrak{y}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{1-s_{a}}\sqrt{1+s_{a}}}\end{align}&lt;/math&gt;&lt;/p&gt;

==={{Anchor|Schur}} Schur (1885/86, 1900/02) – Beltrami coordinates===

[[w:Friedrich Schur]] (1885/86) discussed spaces of constant Riemann curvature, and by following [[#Beltrami|Beltrami (1868)]] he used the transformation&lt;ref group=M&gt;Schur (1885/86), p. 167&lt;/ref&gt;

:&lt;math&gt;x_{1}=R^{2}\frac{y_{1}+a_{1}}{R^{2}+a_{1}y_{1}},\ x_{2}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{2}}{R^{2}+a_{1}y_{1}},\dots,\ x_{n}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{n}}{R^{2}+a_{1}y_{1}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in arbitrary dimensions by setting ''R=c'' as the speed of light and ''a&lt;sub&gt;1&lt;/sub&gt;=v'' as relative velocity.&lt;/p&gt;

In (1900/02) he derived basic formulas of non-Eucliden geometry, including the case of translation for which he obtained the transformation similar to his previous one:&lt;ref group=M&gt;Schur (1900/02), p. 290; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;x'=\frac{x-a}{1-\mathfrak{k}ax},\quad y'=\frac{y\sqrt{1-\mathfrak{k}a^{2}}}{1-\mathfrak{k}ax}&lt;/math&gt;

where &lt;math&gt;\mathfrak{k}&lt;/math&gt; can have values &gt;0, &lt;0 or ∞.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting ''a=v'' and &lt;math&gt;\mathfrak{k}=\tfrac{1}{c^{2}}&lt;/math&gt;.&lt;/p&gt;

He also defined the triangle&lt;ref group=M&gt;Schur (1900/02), p. 291; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;\frac{1}{\sqrt{1-\mathfrak{k}c^{2}}}=\frac{1}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{1}{\sqrt{1-\mathfrak{k}b^{2}}}-\frac{a}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{b}{\sqrt{1-\mathfrak{k}b^{2}}}\cos\gamma&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to the hyperbolic law of cosines and the relativistic velocity addition ({{equationNote|3f}}, b) or [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']] by setting &lt;math&gt;[\mathfrak{k},\ c,\ a,\ b]=\left[\tfrac{1}{c^{2}},\ \sqrt{u_{x}^{\prime2}+u_{y}^{\prime2}},\ v,\ \sqrt{u_{x}^{2}+u_{y}^{2}}\right]&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Lindemann}} Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute===

[[w:Ferdinand von Lindemann]] discussed hyperbolic geometry in terms of the [[w:Cayley–Klein metric]] in his (1890/91) edition of the lectures on geometry of [[w:Alfred Clebsch]]. Citing [[../Lorentz transformation (general)#Killing|E:Killing (1885)]] and [[../Lorentz transformation (general)#Poincare|Poincaré (1887)]] in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 477–478, 524&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\Omega_{xx}=x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}-4k^{2}dx_{3}^{2}\\
\Omega_{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}
\end{matrix}&lt;/math&gt;

and used the following transformation&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 361–362&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
X_{1}X_{4}+X_{2}X_{3}=\Xi_{1}\Xi_{4}+\Xi_{2}\Xi_{3}\\
\hline \begin{align}X_{1} &amp; =\left(\lambda+\lambda_{1}\right)U_{4} &amp; \Xi_{1} &amp; =\left(\lambda-\lambda_{1}\right)U_{4} &amp; X_{1} &amp; =\frac{\lambda+\lambda_{1}}{\lambda-\lambda_{1}}\Xi_{1}\\
X_{2} &amp; =\left(\lambda+\lambda_{3}\right)U_{4} &amp; \Xi_{2} &amp; =\left(\lambda-\lambda_{3}\right)U_{4} &amp; X_{2} &amp; =\frac{\lambda+\lambda_{3}}{\lambda-\lambda_{3}}\Xi_{2}\\
X_{3} &amp; =\left(\lambda-\lambda_{3}\right)U_{2} &amp; \Xi_{3} &amp; =\left(\lambda+\lambda_{3}\right)U_{2} &amp; X_{3} &amp; =\frac{\lambda-\lambda_{3}}{\lambda+\lambda_{3}}\Xi_{3}\\
X_{4} &amp; =\left(\lambda-\lambda_{1}\right)U_{1} &amp; \Xi_{4} &amp; =\left(\lambda+\lambda_{1}\right)U_{1} &amp; X_{4} &amp; =\frac{\lambda-\lambda_{1}}{\lambda+\lambda_{1}}\Xi_{4}
\end{align}
\end{matrix}&lt;/math&gt;

into which he put&lt;ref group=M name=linde&gt;Lindemann &amp; Clebsch (1890/91), p. 496&lt;/ref&gt;

:&lt;math&gt;\begin{align}X_{1} &amp; =x_{1}+2kx_{4}, &amp; X_{2} &amp; =x_{2}+ix_{3}, &amp; \lambda+\lambda_{1} &amp; =\left(\lambda-\lambda_{1}\right)e^{a},\\
X_{4} &amp; =x_{1}-2kx_{4}, &amp; X_{3} &amp; =x_{2}-ix_{3}, &amp; \lambda+\lambda_{3} &amp; =\left(\lambda-\lambda_{3}\right)e^{\alpha i},
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3d}}) with &lt;math&gt;e^{\alpha i}=1&lt;/math&gt; and ''2k=1'' .&lt;/p&gt;

From that, he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations (''a''=0) or translations (α=0):&lt;ref group=M  name=linde /&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=0\\
\hline \begin{align}x_{2} &amp; =\xi_{2}\cos\alpha+\xi_{3}\sin\alpha, &amp; x_{1} &amp; =\xi_{1}\cos\frac{a}{i}+2ki\xi_{4}\sin\frac{a}{i},\\
x_{3} &amp; =-\xi_{2}\sin\alpha+\xi_{3}\cos\alpha, &amp; 2kx_{4} &amp; =i\xi_{1}\sin\frac{a}{i}+2k\xi_{4}\cos\frac{a}{i}.
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0 and ''2k=1''.&lt;/p&gt;

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane and gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Killing2}} Killing (1893,97) – Weierstrass coordinates===

[[w:Wilhelm Killing]] (1878–1880) gave case of translation in the form&lt;ref group=M name=killtra&gt;Killing (1893), p. 331&lt;/ref&gt;

:&lt;math&gt;y_{0}=x_{0}\operatorname{Ch}a+x_{1}\operatorname{Sh}a,\quad y_{1}=x_{0}\operatorname{Sh}a+x_{1}\operatorname{Ch}a,\quad y_{2}=x_{2}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

In 1898, Killing wrote that relation in a form similar to [[#Escherich|Escherich (1874)]], and derived the corresponding Lorentz transformation for the two cases were ''v'' is unchanged or ''u'' is unchanged:&lt;ref group=M name=kill98&gt;Killing (1898), p. 133&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\xi'=\frac{\xi\operatorname{Ch}\frac{\mu}{l}+l\operatorname{Sh}\frac{\mu}{l}}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}},\ \eta'=\frac{\eta}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}}\\
\hline \frac{u}{p}=\xi,\ \frac{v}{p}=\eta\\
\hline p'=p\operatorname{Ch}\frac{\mu}{l}+\frac{u}{l}\operatorname{Sh}\frac{\mu}{l},\quad u'=pl\operatorname{Sh}\frac{\mu}{l}+u\operatorname{Ch}\frac{\mu}{l},\quad v'=v\\
\text{or}\\
p'=p\operatorname{Ch}\frac{\nu}{l}+\frac{v}{l}\operatorname{Sh}\frac{\nu}{l},\quad u'=u,\quad v'=pl\operatorname{Sh}\frac{\nu}{l}+v\operatorname{Ch}\frac{\nu}{l}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The upper transformation system is equivalent to Lorentz transformation ({{equationNote|3e}}) and the velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] with ''l=c'' and &lt;math&gt;\mu=c\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt;, the system below is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Whitehead}} Whitehead (1897/98) – Universal algebra===

[[w:Alfred North Whitehead]] (1898) discussed the kinematics of hyperbolic space as part of his study of [[w:universal algebra]], and obtained the following transformation:&lt;ref group=M name=white&gt;Whitehead (1898), pp. 459–460&lt;/ref&gt;

:&lt;math&gt;\begin{align}x' &amp; =\left(\eta\cosh\frac{\delta}{\gamma}+\eta_{1}\sinh\frac{\delta}{\gamma}\right)e+\left(\eta\sinh\frac{\delta}{\gamma}+\eta_{1}\cosh\frac{\delta}{\gamma}\right)e_{1}\\
 &amp; \qquad+\left(\eta_{2}\cos\alpha+\eta_{3}\sin\alpha\right)e_{2}+\left(\eta_{3}\cos\alpha-\eta_{2}\sin\alpha\right)e_{3}
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0.&lt;/p&gt;

==={{anchor|Elliott}} Elliott (1903) – Invariant theory ===
[[w:Edwin Bailey Elliott]] (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:&lt;ref group=M&gt;Elliott (1903), p. 109&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=X\cosh\phi+Y\sinh\phi,\quad y=X\sinh\phi+Y\cosh\phi\\
\hline X+Y=e^{-\phi}(x+y),\quad X-Y=e^{\phi}(x-y)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}). The mentioned subgroup corresponds to the one-parameter subgroup generated by Lorentz boosts.&lt;/p&gt;

==={{anchor|Woods2}} Woods (1903) – Weierstrass coordinates ===
[[w:Frederick S. Woods]] (1903, published 1905) gave the case of translation in hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 55&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x_{1}\cos kl+x_{0}\frac{\sin kl}{k},\quad x_{2}^{\prime}=x_{2},\quad x_{2}^{\prime}=x_{3},\quad x_{0}^{\prime}=-x_{1}k\sin kl+x_{0}\cos kl&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.&lt;/p&gt;

and the loxodromic substitution for hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 72&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}x_{1}^{\prime} &amp; =x_{1}\cosh\alpha-x_{0}\sinh\alpha\\
x_{2}^{\prime} &amp; =x_{2}\cos\beta-x_{3}\sin\beta\\
x_{3}^{\prime} &amp; =x_{2}\sin\beta+x_{3}\cos\beta\\
x_{0}^{\prime} &amp; =-x_{1}\sinh\alpha+x_{0}\cosh\alpha
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with β=0.&lt;/p&gt;

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – gave the case of translation in the hyperbolic plane:&lt;ref group=M name=lieb&gt;Liebmann (1904/05), p. 174&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x'\operatorname{ch}a+p'\operatorname{sh}a,\quad y_{1}^{\prime}=y',\quad p_{1}^{\prime}=x'\operatorname{sh}a+p'\operatorname{ch}a&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Frank}} Frank (1909) – Special relativity===
In special relativity, hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity:&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;
 	
=== {{anchor|Herglotz1}} Herglotz (1909/10) – Special relativity===

In special relativity, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic, with the hyperbolic case being:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;
	
:&lt;math&gt;\begin{matrix}Z=Z'e^{\vartheta}\\
	
\begin{aligned}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
	
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
	
\end{aligned}
	
\end{matrix}&lt;/math&gt;	
	
&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3d}}).&lt;/p&gt;

==={{anchor|Varicak}} Varićak (1910) – Special relativity===

In special relativity, hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

He showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bel68sag}} 
*{{#section:History of Topics in Special Relativity/mathsource|bel68fond}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|eli03}} 
*{{#section:History of Topics in Special Relativity/mathsource|esch74}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul35}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul48a}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|glai78}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|guen80}} 
*{{#section:History of Topics in Special Relativity/mathsource|kep09}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|lais74b}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam67}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lind90}}
*{{#section:History of Topics in Special Relativity/mathsource|lip86}}  
*{{#section:History of Topics in Special Relativity/mathsource|ric57}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu85}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu00}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu09}} 
*{{#section:History of Topics in Special Relativity/mathsource|tau26}} 
*{{#section:History of Topics in Special Relativity/mathsource|whit98}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L3}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
      <sha1>4dkw5s3vzt9ckpuqwukvy0rgwe96j0a</sha1>
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    <revision>
      <id>2411819</id>
      <parentid>2411817</parentid>
      <timestamp>2022-08-04T12:30:35Z</timestamp>
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        <id>52339</id>
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      <text bytes="47368" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via hyperbolic functions==
===Translation in the hyperbolic plane===
The case of a Lorentz transformation without spatial rotation is called a [[w:Lorentz boost]]. The simplest case can be given, for instance, by setting ''n=1'' in the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}\\
\\
x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}
\end{align}
\left|\begin{align}g_{01}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{10}^{2} &amp; =1\\
g_{01}g_{11}-g_{00}g_{10} &amp; =0\\
g_{10}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{01}^{2} &amp; =1\\
g_{10}g_{11}-g_{00}g_{01} &amp; =0
\end{align}
\rightarrow\begin{align}g_{00}^{2} &amp; =g_{11}^{2}\\
g_{01}^{2} &amp; =g_{10}^{2}
\end{align}
\right.
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}g_{00} &amp; -g_{10}\\
-g_{01} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\right|\det\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;|{{equationRef|3a}}}}

which resembles precisely the relations of [[w:hyperbolic function]]s in terms of [[w:hyperbolic angle]] &lt;math&gt;\eta&lt;/math&gt;. Thus a Lorentz boost or [[w:hyperbolic rotation]] (being the same as a rotation around an imaginary angle &lt;math&gt;i\eta=\phi&lt;/math&gt; in [[../Lorentz transformation (imaginary)#math_2b|E:'''(2b)''']] or a [[w:Translation (geometry)|translation]] in the hyperbolic plane in terms of the hyperboloid model) is given by

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline g_{00}=g_{11}=\cosh\eta,\ g_{01}=g_{10}=-\sinh\eta\\
\hline \left.\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta &amp;  &amp; =\frac{x_{0}-x_{1}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}-x_{1}v}{\sqrt{1-v^{2}}}\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta &amp;  &amp; =\frac{x_{1}-x_{0}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}-x_{0}v}{\sqrt{1-v^{2}}}\\
x_{2}^{\prime} &amp; =x_{2} &amp;  &amp; =x_{2} &amp;  &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta &amp;  &amp; =\frac{x_{0}^{\prime}-x_{1}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}^{\prime}-x_{1}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta &amp;  &amp; =\frac{x_{1}^{\prime}-x_{0}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}^{\prime}-x_{0}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{2} &amp; =x_{2}^{\prime} &amp;  &amp; =x_{2}^{\prime} &amp;  &amp; =x_{2}^{\prime}
\end{align}
\right|{\scriptstyle \begin{align}\sinh^{2}\eta-\cosh^{2}\eta &amp; =-1 &amp; (a)\\
\cosh^{2}\eta-\sinh^{2}\eta &amp; =1 &amp; (b)\\
\frac{\sinh\eta}{\cosh\eta} &amp; =\tanh\eta=v &amp; (c)\\
\frac{1}{\sqrt{1-\tanh^{2}\eta}} &amp; =\cosh\eta &amp; (d)\\
\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\sinh\eta &amp; (e)\\
\frac{\tanh q\pm\tanh\eta}{1\pm\tanh q\tanh\eta} &amp; =\tanh\left(q\pm\eta\right) &amp; (f)
\end{align}
}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\right|\det\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;|{{equationRef|3b}}}}

Hyperbolic identities (a,b) on the right of ({{equationNote|3b}}) were given by [[#Riccati|Riccati (1757)]], all identities (a,b,c,d,e,f) by [[#Lambert|Lambert (1768–1770)]]. Lorentz transformations ({{equationNote|3b}}-A) were given by [[#Laisant|Laisant (1874)]], [[#Cox|Cox (1882)]], [[#Lindemann|Lindemann (1890/91)]], [[#Gerard|Gérard (1892)]], [[#Killing2|Killing (1893, 1897/98)]], [[#Whitehead|Whitehead (1897/98)]], [[#Woods2|Woods (1903/05)]], [[#Elliott|Elliott (1903)]] and [[#Liebmann|Liebmann (1904/05)]] in terms of Weierstrass coordinates of the [[w:hyperboloid model]], while transformations similar to ({{equationNote|3b}}-C) have been used by [[#Lipschitz1|Lipschitz (1885/86)]].

Rapidity can be composed of arbitrary many rapidities &lt;math&gt;\eta_{1},\eta_{2}\dots&lt;/math&gt; as per the [[w:Hyperbolic functions#Sums of arguments|w:angle sum laws of hyperbolic sines and cosines]], so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between [[w:List of trigonometric identities#Angle sum and difference identities|w:angle sum laws of circular trigonometry]] and spatial rotations. Alternatively, the hyperbolic angle sum laws ''themselves'' can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the [[w:unit hyperbola]]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \left[\eta=\eta_{2}-\eta_{1}\right]\\
\begin{align}x_{0}^{\prime} &amp; =\sinh\eta_{1} &amp;  &amp; =\sinh\left(\eta_{2}-\eta\right) &amp;  &amp; =\sinh\eta_{2}\cosh\eta-\cosh\eta_{2}\sinh\eta &amp;  &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =\cosh\eta_{1} &amp;  &amp; =\cosh\left(\eta_{2}-\eta\right) &amp;  &amp; =-\sinh\eta_{2}\sinh\eta+\cosh\eta_{2}\cosh\eta &amp;  &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta\\
\\
x_{0} &amp; =\sinh\eta_{2} &amp;  &amp; =\sinh\left(\eta_{1}+\eta\right) &amp;  &amp; =\sinh\eta_{1}\cosh\eta+\cosh\eta_{1}\sinh\eta &amp;  &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =\cosh\eta_{2} &amp;  &amp; =\cosh\left(\eta_{1}+\eta\right) &amp;  &amp; =\sinh\eta_{1}\sinh\eta+\cosh\eta_{1}\cosh\eta &amp;  &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \begin{align} &amp; \begin{bmatrix}x_{0}^{\prime}\\
x_{1}^{\prime}
\end{bmatrix}=\begin{bmatrix}\sinh\eta_{1}\\
\cosh\eta_{1}
\end{bmatrix}=\begin{bmatrix}\sinh\left(\eta_{2}-\eta\right)\\
\cosh\left(\eta_{2}-\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}\sinh\eta_{2}\\
\cosh\eta_{2}
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}x_{0}\\
x_{1}
\end{bmatrix}\\
 &amp; \begin{bmatrix}x_{0}\\
x_{1}
\end{bmatrix}=\begin{bmatrix}\sinh\eta_{2}\\
\cosh\eta_{2}
\end{bmatrix}=\begin{bmatrix}\sinh\left(\eta_{1}+\eta\right)\\
\cosh\left(\eta_{1}+\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}\sinh\eta_{1}\\
\cosh\eta_{1}
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}x_{0}^{\prime}\\
x_{1}^{\prime}
\end{bmatrix}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|3c}}}}

Hyperbolic angle sum laws equivalent to Lorentz boost ({{equationNote|3c}}) were given by [[#Riccati|Riccati (1757)]] and [[#Lambert|Lambert (1768–1770)]], while the matrix representation was given by [[#Glaisher|Glaisher (1878)]] and [[#Gunther2|Günther (1880/81)]]. In special relativity, hyperbolic functions were used by [[#Frank|Frank (1909)]] and [[#Varicak|Varićak (1910)]].

Finally, because of &lt;math&gt;\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;, Lorentz boost ({{equationNote|3b}}) assumes a simple form by using [[w:squeeze mapping]]s in analogy to Euler's formula in [[../Lorentz transformation (imaginary)#math_2c|E:'''(2c)''']]:&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \begin{matrix}x_{1}^{\prime}-x_{0}^{\prime}=k\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime}=\frac{1}{k}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime}=x_{2}
\end{matrix}\\
\hline \begin{align}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}-x_{0} &amp; =e^{-\eta}\left(x_{1}^{\prime}-x_{0}^{\prime}\right)\\
x_{1}+x_{0} &amp; =e^{\eta}\left(x_{1}^{\prime}+x_{0}^{\prime}\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\\
\hline k=e^{\eta}=\cosh\eta+\sinh\eta=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}=\sqrt{\frac{1+v}{1-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|3d}}}} 

Lorentz transformations ({{equationNote|3d}}) for arbitrary ''k'' were given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]), while the above exponential form was explicitly used by [[#Lindemann|Lindemann (1890/91)]], [[#Elliott|Elliott (1903)]], [[#Herglotz1|Herglotz (1909)]]. 

===Hyperbolic law of cosines===
In line with equation [[../Lorentz transformation (general)#math_1b|E:'''(1b)''']] one can use coordinates &lt;math&gt;[u_{1},\ u_{2}]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}}\right]&lt;/math&gt; inside the [[w:unit circle]] &lt;math&gt;u_{1}^{2}+u_{2}^{2}=1&lt;/math&gt;, thus the corresponding Lorentz transformations ({{equationNote|3b}}) obtain the form:

{{NumBlk|:|&lt;math&gt;\begin{matrix}u_{x}^{2}+u_{y}^{2}=u_{x}^{\prime2}+u_{y}^{\prime2}=1\\
\hline \begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
u_{1}^{\prime} &amp; =\frac{-\sinh\eta+u_{1}\cosh\eta}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{1}-\tanh\eta}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{1}-v}{1-u_{1}v}\\
u_{2}^{\prime} &amp; =\frac{u_{2}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-v^{2}}}{1-u_{1}v}\\
\\
u_{1} &amp; =\frac{\sinh\eta+u_{1}^{\prime}\cosh\eta}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+\tanh\eta}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+v}{1+u_{1}^{\prime}v}\\
u_{2} &amp; =\frac{u_{2}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|3e}}}}

Transformations (A) were given by [[#Escherich|Escherich (1874)]] and [[#Killing2|Killing (1898)]], and transformations (B) by [[#Beltrami|Beltrami (1868)]] and [[#Schur|Schur (1885/86, 1900/02)]] in terms of [[w:Beltrami–Klein model|Beltrami coordinates]]&lt;ref&gt;Rosenfeld (1988), p. 231&lt;/ref&gt; of hyperbolic geometry. By using the scalar product of &lt;math&gt;\left[u_{1},u_{2}\right]&lt;/math&gt;, the resulting Lorentz transformation can be seen as equivalent to the [[w:hyperbolic law of cosines]]:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{matrix}u^{2}=u_{1}^{2}+u_{2}^{2}\\
u'^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}
\end{matrix}\left|\begin{matrix}u_{1}=u\cos\alpha\\
u_{2}=u\sin\alpha\\
\\
u_{1}^{\prime}=u'\cos\alpha'\\
u_{2}^{\prime}=u'\sin\alpha'
\end{matrix}\right|\begin{align}u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+vu'\cos\alpha'}, &amp; u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-vu\cos\alpha}\\
u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{1+vu'\cos\alpha'}, &amp; u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{1-vu\cos\alpha}\\
\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{u'\cos\alpha'+v}, &amp; \tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{u\cos\alpha-v}
\end{align}
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(vu'\sin\alpha'\right){}^{2}}}{1+vu'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(vu\sin\alpha\right){}^{2}}}{1-vu\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-u^{\prime2}}}=\frac{1}{\sqrt{1-v^{2}}}\frac{1}{\sqrt{1-u^{2}}}-\frac{v}{\sqrt{1-v^{2}}}\frac{u}{\sqrt{1-u^{2}}}\cos\alpha &amp; (b)\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha &amp; (a)
\end{matrix}&lt;/math&gt;|{{equationRef|3f}}}}

The hyperbolic law of cosines (a) was given by [[#Taurinus|Taurinus (1826) and Lobachevsky (1829/30)]] and others, while variant (b) was given by [[#Schur|Schur (1900/02)]]. By further setting ''u=u′'' it follows:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\cos\alpha=\frac{\cos\alpha'+v}{1+v\cos\alpha'},\ \sin\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{1+v\cos\alpha'},\ \tan\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{\cos\alpha'+v},\ \tan\frac{\alpha}{2}=\sqrt{\frac{1-v}{1+v}}\tan\frac{\alpha'}{2}\\
\cos\alpha'=\frac{\cos\alpha-v}{1-v\cos\alpha},\ \sin\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{1-v\cos\alpha},\ \tan\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{\cos\alpha-v},\ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+v}{1-v}}\tan\frac{\alpha}{2}
\end{matrix}&lt;/math&gt;|{{equationRef|3g}}}}

Formulas (3b) resemble the equations of an [[w:ellipse]] of [[w:Orbital eccentricity|eccentricity]] ''v/c'', [[w:eccentric anomaly]] α' and [[w:true anomaly]] α, first geometrically formulated by [[#Euler|Kepler (1609)]] and explicitly written down by [[#Euler|Euler (1735, 1748), Lagrange (1770)]] and many others in relation to planetary motions. They were also used by [[../Lorentz transformation (conformal)#Darboux|E:Darboux (1873)]] as a sphere transformation. In special relativity, these formulas describes the aberration of light, see [[../Lorentz transformation (velocity)#Velocity addition and aberration|E:velocity addition and aberration]].

==Historical notation==
==={{anchor|Euler}} Euler (1735) – True and eccentric anomaly===

[[w:Johannes Kepler]] (1609) geometrically formulated [[w:Kepler's equation]] and the relations between the [[w:mean anomaly]], [[w:true anomaly]], and [[w:eccentric anomaly]].&lt;ref group=M&gt;Kepler (1609), chapter 60. The editors of Kepler's collected papers remark (p. 482), that Kepler's relations correspond to &lt;math&gt;{\scriptstyle \alpha=\beta+e\sin\beta}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\nu=\frac{e+\cos\beta}{1+e\cos\beta}}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\beta=\frac{\cos\nu-e}{1-e\cos\nu}}&lt;/math&gt;&lt;/ref&gt;&lt;ref&gt;Volk (1976), p. 366&lt;/ref&gt; The relation between the true anomaly ''z'' and the eccentric anomaly ''P'' was algebraically expressed by [[w:Leonhard Euler]] (1735/40) as follows:&lt;ref group=M&gt;Euler (1735/40), § 19&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{\cos P+v}{1+v\cos P},\ \cos P=\frac{\cos z-v}{1-v\cos z},\ \int P=\frac{\int z\sqrt{1-v^{2}}}{1-v\cos z}&lt;/math&gt;

and in 1748:&lt;ref group=M&gt;Euler (1748a), section VIII&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{n+\cos y}{1+n\cos y},\ \sin z=\frac{\sin y\sqrt{1-n^{2}}}{1+n\cos y},\ \tan z=\frac{\sin y\sqrt{1-n^{2}}}{n+\cos y}&lt;/math&gt;

while [[w:Joseph-Louis Lagrange]] (1770/71) expressed them as follows&lt;ref group=M&gt;Lagrange (1770/71), section I&lt;/ref&gt;

:&lt;math&gt;\sin u=\frac{m\sin x}{1+n\cos x},\ \cos u=\frac{n+\cos x}{1+n\cos x},\ \operatorname{tang}\frac{1}{2}u=\frac{m}{1+n}\operatorname{tang}\frac{1}{2}x,\ \left(m^{2}=1-n^{2}\right)&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt; These relations resemble formulas ({{equationNote|3g}}), while ({{equationNote|3f}}) follows by setting &lt;math&gt;[\cos z,\sin z]=\left[u_{x},u_{y}\right]&lt;/math&gt; in Euler's formulas or &lt;math&gt;[\cos u,\sin u]=\left[u_{x},u_{y}\right]&lt;/math&gt; in Lagrange's formulas.&lt;/p&gt;

==={{anchor|Riccati}} Riccati (1757) – hyperbolic addition===

[[w:Vincenzo Riccati]] introduced hyperbolic functions in 1757,&lt;ref group=M&gt;Riccati (1757), p. 71&lt;/ref&gt;&lt;ref group=M&gt;Günther (1880/81), pp. 7–13&lt;/ref&gt; in particular he formulated the angle sum laws for hyperbolic sine and cosine:

:&lt;math&gt;\begin{matrix}\mathrm{Ch}(\varphi+\pi)=\frac{\mathrm{Ch}\varphi\mathrm{Ch}\pi+\mathrm{Sh}\varphi\mathrm{Sh}\pi}{r}\\
\mathrm{Sh}(\varphi+\pi)=\frac{\mathrm{Ch}\varphi\mathrm{Sh}\pi+\mathrm{Ch}\pi\mathrm{Sh}\varphi}{r}\\
\left[\mathrm{Ch}^{2}-\mathrm{Sh}^{2}=rr\right]
\end{matrix}&lt;/math&gt;

He furthermore showed that &lt;math&gt;\mathrm{Ch}(\varphi-\pi)&lt;/math&gt; and &lt;math&gt;\mathrm{Sh}(\varphi-\pi)&lt;/math&gt; follow by setting &lt;math&gt;\mathrm{Ch}(\pi)\Rightarrow\mathrm{Ch}(-\pi)&lt;/math&gt; and &lt;math&gt;\mathrm{Sh}(\pi)\Rightarrow\mathrm{Sh}(-\pi)&lt;/math&gt; in the above formulas.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3c}}). (In modern publications, Riccati's additional factor ''r'' is set to unity.)&lt;/p&gt;

==={{anchor|Lambert}} Lambert (1768–1770) – hyperbolic addition===

While [[#Riccati|Riccati (1757)]] discussed the hyperbolic sine and cosine, [[w:Johann Heinrich Lambert]] (read 1767, published 1768) introduced the expression ''tang φ'' or abbreviated ''tφ'' as the [[w:tangens hyperbolicus]] &lt;math&gt;{\scriptstyle \frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}}&lt;/math&gt; of a variable ''u'', or in modern notation ''tφ=tanh(u)'':&lt;ref group=M&gt;Lambert (1761/68), pp. 309–318&lt;/ref&gt;&lt;ref&gt;Barnett (2004), pp. 22–23&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}\xi\xi-1 &amp; =\eta\eta &amp; (a)\\
1+\eta\eta &amp; =\xi\xi &amp; (b)\\
\frac{\eta}{\xi} &amp; =tang\ \phi=t\phi &amp; (c)\\
\xi &amp; =\frac{1}{\sqrt{1-t\phi^{2}}} &amp; (d)\\
\eta &amp; =\frac{t\phi}{\sqrt{1-t\phi^{2}}} &amp; (e)\\
t\phi'' &amp; =\frac{t\phi+t\phi'}{1+t\phi\cdot t\phi'} &amp; (f)\\
t\phi' &amp; =\frac{t\phi''-t\phi}{1-t\phi\cdot t\phi''} &amp; (g)
\end{align}
\right|\begin{align}2u &amp; =\log\frac{1+t\phi}{1-t\phi}\\
\xi &amp; =\frac{e^{u}+e^{-u}}{2}\\
\eta &amp; =\frac{e^{u}-e^{-u}}{2}\\
t\phi &amp; =\frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}\\
e^{u} &amp; =\xi+\eta\\
e^{-u} &amp; =\xi-\eta
\end{align}&lt;/math&gt;

In (1770) he rewrote the addition law for the hyperbolic tangens (f) or (g) as:&lt;ref group=M&gt;Lambert (1770), p. 335&lt;/ref&gt;

:&lt;math&gt;\begin{align}t(y+z) &amp; =(ty+tz):(1+ty\cdot tz) &amp; (f)\\
t(y-z) &amp; =(ty-tz):(1-ty\cdot tz) &amp; (g)
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The hyperbolic relations (a,b,c,d,e,f) are equivalent to the hyperbolic relations on the right of ({{equationNote|3b}}). Relations (f,g) can also be found in ({{equationNote|3e}}). By setting ''tφ=v/c'', formula (c) becomes the relative velocity between two frames, (d) the [[w:Lorentz factor]], (e) the [[w:proper velocity]], (f) or (g) becomes the Lorentz transformation of velocity (or relativistic [[w:velocity addition formula]]) for collinear velocities in [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] and [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']].&lt;/p&gt;

Lambert also formulated the addition laws for the hyperbolic cosine and sine (Lambert's &quot;cos&quot; and &quot;sin&quot; actually mean &quot;cosh&quot; and &quot;sinh&quot;):

:&lt;math&gt;\begin{align}\sin(y+z) &amp; =\sin y\cos z+\cos y\sin z\\
\sin(y-z) &amp; =\sin y\cos z-\cos y\sin z\\
\cos(y+z) &amp; =\cos y\cos z+\sin y\sin z\\
\cos(y-z) &amp; =\cos y\cos z-\sin y\sin z
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{Anchor|Taurinus}} Taurinus (1826) – Hyperbolic law of cosines===

After the addition theorem for the tangens hyperbolicus was given by [[#Lambert|Lambert (1768)]], [[w:hyperbolic geometry]] was used by [[w:Franz Taurinus]] (1826), and later by [[w:Nikolai Lobachevsky]] (1829/30) and others, to formulate the [[w:hyperbolic law of cosines]]:&lt;ref group=M&gt;Taurinus (1826), p. 66; see also p. 272 in the translation by Engel and Stäckel (1899)&lt;/ref&gt;&lt;ref&gt;Bonola (1912), p. 79&lt;/ref&gt;&lt;ref&gt;Gray (1979), p. 242&lt;/ref&gt;

:&lt;math&gt;A=\operatorname{arccos}\frac{\cos\left(\alpha\sqrt{-1}\right)-\cos\left(\beta\sqrt{-1}\right)\cos\left(\gamma\sqrt{-1}\right)}{\sin\left(\beta\sqrt{-1}\right)\sin\left(\gamma\sqrt{-1}\right)}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;When solved for &lt;math&gt;\cos\left(\alpha\sqrt{-1}\right)&lt;/math&gt; it corresponds to the Lorentz transformation in Beltrami coordinates ({{equationNote|3f}}), and by defining the rapidities &lt;math&gt;{\scriptstyle \left(\left[\frac{U}{c},\ \frac{v}{c},\ \frac{u}{c}\right]=\left[\tanh\alpha,\ \tanh\beta,\ \tanh\gamma\right]\right)}&lt;/math&gt; it corresponds to the relativistic velocity addition formula [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']].&lt;/p&gt;

==={{anchor|Beltrami}} Beltrami (1868) – Beltrami coordinates===

[[w:Eugenio Beltrami]] (1868a) introduced coordinates of the [[w:Beltrami–Klein model]] of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:&lt;ref group=M&gt;Beltrami (1868a), pp. 287-288; Note I; Note II&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds^{2}=R^{2}\frac{\left(a^{2}+v^{2}\right)du^{2}-2uv\,du\,dv+\left(a^{2}+v^{2}\right)dv^{2}}{\left(a^{2}+u^{2}+v^{2}\right)^{2}}\\
u^{2}+v^{2}=a^{2}\\
\hline u''=\frac{aa_{0}\left(u'-r_{0}\right)}{a^{2}-r_{0}u'},\ v''=\frac{a_{0}w_{0}v'}{a^{2}-r_{0}u'},\\
\left(r_{0}=\sqrt{u_{0}^{2}+v_{0}^{2}},\ w_{0}=\sqrt{a^{2}-r_{0}^{2}}\right)\\
\hline ds^{2}=R^{2}\frac{\left(a^{2}-v^{2}\right)du^{2}+2uv\,du\,dv+\left(a^{2}-v^{2}\right)dv^{2}}{\left(a^{2}-u^{2}-v^{2}\right)^{2}}\\
(R=R\sqrt{-1},\ a=a\sqrt{-1})
\end{matrix}&lt;/math&gt;

(where the disk radius ''a'' and the [[w:radius of curvature]] ''R'' are real in spherical geometry, in hyperbolic geometry they are imaginary), and for arbitrary dimensions in (1868b)&lt;ref group=M&gt;Beltrami (1868b), pp. 232, 240–241, 253–254&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds=R\frac{\sqrt{dx^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}}}{x}\\
x^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=a^{2}\\
\hline y_{1}=\frac{ab\left(x_{1}-a_{1}\right)}{a^{2}-a_{1}x_{1}}\ \text{or}\ x_{1}=\frac{a\left(ay_{1}+a_{1}b\right)}{ab+a_{1}y_{1}},\ x_{r}=\pm\frac{ay_{r}\sqrt{a^{2}-a_{1}^{2}}}{ab+a_{1}y_{1}}\ (r=2,3,\dots,n)\\
\hline ds=R\frac{\sqrt{dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}-dx^{2}}}{x}\\
x^{2}=a^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\\
\left(R=R\sqrt{-1},\ x=x\sqrt{-1},\ a=a\sqrt{-1}\right)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Setting ''a=a&lt;sub&gt;0&lt;/sub&gt;'' Beltrami's (1868a) formulas become formulas ({{equationNote|3e}}), or in his (1868b) formulas one sets ''a=b'' for arbitrary dimensions.&lt;/p&gt;

=== {{anchor|Laisant2}} Laisant (1874) – Equipollences===
In his French translation of [[w:Giusto Bellavitis]]' principal work on [[w:Equipollence (geometry)|w:equipollences]], [[w:Charles-Ange Laisant]] (1874) added a chapter related to hyperbolas. The equipollence OM and its tangent MT of a hyperbola is defined by Laisant as&lt;ref group=M&gt;Laisant (1874b), pp. 134–135&lt;/ref&gt;

:(1) &lt;math&gt;\begin{matrix} &amp; \mathrm{OM}\bumpeq x\mathrm{OA}+y\mathrm{OB}\\
 &amp; \mathrm{MT}\bumpeq y\mathrm{OA}+x\mathrm{OB}\\
 &amp; \left[x^{2}-y^{2}=1;\ x=\cosh t,\ y=\sinh t\right]\\
\Rightarrow &amp; \mathrm{OM}\bumpeq\cosh t\cdot\mathrm{OA}+\sinh t\cdot\mathrm{OB}
\end{matrix}&lt;/math&gt;

Here, OA and OB are [[w:Conjugate diameters|conjugate semi-diameters]] of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation:

:&lt;math&gt;\begin{matrix}\begin{align}\mathrm{OC} &amp; \bumpeq c\mathrm{OA}+d\mathrm{OB} &amp; \qquad &amp;  &amp; \mathrm{OA} &amp; \bumpeq c\mathrm{OC}-d\mathrm{OD}\\
\mathrm{OD} &amp; \bumpeq d\mathrm{OA}+c\mathrm{OB} &amp;  &amp;  &amp; \mathrm{OB} &amp; \bumpeq-d\mathrm{OC}+c\mathrm{OD}
\end{align}
\\
\left[c^{2}-d^{2}=1\right]
\end{matrix}&lt;/math&gt;

producing the invariant relation

:&lt;math&gt;(\mathrm{OC})^{2}-(\mathrm{OD})^{2}\bumpeq(\mathrm{OA})^{2}-(\mathrm{OB})^{2}&lt;/math&gt;.

Substituting into (1), he showed that OM retains its form

:&lt;math&gt;\begin{matrix}\mathrm{OM}\bumpeq(cx-dy)\mathrm{OC}+(cy-dx)\mathrm{OD}\\
\left[(cx-dy)^{2}-(cy-dx)^{2}=1\right]
\end{matrix}&lt;/math&gt;

He also defined velocity and acceleration by differentiation of (1).

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These relations are equivalent to several Lorentz boosts or hyperbolic rotations producing the invariant Lorentz interval in line with ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Escherich}} Escherich (1874) – Beltrami coordinates===

[[w:Gustav von Escherich]] (1874) discussed the plane of constant negative curvature&lt;ref&gt;Sommerville (1911), p. 297&lt;/ref&gt; based on the [[w:Beltrami–Klein model]] of hyperbolic geometry by [[#Beltrami|Beltrami (1868)]]. Similar to [[w:Christoph Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt;  who introduced axial coordinates ''x''=tan(a) and ''y''=tan(b) in sphere geometry in order to perform coordinate transformations in the case of rotation and translation, Escherich used hyperbolic functions ''x''=tanh(a/k) and ''y''=tanh(b/k)&lt;ref group=M&gt;Escherich (1874), p. 508&lt;/ref&gt; in order to give the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation have the form:&lt;ref group=M name=escher&gt;Escherich (1874), p. 510&lt;/ref&gt;

:&lt;math&gt;x=\frac{\sinh\frac{a}{k}+x'\cosh\frac{a}{k}}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt; and &lt;math&gt;y=\frac{y'}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}), also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting &lt;math&gt;\tfrac{a}{k}=\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt; and multiplying ''[x,y,x′,y′]'' by 1/''c'', and equivalent to Lorentz boost ({{equationNote|3b}}) by setting &lt;math&gt;\scriptstyle (x,\ y,\ x',\ y')=\left(\frac{x_{1}}{x_{0}},\ \frac{x_{2}}{x_{0}},\ \frac{x_{1}^{\prime}}{x_{0}^{\prime}},\ \frac{x_{2}^{\prime}}{x_{0}^{\prime}}\right)&lt;/math&gt;. This is the relation between the [[w:Beltrami–Klein model|Beltrami coordinates]] in terms of Gudermann-Escherich coordinates, and the Weierstrass coordinates of the [[w:hyperboloid model]] introduced by [[../Lorentz transformation (general)#Killing1|E:Killing (1878–1893)]], [[../Lorentz transformation (general)#Poincare|E:Poincaré (1881)]], and [[../Lorentz transformation (general)#Cox|E:Cox (1881)]]. Both coordinate systems were compared by Cox (1881).&lt;ref group=M&gt;Cox (1881), p. 186&lt;/ref&gt;&lt;/p&gt;

==={{anchor|Glaisher}} Glaisher (1878) – hyperbolic addition===

It was shown by [[w:James Whitbread Lee Glaisher]] (1878) that the hyperbolic addition laws can be written as matrix multiplication&lt;ref group=M&gt;Glaisher (1878), p. 30&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\cosh x, &amp; \sinh x\\
\sinh x, &amp; \cosh x
\end{vmatrix}=1,\ \begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y
\end{vmatrix}=1\\
\left[c_{1},c_{2},c_{3},c_{4}\right]=\left[\cosh x,\cosh y,\sinh x,\sinh y\right]\\
\Rightarrow\begin{vmatrix}c_{1}c_{2}+s_{1}s_{2}, &amp; s_{1}c_{2}+c_{1}s_{2}\\
c_{1}s_{2}+s_{1}c_{2}, &amp; s_{1}s_{2}+c_{1}c_{2}
\end{vmatrix}=1\ \Rightarrow\begin{vmatrix}\cosh(x+y), &amp; \sinh(x+y)\\
\sinh(x+y), &amp; \cosh(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{anchor|Gunther1}} Günther (1880/81) – hyperbolic addition ===

Following [[#Glaisher|Glaisher (1878)]], [[w:Siegmund Günther]] (1880/81) formulated the hyperbolic addition laws in matrix form as&lt;ref group=M&gt;Günther (1880/81), p. 405&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\mathfrak{Cos}x, &amp; \mathfrak{Sin}x\\
\mathfrak{Sin}x, &amp; \mathfrak{Cos}x
\end{vmatrix}\cdot\begin{vmatrix}\mathfrak{Cos}y, &amp; \mathfrak{Sin}y\\
\mathfrak{Sin}y, &amp; \mathfrak{Cos}y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}x\mathfrak{Cos}y+\mathfrak{Sin}x\mathfrak{Sin}y, &amp; \mathfrak{Cos}x\mathfrak{Sin}y+\mathfrak{Sin}x\mathfrak{Cos}y\\
\mathfrak{Sin}x\mathfrak{Cos}y+\mathfrak{Cos}x\mathfrak{Sin}y, &amp; \mathfrak{Sin}x\mathfrak{Sin}y+\mathfrak{Cos}x\mathfrak{Cos}y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}(x+y), &amp; \mathfrak{Sin}(x+y)\\
\mathfrak{Sin}(x+y), &amp; \mathfrak{Cos}(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{anchor|Lipschitz1}} Lipschitz (1885/86) ===
[[w:Rudolf Lipschitz]] (1885/86) formulated transformations leaving invariant the sum of squares &lt;math&gt;x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}&lt;/math&gt;, which he rewrote as &lt;math&gt;x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots+x_{n}^{2}-y_{n}^{2}=0&lt;/math&gt;. This led to the problem of finding transformations leaving invariant the pairs &lt;math&gt;x_{a}^{2}-y_{a}^{2}&lt;/math&gt; (''a''=1...n) for which he gave the following solution:&lt;ref group=M&gt;Lipschitz (1886), pp. 90–92&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{a}^{2}-y_{a}^{2}=\mathfrak{x}_{a}^{2}-\mathfrak{y}_{a}^{2}\\
\hline \begin{align}x_{a}-y_{a} &amp; =\left(\mathfrak{x}_{a}-\mathfrak{y}_{a}\right)r_{a}\\
x_{a}+y_{a} &amp; =\left(\mathfrak{x}_{a}+\mathfrak{y}_{a}\right)\frac{1}{r_{a}}
\end{align}
\quad(a)\\
\hline \begin{matrix}\begin{align}2\mathfrak{x}_{a} &amp; =\left(r_{a}+\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}-\frac{1}{r_{a}}\right)y_{a}\\
2\mathfrak{y}_{a} &amp; =\left(r_{a}-\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}+\frac{1}{r_{a}}\right)y_{a}
\end{align}
\quad(b)\end{matrix}\\
\hline \left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}\Rightarrow\begin{align}\mathfrak{x}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}\\
\mathfrak{y}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}
\end{align}
\quad(c)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Equation (a) is identical to Lorentz boost ({{equationNote|3d}}), while (c) is similar, though not identical, to Lorentz boost ({{equationNote|3b}}-C). The difference stems from his definition&lt;br&gt;&lt;math&gt;\qquad\left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}&lt;/math&gt;&lt;br&gt;whereas in accordance to expression &lt;math&gt;\sqrt{\tfrac{1+v}{1-v}}&lt;/math&gt; in ({{equationNote|3d}}) he should have stated &lt;br&gt;&lt;math&gt;\qquad\left\{ \begin{matrix}r_{a}=\frac{\sqrt{1+s_{a}}}{\sqrt{1-s_{a}}}\\
s_{a}&lt;1
\end{matrix}\right\}&lt;/math&gt;.&lt;br&gt;Using the latter choice, equations (c) would assume a form equivalent to ({{equationNote|3b}}):&lt;br&gt;&lt;math&gt;\qquad\begin{align}\mathfrak{x}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{1-s_{a}}\sqrt{1+s_{a}}}\\
\mathfrak{y}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{1-s_{a}}\sqrt{1+s_{a}}}\end{align}&lt;/math&gt;&lt;/p&gt;

==={{Anchor|Schur}} Schur (1885/86, 1900/02) – Beltrami coordinates===

[[w:Friedrich Schur]] (1885/86) discussed spaces of constant Riemann curvature, and by following [[#Beltrami|Beltrami (1868)]] he used the transformation&lt;ref group=M&gt;Schur (1885/86), p. 167&lt;/ref&gt;

:&lt;math&gt;x_{1}=R^{2}\frac{y_{1}+a_{1}}{R^{2}+a_{1}y_{1}},\ x_{2}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{2}}{R^{2}+a_{1}y_{1}},\dots,\ x_{n}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{n}}{R^{2}+a_{1}y_{1}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in arbitrary dimensions by setting ''R=c'' as the speed of light and ''a&lt;sub&gt;1&lt;/sub&gt;=v'' as relative velocity.&lt;/p&gt;

In (1900/02) he derived basic formulas of non-Eucliden geometry, including the case of translation for which he obtained the transformation similar to his previous one:&lt;ref group=M&gt;Schur (1900/02), p. 290; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;x'=\frac{x-a}{1-\mathfrak{k}ax},\quad y'=\frac{y\sqrt{1-\mathfrak{k}a^{2}}}{1-\mathfrak{k}ax}&lt;/math&gt;

where &lt;math&gt;\mathfrak{k}&lt;/math&gt; can have values &gt;0, &lt;0 or ∞.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting ''a=v'' and &lt;math&gt;\mathfrak{k}=\tfrac{1}{c^{2}}&lt;/math&gt;.&lt;/p&gt;

He also defined the triangle&lt;ref group=M&gt;Schur (1900/02), p. 291; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;\frac{1}{\sqrt{1-\mathfrak{k}c^{2}}}=\frac{1}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{1}{\sqrt{1-\mathfrak{k}b^{2}}}-\frac{a}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{b}{\sqrt{1-\mathfrak{k}b^{2}}}\cos\gamma&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to the hyperbolic law of cosines and the relativistic velocity addition ({{equationNote|3f}}, b) or [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']] by setting &lt;math&gt;[\mathfrak{k},\ c,\ a,\ b]=\left[\tfrac{1}{c^{2}},\ \sqrt{u_{x}^{\prime2}+u_{y}^{\prime2}},\ v,\ \sqrt{u_{x}^{2}+u_{y}^{2}}\right]&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Lindemann}} Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute===

[[w:Ferdinand von Lindemann]] discussed hyperbolic geometry in terms of the [[w:Cayley–Klein metric]] in his (1890/91) edition of the lectures on geometry of [[w:Alfred Clebsch]]. Citing [[../Lorentz transformation (general)#Killing|E:Killing (1885)]] and [[../Lorentz transformation (general)#Poincare|Poincaré (1887)]] in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 477–478, 524&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\Omega_{xx}=x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}-4k^{2}dx_{3}^{2}\\
\Omega_{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}
\end{matrix}&lt;/math&gt;

and used the following transformation&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 361–362&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
X_{1}X_{4}+X_{2}X_{3}=\Xi_{1}\Xi_{4}+\Xi_{2}\Xi_{3}\\
\hline \begin{align}X_{1} &amp; =\left(\lambda+\lambda_{1}\right)U_{4} &amp; \Xi_{1} &amp; =\left(\lambda-\lambda_{1}\right)U_{4} &amp; X_{1} &amp; =\frac{\lambda+\lambda_{1}}{\lambda-\lambda_{1}}\Xi_{1}\\
X_{2} &amp; =\left(\lambda+\lambda_{3}\right)U_{4} &amp; \Xi_{2} &amp; =\left(\lambda-\lambda_{3}\right)U_{4} &amp; X_{2} &amp; =\frac{\lambda+\lambda_{3}}{\lambda-\lambda_{3}}\Xi_{2}\\
X_{3} &amp; =\left(\lambda-\lambda_{3}\right)U_{2} &amp; \Xi_{3} &amp; =\left(\lambda+\lambda_{3}\right)U_{2} &amp; X_{3} &amp; =\frac{\lambda-\lambda_{3}}{\lambda+\lambda_{3}}\Xi_{3}\\
X_{4} &amp; =\left(\lambda-\lambda_{1}\right)U_{1} &amp; \Xi_{4} &amp; =\left(\lambda+\lambda_{1}\right)U_{1} &amp; X_{4} &amp; =\frac{\lambda-\lambda_{1}}{\lambda+\lambda_{1}}\Xi_{4}
\end{align}
\end{matrix}&lt;/math&gt;

into which he put&lt;ref group=M name=linde&gt;Lindemann &amp; Clebsch (1890/91), p. 496&lt;/ref&gt;

:&lt;math&gt;\begin{align}X_{1} &amp; =x_{1}+2kx_{4}, &amp; X_{2} &amp; =x_{2}+ix_{3}, &amp; \lambda+\lambda_{1} &amp; =\left(\lambda-\lambda_{1}\right)e^{a},\\
X_{4} &amp; =x_{1}-2kx_{4}, &amp; X_{3} &amp; =x_{2}-ix_{3}, &amp; \lambda+\lambda_{3} &amp; =\left(\lambda-\lambda_{3}\right)e^{\alpha i},
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3d}}) with &lt;math&gt;e^{\alpha i}=1&lt;/math&gt; and ''2k=1'' .&lt;/p&gt;

From that, he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations (''a''=0) or translations (α=0):&lt;ref group=M  name=linde /&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=0\\
\hline \begin{align}x_{2} &amp; =\xi_{2}\cos\alpha+\xi_{3}\sin\alpha, &amp; x_{1} &amp; =\xi_{1}\cos\frac{a}{i}+2ki\xi_{4}\sin\frac{a}{i},\\
x_{3} &amp; =-\xi_{2}\sin\alpha+\xi_{3}\cos\alpha, &amp; 2kx_{4} &amp; =i\xi_{1}\sin\frac{a}{i}+2k\xi_{4}\cos\frac{a}{i}.
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0 and ''2k=1''.&lt;/p&gt;

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane and gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Killing2}} Killing (1893,97) – Weierstrass coordinates===

[[w:Wilhelm Killing]] (1878–1880) gave case of translation in the form&lt;ref group=M name=killtra&gt;Killing (1893), p. 331&lt;/ref&gt;

:&lt;math&gt;y_{0}=x_{0}\operatorname{Ch}a+x_{1}\operatorname{Sh}a,\quad y_{1}=x_{0}\operatorname{Sh}a+x_{1}\operatorname{Ch}a,\quad y_{2}=x_{2}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

In 1898, Killing wrote that relation in a form similar to [[#Escherich|Escherich (1874)]], and derived the corresponding Lorentz transformation for the two cases were ''v'' is unchanged or ''u'' is unchanged:&lt;ref group=M name=kill98&gt;Killing (1898), p. 133&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\xi'=\frac{\xi\operatorname{Ch}\frac{\mu}{l}+l\operatorname{Sh}\frac{\mu}{l}}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}},\ \eta'=\frac{\eta}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}}\\
\hline \frac{u}{p}=\xi,\ \frac{v}{p}=\eta\\
\hline p'=p\operatorname{Ch}\frac{\mu}{l}+\frac{u}{l}\operatorname{Sh}\frac{\mu}{l},\quad u'=pl\operatorname{Sh}\frac{\mu}{l}+u\operatorname{Ch}\frac{\mu}{l},\quad v'=v\\
\text{or}\\
p'=p\operatorname{Ch}\frac{\nu}{l}+\frac{v}{l}\operatorname{Sh}\frac{\nu}{l},\quad u'=u,\quad v'=pl\operatorname{Sh}\frac{\nu}{l}+v\operatorname{Ch}\frac{\nu}{l}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The upper transformation system is equivalent to Lorentz transformation ({{equationNote|3e}}) and the velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] with ''l=c'' and &lt;math&gt;\mu=c\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt;, the system below is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Whitehead}} Whitehead (1897/98) – Universal algebra===

[[w:Alfred North Whitehead]] (1898) discussed the kinematics of hyperbolic space as part of his study of [[w:universal algebra]], and obtained the following transformation:&lt;ref group=M name=white&gt;Whitehead (1898), pp. 459–460&lt;/ref&gt;

:&lt;math&gt;\begin{align}x' &amp; =\left(\eta\cosh\frac{\delta}{\gamma}+\eta_{1}\sinh\frac{\delta}{\gamma}\right)e+\left(\eta\sinh\frac{\delta}{\gamma}+\eta_{1}\cosh\frac{\delta}{\gamma}\right)e_{1}\\
 &amp; \qquad+\left(\eta_{2}\cos\alpha+\eta_{3}\sin\alpha\right)e_{2}+\left(\eta_{3}\cos\alpha-\eta_{2}\sin\alpha\right)e_{3}
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0.&lt;/p&gt;

==={{anchor|Elliott}} Elliott (1903) – Invariant theory ===
[[w:Edwin Bailey Elliott]] (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:&lt;ref group=M&gt;Elliott (1903), p. 109&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=X\cosh\phi+Y\sinh\phi,\quad y=X\sinh\phi+Y\cosh\phi\\
\hline X+Y=e^{-\phi}(x+y),\quad X-Y=e^{\phi}(x-y)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}). The mentioned subgroup corresponds to the one-parameter subgroup generated by Lorentz boosts.&lt;/p&gt;

==={{anchor|Woods2}} Woods (1903) – Weierstrass coordinates ===
[[w:Frederick S. Woods]] (1903, published 1905) gave the case of translation in hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 55&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x_{1}\cos kl+x_{0}\frac{\sin kl}{k},\quad x_{2}^{\prime}=x_{2},\quad x_{2}^{\prime}=x_{3},\quad x_{0}^{\prime}=-x_{1}k\sin kl+x_{0}\cos kl&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.&lt;/p&gt;

and the loxodromic substitution for hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 72&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}x_{1}^{\prime} &amp; =x_{1}\cosh\alpha-x_{0}\sinh\alpha\\
x_{2}^{\prime} &amp; =x_{2}\cos\beta-x_{3}\sin\beta\\
x_{3}^{\prime} &amp; =x_{2}\sin\beta+x_{3}\cos\beta\\
x_{0}^{\prime} &amp; =-x_{1}\sinh\alpha+x_{0}\cosh\alpha
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with β=0.&lt;/p&gt;

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – gave the case of translation in the hyperbolic plane:&lt;ref group=M name=lieb&gt;Liebmann (1904/05), p. 174&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x'\operatorname{ch}a+p'\operatorname{sh}a,\quad y_{1}^{\prime}=y',\quad p_{1}^{\prime}=x'\operatorname{sh}a+p'\operatorname{ch}a&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Frank}} Frank (1909) – Special relativity===
In special relativity, hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity:&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;
 	
=== {{anchor|Herglotz1}} Herglotz (1909/10) – Special relativity===

In special relativity, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic, with the hyperbolic case being:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;
	
:&lt;math&gt;\begin{matrix}Z=Z'e^{\vartheta}\\
	
\begin{aligned}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
	
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
	
\end{aligned}
	
\end{matrix}&lt;/math&gt;	
	
&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3d}}).&lt;/p&gt;

==={{anchor|Varicak}} Varićak (1910) – Special relativity===

In special relativity, hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

He showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bel68sag}} 
*{{#section:History of Topics in Special Relativity/mathsource|bel68fond}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|eli03}} 
*{{#section:History of Topics in Special Relativity/mathsource|esch74}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul35}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul48a}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|glai78}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|guen80}} 
*{{#section:History of Topics in Special Relativity/mathsource|kep09}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lais74b}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam67}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lind90}}
*{{#section:History of Topics in Special Relativity/mathsource|lip86}}  
*{{#section:History of Topics in Special Relativity/mathsource|ric57}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu85}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu00}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu09}} 
*{{#section:History of Topics in Special Relativity/mathsource|tau26}} 
*{{#section:History of Topics in Special Relativity/mathsource|whit98}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L3}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
      <sha1>a5wd7usksbuy4jxpm9o9wzp06cgzbcw</sha1>
    </revision>
    <revision>
      <id>2411825</id>
      <parentid>2411819</parentid>
      <timestamp>2022-08-04T12:58:46Z</timestamp>
      <contributor>
        <username>D.H</username>
        <id>52339</id>
      </contributor>
      <comment>/* Hyperbolic law of cosines */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="47279" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via hyperbolic functions==
===Translation in the hyperbolic plane===
The case of a Lorentz transformation without spatial rotation is called a [[w:Lorentz boost]]. The simplest case can be given, for instance, by setting ''n=1'' in the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}\\
\\
x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}
\end{align}
\left|\begin{align}g_{01}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{10}^{2} &amp; =1\\
g_{01}g_{11}-g_{00}g_{10} &amp; =0\\
g_{10}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{01}^{2} &amp; =1\\
g_{10}g_{11}-g_{00}g_{01} &amp; =0
\end{align}
\rightarrow\begin{align}g_{00}^{2} &amp; =g_{11}^{2}\\
g_{01}^{2} &amp; =g_{10}^{2}
\end{align}
\right.
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}g_{00} &amp; -g_{10}\\
-g_{01} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\right|\det\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;|{{equationRef|3a}}}}

which resembles precisely the relations of [[w:hyperbolic function]]s in terms of [[w:hyperbolic angle]] &lt;math&gt;\eta&lt;/math&gt;. Thus a Lorentz boost or [[w:hyperbolic rotation]] (being the same as a rotation around an imaginary angle &lt;math&gt;i\eta=\phi&lt;/math&gt; in [[../Lorentz transformation (imaginary)#math_2b|E:'''(2b)''']] or a [[w:Translation (geometry)|translation]] in the hyperbolic plane in terms of the hyperboloid model) is given by

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline g_{00}=g_{11}=\cosh\eta,\ g_{01}=g_{10}=-\sinh\eta\\
\hline \left.\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta &amp;  &amp; =\frac{x_{0}-x_{1}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}-x_{1}v}{\sqrt{1-v^{2}}}\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta &amp;  &amp; =\frac{x_{1}-x_{0}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}-x_{0}v}{\sqrt{1-v^{2}}}\\
x_{2}^{\prime} &amp; =x_{2} &amp;  &amp; =x_{2} &amp;  &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta &amp;  &amp; =\frac{x_{0}^{\prime}-x_{1}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}^{\prime}-x_{1}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta &amp;  &amp; =\frac{x_{1}^{\prime}-x_{0}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}^{\prime}-x_{0}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{2} &amp; =x_{2}^{\prime} &amp;  &amp; =x_{2}^{\prime} &amp;  &amp; =x_{2}^{\prime}
\end{align}
\right|{\scriptstyle \begin{align}\sinh^{2}\eta-\cosh^{2}\eta &amp; =-1 &amp; (a)\\
\cosh^{2}\eta-\sinh^{2}\eta &amp; =1 &amp; (b)\\
\frac{\sinh\eta}{\cosh\eta} &amp; =\tanh\eta=v &amp; (c)\\
\frac{1}{\sqrt{1-\tanh^{2}\eta}} &amp; =\cosh\eta &amp; (d)\\
\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\sinh\eta &amp; (e)\\
\frac{\tanh q\pm\tanh\eta}{1\pm\tanh q\tanh\eta} &amp; =\tanh\left(q\pm\eta\right) &amp; (f)
\end{align}
}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\right|\det\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;|{{equationRef|3b}}}}

Hyperbolic identities (a,b) on the right of ({{equationNote|3b}}) were given by [[#Riccati|Riccati (1757)]], all identities (a,b,c,d,e,f) by [[#Lambert|Lambert (1768–1770)]]. Lorentz transformations ({{equationNote|3b}}-A) were given by [[#Laisant|Laisant (1874)]], [[#Cox|Cox (1882)]], [[#Lindemann|Lindemann (1890/91)]], [[#Gerard|Gérard (1892)]], [[#Killing2|Killing (1893, 1897/98)]], [[#Whitehead|Whitehead (1897/98)]], [[#Woods2|Woods (1903/05)]], [[#Elliott|Elliott (1903)]] and [[#Liebmann|Liebmann (1904/05)]] in terms of Weierstrass coordinates of the [[w:hyperboloid model]], while transformations similar to ({{equationNote|3b}}-C) have been used by [[#Lipschitz1|Lipschitz (1885/86)]].

Rapidity can be composed of arbitrary many rapidities &lt;math&gt;\eta_{1},\eta_{2}\dots&lt;/math&gt; as per the [[w:Hyperbolic functions#Sums of arguments|w:angle sum laws of hyperbolic sines and cosines]], so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between [[w:List of trigonometric identities#Angle sum and difference identities|w:angle sum laws of circular trigonometry]] and spatial rotations. Alternatively, the hyperbolic angle sum laws ''themselves'' can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the [[w:unit hyperbola]]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \left[\eta=\eta_{2}-\eta_{1}\right]\\
\begin{align}x_{0}^{\prime} &amp; =\sinh\eta_{1} &amp;  &amp; =\sinh\left(\eta_{2}-\eta\right) &amp;  &amp; =\sinh\eta_{2}\cosh\eta-\cosh\eta_{2}\sinh\eta &amp;  &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =\cosh\eta_{1} &amp;  &amp; =\cosh\left(\eta_{2}-\eta\right) &amp;  &amp; =-\sinh\eta_{2}\sinh\eta+\cosh\eta_{2}\cosh\eta &amp;  &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta\\
\\
x_{0} &amp; =\sinh\eta_{2} &amp;  &amp; =\sinh\left(\eta_{1}+\eta\right) &amp;  &amp; =\sinh\eta_{1}\cosh\eta+\cosh\eta_{1}\sinh\eta &amp;  &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =\cosh\eta_{2} &amp;  &amp; =\cosh\left(\eta_{1}+\eta\right) &amp;  &amp; =\sinh\eta_{1}\sinh\eta+\cosh\eta_{1}\cosh\eta &amp;  &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \begin{align} &amp; \begin{bmatrix}x_{0}^{\prime}\\
x_{1}^{\prime}
\end{bmatrix}=\begin{bmatrix}\sinh\eta_{1}\\
\cosh\eta_{1}
\end{bmatrix}=\begin{bmatrix}\sinh\left(\eta_{2}-\eta\right)\\
\cosh\left(\eta_{2}-\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}\sinh\eta_{2}\\
\cosh\eta_{2}
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}x_{0}\\
x_{1}
\end{bmatrix}\\
 &amp; \begin{bmatrix}x_{0}\\
x_{1}
\end{bmatrix}=\begin{bmatrix}\sinh\eta_{2}\\
\cosh\eta_{2}
\end{bmatrix}=\begin{bmatrix}\sinh\left(\eta_{1}+\eta\right)\\
\cosh\left(\eta_{1}+\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}\sinh\eta_{1}\\
\cosh\eta_{1}
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}x_{0}^{\prime}\\
x_{1}^{\prime}
\end{bmatrix}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|3c}}}}

Hyperbolic angle sum laws equivalent to Lorentz boost ({{equationNote|3c}}) were given by [[#Riccati|Riccati (1757)]] and [[#Lambert|Lambert (1768–1770)]], while the matrix representation was given by [[#Glaisher|Glaisher (1878)]] and [[#Gunther2|Günther (1880/81)]]. In special relativity, hyperbolic functions were used by [[#Frank|Frank (1909)]] and [[#Varicak|Varićak (1910)]].

Finally, because of &lt;math&gt;\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;, Lorentz boost ({{equationNote|3b}}) assumes a simple form by using [[w:squeeze mapping]]s in analogy to Euler's formula in [[../Lorentz transformation (imaginary)#math_2c|E:'''(2c)''']]:&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \begin{matrix}x_{1}^{\prime}-x_{0}^{\prime}=k\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime}=\frac{1}{k}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime}=x_{2}
\end{matrix}\\
\hline \begin{align}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}-x_{0} &amp; =e^{-\eta}\left(x_{1}^{\prime}-x_{0}^{\prime}\right)\\
x_{1}+x_{0} &amp; =e^{\eta}\left(x_{1}^{\prime}+x_{0}^{\prime}\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\\
\hline k=e^{\eta}=\cosh\eta+\sinh\eta=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}=\sqrt{\frac{1+v}{1-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|3d}}}} 

Lorentz transformations ({{equationNote|3d}}) for arbitrary ''k'' were given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]), while the above exponential form was explicitly used by [[#Lindemann|Lindemann (1890/91)]], [[#Elliott|Elliott (1903)]], [[#Herglotz1|Herglotz (1909)]]. 

===Hyperbolic law of cosines===
In line with equation [[../Lorentz transformation (general)#math_1b|E:'''(1b)''']] one can use coordinates &lt;math&gt;[u_{1},\ u_{2}]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}}\right]&lt;/math&gt; inside the [[w:unit circle]] &lt;math&gt;u_{1}^{2}+u_{2}^{2}=1&lt;/math&gt;, thus the corresponding Lorentz transformations ({{equationNote|3b}}) obtain the form:

{{NumBlk|:|&lt;math&gt;\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
u_{1}^{\prime} &amp; =\frac{-\sinh\eta+u_{1}\cosh\eta}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{1}-\tanh\eta}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{1}-v}{1-u_{1}v}\\
u_{2}^{\prime} &amp; =\frac{u_{2}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-v^{2}}}{1-u_{1}v}\\
\\
u_{1} &amp; =\frac{\sinh\eta+u_{1}^{\prime}\cosh\eta}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+\tanh\eta}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+v}{1+u_{1}^{\prime}v}\\
u_{2} &amp; =\frac{u_{2}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}
\end{align}
&lt;/math&gt;|{{equationRef|3e}}}}

Transformations (A) were given by [[#Escherich|Escherich (1874)]] and [[#Killing2|Killing (1898)]], and transformations (B) by [[#Beltrami|Beltrami (1868)]] and [[#Schur|Schur (1885/86, 1900/02)]] in terms of [[w:Beltrami–Klein model|Beltrami coordinates]]&lt;ref&gt;Rosenfeld (1988), p. 231&lt;/ref&gt; of hyperbolic geometry. By using the scalar product of &lt;math&gt;\left[u_{1},u_{2}\right]&lt;/math&gt;, the resulting Lorentz transformation can be seen as equivalent to the [[w:hyperbolic law of cosines]]:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{matrix}u^{2}=u_{1}^{2}+u_{2}^{2}\\
u'^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}
\end{matrix}\left|\begin{matrix}u_{1}=u\cos\alpha\\
u_{2}=u\sin\alpha\\
\\
u_{1}^{\prime}=u'\cos\alpha'\\
u_{2}^{\prime}=u'\sin\alpha'
\end{matrix}\right|\begin{align}u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+vu'\cos\alpha'}, &amp; u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-vu\cos\alpha}\\
u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{1+vu'\cos\alpha'}, &amp; u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{1-vu\cos\alpha}\\
\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{u'\cos\alpha'+v}, &amp; \tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{u\cos\alpha-v}
\end{align}
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(vu'\sin\alpha'\right){}^{2}}}{1+vu'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(vu\sin\alpha\right){}^{2}}}{1-vu\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-u^{\prime2}}}=\frac{1}{\sqrt{1-v^{2}}}\frac{1}{\sqrt{1-u^{2}}}-\frac{v}{\sqrt{1-v^{2}}}\frac{u}{\sqrt{1-u^{2}}}\cos\alpha &amp; (b)\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha &amp; (a)
\end{matrix}&lt;/math&gt;|{{equationRef|3f}}}}

The hyperbolic law of cosines (a) was given by [[#Taurinus|Taurinus (1826) and Lobachevsky (1829/30)]] and others, while variant (b) was given by [[#Schur|Schur (1900/02)]]. By further setting ''u=u′'' it follows:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\cos\alpha=\frac{\cos\alpha'+v}{1+v\cos\alpha'},\ \sin\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{1+v\cos\alpha'},\ \tan\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{\cos\alpha'+v},\ \tan\frac{\alpha}{2}=\sqrt{\frac{1-v}{1+v}}\tan\frac{\alpha'}{2}\\
\cos\alpha'=\frac{\cos\alpha-v}{1-v\cos\alpha},\ \sin\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{1-v\cos\alpha},\ \tan\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{\cos\alpha-v},\ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+v}{1-v}}\tan\frac{\alpha}{2}
\end{matrix}&lt;/math&gt;|{{equationRef|3g}}}}

Formulas (3b) resemble the equations of an [[w:ellipse]] of [[w:Orbital eccentricity|eccentricity]] ''v/c'', [[w:eccentric anomaly]] α' and [[w:true anomaly]] α, first geometrically formulated by [[#Euler|Kepler (1609)]] and explicitly written down by [[#Euler|Euler (1735, 1748), Lagrange (1770)]] and many others in relation to planetary motions. They were also used by [[../Lorentz transformation (conformal)#Darboux|E:Darboux (1873)]] as a sphere transformation. In special relativity, these formulas describes the aberration of light, see [[../Lorentz transformation (velocity)#Velocity addition and aberration|E:velocity addition and aberration]].

==Historical notation==
==={{anchor|Euler}} Euler (1735) – True and eccentric anomaly===

[[w:Johannes Kepler]] (1609) geometrically formulated [[w:Kepler's equation]] and the relations between the [[w:mean anomaly]], [[w:true anomaly]], and [[w:eccentric anomaly]].&lt;ref group=M&gt;Kepler (1609), chapter 60. The editors of Kepler's collected papers remark (p. 482), that Kepler's relations correspond to &lt;math&gt;{\scriptstyle \alpha=\beta+e\sin\beta}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\nu=\frac{e+\cos\beta}{1+e\cos\beta}}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\beta=\frac{\cos\nu-e}{1-e\cos\nu}}&lt;/math&gt;&lt;/ref&gt;&lt;ref&gt;Volk (1976), p. 366&lt;/ref&gt; The relation between the true anomaly ''z'' and the eccentric anomaly ''P'' was algebraically expressed by [[w:Leonhard Euler]] (1735/40) as follows:&lt;ref group=M&gt;Euler (1735/40), § 19&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{\cos P+v}{1+v\cos P},\ \cos P=\frac{\cos z-v}{1-v\cos z},\ \int P=\frac{\int z\sqrt{1-v^{2}}}{1-v\cos z}&lt;/math&gt;

and in 1748:&lt;ref group=M&gt;Euler (1748a), section VIII&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{n+\cos y}{1+n\cos y},\ \sin z=\frac{\sin y\sqrt{1-n^{2}}}{1+n\cos y},\ \tan z=\frac{\sin y\sqrt{1-n^{2}}}{n+\cos y}&lt;/math&gt;

while [[w:Joseph-Louis Lagrange]] (1770/71) expressed them as follows&lt;ref group=M&gt;Lagrange (1770/71), section I&lt;/ref&gt;

:&lt;math&gt;\sin u=\frac{m\sin x}{1+n\cos x},\ \cos u=\frac{n+\cos x}{1+n\cos x},\ \operatorname{tang}\frac{1}{2}u=\frac{m}{1+n}\operatorname{tang}\frac{1}{2}x,\ \left(m^{2}=1-n^{2}\right)&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt; These relations resemble formulas ({{equationNote|3g}}), while ({{equationNote|3f}}) follows by setting &lt;math&gt;[\cos z,\sin z]=\left[u_{x},u_{y}\right]&lt;/math&gt; in Euler's formulas or &lt;math&gt;[\cos u,\sin u]=\left[u_{x},u_{y}\right]&lt;/math&gt; in Lagrange's formulas.&lt;/p&gt;

==={{anchor|Riccati}} Riccati (1757) – hyperbolic addition===

[[w:Vincenzo Riccati]] introduced hyperbolic functions in 1757,&lt;ref group=M&gt;Riccati (1757), p. 71&lt;/ref&gt;&lt;ref group=M&gt;Günther (1880/81), pp. 7–13&lt;/ref&gt; in particular he formulated the angle sum laws for hyperbolic sine and cosine:

:&lt;math&gt;\begin{matrix}\mathrm{Ch}(\varphi+\pi)=\frac{\mathrm{Ch}\varphi\mathrm{Ch}\pi+\mathrm{Sh}\varphi\mathrm{Sh}\pi}{r}\\
\mathrm{Sh}(\varphi+\pi)=\frac{\mathrm{Ch}\varphi\mathrm{Sh}\pi+\mathrm{Ch}\pi\mathrm{Sh}\varphi}{r}\\
\left[\mathrm{Ch}^{2}-\mathrm{Sh}^{2}=rr\right]
\end{matrix}&lt;/math&gt;

He furthermore showed that &lt;math&gt;\mathrm{Ch}(\varphi-\pi)&lt;/math&gt; and &lt;math&gt;\mathrm{Sh}(\varphi-\pi)&lt;/math&gt; follow by setting &lt;math&gt;\mathrm{Ch}(\pi)\Rightarrow\mathrm{Ch}(-\pi)&lt;/math&gt; and &lt;math&gt;\mathrm{Sh}(\pi)\Rightarrow\mathrm{Sh}(-\pi)&lt;/math&gt; in the above formulas.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3c}}). (In modern publications, Riccati's additional factor ''r'' is set to unity.)&lt;/p&gt;

==={{anchor|Lambert}} Lambert (1768–1770) – hyperbolic addition===

While [[#Riccati|Riccati (1757)]] discussed the hyperbolic sine and cosine, [[w:Johann Heinrich Lambert]] (read 1767, published 1768) introduced the expression ''tang φ'' or abbreviated ''tφ'' as the [[w:tangens hyperbolicus]] &lt;math&gt;{\scriptstyle \frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}}&lt;/math&gt; of a variable ''u'', or in modern notation ''tφ=tanh(u)'':&lt;ref group=M&gt;Lambert (1761/68), pp. 309–318&lt;/ref&gt;&lt;ref&gt;Barnett (2004), pp. 22–23&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}\xi\xi-1 &amp; =\eta\eta &amp; (a)\\
1+\eta\eta &amp; =\xi\xi &amp; (b)\\
\frac{\eta}{\xi} &amp; =tang\ \phi=t\phi &amp; (c)\\
\xi &amp; =\frac{1}{\sqrt{1-t\phi^{2}}} &amp; (d)\\
\eta &amp; =\frac{t\phi}{\sqrt{1-t\phi^{2}}} &amp; (e)\\
t\phi'' &amp; =\frac{t\phi+t\phi'}{1+t\phi\cdot t\phi'} &amp; (f)\\
t\phi' &amp; =\frac{t\phi''-t\phi}{1-t\phi\cdot t\phi''} &amp; (g)
\end{align}
\right|\begin{align}2u &amp; =\log\frac{1+t\phi}{1-t\phi}\\
\xi &amp; =\frac{e^{u}+e^{-u}}{2}\\
\eta &amp; =\frac{e^{u}-e^{-u}}{2}\\
t\phi &amp; =\frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}\\
e^{u} &amp; =\xi+\eta\\
e^{-u} &amp; =\xi-\eta
\end{align}&lt;/math&gt;

In (1770) he rewrote the addition law for the hyperbolic tangens (f) or (g) as:&lt;ref group=M&gt;Lambert (1770), p. 335&lt;/ref&gt;

:&lt;math&gt;\begin{align}t(y+z) &amp; =(ty+tz):(1+ty\cdot tz) &amp; (f)\\
t(y-z) &amp; =(ty-tz):(1-ty\cdot tz) &amp; (g)
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The hyperbolic relations (a,b,c,d,e,f) are equivalent to the hyperbolic relations on the right of ({{equationNote|3b}}). Relations (f,g) can also be found in ({{equationNote|3e}}). By setting ''tφ=v/c'', formula (c) becomes the relative velocity between two frames, (d) the [[w:Lorentz factor]], (e) the [[w:proper velocity]], (f) or (g) becomes the Lorentz transformation of velocity (or relativistic [[w:velocity addition formula]]) for collinear velocities in [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] and [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']].&lt;/p&gt;

Lambert also formulated the addition laws for the hyperbolic cosine and sine (Lambert's &quot;cos&quot; and &quot;sin&quot; actually mean &quot;cosh&quot; and &quot;sinh&quot;):

:&lt;math&gt;\begin{align}\sin(y+z) &amp; =\sin y\cos z+\cos y\sin z\\
\sin(y-z) &amp; =\sin y\cos z-\cos y\sin z\\
\cos(y+z) &amp; =\cos y\cos z+\sin y\sin z\\
\cos(y-z) &amp; =\cos y\cos z-\sin y\sin z
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{Anchor|Taurinus}} Taurinus (1826) – Hyperbolic law of cosines===

After the addition theorem for the tangens hyperbolicus was given by [[#Lambert|Lambert (1768)]], [[w:hyperbolic geometry]] was used by [[w:Franz Taurinus]] (1826), and later by [[w:Nikolai Lobachevsky]] (1829/30) and others, to formulate the [[w:hyperbolic law of cosines]]:&lt;ref group=M&gt;Taurinus (1826), p. 66; see also p. 272 in the translation by Engel and Stäckel (1899)&lt;/ref&gt;&lt;ref&gt;Bonola (1912), p. 79&lt;/ref&gt;&lt;ref&gt;Gray (1979), p. 242&lt;/ref&gt;

:&lt;math&gt;A=\operatorname{arccos}\frac{\cos\left(\alpha\sqrt{-1}\right)-\cos\left(\beta\sqrt{-1}\right)\cos\left(\gamma\sqrt{-1}\right)}{\sin\left(\beta\sqrt{-1}\right)\sin\left(\gamma\sqrt{-1}\right)}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;When solved for &lt;math&gt;\cos\left(\alpha\sqrt{-1}\right)&lt;/math&gt; it corresponds to the Lorentz transformation in Beltrami coordinates ({{equationNote|3f}}), and by defining the rapidities &lt;math&gt;{\scriptstyle \left(\left[\frac{U}{c},\ \frac{v}{c},\ \frac{u}{c}\right]=\left[\tanh\alpha,\ \tanh\beta,\ \tanh\gamma\right]\right)}&lt;/math&gt; it corresponds to the relativistic velocity addition formula [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']].&lt;/p&gt;

==={{anchor|Beltrami}} Beltrami (1868) – Beltrami coordinates===

[[w:Eugenio Beltrami]] (1868a) introduced coordinates of the [[w:Beltrami–Klein model]] of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:&lt;ref group=M&gt;Beltrami (1868a), pp. 287-288; Note I; Note II&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds^{2}=R^{2}\frac{\left(a^{2}+v^{2}\right)du^{2}-2uv\,du\,dv+\left(a^{2}+v^{2}\right)dv^{2}}{\left(a^{2}+u^{2}+v^{2}\right)^{2}}\\
u^{2}+v^{2}=a^{2}\\
\hline u''=\frac{aa_{0}\left(u'-r_{0}\right)}{a^{2}-r_{0}u'},\ v''=\frac{a_{0}w_{0}v'}{a^{2}-r_{0}u'},\\
\left(r_{0}=\sqrt{u_{0}^{2}+v_{0}^{2}},\ w_{0}=\sqrt{a^{2}-r_{0}^{2}}\right)\\
\hline ds^{2}=R^{2}\frac{\left(a^{2}-v^{2}\right)du^{2}+2uv\,du\,dv+\left(a^{2}-v^{2}\right)dv^{2}}{\left(a^{2}-u^{2}-v^{2}\right)^{2}}\\
(R=R\sqrt{-1},\ a=a\sqrt{-1})
\end{matrix}&lt;/math&gt;

(where the disk radius ''a'' and the [[w:radius of curvature]] ''R'' are real in spherical geometry, in hyperbolic geometry they are imaginary), and for arbitrary dimensions in (1868b)&lt;ref group=M&gt;Beltrami (1868b), pp. 232, 240–241, 253–254&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds=R\frac{\sqrt{dx^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}}}{x}\\
x^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=a^{2}\\
\hline y_{1}=\frac{ab\left(x_{1}-a_{1}\right)}{a^{2}-a_{1}x_{1}}\ \text{or}\ x_{1}=\frac{a\left(ay_{1}+a_{1}b\right)}{ab+a_{1}y_{1}},\ x_{r}=\pm\frac{ay_{r}\sqrt{a^{2}-a_{1}^{2}}}{ab+a_{1}y_{1}}\ (r=2,3,\dots,n)\\
\hline ds=R\frac{\sqrt{dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}-dx^{2}}}{x}\\
x^{2}=a^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\\
\left(R=R\sqrt{-1},\ x=x\sqrt{-1},\ a=a\sqrt{-1}\right)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Setting ''a=a&lt;sub&gt;0&lt;/sub&gt;'' Beltrami's (1868a) formulas become formulas ({{equationNote|3e}}), or in his (1868b) formulas one sets ''a=b'' for arbitrary dimensions.&lt;/p&gt;

=== {{anchor|Laisant2}} Laisant (1874) – Equipollences===
In his French translation of [[w:Giusto Bellavitis]]' principal work on [[w:Equipollence (geometry)|w:equipollences]], [[w:Charles-Ange Laisant]] (1874) added a chapter related to hyperbolas. The equipollence OM and its tangent MT of a hyperbola is defined by Laisant as&lt;ref group=M&gt;Laisant (1874b), pp. 134–135&lt;/ref&gt;

:(1) &lt;math&gt;\begin{matrix} &amp; \mathrm{OM}\bumpeq x\mathrm{OA}+y\mathrm{OB}\\
 &amp; \mathrm{MT}\bumpeq y\mathrm{OA}+x\mathrm{OB}\\
 &amp; \left[x^{2}-y^{2}=1;\ x=\cosh t,\ y=\sinh t\right]\\
\Rightarrow &amp; \mathrm{OM}\bumpeq\cosh t\cdot\mathrm{OA}+\sinh t\cdot\mathrm{OB}
\end{matrix}&lt;/math&gt;

Here, OA and OB are [[w:Conjugate diameters|conjugate semi-diameters]] of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation:

:&lt;math&gt;\begin{matrix}\begin{align}\mathrm{OC} &amp; \bumpeq c\mathrm{OA}+d\mathrm{OB} &amp; \qquad &amp;  &amp; \mathrm{OA} &amp; \bumpeq c\mathrm{OC}-d\mathrm{OD}\\
\mathrm{OD} &amp; \bumpeq d\mathrm{OA}+c\mathrm{OB} &amp;  &amp;  &amp; \mathrm{OB} &amp; \bumpeq-d\mathrm{OC}+c\mathrm{OD}
\end{align}
\\
\left[c^{2}-d^{2}=1\right]
\end{matrix}&lt;/math&gt;

producing the invariant relation

:&lt;math&gt;(\mathrm{OC})^{2}-(\mathrm{OD})^{2}\bumpeq(\mathrm{OA})^{2}-(\mathrm{OB})^{2}&lt;/math&gt;.

Substituting into (1), he showed that OM retains its form

:&lt;math&gt;\begin{matrix}\mathrm{OM}\bumpeq(cx-dy)\mathrm{OC}+(cy-dx)\mathrm{OD}\\
\left[(cx-dy)^{2}-(cy-dx)^{2}=1\right]
\end{matrix}&lt;/math&gt;

He also defined velocity and acceleration by differentiation of (1).

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These relations are equivalent to several Lorentz boosts or hyperbolic rotations producing the invariant Lorentz interval in line with ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Escherich}} Escherich (1874) – Beltrami coordinates===

[[w:Gustav von Escherich]] (1874) discussed the plane of constant negative curvature&lt;ref&gt;Sommerville (1911), p. 297&lt;/ref&gt; based on the [[w:Beltrami–Klein model]] of hyperbolic geometry by [[#Beltrami|Beltrami (1868)]]. Similar to [[w:Christoph Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt;  who introduced axial coordinates ''x''=tan(a) and ''y''=tan(b) in sphere geometry in order to perform coordinate transformations in the case of rotation and translation, Escherich used hyperbolic functions ''x''=tanh(a/k) and ''y''=tanh(b/k)&lt;ref group=M&gt;Escherich (1874), p. 508&lt;/ref&gt; in order to give the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation have the form:&lt;ref group=M name=escher&gt;Escherich (1874), p. 510&lt;/ref&gt;

:&lt;math&gt;x=\frac{\sinh\frac{a}{k}+x'\cosh\frac{a}{k}}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt; and &lt;math&gt;y=\frac{y'}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}), also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting &lt;math&gt;\tfrac{a}{k}=\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt; and multiplying ''[x,y,x′,y′]'' by 1/''c'', and equivalent to Lorentz boost ({{equationNote|3b}}) by setting &lt;math&gt;\scriptstyle (x,\ y,\ x',\ y')=\left(\frac{x_{1}}{x_{0}},\ \frac{x_{2}}{x_{0}},\ \frac{x_{1}^{\prime}}{x_{0}^{\prime}},\ \frac{x_{2}^{\prime}}{x_{0}^{\prime}}\right)&lt;/math&gt;. This is the relation between the [[w:Beltrami–Klein model|Beltrami coordinates]] in terms of Gudermann-Escherich coordinates, and the Weierstrass coordinates of the [[w:hyperboloid model]] introduced by [[../Lorentz transformation (general)#Killing1|E:Killing (1878–1893)]], [[../Lorentz transformation (general)#Poincare|E:Poincaré (1881)]], and [[../Lorentz transformation (general)#Cox|E:Cox (1881)]]. Both coordinate systems were compared by Cox (1881).&lt;ref group=M&gt;Cox (1881), p. 186&lt;/ref&gt;&lt;/p&gt;

==={{anchor|Glaisher}} Glaisher (1878) – hyperbolic addition===

It was shown by [[w:James Whitbread Lee Glaisher]] (1878) that the hyperbolic addition laws can be written as matrix multiplication&lt;ref group=M&gt;Glaisher (1878), p. 30&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\cosh x, &amp; \sinh x\\
\sinh x, &amp; \cosh x
\end{vmatrix}=1,\ \begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y
\end{vmatrix}=1\\
\left[c_{1},c_{2},c_{3},c_{4}\right]=\left[\cosh x,\cosh y,\sinh x,\sinh y\right]\\
\Rightarrow\begin{vmatrix}c_{1}c_{2}+s_{1}s_{2}, &amp; s_{1}c_{2}+c_{1}s_{2}\\
c_{1}s_{2}+s_{1}c_{2}, &amp; s_{1}s_{2}+c_{1}c_{2}
\end{vmatrix}=1\ \Rightarrow\begin{vmatrix}\cosh(x+y), &amp; \sinh(x+y)\\
\sinh(x+y), &amp; \cosh(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{anchor|Gunther1}} Günther (1880/81) – hyperbolic addition ===

Following [[#Glaisher|Glaisher (1878)]], [[w:Siegmund Günther]] (1880/81) formulated the hyperbolic addition laws in matrix form as&lt;ref group=M&gt;Günther (1880/81), p. 405&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\mathfrak{Cos}x, &amp; \mathfrak{Sin}x\\
\mathfrak{Sin}x, &amp; \mathfrak{Cos}x
\end{vmatrix}\cdot\begin{vmatrix}\mathfrak{Cos}y, &amp; \mathfrak{Sin}y\\
\mathfrak{Sin}y, &amp; \mathfrak{Cos}y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}x\mathfrak{Cos}y+\mathfrak{Sin}x\mathfrak{Sin}y, &amp; \mathfrak{Cos}x\mathfrak{Sin}y+\mathfrak{Sin}x\mathfrak{Cos}y\\
\mathfrak{Sin}x\mathfrak{Cos}y+\mathfrak{Cos}x\mathfrak{Sin}y, &amp; \mathfrak{Sin}x\mathfrak{Sin}y+\mathfrak{Cos}x\mathfrak{Cos}y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}(x+y), &amp; \mathfrak{Sin}(x+y)\\
\mathfrak{Sin}(x+y), &amp; \mathfrak{Cos}(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{anchor|Lipschitz1}} Lipschitz (1885/86) ===
[[w:Rudolf Lipschitz]] (1885/86) formulated transformations leaving invariant the sum of squares &lt;math&gt;x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}&lt;/math&gt;, which he rewrote as &lt;math&gt;x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots+x_{n}^{2}-y_{n}^{2}=0&lt;/math&gt;. This led to the problem of finding transformations leaving invariant the pairs &lt;math&gt;x_{a}^{2}-y_{a}^{2}&lt;/math&gt; (''a''=1...n) for which he gave the following solution:&lt;ref group=M&gt;Lipschitz (1886), pp. 90–92&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{a}^{2}-y_{a}^{2}=\mathfrak{x}_{a}^{2}-\mathfrak{y}_{a}^{2}\\
\hline \begin{align}x_{a}-y_{a} &amp; =\left(\mathfrak{x}_{a}-\mathfrak{y}_{a}\right)r_{a}\\
x_{a}+y_{a} &amp; =\left(\mathfrak{x}_{a}+\mathfrak{y}_{a}\right)\frac{1}{r_{a}}
\end{align}
\quad(a)\\
\hline \begin{matrix}\begin{align}2\mathfrak{x}_{a} &amp; =\left(r_{a}+\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}-\frac{1}{r_{a}}\right)y_{a}\\
2\mathfrak{y}_{a} &amp; =\left(r_{a}-\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}+\frac{1}{r_{a}}\right)y_{a}
\end{align}
\quad(b)\end{matrix}\\
\hline \left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}\Rightarrow\begin{align}\mathfrak{x}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}\\
\mathfrak{y}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}
\end{align}
\quad(c)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Equation (a) is identical to Lorentz boost ({{equationNote|3d}}), while (c) is similar, though not identical, to Lorentz boost ({{equationNote|3b}}-C). The difference stems from his definition&lt;br&gt;&lt;math&gt;\qquad\left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}&lt;/math&gt;&lt;br&gt;whereas in accordance to expression &lt;math&gt;\sqrt{\tfrac{1+v}{1-v}}&lt;/math&gt; in ({{equationNote|3d}}) he should have stated &lt;br&gt;&lt;math&gt;\qquad\left\{ \begin{matrix}r_{a}=\frac{\sqrt{1+s_{a}}}{\sqrt{1-s_{a}}}\\
s_{a}&lt;1
\end{matrix}\right\}&lt;/math&gt;.&lt;br&gt;Using the latter choice, equations (c) would assume a form equivalent to ({{equationNote|3b}}):&lt;br&gt;&lt;math&gt;\qquad\begin{align}\mathfrak{x}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{1-s_{a}}\sqrt{1+s_{a}}}\\
\mathfrak{y}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{1-s_{a}}\sqrt{1+s_{a}}}\end{align}&lt;/math&gt;&lt;/p&gt;

==={{Anchor|Schur}} Schur (1885/86, 1900/02) – Beltrami coordinates===

[[w:Friedrich Schur]] (1885/86) discussed spaces of constant Riemann curvature, and by following [[#Beltrami|Beltrami (1868)]] he used the transformation&lt;ref group=M&gt;Schur (1885/86), p. 167&lt;/ref&gt;

:&lt;math&gt;x_{1}=R^{2}\frac{y_{1}+a_{1}}{R^{2}+a_{1}y_{1}},\ x_{2}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{2}}{R^{2}+a_{1}y_{1}},\dots,\ x_{n}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{n}}{R^{2}+a_{1}y_{1}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in arbitrary dimensions by setting ''R=c'' as the speed of light and ''a&lt;sub&gt;1&lt;/sub&gt;=v'' as relative velocity.&lt;/p&gt;

In (1900/02) he derived basic formulas of non-Eucliden geometry, including the case of translation for which he obtained the transformation similar to his previous one:&lt;ref group=M&gt;Schur (1900/02), p. 290; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;x'=\frac{x-a}{1-\mathfrak{k}ax},\quad y'=\frac{y\sqrt{1-\mathfrak{k}a^{2}}}{1-\mathfrak{k}ax}&lt;/math&gt;

where &lt;math&gt;\mathfrak{k}&lt;/math&gt; can have values &gt;0, &lt;0 or ∞.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting ''a=v'' and &lt;math&gt;\mathfrak{k}=\tfrac{1}{c^{2}}&lt;/math&gt;.&lt;/p&gt;

He also defined the triangle&lt;ref group=M&gt;Schur (1900/02), p. 291; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;\frac{1}{\sqrt{1-\mathfrak{k}c^{2}}}=\frac{1}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{1}{\sqrt{1-\mathfrak{k}b^{2}}}-\frac{a}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{b}{\sqrt{1-\mathfrak{k}b^{2}}}\cos\gamma&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to the hyperbolic law of cosines and the relativistic velocity addition ({{equationNote|3f}}, b) or [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']] by setting &lt;math&gt;[\mathfrak{k},\ c,\ a,\ b]=\left[\tfrac{1}{c^{2}},\ \sqrt{u_{x}^{\prime2}+u_{y}^{\prime2}},\ v,\ \sqrt{u_{x}^{2}+u_{y}^{2}}\right]&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Lindemann}} Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute===

[[w:Ferdinand von Lindemann]] discussed hyperbolic geometry in terms of the [[w:Cayley–Klein metric]] in his (1890/91) edition of the lectures on geometry of [[w:Alfred Clebsch]]. Citing [[../Lorentz transformation (general)#Killing|E:Killing (1885)]] and [[../Lorentz transformation (general)#Poincare|Poincaré (1887)]] in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 477–478, 524&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\Omega_{xx}=x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}-4k^{2}dx_{3}^{2}\\
\Omega_{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}
\end{matrix}&lt;/math&gt;

and used the following transformation&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 361–362&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
X_{1}X_{4}+X_{2}X_{3}=\Xi_{1}\Xi_{4}+\Xi_{2}\Xi_{3}\\
\hline \begin{align}X_{1} &amp; =\left(\lambda+\lambda_{1}\right)U_{4} &amp; \Xi_{1} &amp; =\left(\lambda-\lambda_{1}\right)U_{4} &amp; X_{1} &amp; =\frac{\lambda+\lambda_{1}}{\lambda-\lambda_{1}}\Xi_{1}\\
X_{2} &amp; =\left(\lambda+\lambda_{3}\right)U_{4} &amp; \Xi_{2} &amp; =\left(\lambda-\lambda_{3}\right)U_{4} &amp; X_{2} &amp; =\frac{\lambda+\lambda_{3}}{\lambda-\lambda_{3}}\Xi_{2}\\
X_{3} &amp; =\left(\lambda-\lambda_{3}\right)U_{2} &amp; \Xi_{3} &amp; =\left(\lambda+\lambda_{3}\right)U_{2} &amp; X_{3} &amp; =\frac{\lambda-\lambda_{3}}{\lambda+\lambda_{3}}\Xi_{3}\\
X_{4} &amp; =\left(\lambda-\lambda_{1}\right)U_{1} &amp; \Xi_{4} &amp; =\left(\lambda+\lambda_{1}\right)U_{1} &amp; X_{4} &amp; =\frac{\lambda-\lambda_{1}}{\lambda+\lambda_{1}}\Xi_{4}
\end{align}
\end{matrix}&lt;/math&gt;

into which he put&lt;ref group=M name=linde&gt;Lindemann &amp; Clebsch (1890/91), p. 496&lt;/ref&gt;

:&lt;math&gt;\begin{align}X_{1} &amp; =x_{1}+2kx_{4}, &amp; X_{2} &amp; =x_{2}+ix_{3}, &amp; \lambda+\lambda_{1} &amp; =\left(\lambda-\lambda_{1}\right)e^{a},\\
X_{4} &amp; =x_{1}-2kx_{4}, &amp; X_{3} &amp; =x_{2}-ix_{3}, &amp; \lambda+\lambda_{3} &amp; =\left(\lambda-\lambda_{3}\right)e^{\alpha i},
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3d}}) with &lt;math&gt;e^{\alpha i}=1&lt;/math&gt; and ''2k=1'' .&lt;/p&gt;

From that, he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations (''a''=0) or translations (α=0):&lt;ref group=M  name=linde /&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=0\\
\hline \begin{align}x_{2} &amp; =\xi_{2}\cos\alpha+\xi_{3}\sin\alpha, &amp; x_{1} &amp; =\xi_{1}\cos\frac{a}{i}+2ki\xi_{4}\sin\frac{a}{i},\\
x_{3} &amp; =-\xi_{2}\sin\alpha+\xi_{3}\cos\alpha, &amp; 2kx_{4} &amp; =i\xi_{1}\sin\frac{a}{i}+2k\xi_{4}\cos\frac{a}{i}.
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0 and ''2k=1''.&lt;/p&gt;

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane and gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Killing2}} Killing (1893,97) – Weierstrass coordinates===

[[w:Wilhelm Killing]] (1878–1880) gave case of translation in the form&lt;ref group=M name=killtra&gt;Killing (1893), p. 331&lt;/ref&gt;

:&lt;math&gt;y_{0}=x_{0}\operatorname{Ch}a+x_{1}\operatorname{Sh}a,\quad y_{1}=x_{0}\operatorname{Sh}a+x_{1}\operatorname{Ch}a,\quad y_{2}=x_{2}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

In 1898, Killing wrote that relation in a form similar to [[#Escherich|Escherich (1874)]], and derived the corresponding Lorentz transformation for the two cases were ''v'' is unchanged or ''u'' is unchanged:&lt;ref group=M name=kill98&gt;Killing (1898), p. 133&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\xi'=\frac{\xi\operatorname{Ch}\frac{\mu}{l}+l\operatorname{Sh}\frac{\mu}{l}}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}},\ \eta'=\frac{\eta}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}}\\
\hline \frac{u}{p}=\xi,\ \frac{v}{p}=\eta\\
\hline p'=p\operatorname{Ch}\frac{\mu}{l}+\frac{u}{l}\operatorname{Sh}\frac{\mu}{l},\quad u'=pl\operatorname{Sh}\frac{\mu}{l}+u\operatorname{Ch}\frac{\mu}{l},\quad v'=v\\
\text{or}\\
p'=p\operatorname{Ch}\frac{\nu}{l}+\frac{v}{l}\operatorname{Sh}\frac{\nu}{l},\quad u'=u,\quad v'=pl\operatorname{Sh}\frac{\nu}{l}+v\operatorname{Ch}\frac{\nu}{l}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The upper transformation system is equivalent to Lorentz transformation ({{equationNote|3e}}) and the velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] with ''l=c'' and &lt;math&gt;\mu=c\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt;, the system below is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Whitehead}} Whitehead (1897/98) – Universal algebra===

[[w:Alfred North Whitehead]] (1898) discussed the kinematics of hyperbolic space as part of his study of [[w:universal algebra]], and obtained the following transformation:&lt;ref group=M name=white&gt;Whitehead (1898), pp. 459–460&lt;/ref&gt;

:&lt;math&gt;\begin{align}x' &amp; =\left(\eta\cosh\frac{\delta}{\gamma}+\eta_{1}\sinh\frac{\delta}{\gamma}\right)e+\left(\eta\sinh\frac{\delta}{\gamma}+\eta_{1}\cosh\frac{\delta}{\gamma}\right)e_{1}\\
 &amp; \qquad+\left(\eta_{2}\cos\alpha+\eta_{3}\sin\alpha\right)e_{2}+\left(\eta_{3}\cos\alpha-\eta_{2}\sin\alpha\right)e_{3}
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0.&lt;/p&gt;

==={{anchor|Elliott}} Elliott (1903) – Invariant theory ===
[[w:Edwin Bailey Elliott]] (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:&lt;ref group=M&gt;Elliott (1903), p. 109&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=X\cosh\phi+Y\sinh\phi,\quad y=X\sinh\phi+Y\cosh\phi\\
\hline X+Y=e^{-\phi}(x+y),\quad X-Y=e^{\phi}(x-y)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}). The mentioned subgroup corresponds to the one-parameter subgroup generated by Lorentz boosts.&lt;/p&gt;

==={{anchor|Woods2}} Woods (1903) – Weierstrass coordinates ===
[[w:Frederick S. Woods]] (1903, published 1905) gave the case of translation in hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 55&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x_{1}\cos kl+x_{0}\frac{\sin kl}{k},\quad x_{2}^{\prime}=x_{2},\quad x_{2}^{\prime}=x_{3},\quad x_{0}^{\prime}=-x_{1}k\sin kl+x_{0}\cos kl&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.&lt;/p&gt;

and the loxodromic substitution for hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 72&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}x_{1}^{\prime} &amp; =x_{1}\cosh\alpha-x_{0}\sinh\alpha\\
x_{2}^{\prime} &amp; =x_{2}\cos\beta-x_{3}\sin\beta\\
x_{3}^{\prime} &amp; =x_{2}\sin\beta+x_{3}\cos\beta\\
x_{0}^{\prime} &amp; =-x_{1}\sinh\alpha+x_{0}\cosh\alpha
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with β=0.&lt;/p&gt;

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – gave the case of translation in the hyperbolic plane:&lt;ref group=M name=lieb&gt;Liebmann (1904/05), p. 174&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x'\operatorname{ch}a+p'\operatorname{sh}a,\quad y_{1}^{\prime}=y',\quad p_{1}^{\prime}=x'\operatorname{sh}a+p'\operatorname{ch}a&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Frank}} Frank (1909) – Special relativity===
In special relativity, hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity:&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;
 	
=== {{anchor|Herglotz1}} Herglotz (1909/10) – Special relativity===

In special relativity, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic, with the hyperbolic case being:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;
	
:&lt;math&gt;\begin{matrix}Z=Z'e^{\vartheta}\\
	
\begin{aligned}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
	
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
	
\end{aligned}
	
\end{matrix}&lt;/math&gt;	
	
&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3d}}).&lt;/p&gt;

==={{anchor|Varicak}} Varićak (1910) – Special relativity===

In special relativity, hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

He showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bel68sag}} 
*{{#section:History of Topics in Special Relativity/mathsource|bel68fond}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|eli03}} 
*{{#section:History of Topics in Special Relativity/mathsource|esch74}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul35}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul48a}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|glai78}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|guen80}} 
*{{#section:History of Topics in Special Relativity/mathsource|kep09}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lais74b}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam67}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lind90}}
*{{#section:History of Topics in Special Relativity/mathsource|lip86}}  
*{{#section:History of Topics in Special Relativity/mathsource|ric57}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu85}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu00}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu09}} 
*{{#section:History of Topics in Special Relativity/mathsource|tau26}} 
*{{#section:History of Topics in Special Relativity/mathsource|whit98}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L3}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
      <sha1>oy5aiq4r0mi1qni2pibuivf1v0fm2sm</sha1>
    </revision>
    <revision>
      <id>2411826</id>
      <parentid>2411825</parentid>
      <timestamp>2022-08-04T12:59:14Z</timestamp>
      <contributor>
        <username>D.H</username>
        <id>52339</id>
      </contributor>
      <comment>/* Hyperbolic law of cosines */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="47279" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via hyperbolic functions==
===Translation in the hyperbolic plane===
The case of a Lorentz transformation without spatial rotation is called a [[w:Lorentz boost]]. The simplest case can be given, for instance, by setting ''n=1'' in the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}\\
\\
x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}
\end{align}
\left|\begin{align}g_{01}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{10}^{2} &amp; =1\\
g_{01}g_{11}-g_{00}g_{10} &amp; =0\\
g_{10}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{01}^{2} &amp; =1\\
g_{10}g_{11}-g_{00}g_{01} &amp; =0
\end{align}
\rightarrow\begin{align}g_{00}^{2} &amp; =g_{11}^{2}\\
g_{01}^{2} &amp; =g_{10}^{2}
\end{align}
\right.
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}g_{00} &amp; -g_{10}\\
-g_{01} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\right|\det\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;|{{equationRef|3a}}}}

which resembles precisely the relations of [[w:hyperbolic function]]s in terms of [[w:hyperbolic angle]] &lt;math&gt;\eta&lt;/math&gt;. Thus a Lorentz boost or [[w:hyperbolic rotation]] (being the same as a rotation around an imaginary angle &lt;math&gt;i\eta=\phi&lt;/math&gt; in [[../Lorentz transformation (imaginary)#math_2b|E:'''(2b)''']] or a [[w:Translation (geometry)|translation]] in the hyperbolic plane in terms of the hyperboloid model) is given by

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline g_{00}=g_{11}=\cosh\eta,\ g_{01}=g_{10}=-\sinh\eta\\
\hline \left.\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta &amp;  &amp; =\frac{x_{0}-x_{1}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}-x_{1}v}{\sqrt{1-v^{2}}}\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta &amp;  &amp; =\frac{x_{1}-x_{0}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}-x_{0}v}{\sqrt{1-v^{2}}}\\
x_{2}^{\prime} &amp; =x_{2} &amp;  &amp; =x_{2} &amp;  &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta &amp;  &amp; =\frac{x_{0}^{\prime}-x_{1}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}^{\prime}-x_{1}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta &amp;  &amp; =\frac{x_{1}^{\prime}-x_{0}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}^{\prime}-x_{0}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{2} &amp; =x_{2}^{\prime} &amp;  &amp; =x_{2}^{\prime} &amp;  &amp; =x_{2}^{\prime}
\end{align}
\right|{\scriptstyle \begin{align}\sinh^{2}\eta-\cosh^{2}\eta &amp; =-1 &amp; (a)\\
\cosh^{2}\eta-\sinh^{2}\eta &amp; =1 &amp; (b)\\
\frac{\sinh\eta}{\cosh\eta} &amp; =\tanh\eta=v &amp; (c)\\
\frac{1}{\sqrt{1-\tanh^{2}\eta}} &amp; =\cosh\eta &amp; (d)\\
\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\sinh\eta &amp; (e)\\
\frac{\tanh q\pm\tanh\eta}{1\pm\tanh q\tanh\eta} &amp; =\tanh\left(q\pm\eta\right) &amp; (f)
\end{align}
}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\right|\det\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;|{{equationRef|3b}}}}

Hyperbolic identities (a,b) on the right of ({{equationNote|3b}}) were given by [[#Riccati|Riccati (1757)]], all identities (a,b,c,d,e,f) by [[#Lambert|Lambert (1768–1770)]]. Lorentz transformations ({{equationNote|3b}}-A) were given by [[#Laisant|Laisant (1874)]], [[#Cox|Cox (1882)]], [[#Lindemann|Lindemann (1890/91)]], [[#Gerard|Gérard (1892)]], [[#Killing2|Killing (1893, 1897/98)]], [[#Whitehead|Whitehead (1897/98)]], [[#Woods2|Woods (1903/05)]], [[#Elliott|Elliott (1903)]] and [[#Liebmann|Liebmann (1904/05)]] in terms of Weierstrass coordinates of the [[w:hyperboloid model]], while transformations similar to ({{equationNote|3b}}-C) have been used by [[#Lipschitz1|Lipschitz (1885/86)]].

Rapidity can be composed of arbitrary many rapidities &lt;math&gt;\eta_{1},\eta_{2}\dots&lt;/math&gt; as per the [[w:Hyperbolic functions#Sums of arguments|w:angle sum laws of hyperbolic sines and cosines]], so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between [[w:List of trigonometric identities#Angle sum and difference identities|w:angle sum laws of circular trigonometry]] and spatial rotations. Alternatively, the hyperbolic angle sum laws ''themselves'' can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the [[w:unit hyperbola]]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \left[\eta=\eta_{2}-\eta_{1}\right]\\
\begin{align}x_{0}^{\prime} &amp; =\sinh\eta_{1} &amp;  &amp; =\sinh\left(\eta_{2}-\eta\right) &amp;  &amp; =\sinh\eta_{2}\cosh\eta-\cosh\eta_{2}\sinh\eta &amp;  &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =\cosh\eta_{1} &amp;  &amp; =\cosh\left(\eta_{2}-\eta\right) &amp;  &amp; =-\sinh\eta_{2}\sinh\eta+\cosh\eta_{2}\cosh\eta &amp;  &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta\\
\\
x_{0} &amp; =\sinh\eta_{2} &amp;  &amp; =\sinh\left(\eta_{1}+\eta\right) &amp;  &amp; =\sinh\eta_{1}\cosh\eta+\cosh\eta_{1}\sinh\eta &amp;  &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =\cosh\eta_{2} &amp;  &amp; =\cosh\left(\eta_{1}+\eta\right) &amp;  &amp; =\sinh\eta_{1}\sinh\eta+\cosh\eta_{1}\cosh\eta &amp;  &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \begin{align} &amp; \begin{bmatrix}x_{0}^{\prime}\\
x_{1}^{\prime}
\end{bmatrix}=\begin{bmatrix}\sinh\eta_{1}\\
\cosh\eta_{1}
\end{bmatrix}=\begin{bmatrix}\sinh\left(\eta_{2}-\eta\right)\\
\cosh\left(\eta_{2}-\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}\sinh\eta_{2}\\
\cosh\eta_{2}
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}x_{0}\\
x_{1}
\end{bmatrix}\\
 &amp; \begin{bmatrix}x_{0}\\
x_{1}
\end{bmatrix}=\begin{bmatrix}\sinh\eta_{2}\\
\cosh\eta_{2}
\end{bmatrix}=\begin{bmatrix}\sinh\left(\eta_{1}+\eta\right)\\
\cosh\left(\eta_{1}+\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}\sinh\eta_{1}\\
\cosh\eta_{1}
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}x_{0}^{\prime}\\
x_{1}^{\prime}
\end{bmatrix}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|3c}}}}

Hyperbolic angle sum laws equivalent to Lorentz boost ({{equationNote|3c}}) were given by [[#Riccati|Riccati (1757)]] and [[#Lambert|Lambert (1768–1770)]], while the matrix representation was given by [[#Glaisher|Glaisher (1878)]] and [[#Gunther2|Günther (1880/81)]]. In special relativity, hyperbolic functions were used by [[#Frank|Frank (1909)]] and [[#Varicak|Varićak (1910)]].

Finally, because of &lt;math&gt;\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;, Lorentz boost ({{equationNote|3b}}) assumes a simple form by using [[w:squeeze mapping]]s in analogy to Euler's formula in [[../Lorentz transformation (imaginary)#math_2c|E:'''(2c)''']]:&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \begin{matrix}x_{1}^{\prime}-x_{0}^{\prime}=k\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime}=\frac{1}{k}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime}=x_{2}
\end{matrix}\\
\hline \begin{align}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}-x_{0} &amp; =e^{-\eta}\left(x_{1}^{\prime}-x_{0}^{\prime}\right)\\
x_{1}+x_{0} &amp; =e^{\eta}\left(x_{1}^{\prime}+x_{0}^{\prime}\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\\
\hline k=e^{\eta}=\cosh\eta+\sinh\eta=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}=\sqrt{\frac{1+v}{1-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|3d}}}} 

Lorentz transformations ({{equationNote|3d}}) for arbitrary ''k'' were given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]), while the above exponential form was explicitly used by [[#Lindemann|Lindemann (1890/91)]], [[#Elliott|Elliott (1903)]], [[#Herglotz1|Herglotz (1909)]]. 

===Hyperbolic law of cosines===
In line with equation [[../Lorentz transformation (general)#math_1b|E:'''(1b)''']] one can use coordinates &lt;math&gt;[u_{1},\ u_{2}]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}}\right]&lt;/math&gt; inside the [[w:unit circle]] &lt;math&gt;u_{1}^{2}+u_{2}^{2}=1&lt;/math&gt;, thus the corresponding Lorentz transformations ({{equationNote|3b}}) obtain the form:

{{NumBlk|:|&lt;math&gt;\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
u_{1}^{\prime} &amp; =\frac{-\sinh\eta+u_{1}\cosh\eta}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{1}-\tanh\eta}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{1}-v}{1-u_{1}v}\\
u_{2}^{\prime} &amp; =\frac{u_{2}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-v^{2}}}{1-u_{1}v}\\
\\
u_{1} &amp; =\frac{\sinh\eta+u_{1}^{\prime}\cosh\eta}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+\tanh\eta}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+v}{1+u_{1}^{\prime}v}\\
u_{2} &amp; =\frac{u_{2}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}
\end{align}
&lt;/math&gt;|{{equationRef|3e}}}}

Transformations (A) were given by [[#Escherich|Escherich (1874)]] and [[#Killing2|Killing (1898)]], and transformations (C) by [[#Beltrami|Beltrami (1868)]] and [[#Schur|Schur (1885/86, 1900/02)]] in terms of [[w:Beltrami–Klein model|Beltrami coordinates]]&lt;ref&gt;Rosenfeld (1988), p. 231&lt;/ref&gt; of hyperbolic geometry. By using the scalar product of &lt;math&gt;\left[u_{1},u_{2}\right]&lt;/math&gt;, the resulting Lorentz transformation can be seen as equivalent to the [[w:hyperbolic law of cosines]]:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{matrix}u^{2}=u_{1}^{2}+u_{2}^{2}\\
u'^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}
\end{matrix}\left|\begin{matrix}u_{1}=u\cos\alpha\\
u_{2}=u\sin\alpha\\
\\
u_{1}^{\prime}=u'\cos\alpha'\\
u_{2}^{\prime}=u'\sin\alpha'
\end{matrix}\right|\begin{align}u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+vu'\cos\alpha'}, &amp; u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-vu\cos\alpha}\\
u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{1+vu'\cos\alpha'}, &amp; u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{1-vu\cos\alpha}\\
\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{u'\cos\alpha'+v}, &amp; \tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{u\cos\alpha-v}
\end{align}
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(vu'\sin\alpha'\right){}^{2}}}{1+vu'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(vu\sin\alpha\right){}^{2}}}{1-vu\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-u^{\prime2}}}=\frac{1}{\sqrt{1-v^{2}}}\frac{1}{\sqrt{1-u^{2}}}-\frac{v}{\sqrt{1-v^{2}}}\frac{u}{\sqrt{1-u^{2}}}\cos\alpha &amp; (b)\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha &amp; (a)
\end{matrix}&lt;/math&gt;|{{equationRef|3f}}}}

The hyperbolic law of cosines (a) was given by [[#Taurinus|Taurinus (1826) and Lobachevsky (1829/30)]] and others, while variant (b) was given by [[#Schur|Schur (1900/02)]]. By further setting ''u=u′'' it follows:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\cos\alpha=\frac{\cos\alpha'+v}{1+v\cos\alpha'},\ \sin\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{1+v\cos\alpha'},\ \tan\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{\cos\alpha'+v},\ \tan\frac{\alpha}{2}=\sqrt{\frac{1-v}{1+v}}\tan\frac{\alpha'}{2}\\
\cos\alpha'=\frac{\cos\alpha-v}{1-v\cos\alpha},\ \sin\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{1-v\cos\alpha},\ \tan\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{\cos\alpha-v},\ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+v}{1-v}}\tan\frac{\alpha}{2}
\end{matrix}&lt;/math&gt;|{{equationRef|3g}}}}

Formulas (3b) resemble the equations of an [[w:ellipse]] of [[w:Orbital eccentricity|eccentricity]] ''v/c'', [[w:eccentric anomaly]] α' and [[w:true anomaly]] α, first geometrically formulated by [[#Euler|Kepler (1609)]] and explicitly written down by [[#Euler|Euler (1735, 1748), Lagrange (1770)]] and many others in relation to planetary motions. They were also used by [[../Lorentz transformation (conformal)#Darboux|E:Darboux (1873)]] as a sphere transformation. In special relativity, these formulas describes the aberration of light, see [[../Lorentz transformation (velocity)#Velocity addition and aberration|E:velocity addition and aberration]].

==Historical notation==
==={{anchor|Euler}} Euler (1735) – True and eccentric anomaly===

[[w:Johannes Kepler]] (1609) geometrically formulated [[w:Kepler's equation]] and the relations between the [[w:mean anomaly]], [[w:true anomaly]], and [[w:eccentric anomaly]].&lt;ref group=M&gt;Kepler (1609), chapter 60. The editors of Kepler's collected papers remark (p. 482), that Kepler's relations correspond to &lt;math&gt;{\scriptstyle \alpha=\beta+e\sin\beta}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\nu=\frac{e+\cos\beta}{1+e\cos\beta}}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\beta=\frac{\cos\nu-e}{1-e\cos\nu}}&lt;/math&gt;&lt;/ref&gt;&lt;ref&gt;Volk (1976), p. 366&lt;/ref&gt; The relation between the true anomaly ''z'' and the eccentric anomaly ''P'' was algebraically expressed by [[w:Leonhard Euler]] (1735/40) as follows:&lt;ref group=M&gt;Euler (1735/40), § 19&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{\cos P+v}{1+v\cos P},\ \cos P=\frac{\cos z-v}{1-v\cos z},\ \int P=\frac{\int z\sqrt{1-v^{2}}}{1-v\cos z}&lt;/math&gt;

and in 1748:&lt;ref group=M&gt;Euler (1748a), section VIII&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{n+\cos y}{1+n\cos y},\ \sin z=\frac{\sin y\sqrt{1-n^{2}}}{1+n\cos y},\ \tan z=\frac{\sin y\sqrt{1-n^{2}}}{n+\cos y}&lt;/math&gt;

while [[w:Joseph-Louis Lagrange]] (1770/71) expressed them as follows&lt;ref group=M&gt;Lagrange (1770/71), section I&lt;/ref&gt;

:&lt;math&gt;\sin u=\frac{m\sin x}{1+n\cos x},\ \cos u=\frac{n+\cos x}{1+n\cos x},\ \operatorname{tang}\frac{1}{2}u=\frac{m}{1+n}\operatorname{tang}\frac{1}{2}x,\ \left(m^{2}=1-n^{2}\right)&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt; These relations resemble formulas ({{equationNote|3g}}), while ({{equationNote|3f}}) follows by setting &lt;math&gt;[\cos z,\sin z]=\left[u_{x},u_{y}\right]&lt;/math&gt; in Euler's formulas or &lt;math&gt;[\cos u,\sin u]=\left[u_{x},u_{y}\right]&lt;/math&gt; in Lagrange's formulas.&lt;/p&gt;

==={{anchor|Riccati}} Riccati (1757) – hyperbolic addition===

[[w:Vincenzo Riccati]] introduced hyperbolic functions in 1757,&lt;ref group=M&gt;Riccati (1757), p. 71&lt;/ref&gt;&lt;ref group=M&gt;Günther (1880/81), pp. 7–13&lt;/ref&gt; in particular he formulated the angle sum laws for hyperbolic sine and cosine:

:&lt;math&gt;\begin{matrix}\mathrm{Ch}(\varphi+\pi)=\frac{\mathrm{Ch}\varphi\mathrm{Ch}\pi+\mathrm{Sh}\varphi\mathrm{Sh}\pi}{r}\\
\mathrm{Sh}(\varphi+\pi)=\frac{\mathrm{Ch}\varphi\mathrm{Sh}\pi+\mathrm{Ch}\pi\mathrm{Sh}\varphi}{r}\\
\left[\mathrm{Ch}^{2}-\mathrm{Sh}^{2}=rr\right]
\end{matrix}&lt;/math&gt;

He furthermore showed that &lt;math&gt;\mathrm{Ch}(\varphi-\pi)&lt;/math&gt; and &lt;math&gt;\mathrm{Sh}(\varphi-\pi)&lt;/math&gt; follow by setting &lt;math&gt;\mathrm{Ch}(\pi)\Rightarrow\mathrm{Ch}(-\pi)&lt;/math&gt; and &lt;math&gt;\mathrm{Sh}(\pi)\Rightarrow\mathrm{Sh}(-\pi)&lt;/math&gt; in the above formulas.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3c}}). (In modern publications, Riccati's additional factor ''r'' is set to unity.)&lt;/p&gt;

==={{anchor|Lambert}} Lambert (1768–1770) – hyperbolic addition===

While [[#Riccati|Riccati (1757)]] discussed the hyperbolic sine and cosine, [[w:Johann Heinrich Lambert]] (read 1767, published 1768) introduced the expression ''tang φ'' or abbreviated ''tφ'' as the [[w:tangens hyperbolicus]] &lt;math&gt;{\scriptstyle \frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}}&lt;/math&gt; of a variable ''u'', or in modern notation ''tφ=tanh(u)'':&lt;ref group=M&gt;Lambert (1761/68), pp. 309–318&lt;/ref&gt;&lt;ref&gt;Barnett (2004), pp. 22–23&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}\xi\xi-1 &amp; =\eta\eta &amp; (a)\\
1+\eta\eta &amp; =\xi\xi &amp; (b)\\
\frac{\eta}{\xi} &amp; =tang\ \phi=t\phi &amp; (c)\\
\xi &amp; =\frac{1}{\sqrt{1-t\phi^{2}}} &amp; (d)\\
\eta &amp; =\frac{t\phi}{\sqrt{1-t\phi^{2}}} &amp; (e)\\
t\phi'' &amp; =\frac{t\phi+t\phi'}{1+t\phi\cdot t\phi'} &amp; (f)\\
t\phi' &amp; =\frac{t\phi''-t\phi}{1-t\phi\cdot t\phi''} &amp; (g)
\end{align}
\right|\begin{align}2u &amp; =\log\frac{1+t\phi}{1-t\phi}\\
\xi &amp; =\frac{e^{u}+e^{-u}}{2}\\
\eta &amp; =\frac{e^{u}-e^{-u}}{2}\\
t\phi &amp; =\frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}\\
e^{u} &amp; =\xi+\eta\\
e^{-u} &amp; =\xi-\eta
\end{align}&lt;/math&gt;

In (1770) he rewrote the addition law for the hyperbolic tangens (f) or (g) as:&lt;ref group=M&gt;Lambert (1770), p. 335&lt;/ref&gt;

:&lt;math&gt;\begin{align}t(y+z) &amp; =(ty+tz):(1+ty\cdot tz) &amp; (f)\\
t(y-z) &amp; =(ty-tz):(1-ty\cdot tz) &amp; (g)
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The hyperbolic relations (a,b,c,d,e,f) are equivalent to the hyperbolic relations on the right of ({{equationNote|3b}}). Relations (f,g) can also be found in ({{equationNote|3e}}). By setting ''tφ=v/c'', formula (c) becomes the relative velocity between two frames, (d) the [[w:Lorentz factor]], (e) the [[w:proper velocity]], (f) or (g) becomes the Lorentz transformation of velocity (or relativistic [[w:velocity addition formula]]) for collinear velocities in [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] and [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']].&lt;/p&gt;

Lambert also formulated the addition laws for the hyperbolic cosine and sine (Lambert's &quot;cos&quot; and &quot;sin&quot; actually mean &quot;cosh&quot; and &quot;sinh&quot;):

:&lt;math&gt;\begin{align}\sin(y+z) &amp; =\sin y\cos z+\cos y\sin z\\
\sin(y-z) &amp; =\sin y\cos z-\cos y\sin z\\
\cos(y+z) &amp; =\cos y\cos z+\sin y\sin z\\
\cos(y-z) &amp; =\cos y\cos z-\sin y\sin z
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{Anchor|Taurinus}} Taurinus (1826) – Hyperbolic law of cosines===

After the addition theorem for the tangens hyperbolicus was given by [[#Lambert|Lambert (1768)]], [[w:hyperbolic geometry]] was used by [[w:Franz Taurinus]] (1826), and later by [[w:Nikolai Lobachevsky]] (1829/30) and others, to formulate the [[w:hyperbolic law of cosines]]:&lt;ref group=M&gt;Taurinus (1826), p. 66; see also p. 272 in the translation by Engel and Stäckel (1899)&lt;/ref&gt;&lt;ref&gt;Bonola (1912), p. 79&lt;/ref&gt;&lt;ref&gt;Gray (1979), p. 242&lt;/ref&gt;

:&lt;math&gt;A=\operatorname{arccos}\frac{\cos\left(\alpha\sqrt{-1}\right)-\cos\left(\beta\sqrt{-1}\right)\cos\left(\gamma\sqrt{-1}\right)}{\sin\left(\beta\sqrt{-1}\right)\sin\left(\gamma\sqrt{-1}\right)}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;When solved for &lt;math&gt;\cos\left(\alpha\sqrt{-1}\right)&lt;/math&gt; it corresponds to the Lorentz transformation in Beltrami coordinates ({{equationNote|3f}}), and by defining the rapidities &lt;math&gt;{\scriptstyle \left(\left[\frac{U}{c},\ \frac{v}{c},\ \frac{u}{c}\right]=\left[\tanh\alpha,\ \tanh\beta,\ \tanh\gamma\right]\right)}&lt;/math&gt; it corresponds to the relativistic velocity addition formula [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']].&lt;/p&gt;

==={{anchor|Beltrami}} Beltrami (1868) – Beltrami coordinates===

[[w:Eugenio Beltrami]] (1868a) introduced coordinates of the [[w:Beltrami–Klein model]] of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:&lt;ref group=M&gt;Beltrami (1868a), pp. 287-288; Note I; Note II&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds^{2}=R^{2}\frac{\left(a^{2}+v^{2}\right)du^{2}-2uv\,du\,dv+\left(a^{2}+v^{2}\right)dv^{2}}{\left(a^{2}+u^{2}+v^{2}\right)^{2}}\\
u^{2}+v^{2}=a^{2}\\
\hline u''=\frac{aa_{0}\left(u'-r_{0}\right)}{a^{2}-r_{0}u'},\ v''=\frac{a_{0}w_{0}v'}{a^{2}-r_{0}u'},\\
\left(r_{0}=\sqrt{u_{0}^{2}+v_{0}^{2}},\ w_{0}=\sqrt{a^{2}-r_{0}^{2}}\right)\\
\hline ds^{2}=R^{2}\frac{\left(a^{2}-v^{2}\right)du^{2}+2uv\,du\,dv+\left(a^{2}-v^{2}\right)dv^{2}}{\left(a^{2}-u^{2}-v^{2}\right)^{2}}\\
(R=R\sqrt{-1},\ a=a\sqrt{-1})
\end{matrix}&lt;/math&gt;

(where the disk radius ''a'' and the [[w:radius of curvature]] ''R'' are real in spherical geometry, in hyperbolic geometry they are imaginary), and for arbitrary dimensions in (1868b)&lt;ref group=M&gt;Beltrami (1868b), pp. 232, 240–241, 253–254&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds=R\frac{\sqrt{dx^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}}}{x}\\
x^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=a^{2}\\
\hline y_{1}=\frac{ab\left(x_{1}-a_{1}\right)}{a^{2}-a_{1}x_{1}}\ \text{or}\ x_{1}=\frac{a\left(ay_{1}+a_{1}b\right)}{ab+a_{1}y_{1}},\ x_{r}=\pm\frac{ay_{r}\sqrt{a^{2}-a_{1}^{2}}}{ab+a_{1}y_{1}}\ (r=2,3,\dots,n)\\
\hline ds=R\frac{\sqrt{dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}-dx^{2}}}{x}\\
x^{2}=a^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\\
\left(R=R\sqrt{-1},\ x=x\sqrt{-1},\ a=a\sqrt{-1}\right)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Setting ''a=a&lt;sub&gt;0&lt;/sub&gt;'' Beltrami's (1868a) formulas become formulas ({{equationNote|3e}}), or in his (1868b) formulas one sets ''a=b'' for arbitrary dimensions.&lt;/p&gt;

=== {{anchor|Laisant2}} Laisant (1874) – Equipollences===
In his French translation of [[w:Giusto Bellavitis]]' principal work on [[w:Equipollence (geometry)|w:equipollences]], [[w:Charles-Ange Laisant]] (1874) added a chapter related to hyperbolas. The equipollence OM and its tangent MT of a hyperbola is defined by Laisant as&lt;ref group=M&gt;Laisant (1874b), pp. 134–135&lt;/ref&gt;

:(1) &lt;math&gt;\begin{matrix} &amp; \mathrm{OM}\bumpeq x\mathrm{OA}+y\mathrm{OB}\\
 &amp; \mathrm{MT}\bumpeq y\mathrm{OA}+x\mathrm{OB}\\
 &amp; \left[x^{2}-y^{2}=1;\ x=\cosh t,\ y=\sinh t\right]\\
\Rightarrow &amp; \mathrm{OM}\bumpeq\cosh t\cdot\mathrm{OA}+\sinh t\cdot\mathrm{OB}
\end{matrix}&lt;/math&gt;

Here, OA and OB are [[w:Conjugate diameters|conjugate semi-diameters]] of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation:

:&lt;math&gt;\begin{matrix}\begin{align}\mathrm{OC} &amp; \bumpeq c\mathrm{OA}+d\mathrm{OB} &amp; \qquad &amp;  &amp; \mathrm{OA} &amp; \bumpeq c\mathrm{OC}-d\mathrm{OD}\\
\mathrm{OD} &amp; \bumpeq d\mathrm{OA}+c\mathrm{OB} &amp;  &amp;  &amp; \mathrm{OB} &amp; \bumpeq-d\mathrm{OC}+c\mathrm{OD}
\end{align}
\\
\left[c^{2}-d^{2}=1\right]
\end{matrix}&lt;/math&gt;

producing the invariant relation

:&lt;math&gt;(\mathrm{OC})^{2}-(\mathrm{OD})^{2}\bumpeq(\mathrm{OA})^{2}-(\mathrm{OB})^{2}&lt;/math&gt;.

Substituting into (1), he showed that OM retains its form

:&lt;math&gt;\begin{matrix}\mathrm{OM}\bumpeq(cx-dy)\mathrm{OC}+(cy-dx)\mathrm{OD}\\
\left[(cx-dy)^{2}-(cy-dx)^{2}=1\right]
\end{matrix}&lt;/math&gt;

He also defined velocity and acceleration by differentiation of (1).

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These relations are equivalent to several Lorentz boosts or hyperbolic rotations producing the invariant Lorentz interval in line with ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Escherich}} Escherich (1874) – Beltrami coordinates===

[[w:Gustav von Escherich]] (1874) discussed the plane of constant negative curvature&lt;ref&gt;Sommerville (1911), p. 297&lt;/ref&gt; based on the [[w:Beltrami–Klein model]] of hyperbolic geometry by [[#Beltrami|Beltrami (1868)]]. Similar to [[w:Christoph Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt;  who introduced axial coordinates ''x''=tan(a) and ''y''=tan(b) in sphere geometry in order to perform coordinate transformations in the case of rotation and translation, Escherich used hyperbolic functions ''x''=tanh(a/k) and ''y''=tanh(b/k)&lt;ref group=M&gt;Escherich (1874), p. 508&lt;/ref&gt; in order to give the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation have the form:&lt;ref group=M name=escher&gt;Escherich (1874), p. 510&lt;/ref&gt;

:&lt;math&gt;x=\frac{\sinh\frac{a}{k}+x'\cosh\frac{a}{k}}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt; and &lt;math&gt;y=\frac{y'}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}), also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting &lt;math&gt;\tfrac{a}{k}=\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt; and multiplying ''[x,y,x′,y′]'' by 1/''c'', and equivalent to Lorentz boost ({{equationNote|3b}}) by setting &lt;math&gt;\scriptstyle (x,\ y,\ x',\ y')=\left(\frac{x_{1}}{x_{0}},\ \frac{x_{2}}{x_{0}},\ \frac{x_{1}^{\prime}}{x_{0}^{\prime}},\ \frac{x_{2}^{\prime}}{x_{0}^{\prime}}\right)&lt;/math&gt;. This is the relation between the [[w:Beltrami–Klein model|Beltrami coordinates]] in terms of Gudermann-Escherich coordinates, and the Weierstrass coordinates of the [[w:hyperboloid model]] introduced by [[../Lorentz transformation (general)#Killing1|E:Killing (1878–1893)]], [[../Lorentz transformation (general)#Poincare|E:Poincaré (1881)]], and [[../Lorentz transformation (general)#Cox|E:Cox (1881)]]. Both coordinate systems were compared by Cox (1881).&lt;ref group=M&gt;Cox (1881), p. 186&lt;/ref&gt;&lt;/p&gt;

==={{anchor|Glaisher}} Glaisher (1878) – hyperbolic addition===

It was shown by [[w:James Whitbread Lee Glaisher]] (1878) that the hyperbolic addition laws can be written as matrix multiplication&lt;ref group=M&gt;Glaisher (1878), p. 30&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\cosh x, &amp; \sinh x\\
\sinh x, &amp; \cosh x
\end{vmatrix}=1,\ \begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y
\end{vmatrix}=1\\
\left[c_{1},c_{2},c_{3},c_{4}\right]=\left[\cosh x,\cosh y,\sinh x,\sinh y\right]\\
\Rightarrow\begin{vmatrix}c_{1}c_{2}+s_{1}s_{2}, &amp; s_{1}c_{2}+c_{1}s_{2}\\
c_{1}s_{2}+s_{1}c_{2}, &amp; s_{1}s_{2}+c_{1}c_{2}
\end{vmatrix}=1\ \Rightarrow\begin{vmatrix}\cosh(x+y), &amp; \sinh(x+y)\\
\sinh(x+y), &amp; \cosh(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{anchor|Gunther1}} Günther (1880/81) – hyperbolic addition ===

Following [[#Glaisher|Glaisher (1878)]], [[w:Siegmund Günther]] (1880/81) formulated the hyperbolic addition laws in matrix form as&lt;ref group=M&gt;Günther (1880/81), p. 405&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\mathfrak{Cos}x, &amp; \mathfrak{Sin}x\\
\mathfrak{Sin}x, &amp; \mathfrak{Cos}x
\end{vmatrix}\cdot\begin{vmatrix}\mathfrak{Cos}y, &amp; \mathfrak{Sin}y\\
\mathfrak{Sin}y, &amp; \mathfrak{Cos}y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}x\mathfrak{Cos}y+\mathfrak{Sin}x\mathfrak{Sin}y, &amp; \mathfrak{Cos}x\mathfrak{Sin}y+\mathfrak{Sin}x\mathfrak{Cos}y\\
\mathfrak{Sin}x\mathfrak{Cos}y+\mathfrak{Cos}x\mathfrak{Sin}y, &amp; \mathfrak{Sin}x\mathfrak{Sin}y+\mathfrak{Cos}x\mathfrak{Cos}y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}(x+y), &amp; \mathfrak{Sin}(x+y)\\
\mathfrak{Sin}(x+y), &amp; \mathfrak{Cos}(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{anchor|Lipschitz1}} Lipschitz (1885/86) ===
[[w:Rudolf Lipschitz]] (1885/86) formulated transformations leaving invariant the sum of squares &lt;math&gt;x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}&lt;/math&gt;, which he rewrote as &lt;math&gt;x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots+x_{n}^{2}-y_{n}^{2}=0&lt;/math&gt;. This led to the problem of finding transformations leaving invariant the pairs &lt;math&gt;x_{a}^{2}-y_{a}^{2}&lt;/math&gt; (''a''=1...n) for which he gave the following solution:&lt;ref group=M&gt;Lipschitz (1886), pp. 90–92&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{a}^{2}-y_{a}^{2}=\mathfrak{x}_{a}^{2}-\mathfrak{y}_{a}^{2}\\
\hline \begin{align}x_{a}-y_{a} &amp; =\left(\mathfrak{x}_{a}-\mathfrak{y}_{a}\right)r_{a}\\
x_{a}+y_{a} &amp; =\left(\mathfrak{x}_{a}+\mathfrak{y}_{a}\right)\frac{1}{r_{a}}
\end{align}
\quad(a)\\
\hline \begin{matrix}\begin{align}2\mathfrak{x}_{a} &amp; =\left(r_{a}+\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}-\frac{1}{r_{a}}\right)y_{a}\\
2\mathfrak{y}_{a} &amp; =\left(r_{a}-\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}+\frac{1}{r_{a}}\right)y_{a}
\end{align}
\quad(b)\end{matrix}\\
\hline \left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}\Rightarrow\begin{align}\mathfrak{x}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}\\
\mathfrak{y}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}
\end{align}
\quad(c)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Equation (a) is identical to Lorentz boost ({{equationNote|3d}}), while (c) is similar, though not identical, to Lorentz boost ({{equationNote|3b}}-C). The difference stems from his definition&lt;br&gt;&lt;math&gt;\qquad\left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}&lt;/math&gt;&lt;br&gt;whereas in accordance to expression &lt;math&gt;\sqrt{\tfrac{1+v}{1-v}}&lt;/math&gt; in ({{equationNote|3d}}) he should have stated &lt;br&gt;&lt;math&gt;\qquad\left\{ \begin{matrix}r_{a}=\frac{\sqrt{1+s_{a}}}{\sqrt{1-s_{a}}}\\
s_{a}&lt;1
\end{matrix}\right\}&lt;/math&gt;.&lt;br&gt;Using the latter choice, equations (c) would assume a form equivalent to ({{equationNote|3b}}):&lt;br&gt;&lt;math&gt;\qquad\begin{align}\mathfrak{x}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{1-s_{a}}\sqrt{1+s_{a}}}\\
\mathfrak{y}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{1-s_{a}}\sqrt{1+s_{a}}}\end{align}&lt;/math&gt;&lt;/p&gt;

==={{Anchor|Schur}} Schur (1885/86, 1900/02) – Beltrami coordinates===

[[w:Friedrich Schur]] (1885/86) discussed spaces of constant Riemann curvature, and by following [[#Beltrami|Beltrami (1868)]] he used the transformation&lt;ref group=M&gt;Schur (1885/86), p. 167&lt;/ref&gt;

:&lt;math&gt;x_{1}=R^{2}\frac{y_{1}+a_{1}}{R^{2}+a_{1}y_{1}},\ x_{2}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{2}}{R^{2}+a_{1}y_{1}},\dots,\ x_{n}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{n}}{R^{2}+a_{1}y_{1}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in arbitrary dimensions by setting ''R=c'' as the speed of light and ''a&lt;sub&gt;1&lt;/sub&gt;=v'' as relative velocity.&lt;/p&gt;

In (1900/02) he derived basic formulas of non-Eucliden geometry, including the case of translation for which he obtained the transformation similar to his previous one:&lt;ref group=M&gt;Schur (1900/02), p. 290; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;x'=\frac{x-a}{1-\mathfrak{k}ax},\quad y'=\frac{y\sqrt{1-\mathfrak{k}a^{2}}}{1-\mathfrak{k}ax}&lt;/math&gt;

where &lt;math&gt;\mathfrak{k}&lt;/math&gt; can have values &gt;0, &lt;0 or ∞.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting ''a=v'' and &lt;math&gt;\mathfrak{k}=\tfrac{1}{c^{2}}&lt;/math&gt;.&lt;/p&gt;

He also defined the triangle&lt;ref group=M&gt;Schur (1900/02), p. 291; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;\frac{1}{\sqrt{1-\mathfrak{k}c^{2}}}=\frac{1}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{1}{\sqrt{1-\mathfrak{k}b^{2}}}-\frac{a}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{b}{\sqrt{1-\mathfrak{k}b^{2}}}\cos\gamma&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to the hyperbolic law of cosines and the relativistic velocity addition ({{equationNote|3f}}, b) or [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']] by setting &lt;math&gt;[\mathfrak{k},\ c,\ a,\ b]=\left[\tfrac{1}{c^{2}},\ \sqrt{u_{x}^{\prime2}+u_{y}^{\prime2}},\ v,\ \sqrt{u_{x}^{2}+u_{y}^{2}}\right]&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Lindemann}} Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute===

[[w:Ferdinand von Lindemann]] discussed hyperbolic geometry in terms of the [[w:Cayley–Klein metric]] in his (1890/91) edition of the lectures on geometry of [[w:Alfred Clebsch]]. Citing [[../Lorentz transformation (general)#Killing|E:Killing (1885)]] and [[../Lorentz transformation (general)#Poincare|Poincaré (1887)]] in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 477–478, 524&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\Omega_{xx}=x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}-4k^{2}dx_{3}^{2}\\
\Omega_{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}
\end{matrix}&lt;/math&gt;

and used the following transformation&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 361–362&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
X_{1}X_{4}+X_{2}X_{3}=\Xi_{1}\Xi_{4}+\Xi_{2}\Xi_{3}\\
\hline \begin{align}X_{1} &amp; =\left(\lambda+\lambda_{1}\right)U_{4} &amp; \Xi_{1} &amp; =\left(\lambda-\lambda_{1}\right)U_{4} &amp; X_{1} &amp; =\frac{\lambda+\lambda_{1}}{\lambda-\lambda_{1}}\Xi_{1}\\
X_{2} &amp; =\left(\lambda+\lambda_{3}\right)U_{4} &amp; \Xi_{2} &amp; =\left(\lambda-\lambda_{3}\right)U_{4} &amp; X_{2} &amp; =\frac{\lambda+\lambda_{3}}{\lambda-\lambda_{3}}\Xi_{2}\\
X_{3} &amp; =\left(\lambda-\lambda_{3}\right)U_{2} &amp; \Xi_{3} &amp; =\left(\lambda+\lambda_{3}\right)U_{2} &amp; X_{3} &amp; =\frac{\lambda-\lambda_{3}}{\lambda+\lambda_{3}}\Xi_{3}\\
X_{4} &amp; =\left(\lambda-\lambda_{1}\right)U_{1} &amp; \Xi_{4} &amp; =\left(\lambda+\lambda_{1}\right)U_{1} &amp; X_{4} &amp; =\frac{\lambda-\lambda_{1}}{\lambda+\lambda_{1}}\Xi_{4}
\end{align}
\end{matrix}&lt;/math&gt;

into which he put&lt;ref group=M name=linde&gt;Lindemann &amp; Clebsch (1890/91), p. 496&lt;/ref&gt;

:&lt;math&gt;\begin{align}X_{1} &amp; =x_{1}+2kx_{4}, &amp; X_{2} &amp; =x_{2}+ix_{3}, &amp; \lambda+\lambda_{1} &amp; =\left(\lambda-\lambda_{1}\right)e^{a},\\
X_{4} &amp; =x_{1}-2kx_{4}, &amp; X_{3} &amp; =x_{2}-ix_{3}, &amp; \lambda+\lambda_{3} &amp; =\left(\lambda-\lambda_{3}\right)e^{\alpha i},
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3d}}) with &lt;math&gt;e^{\alpha i}=1&lt;/math&gt; and ''2k=1'' .&lt;/p&gt;

From that, he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations (''a''=0) or translations (α=0):&lt;ref group=M  name=linde /&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=0\\
\hline \begin{align}x_{2} &amp; =\xi_{2}\cos\alpha+\xi_{3}\sin\alpha, &amp; x_{1} &amp; =\xi_{1}\cos\frac{a}{i}+2ki\xi_{4}\sin\frac{a}{i},\\
x_{3} &amp; =-\xi_{2}\sin\alpha+\xi_{3}\cos\alpha, &amp; 2kx_{4} &amp; =i\xi_{1}\sin\frac{a}{i}+2k\xi_{4}\cos\frac{a}{i}.
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0 and ''2k=1''.&lt;/p&gt;

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane and gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Killing2}} Killing (1893,97) – Weierstrass coordinates===

[[w:Wilhelm Killing]] (1878–1880) gave case of translation in the form&lt;ref group=M name=killtra&gt;Killing (1893), p. 331&lt;/ref&gt;

:&lt;math&gt;y_{0}=x_{0}\operatorname{Ch}a+x_{1}\operatorname{Sh}a,\quad y_{1}=x_{0}\operatorname{Sh}a+x_{1}\operatorname{Ch}a,\quad y_{2}=x_{2}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

In 1898, Killing wrote that relation in a form similar to [[#Escherich|Escherich (1874)]], and derived the corresponding Lorentz transformation for the two cases were ''v'' is unchanged or ''u'' is unchanged:&lt;ref group=M name=kill98&gt;Killing (1898), p. 133&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\xi'=\frac{\xi\operatorname{Ch}\frac{\mu}{l}+l\operatorname{Sh}\frac{\mu}{l}}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}},\ \eta'=\frac{\eta}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}}\\
\hline \frac{u}{p}=\xi,\ \frac{v}{p}=\eta\\
\hline p'=p\operatorname{Ch}\frac{\mu}{l}+\frac{u}{l}\operatorname{Sh}\frac{\mu}{l},\quad u'=pl\operatorname{Sh}\frac{\mu}{l}+u\operatorname{Ch}\frac{\mu}{l},\quad v'=v\\
\text{or}\\
p'=p\operatorname{Ch}\frac{\nu}{l}+\frac{v}{l}\operatorname{Sh}\frac{\nu}{l},\quad u'=u,\quad v'=pl\operatorname{Sh}\frac{\nu}{l}+v\operatorname{Ch}\frac{\nu}{l}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The upper transformation system is equivalent to Lorentz transformation ({{equationNote|3e}}) and the velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] with ''l=c'' and &lt;math&gt;\mu=c\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt;, the system below is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Whitehead}} Whitehead (1897/98) – Universal algebra===

[[w:Alfred North Whitehead]] (1898) discussed the kinematics of hyperbolic space as part of his study of [[w:universal algebra]], and obtained the following transformation:&lt;ref group=M name=white&gt;Whitehead (1898), pp. 459–460&lt;/ref&gt;

:&lt;math&gt;\begin{align}x' &amp; =\left(\eta\cosh\frac{\delta}{\gamma}+\eta_{1}\sinh\frac{\delta}{\gamma}\right)e+\left(\eta\sinh\frac{\delta}{\gamma}+\eta_{1}\cosh\frac{\delta}{\gamma}\right)e_{1}\\
 &amp; \qquad+\left(\eta_{2}\cos\alpha+\eta_{3}\sin\alpha\right)e_{2}+\left(\eta_{3}\cos\alpha-\eta_{2}\sin\alpha\right)e_{3}
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0.&lt;/p&gt;

==={{anchor|Elliott}} Elliott (1903) – Invariant theory ===
[[w:Edwin Bailey Elliott]] (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:&lt;ref group=M&gt;Elliott (1903), p. 109&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=X\cosh\phi+Y\sinh\phi,\quad y=X\sinh\phi+Y\cosh\phi\\
\hline X+Y=e^{-\phi}(x+y),\quad X-Y=e^{\phi}(x-y)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}). The mentioned subgroup corresponds to the one-parameter subgroup generated by Lorentz boosts.&lt;/p&gt;

==={{anchor|Woods2}} Woods (1903) – Weierstrass coordinates ===
[[w:Frederick S. Woods]] (1903, published 1905) gave the case of translation in hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 55&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x_{1}\cos kl+x_{0}\frac{\sin kl}{k},\quad x_{2}^{\prime}=x_{2},\quad x_{2}^{\prime}=x_{3},\quad x_{0}^{\prime}=-x_{1}k\sin kl+x_{0}\cos kl&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.&lt;/p&gt;

and the loxodromic substitution for hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 72&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}x_{1}^{\prime} &amp; =x_{1}\cosh\alpha-x_{0}\sinh\alpha\\
x_{2}^{\prime} &amp; =x_{2}\cos\beta-x_{3}\sin\beta\\
x_{3}^{\prime} &amp; =x_{2}\sin\beta+x_{3}\cos\beta\\
x_{0}^{\prime} &amp; =-x_{1}\sinh\alpha+x_{0}\cosh\alpha
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with β=0.&lt;/p&gt;

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – gave the case of translation in the hyperbolic plane:&lt;ref group=M name=lieb&gt;Liebmann (1904/05), p. 174&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x'\operatorname{ch}a+p'\operatorname{sh}a,\quad y_{1}^{\prime}=y',\quad p_{1}^{\prime}=x'\operatorname{sh}a+p'\operatorname{ch}a&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Frank}} Frank (1909) – Special relativity===
In special relativity, hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity:&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;
 	
=== {{anchor|Herglotz1}} Herglotz (1909/10) – Special relativity===

In special relativity, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic, with the hyperbolic case being:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;
	
:&lt;math&gt;\begin{matrix}Z=Z'e^{\vartheta}\\
	
\begin{aligned}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
	
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
	
\end{aligned}
	
\end{matrix}&lt;/math&gt;	
	
&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3d}}).&lt;/p&gt;

==={{anchor|Varicak}} Varićak (1910) – Special relativity===

In special relativity, hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

He showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bel68sag}} 
*{{#section:History of Topics in Special Relativity/mathsource|bel68fond}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|eli03}} 
*{{#section:History of Topics in Special Relativity/mathsource|esch74}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul35}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul48a}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|glai78}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|guen80}} 
*{{#section:History of Topics in Special Relativity/mathsource|kep09}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lais74b}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam67}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lind90}}
*{{#section:History of Topics in Special Relativity/mathsource|lip86}}  
*{{#section:History of Topics in Special Relativity/mathsource|ric57}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu85}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu00}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu09}} 
*{{#section:History of Topics in Special Relativity/mathsource|tau26}} 
*{{#section:History of Topics in Special Relativity/mathsource|whit98}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L3}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
      <sha1>9f5nppvrcnd0ycm16c4bek4c8m96g8v</sha1>
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      <text bytes="47155" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via hyperbolic functions==
===Translation in the hyperbolic plane===
The case of a Lorentz transformation without spatial rotation is called a [[w:Lorentz boost]]. The simplest case can be given, for instance, by setting ''n=1'' in the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}\\
\\
x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}
\end{align}
\left|\begin{align}g_{01}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{10}^{2} &amp; =1\\
g_{01}g_{11}-g_{00}g_{10} &amp; =0\\
g_{10}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{01}^{2} &amp; =1\\
g_{10}g_{11}-g_{00}g_{01} &amp; =0
\end{align}
\rightarrow\begin{align}g_{00}^{2} &amp; =g_{11}^{2}\\
g_{01}^{2} &amp; =g_{10}^{2}
\end{align}
\right.
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}g_{00} &amp; -g_{10}\\
-g_{01} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\right|\det\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;|{{equationRef|3a}}}}

which resembles precisely the relations of [[w:hyperbolic function]]s in terms of [[w:hyperbolic angle]] &lt;math&gt;\eta&lt;/math&gt;. Thus a Lorentz boost or [[w:hyperbolic rotation]] (being the same as a rotation around an imaginary angle &lt;math&gt;i\eta=\phi&lt;/math&gt; in [[../Lorentz transformation (imaginary)#math_2b|E:'''(2b)''']] or a [[w:Translation (geometry)|translation]] in the hyperbolic plane in terms of the hyperboloid model) is given by

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline g_{00}=g_{11}=\cosh\eta,\ g_{01}=g_{10}=-\sinh\eta\\
\hline \left.\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta &amp;  &amp; =\frac{x_{0}-x_{1}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}-x_{1}v}{\sqrt{1-v^{2}}}\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta &amp;  &amp; =\frac{x_{1}-x_{0}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}-x_{0}v}{\sqrt{1-v^{2}}}\\
x_{2}^{\prime} &amp; =x_{2} &amp;  &amp; =x_{2} &amp;  &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta &amp;  &amp; =\frac{x_{0}^{\prime}-x_{1}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}^{\prime}-x_{1}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta &amp;  &amp; =\frac{x_{1}^{\prime}-x_{0}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}^{\prime}-x_{0}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{2} &amp; =x_{2}^{\prime} &amp;  &amp; =x_{2}^{\prime} &amp;  &amp; =x_{2}^{\prime}
\end{align}
\right|{\scriptstyle \begin{align}\sinh^{2}\eta-\cosh^{2}\eta &amp; =-1 &amp; (a)\\
\cosh^{2}\eta-\sinh^{2}\eta &amp; =1 &amp; (b)\\
\frac{\sinh\eta}{\cosh\eta} &amp; =\tanh\eta=v &amp; (c)\\
\frac{1}{\sqrt{1-\tanh^{2}\eta}} &amp; =\cosh\eta &amp; (d)\\
\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\sinh\eta &amp; (e)\\
\frac{\tanh q\pm\tanh\eta}{1\pm\tanh q\tanh\eta} &amp; =\tanh\left(q\pm\eta\right) &amp; (f)
\end{align}
}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\right|\det\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;|{{equationRef|3b}}}}

Hyperbolic identities (a,b) on the right of ({{equationNote|3b}}) were given by [[#Riccati|Riccati (1757)]], all identities (a,b,c,d,e,f) by [[#Lambert|Lambert (1768–1770)]]. Lorentz transformations ({{equationNote|3b}}-A) were given by [[#Laisant|Laisant (1874)]], [[#Cox|Cox (1882)]], [[#Lindemann|Lindemann (1890/91)]], [[#Gerard|Gérard (1892)]], [[#Killing2|Killing (1893, 1897/98)]], [[#Whitehead|Whitehead (1897/98)]], [[#Woods2|Woods (1903/05)]], [[#Elliott|Elliott (1903)]] and [[#Liebmann|Liebmann (1904/05)]] in terms of Weierstrass coordinates of the [[w:hyperboloid model]], while transformations similar to ({{equationNote|3b}}-C) have been used by [[#Lipschitz1|Lipschitz (1885/86)]].

Rapidity can be composed of arbitrary many rapidities &lt;math&gt;\eta_{1},\eta_{2}\dots&lt;/math&gt; as per the [[w:Hyperbolic functions#Sums of arguments|w:angle sum laws of hyperbolic sines and cosines]], so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between [[w:List of trigonometric identities#Angle sum and difference identities|w:angle sum laws of circular trigonometry]] and spatial rotations. Alternatively, the hyperbolic angle sum laws ''themselves'' can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the [[w:unit hyperbola]]:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \left[\eta=\eta_{2}-\eta_{1}\right]\\
\begin{align}x_{0}^{\prime} &amp; =\sinh\eta_{1} &amp;  &amp; =\sinh\left(\eta_{2}-\eta\right) &amp;  &amp; =\sinh\eta_{2}\cosh\eta-\cosh\eta_{2}\sinh\eta &amp;  &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =\cosh\eta_{1} &amp;  &amp; =\cosh\left(\eta_{2}-\eta\right) &amp;  &amp; =-\sinh\eta_{2}\sinh\eta+\cosh\eta_{2}\cosh\eta &amp;  &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta\\
\\
x_{0} &amp; =\sinh\eta_{2} &amp;  &amp; =\sinh\left(\eta_{1}+\eta\right) &amp;  &amp; =\sinh\eta_{1}\cosh\eta+\cosh\eta_{1}\sinh\eta &amp;  &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =\cosh\eta_{2} &amp;  &amp; =\cosh\left(\eta_{1}+\eta\right) &amp;  &amp; =\sinh\eta_{1}\sinh\eta+\cosh\eta_{1}\cosh\eta &amp;  &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \begin{align} &amp; \begin{bmatrix}x_{0}^{\prime}\\
x_{1}^{\prime}
\end{bmatrix}=\begin{bmatrix}\sinh\eta_{1}\\
\cosh\eta_{1}
\end{bmatrix}=\begin{bmatrix}\sinh\left(\eta_{2}-\eta\right)\\
\cosh\left(\eta_{2}-\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}\sinh\eta_{2}\\
\cosh\eta_{2}
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}x_{0}\\
x_{1}
\end{bmatrix}\\
 &amp; \begin{bmatrix}x_{0}\\
x_{1}
\end{bmatrix}=\begin{bmatrix}\sinh\eta_{2}\\
\cosh\eta_{2}
\end{bmatrix}=\begin{bmatrix}\sinh\left(\eta_{1}+\eta\right)\\
\cosh\left(\eta_{1}+\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}\sinh\eta_{1}\\
\cosh\eta_{1}
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\begin{bmatrix}x_{0}^{\prime}\\
x_{1}^{\prime}
\end{bmatrix}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|3c}}}}

Hyperbolic angle sum laws equivalent to Lorentz boost ({{equationNote|3c}}) were given by [[#Riccati|Riccati (1757)]] and [[#Lambert|Lambert (1768–1770)]], while the matrix representation was given by [[#Glaisher|Glaisher (1878)]] and [[#Gunther2|Günther (1880/81)]]. In special relativity, hyperbolic functions were used by [[#Frank|Frank (1909)]] and [[#Varicak|Varićak (1910)]].

Finally, because of &lt;math&gt;\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;, Lorentz boost ({{equationNote|3b}}) assumes a simple form by using [[w:squeeze mapping]]s in analogy to Euler's formula in [[../Lorentz transformation (imaginary)#math_2c|E:'''(2c)''']]:&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \begin{matrix}\begin{aligned}x_{1}^{\prime}-x_{0}^{\prime} &amp; =k\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =\frac{1}{k}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}
\end{aligned}
 &amp; \Rightarrow &amp; \begin{aligned}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}
\end{aligned}
\end{matrix}\\
\hline k=e^{\eta}=\cosh\eta+\sinh\eta=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}=\sqrt{\frac{1+v}{1-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|3d}}}} 

Lorentz transformations ({{equationNote|3d}}) for arbitrary ''k'' were given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]), while the exponential form was explicitly used by [[#Lindemann|Lindemann (1890/91)]], [[#Elliott|Elliott (1903)]], [[#Herglotz1|Herglotz (1909)]].

===Hyperbolic law of cosines===
In line with equation [[../Lorentz transformation (general)#math_1b|E:'''(1b)''']] one can use coordinates &lt;math&gt;[u_{1},\ u_{2}]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}}\right]&lt;/math&gt; inside the [[w:unit circle]] &lt;math&gt;u_{1}^{2}+u_{2}^{2}=1&lt;/math&gt;, thus the corresponding Lorentz transformations ({{equationNote|3b}}) obtain the form:

{{NumBlk|:|&lt;math&gt;\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
u_{1}^{\prime} &amp; =\frac{-\sinh\eta+u_{1}\cosh\eta}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{1}-\tanh\eta}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{1}-v}{1-u_{1}v}\\
u_{2}^{\prime} &amp; =\frac{u_{2}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-v^{2}}}{1-u_{1}v}\\
\\
u_{1} &amp; =\frac{\sinh\eta+u_{1}^{\prime}\cosh\eta}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+\tanh\eta}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+v}{1+u_{1}^{\prime}v}\\
u_{2} &amp; =\frac{u_{2}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}
\end{align}
&lt;/math&gt;|{{equationRef|3e}}}}

Transformations (A) were given by [[#Escherich|Escherich (1874)]] and [[#Killing2|Killing (1898)]], and transformations (C) by [[#Beltrami|Beltrami (1868)]] and [[#Schur|Schur (1885/86, 1900/02)]] in terms of [[w:Beltrami–Klein model|Beltrami coordinates]]&lt;ref&gt;Rosenfeld (1988), p. 231&lt;/ref&gt; of hyperbolic geometry. By using the scalar product of &lt;math&gt;\left[u_{1},u_{2}\right]&lt;/math&gt;, the resulting Lorentz transformation can be seen as equivalent to the [[w:hyperbolic law of cosines]]:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{matrix}u^{2}=u_{1}^{2}+u_{2}^{2}\\
u'^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}
\end{matrix}\left|\begin{matrix}u_{1}=u\cos\alpha\\
u_{2}=u\sin\alpha\\
\\
u_{1}^{\prime}=u'\cos\alpha'\\
u_{2}^{\prime}=u'\sin\alpha'
\end{matrix}\right|\begin{align}u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+vu'\cos\alpha'}, &amp; u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-vu\cos\alpha}\\
u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{1+vu'\cos\alpha'}, &amp; u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{1-vu\cos\alpha}\\
\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{u'\cos\alpha'+v}, &amp; \tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{u\cos\alpha-v}
\end{align}
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(vu'\sin\alpha'\right){}^{2}}}{1+vu'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(vu\sin\alpha\right){}^{2}}}{1-vu\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-u^{\prime2}}}=\frac{1}{\sqrt{1-v^{2}}}\frac{1}{\sqrt{1-u^{2}}}-\frac{v}{\sqrt{1-v^{2}}}\frac{u}{\sqrt{1-u^{2}}}\cos\alpha &amp; (b)\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha &amp; (a)
\end{matrix}&lt;/math&gt;|{{equationRef|3f}}}}

The hyperbolic law of cosines (a) was given by [[#Taurinus|Taurinus (1826) and Lobachevsky (1829/30)]] and others, while variant (b) was given by [[#Schur|Schur (1900/02)]]. By further setting ''u=u′'' it follows:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\cos\alpha=\frac{\cos\alpha'+v}{1+v\cos\alpha'},\ \sin\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{1+v\cos\alpha'},\ \tan\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{\cos\alpha'+v},\ \tan\frac{\alpha}{2}=\sqrt{\frac{1-v}{1+v}}\tan\frac{\alpha'}{2}\\
\cos\alpha'=\frac{\cos\alpha-v}{1-v\cos\alpha},\ \sin\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{1-v\cos\alpha},\ \tan\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{\cos\alpha-v},\ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+v}{1-v}}\tan\frac{\alpha}{2}
\end{matrix}&lt;/math&gt;|{{equationRef|3g}}}}

Formulas (3b) resemble the equations of an [[w:ellipse]] of [[w:Orbital eccentricity|eccentricity]] ''v/c'', [[w:eccentric anomaly]] α' and [[w:true anomaly]] α, first geometrically formulated by [[#Euler|Kepler (1609)]] and explicitly written down by [[#Euler|Euler (1735, 1748), Lagrange (1770)]] and many others in relation to planetary motions. They were also used by [[../Lorentz transformation (conformal)#Darboux|E:Darboux (1873)]] as a sphere transformation. In special relativity, these formulas describes the aberration of light, see [[../Lorentz transformation (velocity)#Velocity addition and aberration|E:velocity addition and aberration]].

==Historical notation==
==={{anchor|Euler}} Euler (1735) – True and eccentric anomaly===

[[w:Johannes Kepler]] (1609) geometrically formulated [[w:Kepler's equation]] and the relations between the [[w:mean anomaly]], [[w:true anomaly]], and [[w:eccentric anomaly]].&lt;ref group=M&gt;Kepler (1609), chapter 60. The editors of Kepler's collected papers remark (p. 482), that Kepler's relations correspond to &lt;math&gt;{\scriptstyle \alpha=\beta+e\sin\beta}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\nu=\frac{e+\cos\beta}{1+e\cos\beta}}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\beta=\frac{\cos\nu-e}{1-e\cos\nu}}&lt;/math&gt;&lt;/ref&gt;&lt;ref&gt;Volk (1976), p. 366&lt;/ref&gt; The relation between the true anomaly ''z'' and the eccentric anomaly ''P'' was algebraically expressed by [[w:Leonhard Euler]] (1735/40) as follows:&lt;ref group=M&gt;Euler (1735/40), § 19&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{\cos P+v}{1+v\cos P},\ \cos P=\frac{\cos z-v}{1-v\cos z},\ \int P=\frac{\int z\sqrt{1-v^{2}}}{1-v\cos z}&lt;/math&gt;

and in 1748:&lt;ref group=M&gt;Euler (1748a), section VIII&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{n+\cos y}{1+n\cos y},\ \sin z=\frac{\sin y\sqrt{1-n^{2}}}{1+n\cos y},\ \tan z=\frac{\sin y\sqrt{1-n^{2}}}{n+\cos y}&lt;/math&gt;

while [[w:Joseph-Louis Lagrange]] (1770/71) expressed them as follows&lt;ref group=M&gt;Lagrange (1770/71), section I&lt;/ref&gt;

:&lt;math&gt;\sin u=\frac{m\sin x}{1+n\cos x},\ \cos u=\frac{n+\cos x}{1+n\cos x},\ \operatorname{tang}\frac{1}{2}u=\frac{m}{1+n}\operatorname{tang}\frac{1}{2}x,\ \left(m^{2}=1-n^{2}\right)&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt; These relations resemble formulas ({{equationNote|3g}}), while ({{equationNote|3f}}) follows by setting &lt;math&gt;[\cos z,\sin z]=\left[u_{x},u_{y}\right]&lt;/math&gt; in Euler's formulas or &lt;math&gt;[\cos u,\sin u]=\left[u_{x},u_{y}\right]&lt;/math&gt; in Lagrange's formulas.&lt;/p&gt;

==={{anchor|Riccati}} Riccati (1757) – hyperbolic addition===

[[w:Vincenzo Riccati]] introduced hyperbolic functions in 1757,&lt;ref group=M&gt;Riccati (1757), p. 71&lt;/ref&gt;&lt;ref group=M&gt;Günther (1880/81), pp. 7–13&lt;/ref&gt; in particular he formulated the angle sum laws for hyperbolic sine and cosine:

:&lt;math&gt;\begin{matrix}\mathrm{Ch}(\varphi+\pi)=\frac{\mathrm{Ch}\varphi\mathrm{Ch}\pi+\mathrm{Sh}\varphi\mathrm{Sh}\pi}{r}\\
\mathrm{Sh}(\varphi+\pi)=\frac{\mathrm{Ch}\varphi\mathrm{Sh}\pi+\mathrm{Ch}\pi\mathrm{Sh}\varphi}{r}\\
\left[\mathrm{Ch}^{2}-\mathrm{Sh}^{2}=rr\right]
\end{matrix}&lt;/math&gt;

He furthermore showed that &lt;math&gt;\mathrm{Ch}(\varphi-\pi)&lt;/math&gt; and &lt;math&gt;\mathrm{Sh}(\varphi-\pi)&lt;/math&gt; follow by setting &lt;math&gt;\mathrm{Ch}(\pi)\Rightarrow\mathrm{Ch}(-\pi)&lt;/math&gt; and &lt;math&gt;\mathrm{Sh}(\pi)\Rightarrow\mathrm{Sh}(-\pi)&lt;/math&gt; in the above formulas.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3c}}). (In modern publications, Riccati's additional factor ''r'' is set to unity.)&lt;/p&gt;

==={{anchor|Lambert}} Lambert (1768–1770) – hyperbolic addition===

While [[#Riccati|Riccati (1757)]] discussed the hyperbolic sine and cosine, [[w:Johann Heinrich Lambert]] (read 1767, published 1768) introduced the expression ''tang φ'' or abbreviated ''tφ'' as the [[w:tangens hyperbolicus]] &lt;math&gt;{\scriptstyle \frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}}&lt;/math&gt; of a variable ''u'', or in modern notation ''tφ=tanh(u)'':&lt;ref group=M&gt;Lambert (1761/68), pp. 309–318&lt;/ref&gt;&lt;ref&gt;Barnett (2004), pp. 22–23&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}\xi\xi-1 &amp; =\eta\eta &amp; (a)\\
1+\eta\eta &amp; =\xi\xi &amp; (b)\\
\frac{\eta}{\xi} &amp; =tang\ \phi=t\phi &amp; (c)\\
\xi &amp; =\frac{1}{\sqrt{1-t\phi^{2}}} &amp; (d)\\
\eta &amp; =\frac{t\phi}{\sqrt{1-t\phi^{2}}} &amp; (e)\\
t\phi'' &amp; =\frac{t\phi+t\phi'}{1+t\phi\cdot t\phi'} &amp; (f)\\
t\phi' &amp; =\frac{t\phi''-t\phi}{1-t\phi\cdot t\phi''} &amp; (g)
\end{align}
\right|\begin{align}2u &amp; =\log\frac{1+t\phi}{1-t\phi}\\
\xi &amp; =\frac{e^{u}+e^{-u}}{2}\\
\eta &amp; =\frac{e^{u}-e^{-u}}{2}\\
t\phi &amp; =\frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}\\
e^{u} &amp; =\xi+\eta\\
e^{-u} &amp; =\xi-\eta
\end{align}&lt;/math&gt;

In (1770) he rewrote the addition law for the hyperbolic tangens (f) or (g) as:&lt;ref group=M&gt;Lambert (1770), p. 335&lt;/ref&gt;

:&lt;math&gt;\begin{align}t(y+z) &amp; =(ty+tz):(1+ty\cdot tz) &amp; (f)\\
t(y-z) &amp; =(ty-tz):(1-ty\cdot tz) &amp; (g)
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The hyperbolic relations (a,b,c,d,e,f) are equivalent to the hyperbolic relations on the right of ({{equationNote|3b}}). Relations (f,g) can also be found in ({{equationNote|3e}}). By setting ''tφ=v/c'', formula (c) becomes the relative velocity between two frames, (d) the [[w:Lorentz factor]], (e) the [[w:proper velocity]], (f) or (g) becomes the Lorentz transformation of velocity (or relativistic [[w:velocity addition formula]]) for collinear velocities in [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] and [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']].&lt;/p&gt;

Lambert also formulated the addition laws for the hyperbolic cosine and sine (Lambert's &quot;cos&quot; and &quot;sin&quot; actually mean &quot;cosh&quot; and &quot;sinh&quot;):

:&lt;math&gt;\begin{align}\sin(y+z) &amp; =\sin y\cos z+\cos y\sin z\\
\sin(y-z) &amp; =\sin y\cos z-\cos y\sin z\\
\cos(y+z) &amp; =\cos y\cos z+\sin y\sin z\\
\cos(y-z) &amp; =\cos y\cos z-\sin y\sin z
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{Anchor|Taurinus}} Taurinus (1826) – Hyperbolic law of cosines===

After the addition theorem for the tangens hyperbolicus was given by [[#Lambert|Lambert (1768)]], [[w:hyperbolic geometry]] was used by [[w:Franz Taurinus]] (1826), and later by [[w:Nikolai Lobachevsky]] (1829/30) and others, to formulate the [[w:hyperbolic law of cosines]]:&lt;ref group=M&gt;Taurinus (1826), p. 66; see also p. 272 in the translation by Engel and Stäckel (1899)&lt;/ref&gt;&lt;ref&gt;Bonola (1912), p. 79&lt;/ref&gt;&lt;ref&gt;Gray (1979), p. 242&lt;/ref&gt;

:&lt;math&gt;A=\operatorname{arccos}\frac{\cos\left(\alpha\sqrt{-1}\right)-\cos\left(\beta\sqrt{-1}\right)\cos\left(\gamma\sqrt{-1}\right)}{\sin\left(\beta\sqrt{-1}\right)\sin\left(\gamma\sqrt{-1}\right)}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;When solved for &lt;math&gt;\cos\left(\alpha\sqrt{-1}\right)&lt;/math&gt; it corresponds to the Lorentz transformation in Beltrami coordinates ({{equationNote|3f}}), and by defining the rapidities &lt;math&gt;{\scriptstyle \left(\left[\frac{U}{c},\ \frac{v}{c},\ \frac{u}{c}\right]=\left[\tanh\alpha,\ \tanh\beta,\ \tanh\gamma\right]\right)}&lt;/math&gt; it corresponds to the relativistic velocity addition formula [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']].&lt;/p&gt;

==={{anchor|Beltrami}} Beltrami (1868) – Beltrami coordinates===

[[w:Eugenio Beltrami]] (1868a) introduced coordinates of the [[w:Beltrami–Klein model]] of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:&lt;ref group=M&gt;Beltrami (1868a), pp. 287-288; Note I; Note II&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds^{2}=R^{2}\frac{\left(a^{2}+v^{2}\right)du^{2}-2uv\,du\,dv+\left(a^{2}+v^{2}\right)dv^{2}}{\left(a^{2}+u^{2}+v^{2}\right)^{2}}\\
u^{2}+v^{2}=a^{2}\\
\hline u''=\frac{aa_{0}\left(u'-r_{0}\right)}{a^{2}-r_{0}u'},\ v''=\frac{a_{0}w_{0}v'}{a^{2}-r_{0}u'},\\
\left(r_{0}=\sqrt{u_{0}^{2}+v_{0}^{2}},\ w_{0}=\sqrt{a^{2}-r_{0}^{2}}\right)\\
\hline ds^{2}=R^{2}\frac{\left(a^{2}-v^{2}\right)du^{2}+2uv\,du\,dv+\left(a^{2}-v^{2}\right)dv^{2}}{\left(a^{2}-u^{2}-v^{2}\right)^{2}}\\
(R=R\sqrt{-1},\ a=a\sqrt{-1})
\end{matrix}&lt;/math&gt;

(where the disk radius ''a'' and the [[w:radius of curvature]] ''R'' are real in spherical geometry, in hyperbolic geometry they are imaginary), and for arbitrary dimensions in (1868b)&lt;ref group=M&gt;Beltrami (1868b), pp. 232, 240–241, 253–254&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds=R\frac{\sqrt{dx^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}}}{x}\\
x^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=a^{2}\\
\hline y_{1}=\frac{ab\left(x_{1}-a_{1}\right)}{a^{2}-a_{1}x_{1}}\ \text{or}\ x_{1}=\frac{a\left(ay_{1}+a_{1}b\right)}{ab+a_{1}y_{1}},\ x_{r}=\pm\frac{ay_{r}\sqrt{a^{2}-a_{1}^{2}}}{ab+a_{1}y_{1}}\ (r=2,3,\dots,n)\\
\hline ds=R\frac{\sqrt{dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}-dx^{2}}}{x}\\
x^{2}=a^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\\
\left(R=R\sqrt{-1},\ x=x\sqrt{-1},\ a=a\sqrt{-1}\right)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Setting ''a=a&lt;sub&gt;0&lt;/sub&gt;'' Beltrami's (1868a) formulas become formulas ({{equationNote|3e}}), or in his (1868b) formulas one sets ''a=b'' for arbitrary dimensions.&lt;/p&gt;

=== {{anchor|Laisant2}} Laisant (1874) – Equipollences===
In his French translation of [[w:Giusto Bellavitis]]' principal work on [[w:Equipollence (geometry)|w:equipollences]], [[w:Charles-Ange Laisant]] (1874) added a chapter related to hyperbolas. The equipollence OM and its tangent MT of a hyperbola is defined by Laisant as&lt;ref group=M&gt;Laisant (1874b), pp. 134–135&lt;/ref&gt;

:(1) &lt;math&gt;\begin{matrix} &amp; \mathrm{OM}\bumpeq x\mathrm{OA}+y\mathrm{OB}\\
 &amp; \mathrm{MT}\bumpeq y\mathrm{OA}+x\mathrm{OB}\\
 &amp; \left[x^{2}-y^{2}=1;\ x=\cosh t,\ y=\sinh t\right]\\
\Rightarrow &amp; \mathrm{OM}\bumpeq\cosh t\cdot\mathrm{OA}+\sinh t\cdot\mathrm{OB}
\end{matrix}&lt;/math&gt;

Here, OA and OB are [[w:Conjugate diameters|conjugate semi-diameters]] of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation:

:&lt;math&gt;\begin{matrix}\begin{align}\mathrm{OC} &amp; \bumpeq c\mathrm{OA}+d\mathrm{OB} &amp; \qquad &amp;  &amp; \mathrm{OA} &amp; \bumpeq c\mathrm{OC}-d\mathrm{OD}\\
\mathrm{OD} &amp; \bumpeq d\mathrm{OA}+c\mathrm{OB} &amp;  &amp;  &amp; \mathrm{OB} &amp; \bumpeq-d\mathrm{OC}+c\mathrm{OD}
\end{align}
\\
\left[c^{2}-d^{2}=1\right]
\end{matrix}&lt;/math&gt;

producing the invariant relation

:&lt;math&gt;(\mathrm{OC})^{2}-(\mathrm{OD})^{2}\bumpeq(\mathrm{OA})^{2}-(\mathrm{OB})^{2}&lt;/math&gt;.

Substituting into (1), he showed that OM retains its form

:&lt;math&gt;\begin{matrix}\mathrm{OM}\bumpeq(cx-dy)\mathrm{OC}+(cy-dx)\mathrm{OD}\\
\left[(cx-dy)^{2}-(cy-dx)^{2}=1\right]
\end{matrix}&lt;/math&gt;

He also defined velocity and acceleration by differentiation of (1).

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These relations are equivalent to several Lorentz boosts or hyperbolic rotations producing the invariant Lorentz interval in line with ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Escherich}} Escherich (1874) – Beltrami coordinates===

[[w:Gustav von Escherich]] (1874) discussed the plane of constant negative curvature&lt;ref&gt;Sommerville (1911), p. 297&lt;/ref&gt; based on the [[w:Beltrami–Klein model]] of hyperbolic geometry by [[#Beltrami|Beltrami (1868)]]. Similar to [[w:Christoph Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt;  who introduced axial coordinates ''x''=tan(a) and ''y''=tan(b) in sphere geometry in order to perform coordinate transformations in the case of rotation and translation, Escherich used hyperbolic functions ''x''=tanh(a/k) and ''y''=tanh(b/k)&lt;ref group=M&gt;Escherich (1874), p. 508&lt;/ref&gt; in order to give the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation have the form:&lt;ref group=M name=escher&gt;Escherich (1874), p. 510&lt;/ref&gt;

:&lt;math&gt;x=\frac{\sinh\frac{a}{k}+x'\cosh\frac{a}{k}}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt; and &lt;math&gt;y=\frac{y'}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}), also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting &lt;math&gt;\tfrac{a}{k}=\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt; and multiplying ''[x,y,x′,y′]'' by 1/''c'', and equivalent to Lorentz boost ({{equationNote|3b}}) by setting &lt;math&gt;\scriptstyle (x,\ y,\ x',\ y')=\left(\frac{x_{1}}{x_{0}},\ \frac{x_{2}}{x_{0}},\ \frac{x_{1}^{\prime}}{x_{0}^{\prime}},\ \frac{x_{2}^{\prime}}{x_{0}^{\prime}}\right)&lt;/math&gt;. This is the relation between the [[w:Beltrami–Klein model|Beltrami coordinates]] in terms of Gudermann-Escherich coordinates, and the Weierstrass coordinates of the [[w:hyperboloid model]] introduced by [[../Lorentz transformation (general)#Killing1|E:Killing (1878–1893)]], [[../Lorentz transformation (general)#Poincare|E:Poincaré (1881)]], and [[../Lorentz transformation (general)#Cox|E:Cox (1881)]]. Both coordinate systems were compared by Cox (1881).&lt;ref group=M&gt;Cox (1881), p. 186&lt;/ref&gt;&lt;/p&gt;

==={{anchor|Glaisher}} Glaisher (1878) – hyperbolic addition===

It was shown by [[w:James Whitbread Lee Glaisher]] (1878) that the hyperbolic addition laws can be written as matrix multiplication&lt;ref group=M&gt;Glaisher (1878), p. 30&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\cosh x, &amp; \sinh x\\
\sinh x, &amp; \cosh x
\end{vmatrix}=1,\ \begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y
\end{vmatrix}=1\\
\left[c_{1},c_{2},c_{3},c_{4}\right]=\left[\cosh x,\cosh y,\sinh x,\sinh y\right]\\
\Rightarrow\begin{vmatrix}c_{1}c_{2}+s_{1}s_{2}, &amp; s_{1}c_{2}+c_{1}s_{2}\\
c_{1}s_{2}+s_{1}c_{2}, &amp; s_{1}s_{2}+c_{1}c_{2}
\end{vmatrix}=1\ \Rightarrow\begin{vmatrix}\cosh(x+y), &amp; \sinh(x+y)\\
\sinh(x+y), &amp; \cosh(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{anchor|Gunther1}} Günther (1880/81) – hyperbolic addition ===

Following [[#Glaisher|Glaisher (1878)]], [[w:Siegmund Günther]] (1880/81) formulated the hyperbolic addition laws in matrix form as&lt;ref group=M&gt;Günther (1880/81), p. 405&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\mathfrak{Cos}x, &amp; \mathfrak{Sin}x\\
\mathfrak{Sin}x, &amp; \mathfrak{Cos}x
\end{vmatrix}\cdot\begin{vmatrix}\mathfrak{Cos}y, &amp; \mathfrak{Sin}y\\
\mathfrak{Sin}y, &amp; \mathfrak{Cos}y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}x\mathfrak{Cos}y+\mathfrak{Sin}x\mathfrak{Sin}y, &amp; \mathfrak{Cos}x\mathfrak{Sin}y+\mathfrak{Sin}x\mathfrak{Cos}y\\
\mathfrak{Sin}x\mathfrak{Cos}y+\mathfrak{Cos}x\mathfrak{Sin}y, &amp; \mathfrak{Sin}x\mathfrak{Sin}y+\mathfrak{Cos}x\mathfrak{Cos}y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}(x+y), &amp; \mathfrak{Sin}(x+y)\\
\mathfrak{Sin}(x+y), &amp; \mathfrak{Cos}(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3c}}).&lt;/p&gt;

==={{anchor|Lipschitz1}} Lipschitz (1885/86) ===
[[w:Rudolf Lipschitz]] (1885/86) formulated transformations leaving invariant the sum of squares &lt;math&gt;x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}&lt;/math&gt;, which he rewrote as &lt;math&gt;x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots+x_{n}^{2}-y_{n}^{2}=0&lt;/math&gt;. This led to the problem of finding transformations leaving invariant the pairs &lt;math&gt;x_{a}^{2}-y_{a}^{2}&lt;/math&gt; (''a''=1...n) for which he gave the following solution:&lt;ref group=M&gt;Lipschitz (1886), pp. 90–92&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{a}^{2}-y_{a}^{2}=\mathfrak{x}_{a}^{2}-\mathfrak{y}_{a}^{2}\\
\hline \begin{align}x_{a}-y_{a} &amp; =\left(\mathfrak{x}_{a}-\mathfrak{y}_{a}\right)r_{a}\\
x_{a}+y_{a} &amp; =\left(\mathfrak{x}_{a}+\mathfrak{y}_{a}\right)\frac{1}{r_{a}}
\end{align}
\quad(a)\\
\hline \begin{matrix}\begin{align}2\mathfrak{x}_{a} &amp; =\left(r_{a}+\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}-\frac{1}{r_{a}}\right)y_{a}\\
2\mathfrak{y}_{a} &amp; =\left(r_{a}-\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}+\frac{1}{r_{a}}\right)y_{a}
\end{align}
\quad(b)\end{matrix}\\
\hline \left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}\Rightarrow\begin{align}\mathfrak{x}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}\\
\mathfrak{y}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}
\end{align}
\quad(c)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;Equation (a) is identical to Lorentz boost ({{equationNote|3d}}), while (c) is similar, though not identical, to Lorentz boost ({{equationNote|3b}}-C). The difference stems from his definition&lt;br&gt;&lt;math&gt;\qquad\left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}&lt;/math&gt;&lt;br&gt;whereas in accordance to expression &lt;math&gt;\sqrt{\tfrac{1+v}{1-v}}&lt;/math&gt; in ({{equationNote|3d}}) he should have stated &lt;br&gt;&lt;math&gt;\qquad\left\{ \begin{matrix}r_{a}=\frac{\sqrt{1+s_{a}}}{\sqrt{1-s_{a}}}\\
s_{a}&lt;1
\end{matrix}\right\}&lt;/math&gt;.&lt;br&gt;Using the latter choice, equations (c) would assume a form equivalent to ({{equationNote|3b}}):&lt;br&gt;&lt;math&gt;\qquad\begin{align}\mathfrak{x}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{1-s_{a}}\sqrt{1+s_{a}}}\\
\mathfrak{y}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{1-s_{a}}\sqrt{1+s_{a}}}\end{align}&lt;/math&gt;&lt;/p&gt;

==={{Anchor|Schur}} Schur (1885/86, 1900/02) – Beltrami coordinates===

[[w:Friedrich Schur]] (1885/86) discussed spaces of constant Riemann curvature, and by following [[#Beltrami|Beltrami (1868)]] he used the transformation&lt;ref group=M&gt;Schur (1885/86), p. 167&lt;/ref&gt;

:&lt;math&gt;x_{1}=R^{2}\frac{y_{1}+a_{1}}{R^{2}+a_{1}y_{1}},\ x_{2}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{2}}{R^{2}+a_{1}y_{1}},\dots,\ x_{n}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{n}}{R^{2}+a_{1}y_{1}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in arbitrary dimensions by setting ''R=c'' as the speed of light and ''a&lt;sub&gt;1&lt;/sub&gt;=v'' as relative velocity.&lt;/p&gt;

In (1900/02) he derived basic formulas of non-Eucliden geometry, including the case of translation for which he obtained the transformation similar to his previous one:&lt;ref group=M&gt;Schur (1900/02), p. 290; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;x'=\frac{x-a}{1-\mathfrak{k}ax},\quad y'=\frac{y\sqrt{1-\mathfrak{k}a^{2}}}{1-\mathfrak{k}ax}&lt;/math&gt;

where &lt;math&gt;\mathfrak{k}&lt;/math&gt; can have values &gt;0, &lt;0 or ∞.

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting ''a=v'' and &lt;math&gt;\mathfrak{k}=\tfrac{1}{c^{2}}&lt;/math&gt;.&lt;/p&gt;

He also defined the triangle&lt;ref group=M&gt;Schur (1900/02), p. 291; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;\frac{1}{\sqrt{1-\mathfrak{k}c^{2}}}=\frac{1}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{1}{\sqrt{1-\mathfrak{k}b^{2}}}-\frac{a}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{b}{\sqrt{1-\mathfrak{k}b^{2}}}\cos\gamma&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to the hyperbolic law of cosines and the relativistic velocity addition ({{equationNote|3f}}, b) or [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']] by setting &lt;math&gt;[\mathfrak{k},\ c,\ a,\ b]=\left[\tfrac{1}{c^{2}},\ \sqrt{u_{x}^{\prime2}+u_{y}^{\prime2}},\ v,\ \sqrt{u_{x}^{2}+u_{y}^{2}}\right]&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Lindemann}} Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute===

[[w:Ferdinand von Lindemann]] discussed hyperbolic geometry in terms of the [[w:Cayley–Klein metric]] in his (1890/91) edition of the lectures on geometry of [[w:Alfred Clebsch]]. Citing [[../Lorentz transformation (general)#Killing|E:Killing (1885)]] and [[../Lorentz transformation (general)#Poincare|Poincaré (1887)]] in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 477–478, 524&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\Omega_{xx}=x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}-4k^{2}dx_{3}^{2}\\
\Omega_{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}
\end{matrix}&lt;/math&gt;

and used the following transformation&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 361–362&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
X_{1}X_{4}+X_{2}X_{3}=\Xi_{1}\Xi_{4}+\Xi_{2}\Xi_{3}\\
\hline \begin{align}X_{1} &amp; =\left(\lambda+\lambda_{1}\right)U_{4} &amp; \Xi_{1} &amp; =\left(\lambda-\lambda_{1}\right)U_{4} &amp; X_{1} &amp; =\frac{\lambda+\lambda_{1}}{\lambda-\lambda_{1}}\Xi_{1}\\
X_{2} &amp; =\left(\lambda+\lambda_{3}\right)U_{4} &amp; \Xi_{2} &amp; =\left(\lambda-\lambda_{3}\right)U_{4} &amp; X_{2} &amp; =\frac{\lambda+\lambda_{3}}{\lambda-\lambda_{3}}\Xi_{2}\\
X_{3} &amp; =\left(\lambda-\lambda_{3}\right)U_{2} &amp; \Xi_{3} &amp; =\left(\lambda+\lambda_{3}\right)U_{2} &amp; X_{3} &amp; =\frac{\lambda-\lambda_{3}}{\lambda+\lambda_{3}}\Xi_{3}\\
X_{4} &amp; =\left(\lambda-\lambda_{1}\right)U_{1} &amp; \Xi_{4} &amp; =\left(\lambda+\lambda_{1}\right)U_{1} &amp; X_{4} &amp; =\frac{\lambda-\lambda_{1}}{\lambda+\lambda_{1}}\Xi_{4}
\end{align}
\end{matrix}&lt;/math&gt;

into which he put&lt;ref group=M name=linde&gt;Lindemann &amp; Clebsch (1890/91), p. 496&lt;/ref&gt;

:&lt;math&gt;\begin{align}X_{1} &amp; =x_{1}+2kx_{4}, &amp; X_{2} &amp; =x_{2}+ix_{3}, &amp; \lambda+\lambda_{1} &amp; =\left(\lambda-\lambda_{1}\right)e^{a},\\
X_{4} &amp; =x_{1}-2kx_{4}, &amp; X_{3} &amp; =x_{2}-ix_{3}, &amp; \lambda+\lambda_{3} &amp; =\left(\lambda-\lambda_{3}\right)e^{\alpha i},
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3d}}) with &lt;math&gt;e^{\alpha i}=1&lt;/math&gt; and ''2k=1'' .&lt;/p&gt;

From that, he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations (''a''=0) or translations (α=0):&lt;ref group=M  name=linde /&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=0\\
\hline \begin{align}x_{2} &amp; =\xi_{2}\cos\alpha+\xi_{3}\sin\alpha, &amp; x_{1} &amp; =\xi_{1}\cos\frac{a}{i}+2ki\xi_{4}\sin\frac{a}{i},\\
x_{3} &amp; =-\xi_{2}\sin\alpha+\xi_{3}\cos\alpha, &amp; 2kx_{4} &amp; =i\xi_{1}\sin\frac{a}{i}+2k\xi_{4}\cos\frac{a}{i}.
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0 and ''2k=1''.&lt;/p&gt;

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane and gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Killing2}} Killing (1893,97) – Weierstrass coordinates===

[[w:Wilhelm Killing]] (1878–1880) gave case of translation in the form&lt;ref group=M name=killtra&gt;Killing (1893), p. 331&lt;/ref&gt;

:&lt;math&gt;y_{0}=x_{0}\operatorname{Ch}a+x_{1}\operatorname{Sh}a,\quad y_{1}=x_{0}\operatorname{Sh}a+x_{1}\operatorname{Ch}a,\quad y_{2}=x_{2}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

In 1898, Killing wrote that relation in a form similar to [[#Escherich|Escherich (1874)]], and derived the corresponding Lorentz transformation for the two cases were ''v'' is unchanged or ''u'' is unchanged:&lt;ref group=M name=kill98&gt;Killing (1898), p. 133&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\xi'=\frac{\xi\operatorname{Ch}\frac{\mu}{l}+l\operatorname{Sh}\frac{\mu}{l}}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}},\ \eta'=\frac{\eta}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}}\\
\hline \frac{u}{p}=\xi,\ \frac{v}{p}=\eta\\
\hline p'=p\operatorname{Ch}\frac{\mu}{l}+\frac{u}{l}\operatorname{Sh}\frac{\mu}{l},\quad u'=pl\operatorname{Sh}\frac{\mu}{l}+u\operatorname{Ch}\frac{\mu}{l},\quad v'=v\\
\text{or}\\
p'=p\operatorname{Ch}\frac{\nu}{l}+\frac{v}{l}\operatorname{Sh}\frac{\nu}{l},\quad u'=u,\quad v'=pl\operatorname{Sh}\frac{\nu}{l}+v\operatorname{Ch}\frac{\nu}{l}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The upper transformation system is equivalent to Lorentz transformation ({{equationNote|3e}}) and the velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] with ''l=c'' and &lt;math&gt;\mu=c\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt;, the system below is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Whitehead}} Whitehead (1897/98) – Universal algebra===

[[w:Alfred North Whitehead]] (1898) discussed the kinematics of hyperbolic space as part of his study of [[w:universal algebra]], and obtained the following transformation:&lt;ref group=M name=white&gt;Whitehead (1898), pp. 459–460&lt;/ref&gt;

:&lt;math&gt;\begin{align}x' &amp; =\left(\eta\cosh\frac{\delta}{\gamma}+\eta_{1}\sinh\frac{\delta}{\gamma}\right)e+\left(\eta\sinh\frac{\delta}{\gamma}+\eta_{1}\cosh\frac{\delta}{\gamma}\right)e_{1}\\
 &amp; \qquad+\left(\eta_{2}\cos\alpha+\eta_{3}\sin\alpha\right)e_{2}+\left(\eta_{3}\cos\alpha-\eta_{2}\sin\alpha\right)e_{3}
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0.&lt;/p&gt;

==={{anchor|Elliott}} Elliott (1903) – Invariant theory ===
[[w:Edwin Bailey Elliott]] (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:&lt;ref group=M&gt;Elliott (1903), p. 109&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=X\cosh\phi+Y\sinh\phi,\quad y=X\sinh\phi+Y\cosh\phi\\
\hline X+Y=e^{-\phi}(x+y),\quad X-Y=e^{\phi}(x-y)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}). The mentioned subgroup corresponds to the one-parameter subgroup generated by Lorentz boosts.&lt;/p&gt;

==={{anchor|Woods2}} Woods (1903) – Weierstrass coordinates ===
[[w:Frederick S. Woods]] (1903, published 1905) gave the case of translation in hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 55&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x_{1}\cos kl+x_{0}\frac{\sin kl}{k},\quad x_{2}^{\prime}=x_{2},\quad x_{2}^{\prime}=x_{3},\quad x_{0}^{\prime}=-x_{1}k\sin kl+x_{0}\cos kl&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.&lt;/p&gt;

and the loxodromic substitution for hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 72&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}x_{1}^{\prime} &amp; =x_{1}\cosh\alpha-x_{0}\sinh\alpha\\
x_{2}^{\prime} &amp; =x_{2}\cos\beta-x_{3}\sin\beta\\
x_{3}^{\prime} &amp; =x_{2}\sin\beta+x_{3}\cos\beta\\
x_{0}^{\prime} &amp; =-x_{1}\sinh\alpha+x_{0}\cosh\alpha
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}) with β=0.&lt;/p&gt;

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – gave the case of translation in the hyperbolic plane:&lt;ref group=M name=lieb&gt;Liebmann (1904/05), p. 174&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x'\operatorname{ch}a+p'\operatorname{sh}a,\quad y_{1}^{\prime}=y',\quad p_{1}^{\prime}=x'\operatorname{sh}a+p'\operatorname{ch}a&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

==={{anchor|Frank}} Frank (1909) – Special relativity===
In special relativity, hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity:&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;
 	
=== {{anchor|Herglotz1}} Herglotz (1909/10) – Special relativity===

In special relativity, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic, with the hyperbolic case being:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;
	
:&lt;math&gt;\begin{matrix}Z=Z'e^{\vartheta}\\
	
\begin{aligned}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
	
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
	
\end{aligned}
	
\end{matrix}&lt;/math&gt;	
	
&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3d}}).&lt;/p&gt;

==={{anchor|Varicak}} Varićak (1910) – Special relativity===

In special relativity, hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz boost ({{equationNote|3b}}).&lt;/p&gt;

He showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bel68sag}} 
*{{#section:History of Topics in Special Relativity/mathsource|bel68fond}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|eli03}} 
*{{#section:History of Topics in Special Relativity/mathsource|esch74}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul35}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul48a}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|glai78}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|guen80}} 
*{{#section:History of Topics in Special Relativity/mathsource|kep09}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lais74b}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam67}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lind90}}
*{{#section:History of Topics in Special Relativity/mathsource|lip86}}  
*{{#section:History of Topics in Special Relativity/mathsource|ric57}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu85}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu00}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu09}} 
*{{#section:History of Topics in Special Relativity/mathsource|tau26}} 
*{{#section:History of Topics in Special Relativity/mathsource|whit98}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L3}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
      <sha1>jxx031ntap5a86qubfi0jijrtvp669q</sha1>
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    <title>History of Topics in Special Relativity/Lorentz transformation (velocity)</title>
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      <text bytes="64941" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via velocity==
===Boosts===
In the [[w:theory of relativity]], Lorentz transformations exhibit the symmetry of [[w:Minkowski spacetime]] by using a constant ''c'' as the [[w:speed of light]], and a parameter ''v'' as the relative [[w:velocity]] between two [[w:inertial reference frames]]. The corresponding formulas are identical to [[../Lorentz transformation (hyperbolic)|E:Lorentz transformations via hyperbolic functions]] introduced long before relativity was developed. In particular, the hyperbolic angle &lt;math&gt;\eta&lt;/math&gt; can be interpreted as the velocity related [[w:rapidity]] &lt;math&gt;\tanh\eta=\beta=v/c&lt;/math&gt;, so that &lt;math&gt;\gamma=\cosh\eta&lt;/math&gt; is the [[w:Lorentz factor]], &lt;math&gt;\beta\gamma=\sinh\eta&lt;/math&gt; the [[w:proper velocity]], &lt;math&gt;u'=c\tanh q&lt;/math&gt; the velocity of another object, &lt;math&gt;u=c\tanh(q+\eta)&lt;/math&gt; the [[w:velocity-addition formula]], thus transformation [[../Lorentz transformation (hyperbolic)#math_3b|E:'''(3b)''']] becomes:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}\gamma-x_{1}\beta\gamma\\
x_{1}^{\prime} &amp; =-x_{0}\beta\gamma+x_{1}\gamma\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\gamma+x_{1}^{\prime}\beta\gamma\\
x_{1} &amp; =x_{0}^{\prime}\beta\gamma+x_{1}^{\prime}\gamma\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\left|{\scriptstyle \begin{align}\beta^{2}\gamma^{2}-\gamma^{2} &amp; =-1 &amp; (a)\\
\gamma^{2}-\beta^{2}\gamma^{2} &amp; =1 &amp; (b)\\
\frac{\beta\gamma}{\gamma} &amp; =\beta &amp; (c)\\
\frac{1}{\sqrt{1-\beta^{2}}} &amp; =\gamma &amp; (d)\\
\frac{\beta}{\sqrt{1-\beta^{2}}} &amp; =\beta\gamma &amp; (e)\\
\frac{u'+v}{1+\frac{u'v}{c^{2}}} &amp; =u &amp; (f)
\end{align}
}\right.
\end{matrix}&lt;/math&gt;|{{equationRef|4a}}}}

Or in four dimensions and by setting &lt;math&gt;x_{0}=ct,\ x_{1}=x,\ x_{2}=y&lt;/math&gt; and adding an unchanged ''z'' the familiar form follows

{{NumBlk|:|&lt;math&gt;\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}+z^{2}=-c^{2}t^{\prime2}+x^{\prime2}+y^{\prime2}+z^{\prime2}\\
\hline \left.\begin{align}t' &amp; =\gamma\left(t-x\frac{v}{c^{2}}\right)\\
x' &amp; =\gamma(x-vt)\\
y' &amp; =y\\
z' &amp; =z
\end{align}
\right|\begin{align}t &amp; =\gamma\left(t'+x\frac{v}{c^{2}}\right)\\
x &amp; =\gamma(x'+vt')\\
y &amp; =y'\\
z &amp; =z'
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|4b}}}}

Analogous transformations have been introduced by [[#Voigt|Voigt (1887)]] in terms of an incompessible medium, and by [[#Lorentz1|Lorentz (1892, 1895)]] who analyzed [[w:Maxwell's equations]], they were completed by [[#Larmor|Larmor (1897, 1900)]] and [[#Lorentz2|Lorentz (1899, 1904)]], and brought into their modern form by [[#Poincare3|Poincaré (1905)]] who gave the transformation the name of Lorentz.&lt;ref&gt;Miller (1981), chapter 1&lt;/ref&gt; Eventually, [[#Einstein|Einstein (1905)]] showed in his development of [[w:special relativity]] that the transformations follow from the [[w:principle of relativity]] and constant light speed alone by modifying the traditional concepts of space and time, without requiring a [[w:Lorentz ether theory|mechanical aether]] in contradistinction to Lorentz and Poincaré.&lt;ref&gt;Miller (1981), chapter 4–7&lt;/ref&gt; [[#Minkowski|Minkowski (1907–1908)]] used them to argue that space and time are inseparably connected as [[w:spacetime]]. [[#Minkowski|Minkowski (1907–1908)]], [[#Frank|Frank (1909)]]  and [[#Varicak|Varićak (1910)]] showed the relation to imaginary and hyperbolic functions. Important contributions to the mathematical understanding of the Lorentz transformation were also made by other authors such as [[#Herglotz1|Herglotz (1909/10)]], [[#Ignatowski|Ignatowski (1910)]], [[#Noether|Noether (1910) and Klein (1910)]], [[#Borel|Borel (1913–14)]].

===Velocity addition and aberration===
In exact analogy to Beltrami coordinates in equation [[../Lorentz transformation (hyperbolic)#math_3e|E:'''(3e)''']], one can substitute &lt;math&gt;\left[\tfrac{u_{x}}{c},\ \tfrac{u_{y}}{c},\ 1\right]=\left[\tfrac{x}{ct},\ \tfrac{y}{ct},\ \tfrac{ct}{ct}\right]&lt;/math&gt; in ({{equationNote|4a}}), producing the Lorentz transformation of velocities (or [[w:velocity addition formula]]) :

{{NumBlk|:|&lt;math&gt;\begin{matrix}\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}=-c^{2}t^{\prime2}+x^{\prime2}+y^{\prime2} &amp; \rightarrow &amp; \begin{align}-c^{2}+u_{x}^{2}+u_{y}^{2} &amp; =\frac{-c^{2}+u_{x}^{\prime2}+u_{y}^{\prime2}}{\gamma^{2}\left(1+\frac{v}{c^{2}}u_{x}^{\prime}\right)^{2}}\\
\frac{-c^{2}+u_{x}^{2}+u_{y}^{2}}{\gamma^{2}\left(1-\frac{v}{c^{2}}u_{x}\right)^{2}} &amp; =-c^{2}+u_{x}^{\prime2}+u_{y}^{\prime2}
\end{align}
\\
\hline -c^{2}t^{2}+x^{2}+y^{2}=-c^{2}t^{\prime2}+x^{\prime2}+y^{\prime2}=0 &amp; \rightarrow &amp; -c^{2}+u_{x}^{2}+u_{y}^{2}=-c^{2}+u_{x}^{\prime2}+u_{y}^{\prime2}=0
\end{matrix}\\
\hline {\scriptstyle \begin{align}\frac{\sinh\eta}{\cosh\eta} &amp; =\tanh\eta=\frac{v}{c}\\
\cosh\eta &amp; =\frac{1}{\sqrt{1-\tanh^{2}\eta}}
\end{align}
}\left|\begin{align}u_{x}^{\prime} &amp; =\frac{-c^{2}\sinh\eta+u_{x}c\cosh\eta}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{x}-c\tanh\eta}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}-v}{1-\frac{v}{c^{2}}u{}_{x}}\\
u_{y}^{\prime} &amp; =\frac{cu_{y}}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\tanh^{2}\eta}}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u{}_{x}}\\
\\
u_{x} &amp; =\frac{c^{2}\sinh\eta+u_{x}^{\prime}c\cosh\eta}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+c\tanh\eta}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+v}{1+\frac{v}{c^{2}}u_{x}^{\prime}}\\
u_{y} &amp; =\frac{cy'}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u_{x}^{\prime}}
\end{align}
\right.
\end{matrix}&lt;/math&gt;|{{equationRef|4c}}}}

or using trigonometric and hyperbolic identities as in equation [[../Lorentz transformation (hyperbolic)#math_3f|E:'''(3f)''']], it becomes the hyperbolic law of cosines:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{matrix}u^{2}=u_{x}^{2}+u_{y}^{2}\\
u'^{2}=u_{x}^{\prime2}+u_{y}^{\prime2}
\end{matrix}\left|\begin{matrix}u_{x}=u\cos\alpha\\
u_{y}=u\sin\alpha\\
\\
u_{x}^{\prime}=u'\cos\alpha'\\
u_{y}^{\prime}=u'\sin\alpha'
\end{matrix}\right|\begin{align}u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-\frac{v}{c^{2}}u\cos\alpha}\\
u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{u'\cos\alpha'+v}, &amp; \tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{u\cos\alpha-v}
\end{align}
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(\frac{vu'\sin\alpha'}{c}\right){}^{2}}}{1+\frac{v}{c^{2}}u'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(\frac{vu\sin\alpha}{c}\right){}^{2}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\frac{u^{\prime2}}{c^{2}}}}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{1}{\sqrt{1-\frac{u^{2}}{c^{2}}}}-\frac{v/c}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{u/c}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\cos\alpha\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha
\end{matrix}&lt;/math&gt;|{{equationRef|4d}}}}

and by further setting ''u=u′=c'' one gets the old [[../Lorentz transformation (hyperbolic)#math_3g|E:Kepler formulas '''(3g)''']], which  express the relativistic [[w:aberration of light]]:&lt;ref&gt;Pauli (1921), pp. 562; 565–566&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\cos\alpha=\frac{\cos\alpha'+\frac{v}{c}}{1+\frac{v}{c}\cos\alpha'},\ \sin\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c}\cos\alpha'},\ \tan\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha'+\frac{v}{c}},\ \tan\frac{\alpha}{2}=\sqrt{\frac{c-v}{c+v}}\tan\frac{\alpha'}{2}\\
\cos\alpha'=\frac{\cos\alpha-\frac{v}{c}}{1-\frac{v}{c}\cos\alpha},\ \sin\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c}\cos\alpha},\ \tan\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha-\frac{v}{c}},\ \tan\frac{\alpha'}{2}=\sqrt{\frac{c+v}{c-v}}\tan\frac{\alpha}{2}
\end{matrix}&lt;/math&gt;|{{equationRef|4e}}}}

Formulas ({{equationNote|4c}}, {{equationNote|4d}}) were given by [[#Einstein|Einstein (1905)]] and [[#Poincare3|Poincaré (1905/06)]], while the relations to the spherical and hyperbolic law of cosines were given by [[#Sommerfeld|Sommerfeld (1909)]] and [[#Varicak|Varićak (1910)]]. The aberration formula for cos(α) was given by [[#Einstein|Einstein (1905)]].&lt;ref group=R name=plum&gt;Plummer (1910), pp. 258-259: After deriving the relativistic expressions for the aberration angles φ' and φ, Plummer remarked on p. 259: ''Another geometrical representation is obtained by assimilating φ' to the eccentric and φ to the true anomaly in an ellipse whose eccentricity is v/U = sin β.''&lt;/ref&gt;&lt;ref name=robin&gt;Robinson (1990), chapter 3-4, analyzed the relation between &quot;Kepler's formula&quot; and the &quot;physical velocity addition formula&quot; in special relativity.&lt;/ref&gt;

===Vectorial Lorentz transformation===
Also Lorentz boosts for arbitrary directions in line with [[../Lorentz transformation (general)#math_1a|E:general Lorentz transformation '''(1a)''']] can be given as:&lt;ref&gt;Møller (1952/55), Chapter II, § 18&lt;/ref&gt;

:&lt;math&gt;\mathbf{x}'=\begin{bmatrix}\gamma &amp; -\gamma\beta n_{x} &amp; -\gamma\beta n_{y} &amp; -\gamma\beta n_{z}\\
-\gamma\beta n_{x} &amp; 1+(\gamma-1)n_{x}^{2} &amp; (\gamma-1)n_{x}n_{y} &amp; (\gamma-1)n_{x}n_{z}\\
-\gamma\beta n_{y} &amp; (\gamma-1)n_{y}n_{x} &amp; 1+(\gamma-1)n_{y}^{2} &amp; (\gamma-1)n_{y}n_{z}\\
-\gamma\beta n_{z} &amp; (\gamma-1)n_{z}n_{x} &amp; (\gamma-1)n_{z}n_{y} &amp; 1+(\gamma-1)n_{z}^{2}
\end{bmatrix}\cdot\mathbf{x},\quad\left[\mathbf{n}=\frac{\mathbf{v}}{v}\right]&lt;/math&gt;

or in vector notation

{{NumBlk|:|&lt;math&gt;\begin{align}t' &amp; =\gamma\left(t-\frac{v\mathbf{n}\cdot\mathbf{r}}{c^{2}}\right)\\
\mathbf{r}' &amp; =\mathbf{r}+(\gamma-1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n}-\gamma tv\mathbf{n}
\end{align}
&lt;/math&gt;|{{equationRef|4f}}}}

The corresponding vectorial velocity addition formula in line with [[../Lorentz transformation (general)#math_1b|E:general Lorentz transformation '''(1b)''']] is given by

{{NumBlk|:|&lt;math&gt;\mathbf{u}'=\frac{1}{1+\frac{\mathbf{v}\cdot\mathbf{u}}{c^{2}}}\left[\frac{\mathbf{u}}{\gamma_{\mathbf{v}}}+\mathbf{v}+\frac{1}{c^{2}}\frac{\gamma_{\mathbf{v}}}{\gamma_{\mathbf{v}}+1}(\mathbf{u}\cdot\mathbf{v})\mathbf{v}\right]&lt;/math&gt;|{{equationRef|4g}}}}

Such transformations were formulated by [[#Herglotz2|Ignatowski (1910), Herglotz (1911)]] and others.

==Historical notation=={{aor|Voigt}} Voigt (1887) ===

[[w:Woldemar Voigt]] (1887)&lt;ref group=R&gt;Voigt (1887), p. 45&lt;/ref&gt; developed a transformation in connection with the [[w:Doppler effect]] and an incompressible medium, being in modern notation:&lt;ref&gt;Miller (1981), 114–115&lt;/ref&gt;&lt;ref name=pais&gt;Pais (1982), Kap. 6b&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\xi_{1} &amp; =x_{1}-\varkappa t\\
\eta_{1} &amp; =y_{1}q\\
\zeta_{1} &amp; =z_{1}q\\
\tau &amp; =t-\frac{\varkappa x_{1}}{\omega^{2}}\\
q &amp; =\sqrt{1-\frac{\varkappa^{2}}{\omega^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =\frac{y}{\gamma}\\
z^{\prime} &amp; =\frac{z}{\gamma}\\
t^{\prime} &amp; =t-\frac{vx}{c^{2}}\\
\frac{1}{\gamma} &amp; =\sqrt{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

If the right-hand sides of his equations are multiplied by γ they are the modern Lorentz transformation ({{equationNote|4b}}). In Voigt's theory the speed of light is invariant, but his transformations mix up a relativistic boost together with a rescaling of space-time. Optical phenomena in free space are [[w:Scale invariance|scale]], [[w:Conformal map|conformal]] (using the factor λ discussed [[#Lorsph|above]]), and [[w:Lorentz covariance|Lorentz invariant]], so the combination is invariant too.&lt;ref name=pais /&gt; For instance, Lorentz transformations can be extended by using &lt;math&gt;l=\sqrt{\lambda}&lt;/math&gt;:&lt;ref group=R&gt;Lorentz (1915/16), p. 197&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\quad y^{\prime}=ly,\quad z^{\prime}=lz,\quad t^{\prime}=\gamma l\left(t-x\frac{v}{c^{2}}\right)&lt;/math&gt;.

''l''=1/γ gives the Voigt transformation, ''l''=1 the Lorentz transformation. But scale transformations are not a symmetry of all the laws of nature, only of electromagnetism, so these transformations cannot be used to formulate a [[w:principle of relativity]] in general. It was demonstrated by Poincaré and Einstein that one has to set ''l''=1 in order to make the above transformation symmetric and to form a group as required by the relativity principle, therefore the Lorentz transformation is the only viable choice.

Voigt sent his 1887 paper to Lorentz in 1908,&lt;ref&gt;Voigt's transformations and the beginning of the relativistic revolution, Ricardo Heras, arXiv:1411.2559 [https://arxiv.org/abs/1411.2559]&lt;/ref&gt; and that was acknowledged in 1909: {{Quote|In a paper &quot;Über das Doppler'sche Princip&quot;, published in 1887 (Gött. Nachrichten, p. 41) and which to my regret has escaped my notice all these years, Voigt has applied to equations of the form (7) (§ 3 of this book) [namely &lt;math&gt;\Delta\Psi-\tfrac{1}{c^{2}}\tfrac{\partial^{2}\Psi}{\partial t^{2}}=0&lt;/math&gt;] a transformation equivalent to the formulae (287) and (288) [namely &lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\ y^{\prime}=ly,\ z^{\prime}=lz,\ t^{\prime}=\gamma l\left(t-\tfrac{v}{c^{2}}x\right)&lt;/math&gt;]. The idea of the transformations used above (and in § 44) might therefore have been borrowed from Voigt and the proof that it does not alter the form of the equations for the ''free'' ether is contained in his paper.&lt;ref group=R&gt;Lorentz (1915/16), p. 198&lt;/ref&gt;}}

Also [[w:Hermann Minkowski]] said in 1908 that the transformations which play the main role in the principle of relativity were first examined by Voigt in 1887. Voigt responded in the same paper by saying that his theory was based on an elastic theory of light, not an electromagnetic one. However, he concluded that some results were actually the same.&lt;ref group=R&gt;Bucherer (1908), p. 762&lt;/ref&gt;

==={{anchor|Heaviside}} Heaviside (1888), Thomson (1889), Searle (1896)===

In 1888, [[w:Oliver Heaviside]]&lt;ref group=R&gt;Heaviside (1888), p. 324&lt;/ref&gt; investigated the properties of [[w:Relativistic electromagnetism|charges in motion]] according to Maxwell's electrodynamics. He calculated, among other things, anisotropies in the electric field of moving bodies represented by this formula:&lt;ref&gt;Brown (2003)&lt;/ref&gt;

:&lt;math&gt;\mathrm{E}=\left(\frac{q\mathrm{r}}{r^{2}}\right)\left(1-\frac{v^{2}\sin^{2}\theta}{c^{2}}\right)^{-3/2}&lt;/math&gt;.

Consequently, [[w:Joseph John Thomson]] (1889)&lt;ref group=R&gt;Thomson (1889), p. 12&lt;/ref&gt; found a way to substantially simplify calculations concerning moving charges by using the following mathematical transformation (like other authors such as Lorentz or Larmor, also Thomson implicitly used the [[w:Galilean transformation]] ''z-vt'' in his equation&lt;ref name=mil /&gt;):

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}z &amp; =\left\{ 1-\frac{\omega^{2}}{v^{2}}\right\} ^{\frac{1}{2}}z'\end{align}
\right| &amp; \begin{align}z^{\ast}=z-vt &amp; =\frac{z'}{\gamma}\end{align}
\end{matrix}&lt;/math&gt;

Thereby, [[w:inhomogeneous electromagnetic wave equation]]s are transformed into a [[w:Poisson equation]].&lt;ref name=mil&gt;Miller (1981), 98–99&lt;/ref&gt; Eventually, [[w:George Frederick Charles Searle]]&lt;ref group=R&gt;Searle (1886), p. 333&lt;/ref&gt; noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called &quot;Heaviside-Ellipsoid&quot; of [[w:axial ratio]]

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align} &amp; \sqrt{\alpha}:1:1\\
\alpha= &amp; 1-\frac{u^{2}}{v^{2}}
\end{align}
\right| &amp; \begin{align} &amp; \frac{1}{\gamma}:1:1\\
\frac{1}{\gamma^{2}} &amp; =1-\frac{v^{2}}{c^{2}}
\end{align}
\end{matrix}&lt;/math&gt;&lt;ref name=mil /&gt;

=== {{anchor|Lorentz1}} Lorentz (1892, 1895) ===

In order to explain the [[w:aberration of light]] and the result of the [[w:Fizeau experiment]] in accordance with [[w:Maxwell's equations]], Lorentz in 1892 developed a model (&quot;[[w:Lorentz ether theory]]&quot;) in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. In order to calculate the optics of moving bodies, Lorentz introduced the following quantities to transform from the aether system into a moving system (it's unknown whether he was influenced by Voigt, Heaviside, and Thomson)&lt;ref group=R&gt;Lorentz (1892a), p. 141&lt;/ref&gt;&lt;ref name=milf&gt;Miller (1982), 1.4 &amp; 1.5&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\mathfrak{x} &amp; =\frac{V}{\sqrt{V^{2}-p^{2}}}x\\
t' &amp; =t-\frac{\varepsilon}{V}\mathfrak{x}\\
\varepsilon &amp; =\frac{p}{\sqrt{V^{2}-p^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)\\
\gamma\frac{v}{c} &amp; =\frac{v}{\sqrt{c^{2}-v^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

where ''x&lt;sup&gt;*&lt;/sup&gt;'' is the [[w:Galilean transformation]] ''x-vt''. Except the additional γ in the time transformation, this is the complete Lorentz transformation ({{equationNote|4b}}).&lt;ref name=milf /&gt; While ''t'' is the &quot;true&quot; time for observers resting in the aether, ''t′'' is an auxiliary variable only for calculating processes for moving systems. It is also important that Lorentz and later also Larmor formulated this transformation in two steps. At first an implicit Galilean transformation, and later the expansion into the &quot;fictitious&quot; electromagnetic system with the aid of the Lorentz transformation. In order to explain the negative result of the [[w:Michelson–Morley experiment]], he (1892b)&lt;ref group=R&gt;Lorentz (1892b), p. 141&lt;/ref&gt; introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced [[w:length contraction]] in his theory (without proof as he admitted). The same hypothesis was already made by [[w:George FitzGerald]] in 1889 based on Heaviside's work. While length contraction was a real physical effect for Lorentz, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation.

In 1895, Lorentz further elaborated on his theory and introduced the &quot;theorem of corresponding states&quot;. This theorem states that a moving observer (relative to the ether) in his &quot;fictitious&quot; field makes the same observations as a resting observers in his &quot;real&quot; field for velocities to first order in ''v/c''. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:&lt;ref group=R&gt;Lorentz (1895), p. 37&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =x^{\prime}\sqrt{1-\frac{\mathfrak{p}^{2}}{V^{2}}}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{x^{\prime}}{\gamma}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\end{matrix}&lt;/math&gt;

For solving optical problems Lorentz used the following transformation, in which the modified time variable was called &quot;local time&quot; ({{lang-de|Ortszeit}}) by him:&lt;ref group=R&gt;Lorentz (1895), p. 49 for local time and p. 56 for spatial coordinates.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\mathrm{x}-\mathfrak{p}_{x}t\\
y &amp; =\mathrm{y}-\mathfrak{p}_{y}t\\
z &amp; =\mathrm{z}-\mathfrak{p}_{z}t\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}}x-\frac{\mathfrak{p}_{y}}{V^{2}}y-\frac{\mathfrak{p}_{z}}{V^{2}}z
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-v_{x}t\\
y^{\prime} &amp; =y-v_{y}t\\
z^{\prime} &amp; =z-v_{z}t\\
t^{\prime} &amp; =t-\frac{v_{x}}{c^{2}}x'-\frac{v_{y}}{c^{2}}y'-\frac{v_{z}}{c^{2}}z'
\end{align}
\end{matrix}&lt;/math&gt;

With this concept Lorentz could explain the [[w:Doppler effect]], the [[w:aberration of light]], and the [[w:Fizeau experiment]].&lt;ref&gt;Janssen (1995), 3.1&lt;/ref&gt;

=== {{anchor|Larmor}} Larmor (1897, 1900) ===

In 1897, Larmor extended the work of Lorentz and derived the following transformation&lt;ref group=R&gt;Larmor (1897), p. 229&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =x\varepsilon^{\frac{1}{2}}\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-vx/c^{2}\\
dt_{1} &amp; =dt^{\prime}\varepsilon^{-\frac{1}{2}}\\
\varepsilon &amp; =\left(1-v^{2}/c^{2}\right)^{-1}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-\frac{vx^{\ast}}{c^{2}}=t-\frac{v(x-vt)}{c^{2}}\\
dt_{1} &amp; =\frac{dt^{\prime}}{\gamma}\\
\gamma^{2} &amp; =\frac{1}{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

Larmor noted that if it is assumed that the constitution of molecules is electrical then the FitzGerald–Lorentz contraction is a consequence of this transformation, explaining the [[w:Michelson–Morley experiment]]. It's notable that Larmor was the first who recognized that some sort of [[w:time dilation]] is a consequence of this transformation as well, because &quot;individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 1/γ&quot;.&lt;ref&gt;Darrigol (2000), Chap. 8.5&lt;/ref&gt;&lt;ref&gt;Macrossan (1986)&lt;/ref&gt; Larmor wrote his electrodynamical equations and transformations neglecting terms of higher order than ''(v/c)''&lt;sup&gt;2&lt;/sup&gt; – when his 1897 paper was reprinted in 1929, Larmor added the following comment in which he described how they can be made valid to all orders of ''v/c'':&lt;ref group=R&gt;Larmor (1897/1929), p. 39&lt;/ref&gt;

{{Quote|Nothing need be neglected: the transformation is ''exact'' if ''v/c''&lt;sup&gt;2&lt;/sup&gt; is replaced by ''εv/c''&lt;sup&gt;2&lt;/sup&gt; in the equations and also in the change following from ''t'' to ''t′'', as is worked out in ''Aether and Matter'' (1900), p. 168, and as Lorentz found it to be in 1904, thereby stimulating the modern schemes of intrinsic relational relativity.}}

In line with that comment, in his book Aether and Matter published in 1900, Larmor used a modified local time ''t″=t′-εvx′/c&lt;sup&gt;2&lt;/sup&gt;'' instead of the 1897 expression ''t′=t-vx/c&lt;sup&gt;2&lt;/sup&gt;'' by replacing ''v/c''&lt;sup&gt;2&lt;/sup&gt; with ''εv/c''&lt;sup&gt;2&lt;/sup&gt;, so that ''t″'' is now identical to the one given by Lorentz in 1892, which he combined with a Galilean transformation for the ''x′, y′, z′, t′'' coordinates:&lt;ref group=R&gt;Larmor (1900), p. 168&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime} &amp; =t^{\prime}-\varepsilon vx^{\prime}/c^{2}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime}=t^{\prime}-\frac{\gamma^{2}vx^{\prime}}{c^{2}} &amp; =\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor ''(v/c)''&lt;sup&gt;2&lt;/sup&gt;, and so he sought the transformations which were &quot;accurate to second order&quot; (as he put it). Thus he wrote the final transformations (where ''x′=x-vt'' and ''t″'' as given above) as:&lt;ref group=R&gt;Larmor (1900), p. 174&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =\varepsilon^{\frac{1}{2}}x^{\prime}\\
y_{1} &amp; =y^{\prime}\\
z_{1} &amp; =z^{\prime}\\
dt_{1} &amp; =\varepsilon^{-\frac{1}{2}}dt^{\prime\prime}=\varepsilon^{-\frac{1}{2}}\left(dt^{\prime}-\frac{v}{c^{2}}\varepsilon dx^{\prime}\right)\\
t_{1} &amp; =\varepsilon^{-\frac{1}{2}}t^{\prime}-\frac{v}{c^{2}}\varepsilon^{\frac{1}{2}}x^{\prime}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\prime}=\gamma(x-vt)\\
y_{1} &amp; =y'=y\\
z_{1} &amp; =z'=z\\
dt_{1} &amp; =\frac{dt^{\prime\prime}}{\gamma}=\frac{1}{\gamma}\left(dt^{\prime}-\frac{\gamma^{2}vdx^{\prime}}{c^{2}}\right)=\gamma\left(dt-\frac{vdx}{c^{2}}\right)\\
t_{1} &amp; =\frac{t^{\prime}}{\gamma}-\frac{\gamma vx^{\prime}}{c^{2}}=\gamma\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

by which he arrived at the complete Lorentz transformation ({{equationNote|4b}}). Larmor showed that Maxwell's equations were invariant under this two-step transformation, &quot;to second order in ''v/c''&quot; – it was later shown by Lorentz (1904) and Poincaré (1905) that they are indeed invariant under this transformation to all orders in ''v/c''.

Larmor gave credit to Lorentz in two papers published in 1904, in which he used the term &quot;Lorentz transformation&quot; for Lorentz's first order transformations of coordinates and field configurations:

{{Quote|p. 583: [..] Lorentz's transformation for passing from the field of activity of a stationary electrodynamic material system to that of one moving with uniform velocity of translation through the aether.&lt;br /&gt; p. 585: [..] the Lorentz transformation has shown us what is not so immediately obvious [..]&lt;ref group=R&gt;Larmor (1904a), p. 583, 585&lt;/ref&gt; &lt;br /&gt; p. 622: [..] the transformation first developed by Lorentz: namely, each point in space is to have its own origin from which time is measured, its &quot;local time&quot; in Lorentz's phraseology, and then the values of the electric and magnetic vectors [..] at all points in the aether between the molecules in the system at rest, are the same as those of the vectors [..] at the corresponding points in the convected system at the same local times.&lt;ref group=R&gt;Larmor (1904b), p. 622&lt;/ref&gt;}}

=== {{anchor|Lorentz2}} Lorentz (1899, 1904) ===

Also Lorentz extended his theorem of corresponding states in 1899. First he wrote a transformation equivalent to the one from 1892 (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1899), p. 429&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}x\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}-\mathfrak{p}_{x}^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Then he introduced a factor ε of which he said he has no means of determining it, and modified his transformation as follows (where the above value of ''t′'' has to be inserted):&lt;ref group=R&gt;Lorentz (1899), p. 439&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\frac{\varepsilon}{k}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon x^{\prime\prime}\\
t^{\prime} &amp; =k\varepsilon t^{\prime\prime}\\
k &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{\varepsilon}{\gamma}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon z^{\prime\prime}\\
t^{\prime}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right) &amp; =\gamma\varepsilon t^{\prime\prime}\\
\gamma &amp; =\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
\end{align}
\end{matrix}&lt;/math&gt;

This is equivalent to the complete Lorentz transformation ({{equationNote|4b}}) when solved for ''x″'' and ''t″'' and with ε=1. Like Larmor, Lorentz noticed in 1899&lt;ref group=R&gt;Lorentz (1899), p. 442&lt;/ref&gt; also some sort of time dilation effect in relation to the frequency of oscillating electrons ''&quot;that in ''S'' the time of vibrations be ''kε'' times as great as in ''S&lt;sub&gt;0&lt;/sub&gt;''&quot;'', where ''S&lt;sub&gt;0&lt;/sub&gt;'' is the aether frame.&lt;ref&gt;Janssen (1995), Kap. 3.3&lt;/ref&gt;

In 1904 he rewrote the equations in the following form by setting ''l''=1/ε (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1904), p. 812&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =klx\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =\frac{l}{k}t-kl\frac{w}{c^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma lx^{\ast}=\gamma l(x-vt)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t^{\prime} &amp; =\frac{lt}{\gamma}-\frac{\gamma lvx^{\ast}}{c^{2}}=\gamma l\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Under the assumption that ''l=1'' when ''v''=0, he demonstrated that ''l=1'' must be the case at all velocities, therefore length contraction can only arise in the line of motion. So by setting the factor ''l'' to unity, Lorentz's transformations now assumed the same form as Larmor's and are now completed. Unlike Larmor, who restricted himself to show the covariance of Maxwell's equations to second order, Lorentz tried to widen its covariance to all orders in ''v/c''. He also derived the correct formulas for the velocity dependence of [[w:electromagnetic mass]], and concluded that the transformation formulas must apply to all forces of nature, not only electrical ones.&lt;ref group=R&gt;Lorentz (1904), p. 826&lt;/ref&gt; However, he didn't achieve full covariance of the transformation equations for charge density and velocity.&lt;ref&gt;Miller (1981), Chap. 1.12.2&lt;/ref&gt; When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:&lt;ref&gt;Janssen (1995), Chap. 3.5.6&lt;/ref&gt;

{{Quote|One will notice that in this work the transformation equations of Einstein’s Relativity Theory have not quite been attained. [..] On this circumstance depends the clumsiness of many of the further considerations in this work.}}

Lorentz's 1904 transformation was cited and used by [[w:Alfred Bucherer]] in July 1904:&lt;ref group=R&gt;Bucherer, p. 129; Definition of s on p. 32&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\sqrt{s}x,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{\sqrt{s}}-\sqrt{s}\frac{u}{v^{2}}x,\quad s=1-\frac{u^{2}}{v^{2}}&lt;/math&gt;

or by [[w:Wilhelm Wien]] in July 1904:&lt;ref group=R&gt;Wien (1904), p. 394&lt;/ref&gt;

:&lt;math&gt;x=kx',\quad y=y',\quad z=z',\quad t'=kt-\frac{v}{kc^{2}}x&lt;/math&gt;

or by [[w:Emil Cohn]] in November 1904 (setting the speed of light to unity):&lt;ref group=R&gt;Cohn (1904a), pp. 1296-1297&lt;/ref&gt;

:&lt;math&gt;x=\frac{x_{0}}{k},\quad y=y_{0},\quad z=z_{0},\quad t=kt_{0},\quad t_{1}=t_{0}-w\cdot r_{0},\quad k^{2}=\frac{1}{1-w^{2}}&lt;/math&gt;

or by [[w:Richard Gans]] in February 1905:&lt;ref group=R&gt;Gans (1905), p. 169&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=kx,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{k}-\frac{kwx}{c^{2}},\quad k^{2}=\frac{c^{2}}{c^{2}-w^{2}}&lt;/math&gt;

=== {{anchor|Poincare3}} Poincaré (1900, 1905) ===

==== Local time ====

Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time. However, [[w:Henri Poincaré]] in 1900 commented on the origin of Lorentz's &quot;wonderful invention&quot; of local time.&lt;ref&gt;Darrigol (2005), Kap. 4&lt;/ref&gt; He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed &lt;math&gt;c&lt;/math&gt; in both directions, which lead to what is nowadays called [[w:relativity of simultaneity]], although Poincaré's calculation does not involve length contraction or time dilation.&lt;ref group=R&gt;Poincaré (1900), pp. 272–273&lt;/ref&gt; In order to synchronise the clocks here on Earth (the ''x*, t''* frame) a light signal from one clock (at the origin) is sent to another (at ''x''*), and is sent back. It's supposed that the Earth is moving with speed ''v'' in the ''x''-direction (= ''x''*-direction) in some rest system (''x, t'') (''i.e.'' the [[w:luminiferous aether]] system for Lorentz and Larmor). The time of flight outwards is

:&lt;math&gt;\delta t_{a}=\frac{x^{\ast}}{\left(c-v\right)}&lt;/math&gt;

and the time of flight back is

:&lt;math&gt;\delta t_{b}=\frac{x^{\ast}}{\left(c+v\right)}&lt;/math&gt;.

The elapsed time on the clock when the signal is returned is ''δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;'' and the time ''t*=(δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;)/2'' is ascribed to the moment when the light signal reached the distant clock. In the rest frame the time ''t=δt&lt;sub&gt;a&lt;/sub&gt;'' is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus

:&lt;math&gt;t^{\ast}=t-\frac{\gamma^{2}vx^{*}}{c^{2}}&lt;/math&gt;

identical to Lorentz (1892). By dropping the factor γ&lt;sup&gt;2&lt;/sup&gt; under the assumption that &lt;math&gt;\tfrac{v^{2}}{c^{2}}\ll1&lt;/math&gt;, Poincaré gave the result ''t*=t-vx*/c&lt;sup&gt;2&lt;/sup&gt;'', which is the form used by Lorentz in 1895.

Similar physical interpretations of local time were later given by [[w:Emil Cohn]] (1904)&lt;ref group=R&gt;Cohn (1904b), p. 1408&lt;/ref&gt; and [[w:Max Abraham]] (1905).&lt;ref group=R&gt;Abraham (1905), § 42&lt;/ref&gt;

==== Lorentz transformation ====

On June 5, 1905 (published June 9) Poincaré formulated transformation equations which are algebraically equivalent to those of Larmor and Lorentz and gave them the modern form ({{equationNote|4b}}):&lt;ref group=R&gt;Poincaré (1905), p. 1505&lt;/ref&gt;

:&lt;math&gt;\begin{align}x^{\prime} &amp; =kl(x+\varepsilon t)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =kl(t+\varepsilon x)\\
k &amp; =\frac{1}{\sqrt{1-\varepsilon^{2}}}
\end{align}
&lt;/math&gt;.

Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name &quot;Lorentz transformation&quot;.&lt;ref&gt;Pais (1982), Chap. 6c&lt;/ref&gt;&lt;ref&gt;Katzir (2005), 280–288&lt;/ref&gt; Poincaré set the speed of light to unity, pointed out the group characteristics of the transformation by setting ''l''=1, and modified/corrected Lorentz's derivation of the equations of electrodynamics in some details in order to fully satisfy the principle of relativity, ''i.e.'' making them fully Lorentz covariant.&lt;ref&gt;Miller (1981), Chap. 1.14&lt;/ref&gt;

In July 1905 (published in January 1906)&lt;ref group=R&gt;Poincaré (1905/06), pp. 129ff&lt;/ref&gt; Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of the [[w:principle of least action]]; he demonstrated in more detail the group characteristics of the transformation, which he called [[w:Lorentz group]], and he showed that the combination ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;+z&lt;sup&gt;2&lt;/sup&gt;-t&lt;sup&gt;2&lt;/sup&gt;'' is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing &lt;math&gt;ct\sqrt{-1}&lt;/math&gt; as a fourth imaginary coordinate, and he used an early form of [[w:four-vector]]s. He also formulated the velocity addition formula ({{equationNote|4c}}), which he had already derived in unpublished letters to Lorentz from May 1905:&lt;ref group=R&gt;Poincaré (1905/06), p. 144&lt;/ref&gt;

:&lt;math&gt;\xi'=\frac{\xi+\varepsilon}{1+\xi\varepsilon},\ \eta'=\frac{\eta}{k(1+\xi\varepsilon)}&lt;/math&gt;.

==={{anchor|Einstein}} Einstein (1905) – Special relativity===

On June 30, 1905 (published September 1905) Einstein published what is now called [[w:special relativity]] and gave a new derivation of the transformation, which was based only on the principle on relativity and the principle of the constancy of the speed of light. While Lorentz considered &quot;local time&quot; to be a mathematical stipulation device for explaining the Michelson-Morley experiment, Einstein showed that the coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. For quantities of first order in ''v/c'' this was also done by Poincaré in 1900, while Einstein derived the complete transformation by this method. Unlike Lorentz and Poincaré who still distinguished between real time in the aether and apparent time for moving observers, Einstein showed that the transformations concern the nature of space and time.&lt;ref&gt;Miller (1981), Chap. 6&lt;/ref&gt;&lt;ref&gt;Pais (1982), Kap. 7&lt;/ref&gt;&lt;ref&gt;Darrigol (2005), Chap. 6&lt;/ref&gt;

The notation for this transformation is equivalent to Poincaré's of 1905 and ({{equationNote|4b}}), except that Einstein didn't set the speed of light to unity:&lt;ref group=R&gt;Einstein (1905), p. 902&lt;/ref&gt;

:&lt;math&gt;\begin{align}\tau &amp; =\beta\left(t-\frac{v}{V^{2}}x\right)\\
\xi &amp; =\beta(x-vt)\\
\eta &amp; =y\\
\zeta &amp; =z\\
\beta &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}
\end{align}
&lt;/math&gt;

Einstein also defined the velocity addition formula ({{equationNote|4c}}, {{equationNote|4d}}):&lt;ref group=R&gt;Einstein (1905), § 5 and § 9&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=\frac{w_{\xi}+v}{1+\frac{vw_{\xi}}{V^{2}}}t,\ y=\frac{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}{1+\frac{vw_{\xi}}{V^{2}}}w_{\eta}t\\
U^{2}=\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2},\ w^{2}=w_{\xi}^{2}+w_{\eta}^{2},\ \alpha=\operatorname{arctg}\frac{w_{y}}{w_{x}}\\
U=\frac{\sqrt{\left(v^{2}+w^{2}+2vw\cos\alpha\right)-\left(\frac{vw\sin\alpha}{V}\right)^{2}}}{1+\frac{vw\cos\alpha}{V^{2}}}
\end{matrix}\left|\begin{matrix}\frac{u_{x}-v}{1-\frac{u_{x}v}{V^{2}}}=u_{\xi}\\
\frac{u_{y}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\eta}\\
\frac{u_{z}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\zeta}
\end{matrix}\right.&lt;/math&gt;

and the light aberration formula ({{equationNote|4e}}):&lt;ref group=R&gt;Einstein (1905), § 7&lt;/ref&gt;

:&lt;math&gt;\cos\varphi'=\frac{\cos\varphi-\frac{v}{V}}{1-\frac{v}{V}\cos\varphi}&lt;/math&gt;

=== {{anchor|Minkowski}} Minkowski (1907–1908) – Spacetime ===

The work on the principle of relativity by Lorentz, Einstein, [[w:Max Planck|Planck]], together with Poincaré's four-dimensional approach, were further elaborated and combined with the [[w:hyperboloid model]] by [[w:Hermann Minkowski]] in 1907 and 1908.&lt;ref group=R&gt;Minkowski (1907/15), pp. 927ff&lt;/ref&gt;&lt;ref group=R&gt;Minkowski (1907/08), pp. 53ff&lt;/ref&gt; Minkowski particularly reformulated electrodynamics in a four-dimensional way ([[w:Minkowski spacetime]]).&lt;ref&gt;Walter (1999a), (1999b), (2018)&lt;/ref&gt; For instance, he wrote ''x, y, z, it'' in the form ''x&lt;sub&gt;1&lt;/sub&gt;, x&lt;sub&gt;2&lt;/sub&gt;, x&lt;sub&gt;3&lt;/sub&gt;, x&lt;sub&gt;4&lt;/sub&gt;''. By defining ψ as the angle of rotation around the ''z''-axis, the Lorentz transformation assumes a form (with ''c''=1):&lt;ref group=R name=mink1&gt;Minkowski (1907/08), p. 59&lt;/ref&gt;

:&lt;math&gt;\begin{align}x'_{1} &amp; =x_{1}\\
x'_{2} &amp; =x_{2}\\
x'_{3} &amp; =x_{3}\cos i\psi+x_{4}\sin i\psi\\
x'_{4} &amp; =-x_{3}\sin i\psi+x_{4}\cos i\psi\\
\cos i\psi &amp; =\frac{1}{\sqrt{1-q^{2}}}
\end{align}
&lt;/math&gt;

Even though Minkowski used the imaginary number iψ, he for once&lt;ref group=R name=mink1 /&gt; directly used the [[w:tangens hyperbolicus]] in the equation for velocity

:&lt;math&gt;-i\tan i\psi=\frac{e^{\psi}-e^{-\psi}}{e^{\psi}+e^{-\psi}}=q&lt;/math&gt; with &lt;math&gt;\psi=\frac{1}{2}\ln\frac{1+q}{1-q}&lt;/math&gt;.

Minkowski's expression can also by written as ψ=atanh(q) and was later called [[w:rapidity]]. He also wrote the Lorentz transformation in matrix form:&lt;ref group=R&gt;Minkowski (1907/08), pp. 65–66, 81–82&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}+x_{4}^{\prime2}\\
\left(x_{1}^{\prime}=x',\ x_{2}^{\prime}=y',\ x_{3}^{\prime}=z',\ x_{4}^{\prime}=it'\right)\\
-x^{2}-y^{2}-z^{2}+t^{2}=-x^{\prime2}-y^{\prime2}-z^{\prime2}+t^{\prime2}\\
\hline x_{h}=\alpha_{h1}x_{1}^{\prime}+\alpha_{h2}x_{2}^{\prime}+\alpha_{h3}x_{3}^{\prime}+\alpha_{h4}x_{4}^{\prime}\\
\mathrm{A}=\mathrm{\left|\begin{matrix}\alpha_{11}, &amp; \alpha_{12}, &amp; \alpha_{13}, &amp; \alpha_{14}\\
\alpha_{21}, &amp; \alpha_{22}, &amp; \alpha_{23}, &amp; \alpha_{24}\\
\alpha_{31}, &amp; \alpha_{32}, &amp; \alpha_{33}, &amp; \alpha_{34}\\
\alpha_{41}, &amp; \alpha_{42}, &amp; \alpha_{43}, &amp; \alpha_{44}
\end{matrix}\right|,\ \begin{align}\bar{\mathrm{A}}\mathrm{A} &amp; =1\\
\left(\det \mathrm{A}\right)^{2} &amp; =1\\
\det \mathrm{A} &amp; =1\\
\alpha_{44} &amp; &gt;0
\end{align}
}
\end{matrix}&lt;/math&gt;

As a graphical representation of the Lorentz transformation he introduced the [[w:Minkowski diagram]], which became a standard tool in textbooks and research articles on relativity:&lt;ref group=R&gt;Minkowski (1908/09), p. 77&lt;/ref&gt;

[[File:Minkowski1.png|center|thumb|400px|Original spacetime diagram by Minkowski in 1908.]]

==={{anchor|Frank}} Frank (1909) – Hyperbolic functions===
Hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity in agreement with ({{equationNote|4b}}):&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

==={{Anchor|Sommerfeld}} Sommerfeld (1909) – Spherical trigonometry===

Using an imaginary rapidity such as Minkowski, [[w:Arnold Sommerfeld]] (1909) formulated a transformation equivalent to Lorentz boost ({{equationNote|4b}}), and the relativistc velocity addition ({{equationNote|4c}}) in terms of trigonometric functions and the [[w:spherical law of cosines]]:&lt;ref group=R&gt;Sommerfeld (1909), p. 826ff.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{array}{lrl}
x'= &amp; x\ \cos\varphi+l\ \sin\varphi, &amp; y'=y\\
l'= &amp; -x\ \sin\varphi+l\ \cos\varphi, &amp; z'=z
\end{array}\right\} \\
\left(\operatorname{tg}\varphi=i\beta,\ \cos\varphi=\frac{1}{\sqrt{1-\beta^{2}}},\ \sin\varphi=\frac{i\beta}{\sqrt{1-\beta^{2}}}\right)\\
\hline \beta=\frac{1}{i}\operatorname{tg}\left(\varphi_{1}+\varphi_{2}\right)=\frac{1}{i}\frac{\operatorname{tg}\varphi_{1}+\operatorname{tg}\varphi_{2}}{1-\operatorname{tg}\varphi_{1}\operatorname{tg}\varphi_{2}}=\frac{\beta_{1}+\beta_{2}}{1+\beta_{1}\beta_{2}}\\
\cos\varphi=\cos\varphi_{1}\cos\varphi_{2}-\sin\varphi_{1}\sin\varphi_{2}\cos\alpha\\
v^{2}=\frac{v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos\alpha-\frac{1}{c^{2}}v_{1}^{2}v_{2}^{2}\sin^{2}\alpha}{\left(1+\frac{1}{c^{2}}v_{1}v_{2}\cos\alpha\right)^{2}}
\end{matrix}&lt;/math&gt;

=== {{anchor|Herglotz1}} Herglotz (1909/10) – Möbius transformation ===

Following [[../Lorentz transformation (Möbius)#Klein2|E:Klein (1889–1897)]] as well as [[../Lorentz transformation (Möbius)#Fricke|E:Fricke &amp; Klein (1897)]] concerning the Cayley absolute, hyperbolic motion and its transformation, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic. The general case (on the left) equivalent to Lorentz-Möbius transformation [[../Lorentz transformation (Möbius)#math_6a|E:'''(6a)''']] and the hyperbolic case (on the right) equivalent to Lorentz transformation [[../Lorentz transformation (hyperbolic)#math_3d|E:'''(3d)''']] or squeeze mapping [[../Lorentz transformation (squeeze)#math_9d|E:'''(9d)''']] are as follows:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;

:&lt;math&gt;\left.\begin{matrix}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\\
z_{1}=x,\ z_{2}=y,\ z_{3}=z,\ z_{4}=t\\
Z=\frac{z_{1}+iz_{2}}{z_{4}-z_{3}}=\frac{x+iy}{t-z},\ Z'=\frac{x'+iy'}{t'-z'}\\
Z=\frac{\alpha Z'+\beta}{\gamma Z'+\delta}
\end{matrix}\right|\begin{matrix}Z=Z'e^{\vartheta}\\
\begin{align}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
\end{align}
\end{matrix}&lt;/math&gt;

==={{anchor|Varicak}} Varićak (1910) – Hyperbolic functions===

Following [[#Sommerfeld|Sommerfeld (1909)]], hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

and showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

Subsequently, other authors such as [[w:E. T. Whittaker]] (1910) or [[w:Alfred Robb]] (1911, who coined the name rapidity) used similar expressions, which are still used in modern textbooks.&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

=== {{anchor|Ignatowski}} Ignatowski (1910) ===

While earlier derivations and formulations of the Lorentz transformation relied from the outset on optics, electrodynamics, or the invariance of the speed of light, [[w:Vladimir Ignatowski]] (1910) showed that it is possible to use the principle of relativity (and related [[w:Group theory|group theoretical]] principles) alone, in order to derive the following transformation between two inertial frames:&lt;ref group=R&gt;Ignatowski (1910), pp. 973–974&lt;/ref&gt;&lt;ref group=R&gt;Ignatowski (1910/11ab)&lt;/ref&gt;

:&lt;math&gt;\begin{align}dx' &amp; =p\ dx-pq\ dt\\
dt' &amp; =-pqn\ dx+p\ dt\\
p &amp; =\frac{1}{\sqrt{1-q^{2}n}}
\end{align}
&lt;/math&gt;

The variable ''n'' can be seen as a space-time constant whose value has to be determined by experiment or taken from a known physical law such as electrodynamics. For that purpose, Ignatowski used the above-mentioned Heaviside ellipsoid representing a contraction of electrostatic fields by ''x''/γ in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation when ''n=1/c''&lt;sup&gt;2&lt;/sup&gt;, resulting in ''p''=γ and the Lorentz transformation ({{equationNote|4b}}). With ''n''=0, no length changes arise and the Galilean transformation follows. Ignatowski's method was further developed and improved by [[w:Philipp Frank]] and [[w:Hermann Rothe]] (1911, 1912),&lt;ref group=R&gt;Frank &amp; Rothe (1911), pp. 825ff; (1912), p. 750ff.&lt;/ref&gt; with various authors developing similar methods in subsequent years.&lt;ref name=baccetti&gt;Baccetti (2011), see references 1–25 therein.&lt;/ref&gt;

==={{anchor|Noether}} Noether (1910), Klein (1910) – Quaternions===

[[w:Felix Klein]] (1908) described [[../Lorentz transformation (Quaternions)#Cayley3|E:Cayley's (1854)]] 4D quaternion multiplications as &quot;Drehstreckungen&quot; (orthogonal substitutions in terms of rotations leaving invariant a quadratic form up to a factor), and pointed out that the modern principle of relativity as provided by Minkowski is essentially only the consequent application of such Drehstreckungen, even though he didn't provide details.&lt;ref group=R&gt;Klein (1908), p. 165&lt;/ref&gt;

In an appendix to Klein's and Sommerfeld's &quot;Theory of the top&quot; (1910), [[w:Fritz Noether]] showed how to formulate hyperbolic rotations using biquaternions with &lt;math&gt;\omega=\sqrt{-1}&lt;/math&gt;, which he also related to the speed of light by setting ω&lt;sup&gt;2&lt;/sup&gt;=-''c''&lt;sup&gt;2&lt;/sup&gt;. He concluded that this is the principal ingredient for a rational representation of the group of Lorentz transformations:&lt;ref group=R&gt;Noether (1910), pp. 939–943&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}V=\frac{Q_{1}vQ_{2}}{T_{1}T_{2}}\\
\hline X^{2}+Y^{2}+Z^{2}+\omega^{2}S^{2}=x^{2}+y^{2}+z^{2}+\omega^{2}s^{2}\\
\hline \begin{align}V &amp; =Xi+Yj+Zk+\omega S\\
v &amp; =xi+yj+zk+\omega s\\
Q_{1} &amp; =(+Ai+Bj+Ck+D)+\omega(A'i+B'j+C'k+D')\\
Q_{2} &amp; =(-Ai-Bj-Ck+D)+\omega(A'i+B'j+C'k-D')\\
T_{1}T_{2} &amp; =T_{1}^{2}=T_{2}^{2}=A^{2}+B^{2}+C^{2}+D^{2}+\omega^{2}\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Besides citing quaternion related standard works such as [[#Cayley3|Cayley (1854)]], Noether referred to the entries in Klein's encyclopedia by [[w:Eduard Study]] (1899) and the French version by [[w:Élie Cartan]] (1908).&lt;ref&gt;Cartan &amp; Study (1908), sections 35–36&lt;/ref&gt; Cartan's version contains a description of Study's [[w:dual number]]s, Clifford's biquaternions (including the choice &lt;math&gt;\omega=\sqrt{-1}&lt;/math&gt; for hyperbolic geometry), and Clifford algebra, with references to [[../Lorentz transformation (Quaternions)#Stephanos|Stephanos (1883)]], [[../Lorentz transformation (Quaternions)#Buchheim|Buchheim (1884/85)]], [[../Lorentz transformation (Quaternions)#Vahlen|Vahlen (1901/02)]] and others.

Citing Noether, Klein himself published in August 1910 the following quaternion substitutions forming the group of Lorentz transformations:&lt;ref group=R&gt;Klein (1910), p. 300&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align} &amp; \left(i_{1}x'+i_{2}y'+i_{3}z'+ict'\right)\\
 &amp; \quad-\left(i_{1}x_{0}+i_{2}y_{0}+i_{3}z_{0}+ict_{0}\right)
\end{align}
=\frac{\left[\begin{align} &amp; \left(i_{1}(A+iA')+i_{2}(B+iB')+i_{3}(C+iC')+i_{4}(D+iD')\right)\\
 &amp; \quad\cdot\left(i_{1}x+i_{2}y+i_{3}z+ict\right)\\
 &amp; \quad\quad\cdot\left(i_{1}(A-iA')+i_{2}(B-iB')+i_{3}(C-iC')-(D-iD')\right)
\end{align}
\right]}{\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)-\left(A^{2}+B^{2}+C^{2}+D^{2}\right)}\\
\hline \text{where}\\
AA'+BB'+CC'+DD'=0\\
A^{2}+B^{2}+C^{2}+D^{2}&gt;A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}
\end{matrix}&lt;/math&gt;

or in March 1911&lt;ref group=R&gt;Klein (1911), pp. 602ff.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}g'=\frac{pg\pi}{M}\\
\hline \begin{align}g &amp; =\sqrt{-1}ct+ix+jy+kz\\
g' &amp; =\sqrt{-1}ct'+ix'+jy'+kz'\\
p &amp; =(D+\sqrt{-1}D')+i(A+\sqrt{-1}A')+j(B+\sqrt{-1}B')+k(C+\sqrt{-1}C')\\
\pi &amp; =(D-\sqrt{-1}D')-i(A-\sqrt{-1}A')-j(B-\sqrt{-1}B')-k(C-\sqrt{-1}C')\\
M &amp; =\left(A^{2}+B^{2}+C^{2}+D^{2}\right)-\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)\\
 &amp; AA'+BB'+CC'+DD'=0\\
 &amp; A^{2}+B^{2}+C^{2}+D^{2}&gt;A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}
\end{align}
\end{matrix}&lt;/math&gt;

==={{anchor|Conway}} Conway (1911), Silberstein (1911) – Quaternions===

[[w:Arthur W. Conway]] in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms of velocity λ:&lt;ref group=R&gt;Conway (1911), p. 8&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}\mathtt{D} &amp; =\mathbf{a}^{-1}\mathtt{D}'\mathbf{a}^{-1}\\
\mathtt{\sigma} &amp; =\mathbf{a}\mathtt{\sigma}'\mathbf{a}^{-1}
\end{align}
\\
e=\mathbf{a}^{-1}e'\mathbf{a}^{-1}\\
\hline a=\left(1-hc^{-1}\lambda\right)^{\frac{1}{2}}\left(1+c^{-2}\lambda^{2}\right)^{-\frac{1}{4}}
\end{matrix}&lt;/math&gt;

Also [[w:Ludwik Silberstein]] in November 1911&lt;ref group=R&gt;Silberstein (1911/12), p. 793&lt;/ref&gt; as well as in 1914,&lt;ref&gt;Silberstein (1914), p. 156&lt;/ref&gt; formulated the Lorentz transformation in terms of velocity ''v'':

:&lt;math&gt;\begin{matrix}q'=QqQ\\
\hline \begin{align}q &amp; =\mathbf{r}+l=xi+yj+zk+\iota ct\\
q &amp; '=\mathbf{r}'+l'=x'i+y'j+z'k+\iota ct'\\
Q &amp; =\frac{1}{\sqrt{2}}\left(\sqrt{1+\gamma}+\mathrm{u}\sqrt{1-\gamma}\right)\\
 &amp; =\cos\alpha+\mathrm{u}\sin\alpha=e^{\alpha\mathrm{u}}\\
 &amp; \left\{ \gamma=\left(1-v^{2}/c^{2}\right)^{-1/2},\ 2\alpha=\operatorname{arctg}\ \left(\iota\frac{v}{c}\right)\right\} 
\end{align}
\end{matrix}&lt;/math&gt;

Silberstein cites [[../Lorentz transformation (Quaternions)#Cayley3|E:Cayley (1854, 1855)]] and Study's encyclopedia entry (in the extended French version of Cartan in 1908), as well as the appendix of Klein's and Sommerfeld's book.

==={{anchor|Herglotz2}} Ignatowski (1910/11), Herglotz (1911), and others – Vector transformation===

{{See|w:Lorentz transformation#Vector transformations}}

[[w:Vladimir Ignatowski]] (1910, published 1911) defined the vectorial velocity addition ({{equationNote|4g}}) as well as general Lorentz boost ({{equationNote|4f}}) as&lt;ref group=R&gt;Ignatowski (1910/11a), p. 23; (1910/11b), p. 22&lt;/ref&gt; 

:&lt;math&gt;\begin{matrix}\begin{matrix}\mathfrak{v} =\frac{\mathfrak{v}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{v}'+pq\mathfrak{c}_{0}}{p\left(1+nq\mathfrak{c}_{0}\mathfrak{v}'\right)} &amp; \left|\begin{align}\mathfrak{A}' &amp; =\mathfrak{A}+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}-pqb\mathfrak{c}_{0}\\
b' &amp; =pb-pqn\mathfrak{A}\mathfrak{c}_{0}\\
\\
\mathfrak{A} &amp; =\mathfrak{A}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}'+pqb'\mathfrak{c}_{0}\\
b &amp; =pb'+pqn\mathfrak{A}'\mathfrak{c}_{0}
\end{align}
\right.\end{matrix}\\
\left[\mathfrak{v}=\mathbf{u},\ \mathfrak{A}=\mathbf{x},\ b=t,\ \mathfrak{c}_{0}=\frac{\mathbf{v}}{v},\ p=\gamma,\ n=\frac{1}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

An equivalent transformation was given by [[w:Gustav Herglotz]] (1911)&lt;ref group=R&gt;Herglotz (1911), p. 497&lt;/ref&gt; using '''v'''=''(v&lt;sub&gt;x&lt;/sub&gt;, v&lt;sub&gt;y&lt;/sub&gt;, v&lt;sub&gt;z&lt;/sub&gt;)'' and '''r'''=''(x, y, z)'':

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{0} &amp; =x+\alpha u(ux+vy+wz)-\beta ut\\
y^{0} &amp; =y+\alpha v(ux+vy+wz)-\beta vt\\
z^{0} &amp; =z+\alpha w(ux+vy+wz)-\beta wt\\
t^{0} &amp; =-\beta(ux+vy+wz)+\beta t\\
 &amp; \alpha=\frac{1}{\sqrt{1-s^{2}}\left(1+\sqrt{1-s^{2}}\right)},\ \beta=\frac{1}{\sqrt{1-s^{2}}}
\end{align}
\right| &amp; \begin{align}x' &amp; =x+\alpha v_{x}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{x}t\\
y' &amp; =y+\alpha v_{y}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{y}t\\
z' &amp; =z+\alpha v_{z}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{z}t\\
t' &amp; =-\gamma\left(v_{x}x+v_{y}y+v_{z}z\right)+\gamma t\\
 &amp; \alpha=\frac{\gamma^{2}}{\gamma+1},\ \gamma=\frac{1}{\sqrt{1-v^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

Or by [[w:Ludwik Silberstein]] (1911 on the left, 1914 on the right):&lt;ref group=R&gt;Silberstein (1911/12), p. 792; (1914), p. 123&lt;/ref&gt;

:&lt;math&gt;\begin{array}{c|c}
\begin{align}\mathbf{r}' &amp; =\mathbf{r}+(\gamma-1)(\mathbf{ru})\mathbf{u}+i\beta\gamma lu\\
l' &amp; =\gamma\left[l-i\beta(\mathbf{ru})\right]
\end{align}
 &amp; \begin{align}\mathbf{r}' &amp; =\mathbf{r}+\left[\frac{\gamma-1}{v^{2}}(\mathbf{vr})-\gamma t\right]\mathbf{v}\\
t' &amp; =\gamma\left[t-\frac{1}{c^{2}}(\mathbf{vr})\right]
\end{align}
\end{array}&lt;/math&gt;

[[w:Erwin Madelung]] (1922) provided the matrix form&lt;ref&gt;Madelung (1921), p. 207&lt;/ref&gt;

:&lt;math&gt;\begin{array}{c|c|c|c|c}
 &amp; x &amp; y &amp; z &amp; t\\
\hline x' &amp; 1-\frac{v_{x}^{2}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{x}v_{y}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{x}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; \frac{-v_{x}}{\sqrt{1-\beta^{2}}}\\
y' &amp; -\frac{v_{x}v_{y}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; 1-\frac{v_{y}^{2}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{y}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; \frac{-v_{y}}{\sqrt{1-\beta^{2}}}\\
z' &amp; -\frac{v_{x}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{y}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; 1-\frac{v_{z}^{2}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; \frac{-v_{z}}{\sqrt{1-\beta^{2}}}\\
t' &amp; \frac{-v_{x}}{c^{2}\sqrt{1-\beta^{2}}} &amp; \frac{-v_{y}}{c^{2}\sqrt{1-\beta^{2}}} &amp; \frac{-v_{z}}{c^{2}\sqrt{1-\beta^{2}}} &amp; \frac{1}{\sqrt{1-\beta^{2}}}
\end{array}&lt;/math&gt;

These formulas were called &quot;general Lorentz transformation without rotation&quot; by [[w:Christian Møller]] (1952),&lt;ref&gt;Møller (1952/55), pp. 41–43&lt;/ref&gt; who in addition gave an even more general Lorentz transformation in which the Cartesian axes have different orientations, using a [[w:Three-dimensional rotation operator|rotation operator]] &lt;math&gt;\mathfrak{D}&lt;/math&gt;. In this case, '''v′'''=''(v′&lt;sub&gt;x&lt;/sub&gt;, v′&lt;sub&gt;y&lt;/sub&gt;, v′&lt;sub&gt;z&lt;/sub&gt;)'' is not equal to -'''v'''=''(-v&lt;sub&gt;x&lt;/sub&gt;, -v&lt;sub&gt;y&lt;/sub&gt;, -v&lt;sub&gt;z&lt;/sub&gt;)'', but the relation &lt;math&gt;\mathbf{v}'=-\mathfrak{D}\mathbf{v}&lt;/math&gt; holds instead, with the result

:&lt;math&gt;\begin{array}{c}
\begin{align}\mathbf{x}' &amp; =\mathfrak{D}^{-1}\mathbf{x}-\mathbf{v}'\left\{ \left(\gamma-1\right)(\mathbf{x\cdot v})/v^{2}-\gamma t\right\} \\
t' &amp; =\gamma\left(t-(\mathbf{v}\cdot\mathbf{x})/c^{2}\right)
\end{align}
\end{array}&lt;/math&gt;

==={{anchor|Borel}} Borel (1913–14) – Cayley–Hermite parameter===

Borel (1913) started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and [[../Lorentz transformation (Cayley-Hermite)#Cayley|E:Cayley's (1846)]] parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions:&lt;ref group=R&gt;Borel (1913/14), p. 39&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}-z^{2}-1=0\\
\hline {\scriptstyle \begin{align}\delta a &amp; =\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta b &amp; =2(\lambda\mu+\nu\rho), &amp; \delta c &amp; =-2(\lambda\nu+\mu\rho),\\
\delta a' &amp; =2(\lambda\mu-\nu\rho), &amp; \delta b' &amp; =-\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta c' &amp; =2(\lambda\rho-\mu\nu),\\
\delta a'' &amp; =2(\lambda\nu-\mu\rho), &amp; \delta b'' &amp; =2(\lambda\rho+\mu\nu), &amp; \delta c'' &amp; =-\left(\lambda^{2}+\mu^{2}+\nu^{2}+\rho^{2}\right),
\end{align}
}\\
\left(\delta=\lambda^{2}+\mu^{2}-\rho^{2}-\nu^{2}\right)\\
\lambda=\nu=0\rightarrow\text{Hyperbolic rotation}
\end{matrix}&lt;/math&gt;

In four dimensions:&lt;ref group=R&gt;Borel (1913/14), p. 41&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}-\left(t_{1}-t_{2}\right)^{2}\\
\hline {\scriptstyle \begin{align} &amp; \left(\mu^{2}+\nu^{2}-\alpha^{2}\right)\cos\varphi+\left(\lambda^{2}-\beta^{2}-\gamma^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi-\gamma\operatorname{sh}{\theta}\\
 &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi+\gamma\operatorname{sh}{\theta} &amp;  &amp; \left(\mu^{2}+\nu^{2}-\beta^{2}\right)\cos\varphi+\left(\mu^{2}-\alpha^{2}-\gamma^{2}\right)\operatorname{ch}{\theta}\\
 &amp; -(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi-\beta\operatorname{sh}{\theta} &amp;  &amp; -(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\lambda\sin\varphi+\alpha\operatorname{sh}{\theta}\\
 &amp; (\gamma\mu-\beta\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta} &amp;  &amp; -(\alpha\nu-\lambda\gamma)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
\\
 &amp; \quad-(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi+\beta\operatorname{sh}{\theta} &amp;  &amp; \quad(\beta\nu-\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta}\\
 &amp; \quad-(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})-\lambda\sin\varphi-\alpha\operatorname{sh}{\theta} &amp;  &amp; \quad(\lambda\gamma-\alpha\nu)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
 &amp; \quad\left(\lambda^{2}+\mu^{2}-\gamma^{2}\right)\cos\varphi+\left(\nu^{2}-\alpha^{2}-\beta^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; \quad(\alpha\mu-\beta\lambda)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta}\\
 &amp; \quad(\beta\gamma-\alpha\mu)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta} &amp;  &amp; \quad-\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\cos\varphi+\left(\lambda^{2}+\mu^{2}+\nu^{2}\right)\operatorname{ch}{\theta}
\end{align}
}\\
\left(\alpha^{2}+\beta^{2}+\gamma^{2}-\lambda^{2}-\mu^{2}-\nu^{2}=-1\right)
\end{matrix}&lt;/math&gt;

==={{anchor|Gruner}} Gruner (1921) – Trigonometric Lorentz boosts===

In order to simplify the graphical representation of Minkowski space, [[w:Paul Gruner]] (1921) (with the aid of Josef Sauter) developed what is now called [[w:Loedel diagram]]s, using the following relations:&lt;ref group=R&gt;Gruner (1921a),&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}v=\alpha\cdot c;\quad\beta=\frac{1}{\sqrt{1-\alpha^{2}}}\\
\sin\varphi=\alpha;\quad\beta=\frac{1}{\cos\varphi};\quad\alpha\beta=\tan\varphi\\
\hline x'=\frac{x}{\cos\varphi}-t\cdot\tan\varphi,\quad t'=\frac{t}{\cos\varphi}-x\cdot\tan\varphi
\end{matrix}&lt;/math&gt;

In another paper Gruner used the alternative relations:&lt;ref group=R&gt;Gruner (1921b)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\alpha=\frac{v}{c};\ \beta=\frac{1}{\sqrt{1-\alpha^{2}}};\\
\cos\theta=\alpha=\frac{v}{c};\ \sin\theta=\frac{1}{\beta};\ \cot\theta=\alpha\cdot\beta\\
\hline x'=\frac{x}{\sin\theta}-t\cdot\cot\theta,\quad t'=\frac{t}{\sin\theta}-x\cdot\cot\theta
\end{matrix}&lt;/math&gt;

==References==
===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|abra1905}}
*{{#section:History of Topics in Special Relativity/relsource|bor14}}
*{{#section:History of Topics in Special Relativity/relsource|buch08}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04a}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04b}}
*{{#section:History of Topics in Special Relativity/relsource|con11qua}}
*{{#section:History of Topics in Special Relativity/relsource|einst05elek}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|fra11}}
*{{#section:History of Topics in Special Relativity/relsource|fra12}}
*{{#section:History of Topics in Special Relativity/relsource|gans05}}
*{{#section:History of Topics in Special Relativity/relsource|grun21a}}
*{{#section:History of Topics in Special Relativity/relsource|grun21b}}
*{{#section:History of Topics in Special Relativity/relsource|heav89}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|herg11ela}}
*{{#section:History of Topics in Special Relativity/relsource|igna10}}
*{{#section:History of Topics in Special Relativity/relsource|ignat10prin2}}
*{{#section:History of Topics in Special Relativity/relsource|klein08}}
*{{#section:History of Topics in Special Relativity/relsource|klein10b}}
*{{#section:History of Topics in Special Relativity/relsource|klein10}}
*{{#section:History of Topics in Special Relativity/relsource|klein11}}
*{{#section:History of Topics in Special Relativity/relsource|lar97}}
*{{#section:History of Topics in Special Relativity/relsource|lar29}}
*{{#section:History of Topics in Special Relativity/relsource|lar00}}
*{{#section:History of Topics in Special Relativity/relsource|lar04a}}
*{{#section:History of Topics in Special Relativity/relsource|lar04b}}
*{{#section:History of Topics in Special Relativity/relsource|lor92elek}}
*{{#section:History of Topics in Special Relativity/relsource|lor92b}}
*{{#section:History of Topics in Special Relativity/relsource|lor95}}
*{{#section:History of Topics in Special Relativity/relsource|lor99}}
*{{#section:History of Topics in Special Relativity/relsource|lor04}}
*{{#section:History of Topics in Special Relativity/relsource|lor16}}
*{{#section:History of Topics in Special Relativity/relsource|mink07a}}
*{{#section:History of Topics in Special Relativity/relsource|mink07b}}
*{{#section:History of Topics in Special Relativity/relsource|mink08}}
*{{#section:History of Topics in Special Relativity/relsource|plum10}}
*{{#section:History of Topics in Special Relativity/relsource|poi00}}
*{{#section:History of Topics in Special Relativity/relsource|poi04}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05a}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05b}}
*{{#section:History of Topics in Special Relativity/relsource|sea97}}
*{{#section:History of Topics in Special Relativity/relsource|silber11quat}}
*{{#section:History of Topics in Special Relativity/relsource|silber12quat}}
*{{#section:History of Topics in Special Relativity/relsource|silber14quat}}
*{{#section:History of Topics in Special Relativity/relsource|som09}}
*{{#section:History of Topics in Special Relativity/relsource|thom89}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}
*{{#section:History of Topics in Special Relativity/relsource|voi87}}
*{{#section:History of Topics in Special Relativity/relsource|wien04}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L4}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
      <sha1>8q41fc0wvnovhj904yzv6k2n3lg435o</sha1>
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      <parentid>2411820</parentid>
      <timestamp>2022-08-04T12:32:03Z</timestamp>
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        <username>D.H</username>
        <id>52339</id>
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      <text bytes="64948" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via velocity==
===Boosts===
In the [[w:theory of relativity]], Lorentz transformations exhibit the symmetry of [[w:Minkowski spacetime]] by using a constant ''c'' as the [[w:speed of light]], and a parameter ''v'' as the relative [[w:velocity]] between two [[w:inertial reference frames]]. The corresponding formulas are identical to [[../Lorentz transformation (hyperbolic)|E:Lorentz transformations via hyperbolic functions]] introduced long before relativity was developed. In particular, the hyperbolic angle &lt;math&gt;\eta&lt;/math&gt; can be interpreted as the velocity related [[w:rapidity]] &lt;math&gt;\tanh\eta=\beta=v/c&lt;/math&gt;, so that &lt;math&gt;\gamma=\cosh\eta&lt;/math&gt; is the [[w:Lorentz factor]], &lt;math&gt;\beta\gamma=\sinh\eta&lt;/math&gt; the [[w:proper velocity]], &lt;math&gt;u'=c\tanh q&lt;/math&gt; the velocity of another object, &lt;math&gt;u=c\tanh(q+\eta)&lt;/math&gt; the [[w:velocity-addition formula]], thus transformation [[../Lorentz transformation (hyperbolic)#math_3b|E:'''(3b)''']] becomes:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}\gamma-x_{1}\beta\gamma\\
x_{1}^{\prime} &amp; =-x_{0}\beta\gamma+x_{1}\gamma\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\gamma+x_{1}^{\prime}\beta\gamma\\
x_{1} &amp; =x_{0}^{\prime}\beta\gamma+x_{1}^{\prime}\gamma\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\left|{\scriptstyle \begin{align}\beta^{2}\gamma^{2}-\gamma^{2} &amp; =-1 &amp; (a)\\
\gamma^{2}-\beta^{2}\gamma^{2} &amp; =1 &amp; (b)\\
\frac{\beta\gamma}{\gamma} &amp; =\beta &amp; (c)\\
\frac{1}{\sqrt{1-\beta^{2}}} &amp; =\gamma &amp; (d)\\
\frac{\beta}{\sqrt{1-\beta^{2}}} &amp; =\beta\gamma &amp; (e)\\
\frac{u'+v}{1+\frac{u'v}{c^{2}}} &amp; =u &amp; (f)
\end{align}
}\right.
\end{matrix}&lt;/math&gt;|{{equationRef|4a}}}}

Or in four dimensions and by setting &lt;math&gt;x_{0}=ct,\ x_{1}=x,\ x_{2}=y&lt;/math&gt; and adding an unchanged ''z'' the familiar form follows

{{NumBlk|:|&lt;math&gt;\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}+z^{2}=-c^{2}t^{\prime2}+x^{\prime2}+y^{\prime2}+z^{\prime2}\\
\hline \left.\begin{align}t' &amp; =\gamma\left(t-x\frac{v}{c^{2}}\right)\\
x' &amp; =\gamma(x-vt)\\
y' &amp; =y\\
z' &amp; =z
\end{align}
\right|\begin{align}t &amp; =\gamma\left(t'+x\frac{v}{c^{2}}\right)\\
x &amp; =\gamma(x'+vt')\\
y &amp; =y'\\
z &amp; =z'
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|4b}}}}

Analogous transformations have been introduced by [[#Voigt|Voigt (1887)]] in terms of an incompessible medium, and by [[#Lorentz1|Lorentz (1892, 1895)]] who analyzed [[w:Maxwell's equations]], they were completed by [[#Larmor|Larmor (1897, 1900)]] and [[#Lorentz2|Lorentz (1899, 1904)]], and brought into their modern form by [[#Poincare3|Poincaré (1905)]] who gave the transformation the name of Lorentz.&lt;ref&gt;Miller (1981), chapter 1&lt;/ref&gt; Eventually, [[#Einstein|Einstein (1905)]] showed in his development of [[w:special relativity]] that the transformations follow from the [[w:principle of relativity]] and constant light speed alone by modifying the traditional concepts of space and time, without requiring a [[w:Lorentz ether theory|mechanical aether]] in contradistinction to Lorentz and Poincaré.&lt;ref&gt;Miller (1981), chapter 4–7&lt;/ref&gt; [[#Minkowski|Minkowski (1907–1908)]] used them to argue that space and time are inseparably connected as [[w:spacetime]]. [[#Minkowski|Minkowski (1907–1908)]], [[#Frank|Frank (1909)]]  and [[#Varicak|Varićak (1910)]] showed the relation to imaginary and hyperbolic functions. Important contributions to the mathematical understanding of the Lorentz transformation were also made by other authors such as [[#Herglotz1|Herglotz (1909/10)]], [[#Ignatowski|Ignatowski (1910)]], [[#Noether|Noether (1910) and Klein (1910)]], [[#Borel|Borel (1913–14)]].

===Velocity addition and aberration===
In exact analogy to Beltrami coordinates in equation [[../Lorentz transformation (hyperbolic)#math_3e|E:'''(3e)''']], one can substitute &lt;math&gt;\left[\tfrac{u_{x}}{c},\ \tfrac{u_{y}}{c},\ 1\right]=\left[\tfrac{x}{ct},\ \tfrac{y}{ct},\ \tfrac{ct}{ct}\right]&lt;/math&gt; in ({{equationNote|4a}}), producing the Lorentz transformation of velocities (or [[w:velocity addition formula]]) :

{{NumBlk|:|&lt;math&gt;\begin{matrix}\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}=-c^{2}t^{\prime2}+x^{\prime2}+y^{\prime2} &amp; \rightarrow &amp; \begin{align}-c^{2}+u_{x}^{2}+u_{y}^{2} &amp; =\frac{-c^{2}+u_{x}^{\prime2}+u_{y}^{\prime2}}{\gamma^{2}\left(1+\frac{v}{c^{2}}u_{x}^{\prime}\right)^{2}}\\
\frac{-c^{2}+u_{x}^{2}+u_{y}^{2}}{\gamma^{2}\left(1-\frac{v}{c^{2}}u_{x}\right)^{2}} &amp; =-c^{2}+u_{x}^{\prime2}+u_{y}^{\prime2}
\end{align}
\\
\hline -c^{2}t^{2}+x^{2}+y^{2}=-c^{2}t^{\prime2}+x^{\prime2}+y^{\prime2}=0 &amp; \rightarrow &amp; -c^{2}+u_{x}^{2}+u_{y}^{2}=-c^{2}+u_{x}^{\prime2}+u_{y}^{\prime2}=0
\end{matrix}\\
\hline {\scriptstyle \begin{align}\frac{\sinh\eta}{\cosh\eta} &amp; =\tanh\eta=\frac{v}{c}\\
\cosh\eta &amp; =\frac{1}{\sqrt{1-\tanh^{2}\eta}}
\end{align}
}\left|\begin{align}u_{x}^{\prime} &amp; =\frac{-c^{2}\sinh\eta+u_{x}c\cosh\eta}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{x}-c\tanh\eta}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}-v}{1-\frac{v}{c^{2}}u{}_{x}}\\
u_{y}^{\prime} &amp; =\frac{cu_{y}}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\tanh^{2}\eta}}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u{}_{x}}\\
\\
u_{x} &amp; =\frac{c^{2}\sinh\eta+u_{x}^{\prime}c\cosh\eta}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+c\tanh\eta}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+v}{1+\frac{v}{c^{2}}u_{x}^{\prime}}\\
u_{y} &amp; =\frac{cy'}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u_{x}^{\prime}}
\end{align}
\right.
\end{matrix}&lt;/math&gt;|{{equationRef|4c}}}}

or using trigonometric and hyperbolic identities as in equation [[../Lorentz transformation (hyperbolic)#math_3f|E:'''(3f)''']], it becomes the hyperbolic law of cosines:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{matrix}u^{2}=u_{x}^{2}+u_{y}^{2}\\
u'^{2}=u_{x}^{\prime2}+u_{y}^{\prime2}
\end{matrix}\left|\begin{matrix}u_{x}=u\cos\alpha\\
u_{y}=u\sin\alpha\\
\\
u_{x}^{\prime}=u'\cos\alpha'\\
u_{y}^{\prime}=u'\sin\alpha'
\end{matrix}\right|\begin{align}u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-\frac{v}{c^{2}}u\cos\alpha}\\
u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{u'\cos\alpha'+v}, &amp; \tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{u\cos\alpha-v}
\end{align}
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(\frac{vu'\sin\alpha'}{c}\right){}^{2}}}{1+\frac{v}{c^{2}}u'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(\frac{vu\sin\alpha}{c}\right){}^{2}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\frac{u^{\prime2}}{c^{2}}}}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{1}{\sqrt{1-\frac{u^{2}}{c^{2}}}}-\frac{v/c}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{u/c}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\cos\alpha\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha
\end{matrix}&lt;/math&gt;|{{equationRef|4d}}}}

and by further setting ''u=u′=c'' one gets the old [[../Lorentz transformation (hyperbolic)#math_3g|E:Kepler formulas '''(3g)''']], which  express the relativistic [[w:aberration of light]]:&lt;ref&gt;Pauli (1921), pp. 562; 565–566&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\cos\alpha=\frac{\cos\alpha'+\frac{v}{c}}{1+\frac{v}{c}\cos\alpha'},\ \sin\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c}\cos\alpha'},\ \tan\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha'+\frac{v}{c}},\ \tan\frac{\alpha}{2}=\sqrt{\frac{c-v}{c+v}}\tan\frac{\alpha'}{2}\\
\cos\alpha'=\frac{\cos\alpha-\frac{v}{c}}{1-\frac{v}{c}\cos\alpha},\ \sin\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c}\cos\alpha},\ \tan\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha-\frac{v}{c}},\ \tan\frac{\alpha'}{2}=\sqrt{\frac{c+v}{c-v}}\tan\frac{\alpha}{2}
\end{matrix}&lt;/math&gt;|{{equationRef|4e}}}}

Formulas ({{equationNote|4c}}, {{equationNote|4d}}) were given by [[#Einstein|Einstein (1905)]] and [[#Poincare3|Poincaré (1905/06)]], while the relations to the spherical and hyperbolic law of cosines were given by [[#Sommerfeld|Sommerfeld (1909)]] and [[#Varicak|Varićak (1910)]]. The aberration formula for cos(α) was given by [[#Einstein|Einstein (1905)]].&lt;ref group=R name=plum&gt;Plummer (1910), pp. 258-259: After deriving the relativistic expressions for the aberration angles φ' and φ, Plummer remarked on p. 259: ''Another geometrical representation is obtained by assimilating φ' to the eccentric and φ to the true anomaly in an ellipse whose eccentricity is v/U = sin β.''&lt;/ref&gt;&lt;ref name=robin&gt;Robinson (1990), chapter 3-4, analyzed the relation between &quot;Kepler's formula&quot; and the &quot;physical velocity addition formula&quot; in special relativity.&lt;/ref&gt;

===Vectorial Lorentz transformation===
Also Lorentz boosts for arbitrary directions in line with [[../Lorentz transformation (general)#math_1a|E:general Lorentz transformation '''(1a)''']] can be given as:&lt;ref&gt;Møller (1952/55), Chapter II, § 18&lt;/ref&gt;

:&lt;math&gt;\mathbf{x}'=\begin{bmatrix}\gamma &amp; -\gamma\beta n_{x} &amp; -\gamma\beta n_{y} &amp; -\gamma\beta n_{z}\\
-\gamma\beta n_{x} &amp; 1+(\gamma-1)n_{x}^{2} &amp; (\gamma-1)n_{x}n_{y} &amp; (\gamma-1)n_{x}n_{z}\\
-\gamma\beta n_{y} &amp; (\gamma-1)n_{y}n_{x} &amp; 1+(\gamma-1)n_{y}^{2} &amp; (\gamma-1)n_{y}n_{z}\\
-\gamma\beta n_{z} &amp; (\gamma-1)n_{z}n_{x} &amp; (\gamma-1)n_{z}n_{y} &amp; 1+(\gamma-1)n_{z}^{2}
\end{bmatrix}\cdot\mathbf{x},\quad\left[\mathbf{n}=\frac{\mathbf{v}}{v}\right]&lt;/math&gt;

or in vector notation

{{NumBlk|:|&lt;math&gt;\begin{align}t' &amp; =\gamma\left(t-\frac{v\mathbf{n}\cdot\mathbf{r}}{c^{2}}\right)\\
\mathbf{r}' &amp; =\mathbf{r}+(\gamma-1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n}-\gamma tv\mathbf{n}
\end{align}
&lt;/math&gt;|{{equationRef|4f}}}}

The corresponding vectorial velocity addition formula in line with [[../Lorentz transformation (general)#math_1b|E:general Lorentz transformation '''(1b)''']] is given by

{{NumBlk|:|&lt;math&gt;\mathbf{u}'=\frac{1}{1+\frac{\mathbf{v}\cdot\mathbf{u}}{c^{2}}}\left[\frac{\mathbf{u}}{\gamma_{\mathbf{v}}}+\mathbf{v}+\frac{1}{c^{2}}\frac{\gamma_{\mathbf{v}}}{\gamma_{\mathbf{v}}+1}(\mathbf{u}\cdot\mathbf{v})\mathbf{v}\right]&lt;/math&gt;|{{equationRef|4g}}}}

Such transformations were formulated by [[#Herglotz2|Ignatowski (1910), Herglotz (1911)]] and others.

==Historical notation==
==={{anchor|Voigt}} Voigt (1887) ===

[[w:Woldemar Voigt]] (1887)&lt;ref group=R&gt;Voigt (1887), p. 45&lt;/ref&gt; developed a transformation in connection with the [[w:Doppler effect]] and an incompressible medium, being in modern notation:&lt;ref&gt;Miller (1981), 114–115&lt;/ref&gt;&lt;ref name=pais&gt;Pais (1982), Kap. 6b&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\xi_{1} &amp; =x_{1}-\varkappa t\\
\eta_{1} &amp; =y_{1}q\\
\zeta_{1} &amp; =z_{1}q\\
\tau &amp; =t-\frac{\varkappa x_{1}}{\omega^{2}}\\
q &amp; =\sqrt{1-\frac{\varkappa^{2}}{\omega^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =\frac{y}{\gamma}\\
z^{\prime} &amp; =\frac{z}{\gamma}\\
t^{\prime} &amp; =t-\frac{vx}{c^{2}}\\
\frac{1}{\gamma} &amp; =\sqrt{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

If the right-hand sides of his equations are multiplied by γ they are the modern Lorentz transformation ({{equationNote|4b}}). In Voigt's theory the speed of light is invariant, but his transformations mix up a relativistic boost together with a rescaling of space-time. Optical phenomena in free space are [[w:Scale invariance|scale]], [[w:Conformal map|conformal]] (using the factor λ discussed [[#Lorsph|above]]), and [[w:Lorentz covariance|Lorentz invariant]], so the combination is invariant too.&lt;ref name=pais /&gt; For instance, Lorentz transformations can be extended by using &lt;math&gt;l=\sqrt{\lambda}&lt;/math&gt;:&lt;ref group=R&gt;Lorentz (1915/16), p. 197&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\quad y^{\prime}=ly,\quad z^{\prime}=lz,\quad t^{\prime}=\gamma l\left(t-x\frac{v}{c^{2}}\right)&lt;/math&gt;.

''l''=1/γ gives the Voigt transformation, ''l''=1 the Lorentz transformation. But scale transformations are not a symmetry of all the laws of nature, only of electromagnetism, so these transformations cannot be used to formulate a [[w:principle of relativity]] in general. It was demonstrated by Poincaré and Einstein that one has to set ''l''=1 in order to make the above transformation symmetric and to form a group as required by the relativity principle, therefore the Lorentz transformation is the only viable choice.

Voigt sent his 1887 paper to Lorentz in 1908,&lt;ref&gt;Voigt's transformations and the beginning of the relativistic revolution, Ricardo Heras, arXiv:1411.2559 [https://arxiv.org/abs/1411.2559]&lt;/ref&gt; and that was acknowledged in 1909: {{Quote|In a paper &quot;Über das Doppler'sche Princip&quot;, published in 1887 (Gött. Nachrichten, p. 41) and which to my regret has escaped my notice all these years, Voigt has applied to equations of the form (7) (§ 3 of this book) [namely &lt;math&gt;\Delta\Psi-\tfrac{1}{c^{2}}\tfrac{\partial^{2}\Psi}{\partial t^{2}}=0&lt;/math&gt;] a transformation equivalent to the formulae (287) and (288) [namely &lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\ y^{\prime}=ly,\ z^{\prime}=lz,\ t^{\prime}=\gamma l\left(t-\tfrac{v}{c^{2}}x\right)&lt;/math&gt;]. The idea of the transformations used above (and in § 44) might therefore have been borrowed from Voigt and the proof that it does not alter the form of the equations for the ''free'' ether is contained in his paper.&lt;ref group=R&gt;Lorentz (1915/16), p. 198&lt;/ref&gt;}}

Also [[w:Hermann Minkowski]] said in 1908 that the transformations which play the main role in the principle of relativity were first examined by Voigt in 1887. Voigt responded in the same paper by saying that his theory was based on an elastic theory of light, not an electromagnetic one. However, he concluded that some results were actually the same.&lt;ref group=R&gt;Bucherer (1908), p. 762&lt;/ref&gt;

==={{anchor|Heaviside}} Heaviside (1888), Thomson (1889), Searle (1896)===

In 1888, [[w:Oliver Heaviside]]&lt;ref group=R&gt;Heaviside (1888), p. 324&lt;/ref&gt; investigated the properties of [[w:Relativistic electromagnetism|charges in motion]] according to Maxwell's electrodynamics. He calculated, among other things, anisotropies in the electric field of moving bodies represented by this formula:&lt;ref&gt;Brown (2003)&lt;/ref&gt;

:&lt;math&gt;\mathrm{E}=\left(\frac{q\mathrm{r}}{r^{2}}\right)\left(1-\frac{v^{2}\sin^{2}\theta}{c^{2}}\right)^{-3/2}&lt;/math&gt;.

Consequently, [[w:Joseph John Thomson]] (1889)&lt;ref group=R&gt;Thomson (1889), p. 12&lt;/ref&gt; found a way to substantially simplify calculations concerning moving charges by using the following mathematical transformation (like other authors such as Lorentz or Larmor, also Thomson implicitly used the [[w:Galilean transformation]] ''z-vt'' in his equation&lt;ref name=mil /&gt;):

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}z &amp; =\left\{ 1-\frac{\omega^{2}}{v^{2}}\right\} ^{\frac{1}{2}}z'\end{align}
\right| &amp; \begin{align}z^{\ast}=z-vt &amp; =\frac{z'}{\gamma}\end{align}
\end{matrix}&lt;/math&gt;

Thereby, [[w:inhomogeneous electromagnetic wave equation]]s are transformed into a [[w:Poisson equation]].&lt;ref name=mil&gt;Miller (1981), 98–99&lt;/ref&gt; Eventually, [[w:George Frederick Charles Searle]]&lt;ref group=R&gt;Searle (1886), p. 333&lt;/ref&gt; noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called &quot;Heaviside-Ellipsoid&quot; of [[w:axial ratio]]

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align} &amp; \sqrt{\alpha}:1:1\\
\alpha= &amp; 1-\frac{u^{2}}{v^{2}}
\end{align}
\right| &amp; \begin{align} &amp; \frac{1}{\gamma}:1:1\\
\frac{1}{\gamma^{2}} &amp; =1-\frac{v^{2}}{c^{2}}
\end{align}
\end{matrix}&lt;/math&gt;&lt;ref name=mil /&gt;

=== {{anchor|Lorentz1}} Lorentz (1892, 1895) ===

In order to explain the [[w:aberration of light]] and the result of the [[w:Fizeau experiment]] in accordance with [[w:Maxwell's equations]], Lorentz in 1892 developed a model (&quot;[[w:Lorentz ether theory]]&quot;) in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. In order to calculate the optics of moving bodies, Lorentz introduced the following quantities to transform from the aether system into a moving system (it's unknown whether he was influenced by Voigt, Heaviside, and Thomson)&lt;ref group=R&gt;Lorentz (1892a), p. 141&lt;/ref&gt;&lt;ref name=milf&gt;Miller (1982), 1.4 &amp; 1.5&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\mathfrak{x} &amp; =\frac{V}{\sqrt{V^{2}-p^{2}}}x\\
t' &amp; =t-\frac{\varepsilon}{V}\mathfrak{x}\\
\varepsilon &amp; =\frac{p}{\sqrt{V^{2}-p^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)\\
\gamma\frac{v}{c} &amp; =\frac{v}{\sqrt{c^{2}-v^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

where ''x&lt;sup&gt;*&lt;/sup&gt;'' is the [[w:Galilean transformation]] ''x-vt''. Except the additional γ in the time transformation, this is the complete Lorentz transformation ({{equationNote|4b}}).&lt;ref name=milf /&gt; While ''t'' is the &quot;true&quot; time for observers resting in the aether, ''t′'' is an auxiliary variable only for calculating processes for moving systems. It is also important that Lorentz and later also Larmor formulated this transformation in two steps. At first an implicit Galilean transformation, and later the expansion into the &quot;fictitious&quot; electromagnetic system with the aid of the Lorentz transformation. In order to explain the negative result of the [[w:Michelson–Morley experiment]], he (1892b)&lt;ref group=R&gt;Lorentz (1892b), p. 141&lt;/ref&gt; introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced [[w:length contraction]] in his theory (without proof as he admitted). The same hypothesis was already made by [[w:George FitzGerald]] in 1889 based on Heaviside's work. While length contraction was a real physical effect for Lorentz, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation.

In 1895, Lorentz further elaborated on his theory and introduced the &quot;theorem of corresponding states&quot;. This theorem states that a moving observer (relative to the ether) in his &quot;fictitious&quot; field makes the same observations as a resting observers in his &quot;real&quot; field for velocities to first order in ''v/c''. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:&lt;ref group=R&gt;Lorentz (1895), p. 37&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =x^{\prime}\sqrt{1-\frac{\mathfrak{p}^{2}}{V^{2}}}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{x^{\prime}}{\gamma}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\end{matrix}&lt;/math&gt;

For solving optical problems Lorentz used the following transformation, in which the modified time variable was called &quot;local time&quot; ({{lang-de|Ortszeit}}) by him:&lt;ref group=R&gt;Lorentz (1895), p. 49 for local time and p. 56 for spatial coordinates.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\mathrm{x}-\mathfrak{p}_{x}t\\
y &amp; =\mathrm{y}-\mathfrak{p}_{y}t\\
z &amp; =\mathrm{z}-\mathfrak{p}_{z}t\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}}x-\frac{\mathfrak{p}_{y}}{V^{2}}y-\frac{\mathfrak{p}_{z}}{V^{2}}z
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-v_{x}t\\
y^{\prime} &amp; =y-v_{y}t\\
z^{\prime} &amp; =z-v_{z}t\\
t^{\prime} &amp; =t-\frac{v_{x}}{c^{2}}x'-\frac{v_{y}}{c^{2}}y'-\frac{v_{z}}{c^{2}}z'
\end{align}
\end{matrix}&lt;/math&gt;

With this concept Lorentz could explain the [[w:Doppler effect]], the [[w:aberration of light]], and the [[w:Fizeau experiment]].&lt;ref&gt;Janssen (1995), 3.1&lt;/ref&gt;

=== {{anchor|Larmor}} Larmor (1897, 1900) ===

In 1897, Larmor extended the work of Lorentz and derived the following transformation&lt;ref group=R&gt;Larmor (1897), p. 229&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =x\varepsilon^{\frac{1}{2}}\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-vx/c^{2}\\
dt_{1} &amp; =dt^{\prime}\varepsilon^{-\frac{1}{2}}\\
\varepsilon &amp; =\left(1-v^{2}/c^{2}\right)^{-1}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-\frac{vx^{\ast}}{c^{2}}=t-\frac{v(x-vt)}{c^{2}}\\
dt_{1} &amp; =\frac{dt^{\prime}}{\gamma}\\
\gamma^{2} &amp; =\frac{1}{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

Larmor noted that if it is assumed that the constitution of molecules is electrical then the FitzGerald–Lorentz contraction is a consequence of this transformation, explaining the [[w:Michelson–Morley experiment]]. It's notable that Larmor was the first who recognized that some sort of [[w:time dilation]] is a consequence of this transformation as well, because &quot;individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 1/γ&quot;.&lt;ref&gt;Darrigol (2000), Chap. 8.5&lt;/ref&gt;&lt;ref&gt;Macrossan (1986)&lt;/ref&gt; Larmor wrote his electrodynamical equations and transformations neglecting terms of higher order than ''(v/c)''&lt;sup&gt;2&lt;/sup&gt; – when his 1897 paper was reprinted in 1929, Larmor added the following comment in which he described how they can be made valid to all orders of ''v/c'':&lt;ref group=R&gt;Larmor (1897/1929), p. 39&lt;/ref&gt;

{{Quote|Nothing need be neglected: the transformation is ''exact'' if ''v/c''&lt;sup&gt;2&lt;/sup&gt; is replaced by ''εv/c''&lt;sup&gt;2&lt;/sup&gt; in the equations and also in the change following from ''t'' to ''t′'', as is worked out in ''Aether and Matter'' (1900), p. 168, and as Lorentz found it to be in 1904, thereby stimulating the modern schemes of intrinsic relational relativity.}}

In line with that comment, in his book Aether and Matter published in 1900, Larmor used a modified local time ''t″=t′-εvx′/c&lt;sup&gt;2&lt;/sup&gt;'' instead of the 1897 expression ''t′=t-vx/c&lt;sup&gt;2&lt;/sup&gt;'' by replacing ''v/c''&lt;sup&gt;2&lt;/sup&gt; with ''εv/c''&lt;sup&gt;2&lt;/sup&gt;, so that ''t″'' is now identical to the one given by Lorentz in 1892, which he combined with a Galilean transformation for the ''x′, y′, z′, t′'' coordinates:&lt;ref group=R&gt;Larmor (1900), p. 168&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime} &amp; =t^{\prime}-\varepsilon vx^{\prime}/c^{2}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime}=t^{\prime}-\frac{\gamma^{2}vx^{\prime}}{c^{2}} &amp; =\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor ''(v/c)''&lt;sup&gt;2&lt;/sup&gt;, and so he sought the transformations which were &quot;accurate to second order&quot; (as he put it). Thus he wrote the final transformations (where ''x′=x-vt'' and ''t″'' as given above) as:&lt;ref group=R&gt;Larmor (1900), p. 174&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =\varepsilon^{\frac{1}{2}}x^{\prime}\\
y_{1} &amp; =y^{\prime}\\
z_{1} &amp; =z^{\prime}\\
dt_{1} &amp; =\varepsilon^{-\frac{1}{2}}dt^{\prime\prime}=\varepsilon^{-\frac{1}{2}}\left(dt^{\prime}-\frac{v}{c^{2}}\varepsilon dx^{\prime}\right)\\
t_{1} &amp; =\varepsilon^{-\frac{1}{2}}t^{\prime}-\frac{v}{c^{2}}\varepsilon^{\frac{1}{2}}x^{\prime}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\prime}=\gamma(x-vt)\\
y_{1} &amp; =y'=y\\
z_{1} &amp; =z'=z\\
dt_{1} &amp; =\frac{dt^{\prime\prime}}{\gamma}=\frac{1}{\gamma}\left(dt^{\prime}-\frac{\gamma^{2}vdx^{\prime}}{c^{2}}\right)=\gamma\left(dt-\frac{vdx}{c^{2}}\right)\\
t_{1} &amp; =\frac{t^{\prime}}{\gamma}-\frac{\gamma vx^{\prime}}{c^{2}}=\gamma\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

by which he arrived at the complete Lorentz transformation ({{equationNote|4b}}). Larmor showed that Maxwell's equations were invariant under this two-step transformation, &quot;to second order in ''v/c''&quot; – it was later shown by Lorentz (1904) and Poincaré (1905) that they are indeed invariant under this transformation to all orders in ''v/c''.

Larmor gave credit to Lorentz in two papers published in 1904, in which he used the term &quot;Lorentz transformation&quot; for Lorentz's first order transformations of coordinates and field configurations:

{{Quote|p. 583: [..] Lorentz's transformation for passing from the field of activity of a stationary electrodynamic material system to that of one moving with uniform velocity of translation through the aether.&lt;br /&gt; p. 585: [..] the Lorentz transformation has shown us what is not so immediately obvious [..]&lt;ref group=R&gt;Larmor (1904a), p. 583, 585&lt;/ref&gt; &lt;br /&gt; p. 622: [..] the transformation first developed by Lorentz: namely, each point in space is to have its own origin from which time is measured, its &quot;local time&quot; in Lorentz's phraseology, and then the values of the electric and magnetic vectors [..] at all points in the aether between the molecules in the system at rest, are the same as those of the vectors [..] at the corresponding points in the convected system at the same local times.&lt;ref group=R&gt;Larmor (1904b), p. 622&lt;/ref&gt;}}

=== {{anchor|Lorentz2}} Lorentz (1899, 1904) ===

Also Lorentz extended his theorem of corresponding states in 1899. First he wrote a transformation equivalent to the one from 1892 (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1899), p. 429&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}x\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}-\mathfrak{p}_{x}^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Then he introduced a factor ε of which he said he has no means of determining it, and modified his transformation as follows (where the above value of ''t′'' has to be inserted):&lt;ref group=R&gt;Lorentz (1899), p. 439&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\frac{\varepsilon}{k}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon x^{\prime\prime}\\
t^{\prime} &amp; =k\varepsilon t^{\prime\prime}\\
k &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{\varepsilon}{\gamma}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon z^{\prime\prime}\\
t^{\prime}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right) &amp; =\gamma\varepsilon t^{\prime\prime}\\
\gamma &amp; =\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
\end{align}
\end{matrix}&lt;/math&gt;

This is equivalent to the complete Lorentz transformation ({{equationNote|4b}}) when solved for ''x″'' and ''t″'' and with ε=1. Like Larmor, Lorentz noticed in 1899&lt;ref group=R&gt;Lorentz (1899), p. 442&lt;/ref&gt; also some sort of time dilation effect in relation to the frequency of oscillating electrons ''&quot;that in ''S'' the time of vibrations be ''kε'' times as great as in ''S&lt;sub&gt;0&lt;/sub&gt;''&quot;'', where ''S&lt;sub&gt;0&lt;/sub&gt;'' is the aether frame.&lt;ref&gt;Janssen (1995), Kap. 3.3&lt;/ref&gt;

In 1904 he rewrote the equations in the following form by setting ''l''=1/ε (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1904), p. 812&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =klx\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =\frac{l}{k}t-kl\frac{w}{c^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma lx^{\ast}=\gamma l(x-vt)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t^{\prime} &amp; =\frac{lt}{\gamma}-\frac{\gamma lvx^{\ast}}{c^{2}}=\gamma l\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Under the assumption that ''l=1'' when ''v''=0, he demonstrated that ''l=1'' must be the case at all velocities, therefore length contraction can only arise in the line of motion. So by setting the factor ''l'' to unity, Lorentz's transformations now assumed the same form as Larmor's and are now completed. Unlike Larmor, who restricted himself to show the covariance of Maxwell's equations to second order, Lorentz tried to widen its covariance to all orders in ''v/c''. He also derived the correct formulas for the velocity dependence of [[w:electromagnetic mass]], and concluded that the transformation formulas must apply to all forces of nature, not only electrical ones.&lt;ref group=R&gt;Lorentz (1904), p. 826&lt;/ref&gt; However, he didn't achieve full covariance of the transformation equations for charge density and velocity.&lt;ref&gt;Miller (1981), Chap. 1.12.2&lt;/ref&gt; When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:&lt;ref&gt;Janssen (1995), Chap. 3.5.6&lt;/ref&gt;

{{Quote|One will notice that in this work the transformation equations of Einstein’s Relativity Theory have not quite been attained. [..] On this circumstance depends the clumsiness of many of the further considerations in this work.}}

Lorentz's 1904 transformation was cited and used by [[w:Alfred Bucherer]] in July 1904:&lt;ref group=R&gt;Bucherer, p. 129; Definition of s on p. 32&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\sqrt{s}x,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{\sqrt{s}}-\sqrt{s}\frac{u}{v^{2}}x,\quad s=1-\frac{u^{2}}{v^{2}}&lt;/math&gt;

or by [[w:Wilhelm Wien]] in July 1904:&lt;ref group=R&gt;Wien (1904), p. 394&lt;/ref&gt;

:&lt;math&gt;x=kx',\quad y=y',\quad z=z',\quad t'=kt-\frac{v}{kc^{2}}x&lt;/math&gt;

or by [[w:Emil Cohn]] in November 1904 (setting the speed of light to unity):&lt;ref group=R&gt;Cohn (1904a), pp. 1296-1297&lt;/ref&gt;

:&lt;math&gt;x=\frac{x_{0}}{k},\quad y=y_{0},\quad z=z_{0},\quad t=kt_{0},\quad t_{1}=t_{0}-w\cdot r_{0},\quad k^{2}=\frac{1}{1-w^{2}}&lt;/math&gt;

or by [[w:Richard Gans]] in February 1905:&lt;ref group=R&gt;Gans (1905), p. 169&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=kx,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{k}-\frac{kwx}{c^{2}},\quad k^{2}=\frac{c^{2}}{c^{2}-w^{2}}&lt;/math&gt;

=== {{anchor|Poincare3}} Poincaré (1900, 1905) ===

==== Local time ====

Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time. However, [[w:Henri Poincaré]] in 1900 commented on the origin of Lorentz's &quot;wonderful invention&quot; of local time.&lt;ref&gt;Darrigol (2005), Kap. 4&lt;/ref&gt; He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed &lt;math&gt;c&lt;/math&gt; in both directions, which lead to what is nowadays called [[w:relativity of simultaneity]], although Poincaré's calculation does not involve length contraction or time dilation.&lt;ref group=R&gt;Poincaré (1900), pp. 272–273&lt;/ref&gt; In order to synchronise the clocks here on Earth (the ''x*, t''* frame) a light signal from one clock (at the origin) is sent to another (at ''x''*), and is sent back. It's supposed that the Earth is moving with speed ''v'' in the ''x''-direction (= ''x''*-direction) in some rest system (''x, t'') (''i.e.'' the [[w:luminiferous aether]] system for Lorentz and Larmor). The time of flight outwards is

:&lt;math&gt;\delta t_{a}=\frac{x^{\ast}}{\left(c-v\right)}&lt;/math&gt;

and the time of flight back is

:&lt;math&gt;\delta t_{b}=\frac{x^{\ast}}{\left(c+v\right)}&lt;/math&gt;.

The elapsed time on the clock when the signal is returned is ''δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;'' and the time ''t*=(δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;)/2'' is ascribed to the moment when the light signal reached the distant clock. In the rest frame the time ''t=δt&lt;sub&gt;a&lt;/sub&gt;'' is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus

:&lt;math&gt;t^{\ast}=t-\frac{\gamma^{2}vx^{*}}{c^{2}}&lt;/math&gt;

identical to Lorentz (1892). By dropping the factor γ&lt;sup&gt;2&lt;/sup&gt; under the assumption that &lt;math&gt;\tfrac{v^{2}}{c^{2}}\ll1&lt;/math&gt;, Poincaré gave the result ''t*=t-vx*/c&lt;sup&gt;2&lt;/sup&gt;'', which is the form used by Lorentz in 1895.

Similar physical interpretations of local time were later given by [[w:Emil Cohn]] (1904)&lt;ref group=R&gt;Cohn (1904b), p. 1408&lt;/ref&gt; and [[w:Max Abraham]] (1905).&lt;ref group=R&gt;Abraham (1905), § 42&lt;/ref&gt;

==== Lorentz transformation ====

On June 5, 1905 (published June 9) Poincaré formulated transformation equations which are algebraically equivalent to those of Larmor and Lorentz and gave them the modern form ({{equationNote|4b}}):&lt;ref group=R&gt;Poincaré (1905), p. 1505&lt;/ref&gt;

:&lt;math&gt;\begin{align}x^{\prime} &amp; =kl(x+\varepsilon t)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =kl(t+\varepsilon x)\\
k &amp; =\frac{1}{\sqrt{1-\varepsilon^{2}}}
\end{align}
&lt;/math&gt;.

Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name &quot;Lorentz transformation&quot;.&lt;ref&gt;Pais (1982), Chap. 6c&lt;/ref&gt;&lt;ref&gt;Katzir (2005), 280–288&lt;/ref&gt; Poincaré set the speed of light to unity, pointed out the group characteristics of the transformation by setting ''l''=1, and modified/corrected Lorentz's derivation of the equations of electrodynamics in some details in order to fully satisfy the principle of relativity, ''i.e.'' making them fully Lorentz covariant.&lt;ref&gt;Miller (1981), Chap. 1.14&lt;/ref&gt;

In July 1905 (published in January 1906)&lt;ref group=R&gt;Poincaré (1905/06), pp. 129ff&lt;/ref&gt; Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of the [[w:principle of least action]]; he demonstrated in more detail the group characteristics of the transformation, which he called [[w:Lorentz group]], and he showed that the combination ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;+z&lt;sup&gt;2&lt;/sup&gt;-t&lt;sup&gt;2&lt;/sup&gt;'' is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing &lt;math&gt;ct\sqrt{-1}&lt;/math&gt; as a fourth imaginary coordinate, and he used an early form of [[w:four-vector]]s. He also formulated the velocity addition formula ({{equationNote|4c}}), which he had already derived in unpublished letters to Lorentz from May 1905:&lt;ref group=R&gt;Poincaré (1905/06), p. 144&lt;/ref&gt;

:&lt;math&gt;\xi'=\frac{\xi+\varepsilon}{1+\xi\varepsilon},\ \eta'=\frac{\eta}{k(1+\xi\varepsilon)}&lt;/math&gt;.

==={{anchor|Einstein}} Einstein (1905) – Special relativity===

On June 30, 1905 (published September 1905) Einstein published what is now called [[w:special relativity]] and gave a new derivation of the transformation, which was based only on the principle on relativity and the principle of the constancy of the speed of light. While Lorentz considered &quot;local time&quot; to be a mathematical stipulation device for explaining the Michelson-Morley experiment, Einstein showed that the coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. For quantities of first order in ''v/c'' this was also done by Poincaré in 1900, while Einstein derived the complete transformation by this method. Unlike Lorentz and Poincaré who still distinguished between real time in the aether and apparent time for moving observers, Einstein showed that the transformations concern the nature of space and time.&lt;ref&gt;Miller (1981), Chap. 6&lt;/ref&gt;&lt;ref&gt;Pais (1982), Kap. 7&lt;/ref&gt;&lt;ref&gt;Darrigol (2005), Chap. 6&lt;/ref&gt;

The notation for this transformation is equivalent to Poincaré's of 1905 and ({{equationNote|4b}}), except that Einstein didn't set the speed of light to unity:&lt;ref group=R&gt;Einstein (1905), p. 902&lt;/ref&gt;

:&lt;math&gt;\begin{align}\tau &amp; =\beta\left(t-\frac{v}{V^{2}}x\right)\\
\xi &amp; =\beta(x-vt)\\
\eta &amp; =y\\
\zeta &amp; =z\\
\beta &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}
\end{align}
&lt;/math&gt;

Einstein also defined the velocity addition formula ({{equationNote|4c}}, {{equationNote|4d}}):&lt;ref group=R&gt;Einstein (1905), § 5 and § 9&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=\frac{w_{\xi}+v}{1+\frac{vw_{\xi}}{V^{2}}}t,\ y=\frac{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}{1+\frac{vw_{\xi}}{V^{2}}}w_{\eta}t\\
U^{2}=\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2},\ w^{2}=w_{\xi}^{2}+w_{\eta}^{2},\ \alpha=\operatorname{arctg}\frac{w_{y}}{w_{x}}\\
U=\frac{\sqrt{\left(v^{2}+w^{2}+2vw\cos\alpha\right)-\left(\frac{vw\sin\alpha}{V}\right)^{2}}}{1+\frac{vw\cos\alpha}{V^{2}}}
\end{matrix}\left|\begin{matrix}\frac{u_{x}-v}{1-\frac{u_{x}v}{V^{2}}}=u_{\xi}\\
\frac{u_{y}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\eta}\\
\frac{u_{z}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\zeta}
\end{matrix}\right.&lt;/math&gt;

and the light aberration formula ({{equationNote|4e}}):&lt;ref group=R&gt;Einstein (1905), § 7&lt;/ref&gt;

:&lt;math&gt;\cos\varphi'=\frac{\cos\varphi-\frac{v}{V}}{1-\frac{v}{V}\cos\varphi}&lt;/math&gt;

=== {{anchor|Minkowski}} Minkowski (1907–1908) – Spacetime ===

The work on the principle of relativity by Lorentz, Einstein, [[w:Max Planck|Planck]], together with Poincaré's four-dimensional approach, were further elaborated and combined with the [[w:hyperboloid model]] by [[w:Hermann Minkowski]] in 1907 and 1908.&lt;ref group=R&gt;Minkowski (1907/15), pp. 927ff&lt;/ref&gt;&lt;ref group=R&gt;Minkowski (1907/08), pp. 53ff&lt;/ref&gt; Minkowski particularly reformulated electrodynamics in a four-dimensional way ([[w:Minkowski spacetime]]).&lt;ref&gt;Walter (1999a), (1999b), (2018)&lt;/ref&gt; For instance, he wrote ''x, y, z, it'' in the form ''x&lt;sub&gt;1&lt;/sub&gt;, x&lt;sub&gt;2&lt;/sub&gt;, x&lt;sub&gt;3&lt;/sub&gt;, x&lt;sub&gt;4&lt;/sub&gt;''. By defining ψ as the angle of rotation around the ''z''-axis, the Lorentz transformation assumes a form (with ''c''=1):&lt;ref group=R name=mink1&gt;Minkowski (1907/08), p. 59&lt;/ref&gt;

:&lt;math&gt;\begin{align}x'_{1} &amp; =x_{1}\\
x'_{2} &amp; =x_{2}\\
x'_{3} &amp; =x_{3}\cos i\psi+x_{4}\sin i\psi\\
x'_{4} &amp; =-x_{3}\sin i\psi+x_{4}\cos i\psi\\
\cos i\psi &amp; =\frac{1}{\sqrt{1-q^{2}}}
\end{align}
&lt;/math&gt;

Even though Minkowski used the imaginary number iψ, he for once&lt;ref group=R name=mink1 /&gt; directly used the [[w:tangens hyperbolicus]] in the equation for velocity

:&lt;math&gt;-i\tan i\psi=\frac{e^{\psi}-e^{-\psi}}{e^{\psi}+e^{-\psi}}=q&lt;/math&gt; with &lt;math&gt;\psi=\frac{1}{2}\ln\frac{1+q}{1-q}&lt;/math&gt;.

Minkowski's expression can also by written as ψ=atanh(q) and was later called [[w:rapidity]]. He also wrote the Lorentz transformation in matrix form:&lt;ref group=R&gt;Minkowski (1907/08), pp. 65–66, 81–82&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}+x_{4}^{\prime2}\\
\left(x_{1}^{\prime}=x',\ x_{2}^{\prime}=y',\ x_{3}^{\prime}=z',\ x_{4}^{\prime}=it'\right)\\
-x^{2}-y^{2}-z^{2}+t^{2}=-x^{\prime2}-y^{\prime2}-z^{\prime2}+t^{\prime2}\\
\hline x_{h}=\alpha_{h1}x_{1}^{\prime}+\alpha_{h2}x_{2}^{\prime}+\alpha_{h3}x_{3}^{\prime}+\alpha_{h4}x_{4}^{\prime}\\
\mathrm{A}=\mathrm{\left|\begin{matrix}\alpha_{11}, &amp; \alpha_{12}, &amp; \alpha_{13}, &amp; \alpha_{14}\\
\alpha_{21}, &amp; \alpha_{22}, &amp; \alpha_{23}, &amp; \alpha_{24}\\
\alpha_{31}, &amp; \alpha_{32}, &amp; \alpha_{33}, &amp; \alpha_{34}\\
\alpha_{41}, &amp; \alpha_{42}, &amp; \alpha_{43}, &amp; \alpha_{44}
\end{matrix}\right|,\ \begin{align}\bar{\mathrm{A}}\mathrm{A} &amp; =1\\
\left(\det \mathrm{A}\right)^{2} &amp; =1\\
\det \mathrm{A} &amp; =1\\
\alpha_{44} &amp; &gt;0
\end{align}
}
\end{matrix}&lt;/math&gt;

As a graphical representation of the Lorentz transformation he introduced the [[w:Minkowski diagram]], which became a standard tool in textbooks and research articles on relativity:&lt;ref group=R&gt;Minkowski (1908/09), p. 77&lt;/ref&gt;

[[File:Minkowski1.png|center|thumb|400px|Original spacetime diagram by Minkowski in 1908.]]

==={{anchor|Frank}} Frank (1909) – Hyperbolic functions===
Hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity in agreement with ({{equationNote|4b}}):&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

==={{Anchor|Sommerfeld}} Sommerfeld (1909) – Spherical trigonometry===

Using an imaginary rapidity such as Minkowski, [[w:Arnold Sommerfeld]] (1909) formulated a transformation equivalent to Lorentz boost ({{equationNote|4b}}), and the relativistc velocity addition ({{equationNote|4c}}) in terms of trigonometric functions and the [[w:spherical law of cosines]]:&lt;ref group=R&gt;Sommerfeld (1909), p. 826ff.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{array}{lrl}
x'= &amp; x\ \cos\varphi+l\ \sin\varphi, &amp; y'=y\\
l'= &amp; -x\ \sin\varphi+l\ \cos\varphi, &amp; z'=z
\end{array}\right\} \\
\left(\operatorname{tg}\varphi=i\beta,\ \cos\varphi=\frac{1}{\sqrt{1-\beta^{2}}},\ \sin\varphi=\frac{i\beta}{\sqrt{1-\beta^{2}}}\right)\\
\hline \beta=\frac{1}{i}\operatorname{tg}\left(\varphi_{1}+\varphi_{2}\right)=\frac{1}{i}\frac{\operatorname{tg}\varphi_{1}+\operatorname{tg}\varphi_{2}}{1-\operatorname{tg}\varphi_{1}\operatorname{tg}\varphi_{2}}=\frac{\beta_{1}+\beta_{2}}{1+\beta_{1}\beta_{2}}\\
\cos\varphi=\cos\varphi_{1}\cos\varphi_{2}-\sin\varphi_{1}\sin\varphi_{2}\cos\alpha\\
v^{2}=\frac{v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos\alpha-\frac{1}{c^{2}}v_{1}^{2}v_{2}^{2}\sin^{2}\alpha}{\left(1+\frac{1}{c^{2}}v_{1}v_{2}\cos\alpha\right)^{2}}
\end{matrix}&lt;/math&gt;

=== {{anchor|Herglotz1}} Herglotz (1909/10) – Möbius transformation ===

Following [[../Lorentz transformation (Möbius)#Klein2|E:Klein (1889–1897)]] as well as [[../Lorentz transformation (Möbius)#Fricke|E:Fricke &amp; Klein (1897)]] concerning the Cayley absolute, hyperbolic motion and its transformation, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic. The general case (on the left) equivalent to Lorentz-Möbius transformation [[../Lorentz transformation (Möbius)#math_6a|E:'''(6a)''']] and the hyperbolic case (on the right) equivalent to Lorentz transformation [[../Lorentz transformation (hyperbolic)#math_3d|E:'''(3d)''']] or squeeze mapping [[../Lorentz transformation (squeeze)#math_9d|E:'''(9d)''']] are as follows:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;

:&lt;math&gt;\left.\begin{matrix}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\\
z_{1}=x,\ z_{2}=y,\ z_{3}=z,\ z_{4}=t\\
Z=\frac{z_{1}+iz_{2}}{z_{4}-z_{3}}=\frac{x+iy}{t-z},\ Z'=\frac{x'+iy'}{t'-z'}\\
Z=\frac{\alpha Z'+\beta}{\gamma Z'+\delta}
\end{matrix}\right|\begin{matrix}Z=Z'e^{\vartheta}\\
\begin{align}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
\end{align}
\end{matrix}&lt;/math&gt;

==={{anchor|Varicak}} Varićak (1910) – Hyperbolic functions===

Following [[#Sommerfeld|Sommerfeld (1909)]], hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

and showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

Subsequently, other authors such as [[w:E. T. Whittaker]] (1910) or [[w:Alfred Robb]] (1911, who coined the name rapidity) used similar expressions, which are still used in modern textbooks.&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

=== {{anchor|Ignatowski}} Ignatowski (1910) ===

While earlier derivations and formulations of the Lorentz transformation relied from the outset on optics, electrodynamics, or the invariance of the speed of light, [[w:Vladimir Ignatowski]] (1910) showed that it is possible to use the principle of relativity (and related [[w:Group theory|group theoretical]] principles) alone, in order to derive the following transformation between two inertial frames:&lt;ref group=R&gt;Ignatowski (1910), pp. 973–974&lt;/ref&gt;&lt;ref group=R&gt;Ignatowski (1910/11ab)&lt;/ref&gt;

:&lt;math&gt;\begin{align}dx' &amp; =p\ dx-pq\ dt\\
dt' &amp; =-pqn\ dx+p\ dt\\
p &amp; =\frac{1}{\sqrt{1-q^{2}n}}
\end{align}
&lt;/math&gt;

The variable ''n'' can be seen as a space-time constant whose value has to be determined by experiment or taken from a known physical law such as electrodynamics. For that purpose, Ignatowski used the above-mentioned Heaviside ellipsoid representing a contraction of electrostatic fields by ''x''/γ in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation when ''n=1/c''&lt;sup&gt;2&lt;/sup&gt;, resulting in ''p''=γ and the Lorentz transformation ({{equationNote|4b}}). With ''n''=0, no length changes arise and the Galilean transformation follows. Ignatowski's method was further developed and improved by [[w:Philipp Frank]] and [[w:Hermann Rothe]] (1911, 1912),&lt;ref group=R&gt;Frank &amp; Rothe (1911), pp. 825ff; (1912), p. 750ff.&lt;/ref&gt; with various authors developing similar methods in subsequent years.&lt;ref name=baccetti&gt;Baccetti (2011), see references 1–25 therein.&lt;/ref&gt;

==={{anchor|Noether}} Noether (1910), Klein (1910) – Quaternions===

[[w:Felix Klein]] (1908) described [[../Lorentz transformation (Quaternions)#Cayley3|E:Cayley's (1854)]] 4D quaternion multiplications as &quot;Drehstreckungen&quot; (orthogonal substitutions in terms of rotations leaving invariant a quadratic form up to a factor), and pointed out that the modern principle of relativity as provided by Minkowski is essentially only the consequent application of such Drehstreckungen, even though he didn't provide details.&lt;ref group=R&gt;Klein (1908), p. 165&lt;/ref&gt;

In an appendix to Klein's and Sommerfeld's &quot;Theory of the top&quot; (1910), [[w:Fritz Noether]] showed how to formulate hyperbolic rotations using biquaternions with &lt;math&gt;\omega=\sqrt{-1}&lt;/math&gt;, which he also related to the speed of light by setting ω&lt;sup&gt;2&lt;/sup&gt;=-''c''&lt;sup&gt;2&lt;/sup&gt;. He concluded that this is the principal ingredient for a rational representation of the group of Lorentz transformations:&lt;ref group=R&gt;Noether (1910), pp. 939–943&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}V=\frac{Q_{1}vQ_{2}}{T_{1}T_{2}}\\
\hline X^{2}+Y^{2}+Z^{2}+\omega^{2}S^{2}=x^{2}+y^{2}+z^{2}+\omega^{2}s^{2}\\
\hline \begin{align}V &amp; =Xi+Yj+Zk+\omega S\\
v &amp; =xi+yj+zk+\omega s\\
Q_{1} &amp; =(+Ai+Bj+Ck+D)+\omega(A'i+B'j+C'k+D')\\
Q_{2} &amp; =(-Ai-Bj-Ck+D)+\omega(A'i+B'j+C'k-D')\\
T_{1}T_{2} &amp; =T_{1}^{2}=T_{2}^{2}=A^{2}+B^{2}+C^{2}+D^{2}+\omega^{2}\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Besides citing quaternion related standard works such as [[#Cayley3|Cayley (1854)]], Noether referred to the entries in Klein's encyclopedia by [[w:Eduard Study]] (1899) and the French version by [[w:Élie Cartan]] (1908).&lt;ref&gt;Cartan &amp; Study (1908), sections 35–36&lt;/ref&gt; Cartan's version contains a description of Study's [[w:dual number]]s, Clifford's biquaternions (including the choice &lt;math&gt;\omega=\sqrt{-1}&lt;/math&gt; for hyperbolic geometry), and Clifford algebra, with references to [[../Lorentz transformation (Quaternions)#Stephanos|Stephanos (1883)]], [[../Lorentz transformation (Quaternions)#Buchheim|Buchheim (1884/85)]], [[../Lorentz transformation (Quaternions)#Vahlen|Vahlen (1901/02)]] and others.

Citing Noether, Klein himself published in August 1910 the following quaternion substitutions forming the group of Lorentz transformations:&lt;ref group=R&gt;Klein (1910), p. 300&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align} &amp; \left(i_{1}x'+i_{2}y'+i_{3}z'+ict'\right)\\
 &amp; \quad-\left(i_{1}x_{0}+i_{2}y_{0}+i_{3}z_{0}+ict_{0}\right)
\end{align}
=\frac{\left[\begin{align} &amp; \left(i_{1}(A+iA')+i_{2}(B+iB')+i_{3}(C+iC')+i_{4}(D+iD')\right)\\
 &amp; \quad\cdot\left(i_{1}x+i_{2}y+i_{3}z+ict\right)\\
 &amp; \quad\quad\cdot\left(i_{1}(A-iA')+i_{2}(B-iB')+i_{3}(C-iC')-(D-iD')\right)
\end{align}
\right]}{\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)-\left(A^{2}+B^{2}+C^{2}+D^{2}\right)}\\
\hline \text{where}\\
AA'+BB'+CC'+DD'=0\\
A^{2}+B^{2}+C^{2}+D^{2}&gt;A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}
\end{matrix}&lt;/math&gt;

or in March 1911&lt;ref group=R&gt;Klein (1911), pp. 602ff.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}g'=\frac{pg\pi}{M}\\
\hline \begin{align}g &amp; =\sqrt{-1}ct+ix+jy+kz\\
g' &amp; =\sqrt{-1}ct'+ix'+jy'+kz'\\
p &amp; =(D+\sqrt{-1}D')+i(A+\sqrt{-1}A')+j(B+\sqrt{-1}B')+k(C+\sqrt{-1}C')\\
\pi &amp; =(D-\sqrt{-1}D')-i(A-\sqrt{-1}A')-j(B-\sqrt{-1}B')-k(C-\sqrt{-1}C')\\
M &amp; =\left(A^{2}+B^{2}+C^{2}+D^{2}\right)-\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)\\
 &amp; AA'+BB'+CC'+DD'=0\\
 &amp; A^{2}+B^{2}+C^{2}+D^{2}&gt;A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}
\end{align}
\end{matrix}&lt;/math&gt;

==={{anchor|Conway}} Conway (1911), Silberstein (1911) – Quaternions===

[[w:Arthur W. Conway]] in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms of velocity λ:&lt;ref group=R&gt;Conway (1911), p. 8&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}\mathtt{D} &amp; =\mathbf{a}^{-1}\mathtt{D}'\mathbf{a}^{-1}\\
\mathtt{\sigma} &amp; =\mathbf{a}\mathtt{\sigma}'\mathbf{a}^{-1}
\end{align}
\\
e=\mathbf{a}^{-1}e'\mathbf{a}^{-1}\\
\hline a=\left(1-hc^{-1}\lambda\right)^{\frac{1}{2}}\left(1+c^{-2}\lambda^{2}\right)^{-\frac{1}{4}}
\end{matrix}&lt;/math&gt;

Also [[w:Ludwik Silberstein]] in November 1911&lt;ref group=R&gt;Silberstein (1911/12), p. 793&lt;/ref&gt; as well as in 1914,&lt;ref&gt;Silberstein (1914), p. 156&lt;/ref&gt; formulated the Lorentz transformation in terms of velocity ''v'':

:&lt;math&gt;\begin{matrix}q'=QqQ\\
\hline \begin{align}q &amp; =\mathbf{r}+l=xi+yj+zk+\iota ct\\
q &amp; '=\mathbf{r}'+l'=x'i+y'j+z'k+\iota ct'\\
Q &amp; =\frac{1}{\sqrt{2}}\left(\sqrt{1+\gamma}+\mathrm{u}\sqrt{1-\gamma}\right)\\
 &amp; =\cos\alpha+\mathrm{u}\sin\alpha=e^{\alpha\mathrm{u}}\\
 &amp; \left\{ \gamma=\left(1-v^{2}/c^{2}\right)^{-1/2},\ 2\alpha=\operatorname{arctg}\ \left(\iota\frac{v}{c}\right)\right\} 
\end{align}
\end{matrix}&lt;/math&gt;

Silberstein cites [[../Lorentz transformation (Quaternions)#Cayley3|E:Cayley (1854, 1855)]] and Study's encyclopedia entry (in the extended French version of Cartan in 1908), as well as the appendix of Klein's and Sommerfeld's book.

==={{anchor|Herglotz2}} Ignatowski (1910/11), Herglotz (1911), and others – Vector transformation===

{{See|w:Lorentz transformation#Vector transformations}}

[[w:Vladimir Ignatowski]] (1910, published 1911) defined the vectorial velocity addition ({{equationNote|4g}}) as well as general Lorentz boost ({{equationNote|4f}}) as&lt;ref group=R&gt;Ignatowski (1910/11a), p. 23; (1910/11b), p. 22&lt;/ref&gt; 

:&lt;math&gt;\begin{matrix}\begin{matrix}\mathfrak{v} =\frac{\mathfrak{v}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{v}'+pq\mathfrak{c}_{0}}{p\left(1+nq\mathfrak{c}_{0}\mathfrak{v}'\right)} &amp; \left|\begin{align}\mathfrak{A}' &amp; =\mathfrak{A}+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}-pqb\mathfrak{c}_{0}\\
b' &amp; =pb-pqn\mathfrak{A}\mathfrak{c}_{0}\\
\\
\mathfrak{A} &amp; =\mathfrak{A}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}'+pqb'\mathfrak{c}_{0}\\
b &amp; =pb'+pqn\mathfrak{A}'\mathfrak{c}_{0}
\end{align}
\right.\end{matrix}\\
\left[\mathfrak{v}=\mathbf{u},\ \mathfrak{A}=\mathbf{x},\ b=t,\ \mathfrak{c}_{0}=\frac{\mathbf{v}}{v},\ p=\gamma,\ n=\frac{1}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

An equivalent transformation was given by [[w:Gustav Herglotz]] (1911)&lt;ref group=R&gt;Herglotz (1911), p. 497&lt;/ref&gt; using '''v'''=''(v&lt;sub&gt;x&lt;/sub&gt;, v&lt;sub&gt;y&lt;/sub&gt;, v&lt;sub&gt;z&lt;/sub&gt;)'' and '''r'''=''(x, y, z)'':

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{0} &amp; =x+\alpha u(ux+vy+wz)-\beta ut\\
y^{0} &amp; =y+\alpha v(ux+vy+wz)-\beta vt\\
z^{0} &amp; =z+\alpha w(ux+vy+wz)-\beta wt\\
t^{0} &amp; =-\beta(ux+vy+wz)+\beta t\\
 &amp; \alpha=\frac{1}{\sqrt{1-s^{2}}\left(1+\sqrt{1-s^{2}}\right)},\ \beta=\frac{1}{\sqrt{1-s^{2}}}
\end{align}
\right| &amp; \begin{align}x' &amp; =x+\alpha v_{x}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{x}t\\
y' &amp; =y+\alpha v_{y}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{y}t\\
z' &amp; =z+\alpha v_{z}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{z}t\\
t' &amp; =-\gamma\left(v_{x}x+v_{y}y+v_{z}z\right)+\gamma t\\
 &amp; \alpha=\frac{\gamma^{2}}{\gamma+1},\ \gamma=\frac{1}{\sqrt{1-v^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

Or by [[w:Ludwik Silberstein]] (1911 on the left, 1914 on the right):&lt;ref group=R&gt;Silberstein (1911/12), p. 792; (1914), p. 123&lt;/ref&gt;

:&lt;math&gt;\begin{array}{c|c}
\begin{align}\mathbf{r}' &amp; =\mathbf{r}+(\gamma-1)(\mathbf{ru})\mathbf{u}+i\beta\gamma lu\\
l' &amp; =\gamma\left[l-i\beta(\mathbf{ru})\right]
\end{align}
 &amp; \begin{align}\mathbf{r}' &amp; =\mathbf{r}+\left[\frac{\gamma-1}{v^{2}}(\mathbf{vr})-\gamma t\right]\mathbf{v}\\
t' &amp; =\gamma\left[t-\frac{1}{c^{2}}(\mathbf{vr})\right]
\end{align}
\end{array}&lt;/math&gt;

[[w:Erwin Madelung]] (1922) provided the matrix form&lt;ref&gt;Madelung (1921), p. 207&lt;/ref&gt;

:&lt;math&gt;\begin{array}{c|c|c|c|c}
 &amp; x &amp; y &amp; z &amp; t\\
\hline x' &amp; 1-\frac{v_{x}^{2}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{x}v_{y}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{x}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; \frac{-v_{x}}{\sqrt{1-\beta^{2}}}\\
y' &amp; -\frac{v_{x}v_{y}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; 1-\frac{v_{y}^{2}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{y}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; \frac{-v_{y}}{\sqrt{1-\beta^{2}}}\\
z' &amp; -\frac{v_{x}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{y}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; 1-\frac{v_{z}^{2}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; \frac{-v_{z}}{\sqrt{1-\beta^{2}}}\\
t' &amp; \frac{-v_{x}}{c^{2}\sqrt{1-\beta^{2}}} &amp; \frac{-v_{y}}{c^{2}\sqrt{1-\beta^{2}}} &amp; \frac{-v_{z}}{c^{2}\sqrt{1-\beta^{2}}} &amp; \frac{1}{\sqrt{1-\beta^{2}}}
\end{array}&lt;/math&gt;

These formulas were called &quot;general Lorentz transformation without rotation&quot; by [[w:Christian Møller]] (1952),&lt;ref&gt;Møller (1952/55), pp. 41–43&lt;/ref&gt; who in addition gave an even more general Lorentz transformation in which the Cartesian axes have different orientations, using a [[w:Three-dimensional rotation operator|rotation operator]] &lt;math&gt;\mathfrak{D}&lt;/math&gt;. In this case, '''v′'''=''(v′&lt;sub&gt;x&lt;/sub&gt;, v′&lt;sub&gt;y&lt;/sub&gt;, v′&lt;sub&gt;z&lt;/sub&gt;)'' is not equal to -'''v'''=''(-v&lt;sub&gt;x&lt;/sub&gt;, -v&lt;sub&gt;y&lt;/sub&gt;, -v&lt;sub&gt;z&lt;/sub&gt;)'', but the relation &lt;math&gt;\mathbf{v}'=-\mathfrak{D}\mathbf{v}&lt;/math&gt; holds instead, with the result

:&lt;math&gt;\begin{array}{c}
\begin{align}\mathbf{x}' &amp; =\mathfrak{D}^{-1}\mathbf{x}-\mathbf{v}'\left\{ \left(\gamma-1\right)(\mathbf{x\cdot v})/v^{2}-\gamma t\right\} \\
t' &amp; =\gamma\left(t-(\mathbf{v}\cdot\mathbf{x})/c^{2}\right)
\end{align}
\end{array}&lt;/math&gt;

==={{anchor|Borel}} Borel (1913–14) – Cayley–Hermite parameter===

Borel (1913) started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and [[../Lorentz transformation (Cayley-Hermite)#Cayley|E:Cayley's (1846)]] parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions:&lt;ref group=R&gt;Borel (1913/14), p. 39&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}-z^{2}-1=0\\
\hline {\scriptstyle \begin{align}\delta a &amp; =\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta b &amp; =2(\lambda\mu+\nu\rho), &amp; \delta c &amp; =-2(\lambda\nu+\mu\rho),\\
\delta a' &amp; =2(\lambda\mu-\nu\rho), &amp; \delta b' &amp; =-\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta c' &amp; =2(\lambda\rho-\mu\nu),\\
\delta a'' &amp; =2(\lambda\nu-\mu\rho), &amp; \delta b'' &amp; =2(\lambda\rho+\mu\nu), &amp; \delta c'' &amp; =-\left(\lambda^{2}+\mu^{2}+\nu^{2}+\rho^{2}\right),
\end{align}
}\\
\left(\delta=\lambda^{2}+\mu^{2}-\rho^{2}-\nu^{2}\right)\\
\lambda=\nu=0\rightarrow\text{Hyperbolic rotation}
\end{matrix}&lt;/math&gt;

In four dimensions:&lt;ref group=R&gt;Borel (1913/14), p. 41&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}-\left(t_{1}-t_{2}\right)^{2}\\
\hline {\scriptstyle \begin{align} &amp; \left(\mu^{2}+\nu^{2}-\alpha^{2}\right)\cos\varphi+\left(\lambda^{2}-\beta^{2}-\gamma^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi-\gamma\operatorname{sh}{\theta}\\
 &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi+\gamma\operatorname{sh}{\theta} &amp;  &amp; \left(\mu^{2}+\nu^{2}-\beta^{2}\right)\cos\varphi+\left(\mu^{2}-\alpha^{2}-\gamma^{2}\right)\operatorname{ch}{\theta}\\
 &amp; -(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi-\beta\operatorname{sh}{\theta} &amp;  &amp; -(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\lambda\sin\varphi+\alpha\operatorname{sh}{\theta}\\
 &amp; (\gamma\mu-\beta\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta} &amp;  &amp; -(\alpha\nu-\lambda\gamma)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
\\
 &amp; \quad-(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi+\beta\operatorname{sh}{\theta} &amp;  &amp; \quad(\beta\nu-\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta}\\
 &amp; \quad-(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})-\lambda\sin\varphi-\alpha\operatorname{sh}{\theta} &amp;  &amp; \quad(\lambda\gamma-\alpha\nu)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
 &amp; \quad\left(\lambda^{2}+\mu^{2}-\gamma^{2}\right)\cos\varphi+\left(\nu^{2}-\alpha^{2}-\beta^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; \quad(\alpha\mu-\beta\lambda)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta}\\
 &amp; \quad(\beta\gamma-\alpha\mu)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta} &amp;  &amp; \quad-\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\cos\varphi+\left(\lambda^{2}+\mu^{2}+\nu^{2}\right)\operatorname{ch}{\theta}
\end{align}
}\\
\left(\alpha^{2}+\beta^{2}+\gamma^{2}-\lambda^{2}-\mu^{2}-\nu^{2}=-1\right)
\end{matrix}&lt;/math&gt;

==={{anchor|Gruner}} Gruner (1921) – Trigonometric Lorentz boosts===

In order to simplify the graphical representation of Minkowski space, [[w:Paul Gruner]] (1921) (with the aid of Josef Sauter) developed what is now called [[w:Loedel diagram]]s, using the following relations:&lt;ref group=R&gt;Gruner (1921a),&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}v=\alpha\cdot c;\quad\beta=\frac{1}{\sqrt{1-\alpha^{2}}}\\
\sin\varphi=\alpha;\quad\beta=\frac{1}{\cos\varphi};\quad\alpha\beta=\tan\varphi\\
\hline x'=\frac{x}{\cos\varphi}-t\cdot\tan\varphi,\quad t'=\frac{t}{\cos\varphi}-x\cdot\tan\varphi
\end{matrix}&lt;/math&gt;

In another paper Gruner used the alternative relations:&lt;ref group=R&gt;Gruner (1921b)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\alpha=\frac{v}{c};\ \beta=\frac{1}{\sqrt{1-\alpha^{2}}};\\
\cos\theta=\alpha=\frac{v}{c};\ \sin\theta=\frac{1}{\beta};\ \cot\theta=\alpha\cdot\beta\\
\hline x'=\frac{x}{\sin\theta}-t\cdot\cot\theta,\quad t'=\frac{t}{\sin\theta}-x\cdot\cot\theta
\end{matrix}&lt;/math&gt;

==References==
===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|abra1905}}
*{{#section:History of Topics in Special Relativity/relsource|bor14}}
*{{#section:History of Topics in Special Relativity/relsource|buch08}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04a}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04b}}
*{{#section:History of Topics in Special Relativity/relsource|con11qua}}
*{{#section:History of Topics in Special Relativity/relsource|einst05elek}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|fra11}}
*{{#section:History of Topics in Special Relativity/relsource|fra12}}
*{{#section:History of Topics in Special Relativity/relsource|gans05}}
*{{#section:History of Topics in Special Relativity/relsource|grun21a}}
*{{#section:History of Topics in Special Relativity/relsource|grun21b}}
*{{#section:History of Topics in Special Relativity/relsource|heav89}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|herg11ela}}
*{{#section:History of Topics in Special Relativity/relsource|igna10}}
*{{#section:History of Topics in Special Relativity/relsource|ignat10prin2}}
*{{#section:History of Topics in Special Relativity/relsource|klein08}}
*{{#section:History of Topics in Special Relativity/relsource|klein10b}}
*{{#section:History of Topics in Special Relativity/relsource|klein10}}
*{{#section:History of Topics in Special Relativity/relsource|klein11}}
*{{#section:History of Topics in Special Relativity/relsource|lar97}}
*{{#section:History of Topics in Special Relativity/relsource|lar29}}
*{{#section:History of Topics in Special Relativity/relsource|lar00}}
*{{#section:History of Topics in Special Relativity/relsource|lar04a}}
*{{#section:History of Topics in Special Relativity/relsource|lar04b}}
*{{#section:History of Topics in Special Relativity/relsource|lor92elek}}
*{{#section:History of Topics in Special Relativity/relsource|lor92b}}
*{{#section:History of Topics in Special Relativity/relsource|lor95}}
*{{#section:History of Topics in Special Relativity/relsource|lor99}}
*{{#section:History of Topics in Special Relativity/relsource|lor04}}
*{{#section:History of Topics in Special Relativity/relsource|lor16}}
*{{#section:History of Topics in Special Relativity/relsource|mink07a}}
*{{#section:History of Topics in Special Relativity/relsource|mink07b}}
*{{#section:History of Topics in Special Relativity/relsource|mink08}}
*{{#section:History of Topics in Special Relativity/relsource|plum10}}
*{{#section:History of Topics in Special Relativity/relsource|poi00}}
*{{#section:History of Topics in Special Relativity/relsource|poi04}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05a}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05b}}
*{{#section:History of Topics in Special Relativity/relsource|sea97}}
*{{#section:History of Topics in Special Relativity/relsource|silber11quat}}
*{{#section:History of Topics in Special Relativity/relsource|silber12quat}}
*{{#section:History of Topics in Special Relativity/relsource|silber14quat}}
*{{#section:History of Topics in Special Relativity/relsource|som09}}
*{{#section:History of Topics in Special Relativity/relsource|thom89}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}
*{{#section:History of Topics in Special Relativity/relsource|voi87}}
*{{#section:History of Topics in Special Relativity/relsource|wien04}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L4}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
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      <text bytes="64197" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via velocity==
===Boosts===
In the [[w:theory of relativity]], Lorentz transformations exhibit the symmetry of [[w:Minkowski spacetime]] by using a constant ''c'' as the [[w:speed of light]], and a parameter ''v'' as the relative [[w:velocity]] between two [[w:inertial reference frames]]. The corresponding formulas are identical to [[../Lorentz transformation (hyperbolic)|E:Lorentz transformations via hyperbolic functions]] introduced long before relativity was developed. In particular, the hyperbolic angle &lt;math&gt;\eta&lt;/math&gt; can be interpreted as the velocity related [[w:rapidity]] &lt;math&gt;\tanh\eta=\beta=v/c&lt;/math&gt;, so that &lt;math&gt;\gamma=\cosh\eta&lt;/math&gt; is the [[w:Lorentz factor]], &lt;math&gt;\beta\gamma=\sinh\eta&lt;/math&gt; the [[w:proper velocity]], &lt;math&gt;u'=c\tanh q&lt;/math&gt; the velocity of another object, &lt;math&gt;u=c\tanh(q+\eta)&lt;/math&gt; the [[w:velocity-addition formula]], thus transformation [[../Lorentz transformation (hyperbolic)#math_3b|E:'''(3b)''']] becomes:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}\gamma-x_{1}\beta\gamma\\
x_{1}^{\prime} &amp; =-x_{0}\beta\gamma+x_{1}\gamma\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\gamma+x_{1}^{\prime}\beta\gamma\\
x_{1} &amp; =x_{0}^{\prime}\beta\gamma+x_{1}^{\prime}\gamma\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\left|{\scriptstyle \begin{align}\beta^{2}\gamma^{2}-\gamma^{2} &amp; =-1 &amp; (a)\\
\gamma^{2}-\beta^{2}\gamma^{2} &amp; =1 &amp; (b)\\
\frac{\beta\gamma}{\gamma} &amp; =\beta &amp; (c)\\
\frac{1}{\sqrt{1-\beta^{2}}} &amp; =\gamma &amp; (d)\\
\frac{\beta}{\sqrt{1-\beta^{2}}} &amp; =\beta\gamma &amp; (e)\\
\frac{u'+v}{1+\frac{u'v}{c^{2}}} &amp; =u &amp; (f)
\end{align}
}\right.
\end{matrix}&lt;/math&gt;|{{equationRef|4a}}}}

Or in four dimensions and by setting &lt;math&gt;x_{0}=ct,\ x_{1}=x,\ x_{2}=y&lt;/math&gt; and adding an unchanged ''z'' the familiar form follows

{{NumBlk|:|&lt;math&gt;\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}+z^{2}=-c^{2}t^{\prime2}+x^{\prime2}+y^{\prime2}+z^{\prime2}\\
\hline \left.\begin{align}t' &amp; =\gamma\left(t-x\frac{v}{c^{2}}\right)\\
x' &amp; =\gamma(x-vt)\\
y' &amp; =y\\
z' &amp; =z
\end{align}
\right|\begin{align}t &amp; =\gamma\left(t'+x\frac{v}{c^{2}}\right)\\
x &amp; =\gamma(x'+vt')\\
y &amp; =y'\\
z &amp; =z'
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|4b}}}}

Analogous transformations have been introduced by [[#Voigt|Voigt (1887)]] in terms of an incompessible medium, and by [[#Lorentz1|Lorentz (1892, 1895)]] who analyzed [[w:Maxwell's equations]], they were completed by [[#Larmor|Larmor (1897, 1900)]] and [[#Lorentz2|Lorentz (1899, 1904)]], and brought into their modern form by [[#Poincare3|Poincaré (1905)]] who gave the transformation the name of Lorentz.&lt;ref&gt;Miller (1981), chapter 1&lt;/ref&gt; Eventually, [[#Einstein|Einstein (1905)]] showed in his development of [[w:special relativity]] that the transformations follow from the [[w:principle of relativity]] and constant light speed alone by modifying the traditional concepts of space and time, without requiring a [[w:Lorentz ether theory|mechanical aether]] in contradistinction to Lorentz and Poincaré.&lt;ref&gt;Miller (1981), chapter 4–7&lt;/ref&gt; [[#Minkowski|Minkowski (1907–1908)]] used them to argue that space and time are inseparably connected as [[w:spacetime]]. [[#Minkowski|Minkowski (1907–1908)]], [[#Frank|Frank (1909)]]  and [[#Varicak|Varićak (1910)]] showed the relation to imaginary and hyperbolic functions. Important contributions to the mathematical understanding of the Lorentz transformation were also made by other authors such as [[#Herglotz1|Herglotz (1909/10)]], [[#Ignatowski|Ignatowski (1910)]], [[#Noether|Noether (1910) and Klein (1910)]], [[#Borel|Borel (1913–14)]].

===Velocity addition and aberration===
In exact analogy to Beltrami coordinates in equation [[../Lorentz transformation (hyperbolic)#math_3e|E:'''(3e)''']], one can substitute &lt;math&gt;\left[\tfrac{u_{x}}{c},\ \tfrac{u_{y}}{c},\ 1\right]=\left[\tfrac{x}{ct},\ \tfrac{y}{ct},\ \tfrac{ct}{ct}\right]&lt;/math&gt; in ({{equationNote|4a}}), producing the Lorentz transformation of velocities (or [[w:velocity addition formula]]) :

{{NumBlk|:|&lt;math&gt;\begin{align}u_{x}^{\prime} &amp; =\frac{-c^{2}\sinh\eta+u_{x}c\cosh\eta}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{x}-c\tanh\eta}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}-v}{1-\frac{v}{c^{2}}u{}_{x}}\\
u_{y}^{\prime} &amp; =\frac{cu_{y}}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\tanh^{2}\eta}}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u{}_{x}}\\
\\
u_{x} &amp; =\frac{c^{2}\sinh\eta+u_{x}^{\prime}c\cosh\eta}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+c\tanh\eta}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+v}{1+\frac{v}{c^{2}}u_{x}^{\prime}}\\
u_{y} &amp; =\frac{cy'}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u_{x}^{\prime}}
\end{align}
&lt;/math&gt;|{{equationRef|4c}}}}

or using trigonometric and hyperbolic identities as in equation [[../Lorentz transformation (hyperbolic)#math_3f|E:'''(3f)''']], it becomes the hyperbolic law of cosines:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{matrix}u^{2}=u_{x}^{2}+u_{y}^{2}\\
u'^{2}=u_{x}^{\prime2}+u_{y}^{\prime2}
\end{matrix}\left|\begin{matrix}u_{x}=u\cos\alpha\\
u_{y}=u\sin\alpha\\
\\
u_{x}^{\prime}=u'\cos\alpha'\\
u_{y}^{\prime}=u'\sin\alpha'
\end{matrix}\right|\begin{align}u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-\frac{v}{c^{2}}u\cos\alpha}\\
u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{u'\cos\alpha'+v}, &amp; \tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{u\cos\alpha-v}
\end{align}
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(\frac{vu'\sin\alpha'}{c}\right){}^{2}}}{1+\frac{v}{c^{2}}u'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(\frac{vu\sin\alpha}{c}\right){}^{2}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\frac{u^{\prime2}}{c^{2}}}}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{1}{\sqrt{1-\frac{u^{2}}{c^{2}}}}-\frac{v/c}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{u/c}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\cos\alpha\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha
\end{matrix}&lt;/math&gt;|{{equationRef|4d}}}}

and by further setting ''u=u′=c'' one gets the old [[../Lorentz transformation (hyperbolic)#math_3g|E:Kepler formulas '''(3g)''']], which  express the relativistic [[w:aberration of light]]:&lt;ref&gt;Pauli (1921), pp. 562; 565–566&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\cos\alpha=\frac{\cos\alpha'+\frac{v}{c}}{1+\frac{v}{c}\cos\alpha'},\ \sin\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c}\cos\alpha'},\ \tan\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha'+\frac{v}{c}},\ \tan\frac{\alpha}{2}=\sqrt{\frac{c-v}{c+v}}\tan\frac{\alpha'}{2}\\
\cos\alpha'=\frac{\cos\alpha-\frac{v}{c}}{1-\frac{v}{c}\cos\alpha},\ \sin\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c}\cos\alpha},\ \tan\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha-\frac{v}{c}},\ \tan\frac{\alpha'}{2}=\sqrt{\frac{c+v}{c-v}}\tan\frac{\alpha}{2}
\end{matrix}&lt;/math&gt;|{{equationRef|4e}}}}

Formulas ({{equationNote|4c}}, {{equationNote|4d}}) were given by [[#Einstein|Einstein (1905)]] and [[#Poincare3|Poincaré (1905/06)]], while the relations to the spherical and hyperbolic law of cosines were given by [[#Sommerfeld|Sommerfeld (1909)]] and [[#Varicak|Varićak (1910)]]. The aberration formula for cos(α) was given by [[#Einstein|Einstein (1905)]].&lt;ref group=R name=plum&gt;Plummer (1910), pp. 258-259: After deriving the relativistic expressions for the aberration angles φ' and φ, Plummer remarked on p. 259: ''Another geometrical representation is obtained by assimilating φ' to the eccentric and φ to the true anomaly in an ellipse whose eccentricity is v/U = sin β.''&lt;/ref&gt;&lt;ref name=robin&gt;Robinson (1990), chapter 3-4, analyzed the relation between &quot;Kepler's formula&quot; and the &quot;physical velocity addition formula&quot; in special relativity.&lt;/ref&gt;

===Vectorial Lorentz transformation===
Also Lorentz boosts for arbitrary directions in line with [[../Lorentz transformation (general)#math_1a|E:general Lorentz transformation '''(1a)''']] can be given as:&lt;ref&gt;Møller (1952/55), Chapter II, § 18&lt;/ref&gt;

:&lt;math&gt;\mathbf{x}'=\begin{bmatrix}\gamma &amp; -\gamma\beta n_{x} &amp; -\gamma\beta n_{y} &amp; -\gamma\beta n_{z}\\
-\gamma\beta n_{x} &amp; 1+(\gamma-1)n_{x}^{2} &amp; (\gamma-1)n_{x}n_{y} &amp; (\gamma-1)n_{x}n_{z}\\
-\gamma\beta n_{y} &amp; (\gamma-1)n_{y}n_{x} &amp; 1+(\gamma-1)n_{y}^{2} &amp; (\gamma-1)n_{y}n_{z}\\
-\gamma\beta n_{z} &amp; (\gamma-1)n_{z}n_{x} &amp; (\gamma-1)n_{z}n_{y} &amp; 1+(\gamma-1)n_{z}^{2}
\end{bmatrix}\cdot\mathbf{x},\quad\left[\mathbf{n}=\frac{\mathbf{v}}{v}\right]&lt;/math&gt;

or in vector notation

{{NumBlk|:|&lt;math&gt;\begin{align}t' &amp; =\gamma\left(t-\frac{v\mathbf{n}\cdot\mathbf{r}}{c^{2}}\right)\\
\mathbf{r}' &amp; =\mathbf{r}+(\gamma-1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n}-\gamma tv\mathbf{n}
\end{align}
&lt;/math&gt;|{{equationRef|4f}}}}

The corresponding vectorial velocity addition formula in line with [[../Lorentz transformation (general)#math_1b|E:general Lorentz transformation '''(1b)''']] is given by

{{NumBlk|:|&lt;math&gt;\mathbf{u}'=\frac{1}{1+\frac{\mathbf{v}\cdot\mathbf{u}}{c^{2}}}\left[\frac{\mathbf{u}}{\gamma_{\mathbf{v}}}+\mathbf{v}+\frac{1}{c^{2}}\frac{\gamma_{\mathbf{v}}}{\gamma_{\mathbf{v}}+1}(\mathbf{u}\cdot\mathbf{v})\mathbf{v}\right]&lt;/math&gt;|{{equationRef|4g}}}}

Such transformations were formulated by [[#Herglotz2|Ignatowski (1910), Herglotz (1911)]] and others.

==Historical notation==
==={{anchor|Voigt}} Voigt (1887) ===

[[w:Woldemar Voigt]] (1887)&lt;ref group=R&gt;Voigt (1887), p. 45&lt;/ref&gt; developed a transformation in connection with the [[w:Doppler effect]] and an incompressible medium, being in modern notation:&lt;ref&gt;Miller (1981), 114–115&lt;/ref&gt;&lt;ref name=pais&gt;Pais (1982), Kap. 6b&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\xi_{1} &amp; =x_{1}-\varkappa t\\
\eta_{1} &amp; =y_{1}q\\
\zeta_{1} &amp; =z_{1}q\\
\tau &amp; =t-\frac{\varkappa x_{1}}{\omega^{2}}\\
q &amp; =\sqrt{1-\frac{\varkappa^{2}}{\omega^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =\frac{y}{\gamma}\\
z^{\prime} &amp; =\frac{z}{\gamma}\\
t^{\prime} &amp; =t-\frac{vx}{c^{2}}\\
\frac{1}{\gamma} &amp; =\sqrt{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

If the right-hand sides of his equations are multiplied by γ they are the modern Lorentz transformation ({{equationNote|4b}}). In Voigt's theory the speed of light is invariant, but his transformations mix up a relativistic boost together with a rescaling of space-time. Optical phenomena in free space are [[w:Scale invariance|scale]], [[w:Conformal map|conformal]] (using the factor λ discussed [[#Lorsph|above]]), and [[w:Lorentz covariance|Lorentz invariant]], so the combination is invariant too.&lt;ref name=pais /&gt; For instance, Lorentz transformations can be extended by using &lt;math&gt;l=\sqrt{\lambda}&lt;/math&gt;:&lt;ref group=R&gt;Lorentz (1915/16), p. 197&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\quad y^{\prime}=ly,\quad z^{\prime}=lz,\quad t^{\prime}=\gamma l\left(t-x\frac{v}{c^{2}}\right)&lt;/math&gt;.

''l''=1/γ gives the Voigt transformation, ''l''=1 the Lorentz transformation. But scale transformations are not a symmetry of all the laws of nature, only of electromagnetism, so these transformations cannot be used to formulate a [[w:principle of relativity]] in general. It was demonstrated by Poincaré and Einstein that one has to set ''l''=1 in order to make the above transformation symmetric and to form a group as required by the relativity principle, therefore the Lorentz transformation is the only viable choice.

Voigt sent his 1887 paper to Lorentz in 1908,&lt;ref&gt;Voigt's transformations and the beginning of the relativistic revolution, Ricardo Heras, arXiv:1411.2559 [https://arxiv.org/abs/1411.2559]&lt;/ref&gt; and that was acknowledged in 1909: {{Quote|In a paper &quot;Über das Doppler'sche Princip&quot;, published in 1887 (Gött. Nachrichten, p. 41) and which to my regret has escaped my notice all these years, Voigt has applied to equations of the form (7) (§ 3 of this book) [namely &lt;math&gt;\Delta\Psi-\tfrac{1}{c^{2}}\tfrac{\partial^{2}\Psi}{\partial t^{2}}=0&lt;/math&gt;] a transformation equivalent to the formulae (287) and (288) [namely &lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\ y^{\prime}=ly,\ z^{\prime}=lz,\ t^{\prime}=\gamma l\left(t-\tfrac{v}{c^{2}}x\right)&lt;/math&gt;]. The idea of the transformations used above (and in § 44) might therefore have been borrowed from Voigt and the proof that it does not alter the form of the equations for the ''free'' ether is contained in his paper.&lt;ref group=R&gt;Lorentz (1915/16), p. 198&lt;/ref&gt;}}

Also [[w:Hermann Minkowski]] said in 1908 that the transformations which play the main role in the principle of relativity were first examined by Voigt in 1887. Voigt responded in the same paper by saying that his theory was based on an elastic theory of light, not an electromagnetic one. However, he concluded that some results were actually the same.&lt;ref group=R&gt;Bucherer (1908), p. 762&lt;/ref&gt;

==={{anchor|Heaviside}} Heaviside (1888), Thomson (1889), Searle (1896)===

In 1888, [[w:Oliver Heaviside]]&lt;ref group=R&gt;Heaviside (1888), p. 324&lt;/ref&gt; investigated the properties of [[w:Relativistic electromagnetism|charges in motion]] according to Maxwell's electrodynamics. He calculated, among other things, anisotropies in the electric field of moving bodies represented by this formula:&lt;ref&gt;Brown (2003)&lt;/ref&gt;

:&lt;math&gt;\mathrm{E}=\left(\frac{q\mathrm{r}}{r^{2}}\right)\left(1-\frac{v^{2}\sin^{2}\theta}{c^{2}}\right)^{-3/2}&lt;/math&gt;.

Consequently, [[w:Joseph John Thomson]] (1889)&lt;ref group=R&gt;Thomson (1889), p. 12&lt;/ref&gt; found a way to substantially simplify calculations concerning moving charges by using the following mathematical transformation (like other authors such as Lorentz or Larmor, also Thomson implicitly used the [[w:Galilean transformation]] ''z-vt'' in his equation&lt;ref name=mil /&gt;):

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}z &amp; =\left\{ 1-\frac{\omega^{2}}{v^{2}}\right\} ^{\frac{1}{2}}z'\end{align}
\right| &amp; \begin{align}z^{\ast}=z-vt &amp; =\frac{z'}{\gamma}\end{align}
\end{matrix}&lt;/math&gt;

Thereby, [[w:inhomogeneous electromagnetic wave equation]]s are transformed into a [[w:Poisson equation]].&lt;ref name=mil&gt;Miller (1981), 98–99&lt;/ref&gt; Eventually, [[w:George Frederick Charles Searle]]&lt;ref group=R&gt;Searle (1886), p. 333&lt;/ref&gt; noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called &quot;Heaviside-Ellipsoid&quot; of [[w:axial ratio]]

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align} &amp; \sqrt{\alpha}:1:1\\
\alpha= &amp; 1-\frac{u^{2}}{v^{2}}
\end{align}
\right| &amp; \begin{align} &amp; \frac{1}{\gamma}:1:1\\
\frac{1}{\gamma^{2}} &amp; =1-\frac{v^{2}}{c^{2}}
\end{align}
\end{matrix}&lt;/math&gt;&lt;ref name=mil /&gt;

=== {{anchor|Lorentz1}} Lorentz (1892, 1895) ===

In order to explain the [[w:aberration of light]] and the result of the [[w:Fizeau experiment]] in accordance with [[w:Maxwell's equations]], Lorentz in 1892 developed a model (&quot;[[w:Lorentz ether theory]]&quot;) in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. In order to calculate the optics of moving bodies, Lorentz introduced the following quantities to transform from the aether system into a moving system (it's unknown whether he was influenced by Voigt, Heaviside, and Thomson)&lt;ref group=R&gt;Lorentz (1892a), p. 141&lt;/ref&gt;&lt;ref name=milf&gt;Miller (1982), 1.4 &amp; 1.5&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\mathfrak{x} &amp; =\frac{V}{\sqrt{V^{2}-p^{2}}}x\\
t' &amp; =t-\frac{\varepsilon}{V}\mathfrak{x}\\
\varepsilon &amp; =\frac{p}{\sqrt{V^{2}-p^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)\\
\gamma\frac{v}{c} &amp; =\frac{v}{\sqrt{c^{2}-v^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

where ''x&lt;sup&gt;*&lt;/sup&gt;'' is the [[w:Galilean transformation]] ''x-vt''. Except the additional γ in the time transformation, this is the complete Lorentz transformation ({{equationNote|4b}}).&lt;ref name=milf /&gt; While ''t'' is the &quot;true&quot; time for observers resting in the aether, ''t′'' is an auxiliary variable only for calculating processes for moving systems. It is also important that Lorentz and later also Larmor formulated this transformation in two steps. At first an implicit Galilean transformation, and later the expansion into the &quot;fictitious&quot; electromagnetic system with the aid of the Lorentz transformation. In order to explain the negative result of the [[w:Michelson–Morley experiment]], he (1892b)&lt;ref group=R&gt;Lorentz (1892b), p. 141&lt;/ref&gt; introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced [[w:length contraction]] in his theory (without proof as he admitted). The same hypothesis was already made by [[w:George FitzGerald]] in 1889 based on Heaviside's work. While length contraction was a real physical effect for Lorentz, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation.

In 1895, Lorentz further elaborated on his theory and introduced the &quot;theorem of corresponding states&quot;. This theorem states that a moving observer (relative to the ether) in his &quot;fictitious&quot; field makes the same observations as a resting observers in his &quot;real&quot; field for velocities to first order in ''v/c''. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:&lt;ref group=R&gt;Lorentz (1895), p. 37&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =x^{\prime}\sqrt{1-\frac{\mathfrak{p}^{2}}{V^{2}}}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{x^{\prime}}{\gamma}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\end{matrix}&lt;/math&gt;

For solving optical problems Lorentz used the following transformation, in which the modified time variable was called &quot;local time&quot; ({{lang-de|Ortszeit}}) by him:&lt;ref group=R&gt;Lorentz (1895), p. 49 for local time and p. 56 for spatial coordinates.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\mathrm{x}-\mathfrak{p}_{x}t\\
y &amp; =\mathrm{y}-\mathfrak{p}_{y}t\\
z &amp; =\mathrm{z}-\mathfrak{p}_{z}t\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}}x-\frac{\mathfrak{p}_{y}}{V^{2}}y-\frac{\mathfrak{p}_{z}}{V^{2}}z
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-v_{x}t\\
y^{\prime} &amp; =y-v_{y}t\\
z^{\prime} &amp; =z-v_{z}t\\
t^{\prime} &amp; =t-\frac{v_{x}}{c^{2}}x'-\frac{v_{y}}{c^{2}}y'-\frac{v_{z}}{c^{2}}z'
\end{align}
\end{matrix}&lt;/math&gt;

With this concept Lorentz could explain the [[w:Doppler effect]], the [[w:aberration of light]], and the [[w:Fizeau experiment]].&lt;ref&gt;Janssen (1995), 3.1&lt;/ref&gt;

=== {{anchor|Larmor}} Larmor (1897, 1900) ===

In 1897, Larmor extended the work of Lorentz and derived the following transformation&lt;ref group=R&gt;Larmor (1897), p. 229&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =x\varepsilon^{\frac{1}{2}}\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-vx/c^{2}\\
dt_{1} &amp; =dt^{\prime}\varepsilon^{-\frac{1}{2}}\\
\varepsilon &amp; =\left(1-v^{2}/c^{2}\right)^{-1}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-\frac{vx^{\ast}}{c^{2}}=t-\frac{v(x-vt)}{c^{2}}\\
dt_{1} &amp; =\frac{dt^{\prime}}{\gamma}\\
\gamma^{2} &amp; =\frac{1}{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

Larmor noted that if it is assumed that the constitution of molecules is electrical then the FitzGerald–Lorentz contraction is a consequence of this transformation, explaining the [[w:Michelson–Morley experiment]]. It's notable that Larmor was the first who recognized that some sort of [[w:time dilation]] is a consequence of this transformation as well, because &quot;individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 1/γ&quot;.&lt;ref&gt;Darrigol (2000), Chap. 8.5&lt;/ref&gt;&lt;ref&gt;Macrossan (1986)&lt;/ref&gt; Larmor wrote his electrodynamical equations and transformations neglecting terms of higher order than ''(v/c)''&lt;sup&gt;2&lt;/sup&gt; – when his 1897 paper was reprinted in 1929, Larmor added the following comment in which he described how they can be made valid to all orders of ''v/c'':&lt;ref group=R&gt;Larmor (1897/1929), p. 39&lt;/ref&gt;

{{Quote|Nothing need be neglected: the transformation is ''exact'' if ''v/c''&lt;sup&gt;2&lt;/sup&gt; is replaced by ''εv/c''&lt;sup&gt;2&lt;/sup&gt; in the equations and also in the change following from ''t'' to ''t′'', as is worked out in ''Aether and Matter'' (1900), p. 168, and as Lorentz found it to be in 1904, thereby stimulating the modern schemes of intrinsic relational relativity.}}

In line with that comment, in his book Aether and Matter published in 1900, Larmor used a modified local time ''t″=t′-εvx′/c&lt;sup&gt;2&lt;/sup&gt;'' instead of the 1897 expression ''t′=t-vx/c&lt;sup&gt;2&lt;/sup&gt;'' by replacing ''v/c''&lt;sup&gt;2&lt;/sup&gt; with ''εv/c''&lt;sup&gt;2&lt;/sup&gt;, so that ''t″'' is now identical to the one given by Lorentz in 1892, which he combined with a Galilean transformation for the ''x′, y′, z′, t′'' coordinates:&lt;ref group=R&gt;Larmor (1900), p. 168&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime} &amp; =t^{\prime}-\varepsilon vx^{\prime}/c^{2}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime}=t^{\prime}-\frac{\gamma^{2}vx^{\prime}}{c^{2}} &amp; =\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor ''(v/c)''&lt;sup&gt;2&lt;/sup&gt;, and so he sought the transformations which were &quot;accurate to second order&quot; (as he put it). Thus he wrote the final transformations (where ''x′=x-vt'' and ''t″'' as given above) as:&lt;ref group=R&gt;Larmor (1900), p. 174&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =\varepsilon^{\frac{1}{2}}x^{\prime}\\
y_{1} &amp; =y^{\prime}\\
z_{1} &amp; =z^{\prime}\\
dt_{1} &amp; =\varepsilon^{-\frac{1}{2}}dt^{\prime\prime}=\varepsilon^{-\frac{1}{2}}\left(dt^{\prime}-\frac{v}{c^{2}}\varepsilon dx^{\prime}\right)\\
t_{1} &amp; =\varepsilon^{-\frac{1}{2}}t^{\prime}-\frac{v}{c^{2}}\varepsilon^{\frac{1}{2}}x^{\prime}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\prime}=\gamma(x-vt)\\
y_{1} &amp; =y'=y\\
z_{1} &amp; =z'=z\\
dt_{1} &amp; =\frac{dt^{\prime\prime}}{\gamma}=\frac{1}{\gamma}\left(dt^{\prime}-\frac{\gamma^{2}vdx^{\prime}}{c^{2}}\right)=\gamma\left(dt-\frac{vdx}{c^{2}}\right)\\
t_{1} &amp; =\frac{t^{\prime}}{\gamma}-\frac{\gamma vx^{\prime}}{c^{2}}=\gamma\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

by which he arrived at the complete Lorentz transformation ({{equationNote|4b}}). Larmor showed that Maxwell's equations were invariant under this two-step transformation, &quot;to second order in ''v/c''&quot; – it was later shown by Lorentz (1904) and Poincaré (1905) that they are indeed invariant under this transformation to all orders in ''v/c''.

Larmor gave credit to Lorentz in two papers published in 1904, in which he used the term &quot;Lorentz transformation&quot; for Lorentz's first order transformations of coordinates and field configurations:

{{Quote|p. 583: [..] Lorentz's transformation for passing from the field of activity of a stationary electrodynamic material system to that of one moving with uniform velocity of translation through the aether.&lt;br /&gt; p. 585: [..] the Lorentz transformation has shown us what is not so immediately obvious [..]&lt;ref group=R&gt;Larmor (1904a), p. 583, 585&lt;/ref&gt; &lt;br /&gt; p. 622: [..] the transformation first developed by Lorentz: namely, each point in space is to have its own origin from which time is measured, its &quot;local time&quot; in Lorentz's phraseology, and then the values of the electric and magnetic vectors [..] at all points in the aether between the molecules in the system at rest, are the same as those of the vectors [..] at the corresponding points in the convected system at the same local times.&lt;ref group=R&gt;Larmor (1904b), p. 622&lt;/ref&gt;}}

=== {{anchor|Lorentz2}} Lorentz (1899, 1904) ===

Also Lorentz extended his theorem of corresponding states in 1899. First he wrote a transformation equivalent to the one from 1892 (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1899), p. 429&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}x\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}-\mathfrak{p}_{x}^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Then he introduced a factor ε of which he said he has no means of determining it, and modified his transformation as follows (where the above value of ''t′'' has to be inserted):&lt;ref group=R&gt;Lorentz (1899), p. 439&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\frac{\varepsilon}{k}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon x^{\prime\prime}\\
t^{\prime} &amp; =k\varepsilon t^{\prime\prime}\\
k &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{\varepsilon}{\gamma}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon z^{\prime\prime}\\
t^{\prime}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right) &amp; =\gamma\varepsilon t^{\prime\prime}\\
\gamma &amp; =\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
\end{align}
\end{matrix}&lt;/math&gt;

This is equivalent to the complete Lorentz transformation ({{equationNote|4b}}) when solved for ''x″'' and ''t″'' and with ε=1. Like Larmor, Lorentz noticed in 1899&lt;ref group=R&gt;Lorentz (1899), p. 442&lt;/ref&gt; also some sort of time dilation effect in relation to the frequency of oscillating electrons ''&quot;that in ''S'' the time of vibrations be ''kε'' times as great as in ''S&lt;sub&gt;0&lt;/sub&gt;''&quot;'', where ''S&lt;sub&gt;0&lt;/sub&gt;'' is the aether frame.&lt;ref&gt;Janssen (1995), Kap. 3.3&lt;/ref&gt;

In 1904 he rewrote the equations in the following form by setting ''l''=1/ε (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1904), p. 812&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =klx\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =\frac{l}{k}t-kl\frac{w}{c^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma lx^{\ast}=\gamma l(x-vt)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t^{\prime} &amp; =\frac{lt}{\gamma}-\frac{\gamma lvx^{\ast}}{c^{2}}=\gamma l\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Under the assumption that ''l=1'' when ''v''=0, he demonstrated that ''l=1'' must be the case at all velocities, therefore length contraction can only arise in the line of motion. So by setting the factor ''l'' to unity, Lorentz's transformations now assumed the same form as Larmor's and are now completed. Unlike Larmor, who restricted himself to show the covariance of Maxwell's equations to second order, Lorentz tried to widen its covariance to all orders in ''v/c''. He also derived the correct formulas for the velocity dependence of [[w:electromagnetic mass]], and concluded that the transformation formulas must apply to all forces of nature, not only electrical ones.&lt;ref group=R&gt;Lorentz (1904), p. 826&lt;/ref&gt; However, he didn't achieve full covariance of the transformation equations for charge density and velocity.&lt;ref&gt;Miller (1981), Chap. 1.12.2&lt;/ref&gt; When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:&lt;ref&gt;Janssen (1995), Chap. 3.5.6&lt;/ref&gt;

{{Quote|One will notice that in this work the transformation equations of Einstein’s Relativity Theory have not quite been attained. [..] On this circumstance depends the clumsiness of many of the further considerations in this work.}}

Lorentz's 1904 transformation was cited and used by [[w:Alfred Bucherer]] in July 1904:&lt;ref group=R&gt;Bucherer, p. 129; Definition of s on p. 32&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\sqrt{s}x,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{\sqrt{s}}-\sqrt{s}\frac{u}{v^{2}}x,\quad s=1-\frac{u^{2}}{v^{2}}&lt;/math&gt;

or by [[w:Wilhelm Wien]] in July 1904:&lt;ref group=R&gt;Wien (1904), p. 394&lt;/ref&gt;

:&lt;math&gt;x=kx',\quad y=y',\quad z=z',\quad t'=kt-\frac{v}{kc^{2}}x&lt;/math&gt;

or by [[w:Emil Cohn]] in November 1904 (setting the speed of light to unity):&lt;ref group=R&gt;Cohn (1904a), pp. 1296-1297&lt;/ref&gt;

:&lt;math&gt;x=\frac{x_{0}}{k},\quad y=y_{0},\quad z=z_{0},\quad t=kt_{0},\quad t_{1}=t_{0}-w\cdot r_{0},\quad k^{2}=\frac{1}{1-w^{2}}&lt;/math&gt;

or by [[w:Richard Gans]] in February 1905:&lt;ref group=R&gt;Gans (1905), p. 169&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=kx,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{k}-\frac{kwx}{c^{2}},\quad k^{2}=\frac{c^{2}}{c^{2}-w^{2}}&lt;/math&gt;

=== {{anchor|Poincare3}} Poincaré (1900, 1905) ===

==== Local time ====

Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time. However, [[w:Henri Poincaré]] in 1900 commented on the origin of Lorentz's &quot;wonderful invention&quot; of local time.&lt;ref&gt;Darrigol (2005), Kap. 4&lt;/ref&gt; He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed &lt;math&gt;c&lt;/math&gt; in both directions, which lead to what is nowadays called [[w:relativity of simultaneity]], although Poincaré's calculation does not involve length contraction or time dilation.&lt;ref group=R&gt;Poincaré (1900), pp. 272–273&lt;/ref&gt; In order to synchronise the clocks here on Earth (the ''x*, t''* frame) a light signal from one clock (at the origin) is sent to another (at ''x''*), and is sent back. It's supposed that the Earth is moving with speed ''v'' in the ''x''-direction (= ''x''*-direction) in some rest system (''x, t'') (''i.e.'' the [[w:luminiferous aether]] system for Lorentz and Larmor). The time of flight outwards is

:&lt;math&gt;\delta t_{a}=\frac{x^{\ast}}{\left(c-v\right)}&lt;/math&gt;

and the time of flight back is

:&lt;math&gt;\delta t_{b}=\frac{x^{\ast}}{\left(c+v\right)}&lt;/math&gt;.

The elapsed time on the clock when the signal is returned is ''δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;'' and the time ''t*=(δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;)/2'' is ascribed to the moment when the light signal reached the distant clock. In the rest frame the time ''t=δt&lt;sub&gt;a&lt;/sub&gt;'' is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus

:&lt;math&gt;t^{\ast}=t-\frac{\gamma^{2}vx^{*}}{c^{2}}&lt;/math&gt;

identical to Lorentz (1892). By dropping the factor γ&lt;sup&gt;2&lt;/sup&gt; under the assumption that &lt;math&gt;\tfrac{v^{2}}{c^{2}}\ll1&lt;/math&gt;, Poincaré gave the result ''t*=t-vx*/c&lt;sup&gt;2&lt;/sup&gt;'', which is the form used by Lorentz in 1895.

Similar physical interpretations of local time were later given by [[w:Emil Cohn]] (1904)&lt;ref group=R&gt;Cohn (1904b), p. 1408&lt;/ref&gt; and [[w:Max Abraham]] (1905).&lt;ref group=R&gt;Abraham (1905), § 42&lt;/ref&gt;

==== Lorentz transformation ====

On June 5, 1905 (published June 9) Poincaré formulated transformation equations which are algebraically equivalent to those of Larmor and Lorentz and gave them the modern form ({{equationNote|4b}}):&lt;ref group=R&gt;Poincaré (1905), p. 1505&lt;/ref&gt;

:&lt;math&gt;\begin{align}x^{\prime} &amp; =kl(x+\varepsilon t)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =kl(t+\varepsilon x)\\
k &amp; =\frac{1}{\sqrt{1-\varepsilon^{2}}}
\end{align}
&lt;/math&gt;.

Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name &quot;Lorentz transformation&quot;.&lt;ref&gt;Pais (1982), Chap. 6c&lt;/ref&gt;&lt;ref&gt;Katzir (2005), 280–288&lt;/ref&gt; Poincaré set the speed of light to unity, pointed out the group characteristics of the transformation by setting ''l''=1, and modified/corrected Lorentz's derivation of the equations of electrodynamics in some details in order to fully satisfy the principle of relativity, ''i.e.'' making them fully Lorentz covariant.&lt;ref&gt;Miller (1981), Chap. 1.14&lt;/ref&gt;

In July 1905 (published in January 1906)&lt;ref group=R&gt;Poincaré (1905/06), pp. 129ff&lt;/ref&gt; Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of the [[w:principle of least action]]; he demonstrated in more detail the group characteristics of the transformation, which he called [[w:Lorentz group]], and he showed that the combination ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;+z&lt;sup&gt;2&lt;/sup&gt;-t&lt;sup&gt;2&lt;/sup&gt;'' is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing &lt;math&gt;ct\sqrt{-1}&lt;/math&gt; as a fourth imaginary coordinate, and he used an early form of [[w:four-vector]]s. He also formulated the velocity addition formula ({{equationNote|4c}}), which he had already derived in unpublished letters to Lorentz from May 1905:&lt;ref group=R&gt;Poincaré (1905/06), p. 144&lt;/ref&gt;

:&lt;math&gt;\xi'=\frac{\xi+\varepsilon}{1+\xi\varepsilon},\ \eta'=\frac{\eta}{k(1+\xi\varepsilon)}&lt;/math&gt;.

==={{anchor|Einstein}} Einstein (1905) – Special relativity===

On June 30, 1905 (published September 1905) Einstein published what is now called [[w:special relativity]] and gave a new derivation of the transformation, which was based only on the principle on relativity and the principle of the constancy of the speed of light. While Lorentz considered &quot;local time&quot; to be a mathematical stipulation device for explaining the Michelson-Morley experiment, Einstein showed that the coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. For quantities of first order in ''v/c'' this was also done by Poincaré in 1900, while Einstein derived the complete transformation by this method. Unlike Lorentz and Poincaré who still distinguished between real time in the aether and apparent time for moving observers, Einstein showed that the transformations concern the nature of space and time.&lt;ref&gt;Miller (1981), Chap. 6&lt;/ref&gt;&lt;ref&gt;Pais (1982), Kap. 7&lt;/ref&gt;&lt;ref&gt;Darrigol (2005), Chap. 6&lt;/ref&gt;

The notation for this transformation is equivalent to Poincaré's of 1905 and ({{equationNote|4b}}), except that Einstein didn't set the speed of light to unity:&lt;ref group=R&gt;Einstein (1905), p. 902&lt;/ref&gt;

:&lt;math&gt;\begin{align}\tau &amp; =\beta\left(t-\frac{v}{V^{2}}x\right)\\
\xi &amp; =\beta(x-vt)\\
\eta &amp; =y\\
\zeta &amp; =z\\
\beta &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}
\end{align}
&lt;/math&gt;

Einstein also defined the velocity addition formula ({{equationNote|4c}}, {{equationNote|4d}}):&lt;ref group=R&gt;Einstein (1905), § 5 and § 9&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=\frac{w_{\xi}+v}{1+\frac{vw_{\xi}}{V^{2}}}t,\ y=\frac{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}{1+\frac{vw_{\xi}}{V^{2}}}w_{\eta}t\\
U^{2}=\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2},\ w^{2}=w_{\xi}^{2}+w_{\eta}^{2},\ \alpha=\operatorname{arctg}\frac{w_{y}}{w_{x}}\\
U=\frac{\sqrt{\left(v^{2}+w^{2}+2vw\cos\alpha\right)-\left(\frac{vw\sin\alpha}{V}\right)^{2}}}{1+\frac{vw\cos\alpha}{V^{2}}}
\end{matrix}\left|\begin{matrix}\frac{u_{x}-v}{1-\frac{u_{x}v}{V^{2}}}=u_{\xi}\\
\frac{u_{y}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\eta}\\
\frac{u_{z}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\zeta}
\end{matrix}\right.&lt;/math&gt;

and the light aberration formula ({{equationNote|4e}}):&lt;ref group=R&gt;Einstein (1905), § 7&lt;/ref&gt;

:&lt;math&gt;\cos\varphi'=\frac{\cos\varphi-\frac{v}{V}}{1-\frac{v}{V}\cos\varphi}&lt;/math&gt;

=== {{anchor|Minkowski}} Minkowski (1907–1908) – Spacetime ===

The work on the principle of relativity by Lorentz, Einstein, [[w:Max Planck|Planck]], together with Poincaré's four-dimensional approach, were further elaborated and combined with the [[w:hyperboloid model]] by [[w:Hermann Minkowski]] in 1907 and 1908.&lt;ref group=R&gt;Minkowski (1907/15), pp. 927ff&lt;/ref&gt;&lt;ref group=R&gt;Minkowski (1907/08), pp. 53ff&lt;/ref&gt; Minkowski particularly reformulated electrodynamics in a four-dimensional way ([[w:Minkowski spacetime]]).&lt;ref&gt;Walter (1999a), (1999b), (2018)&lt;/ref&gt; For instance, he wrote ''x, y, z, it'' in the form ''x&lt;sub&gt;1&lt;/sub&gt;, x&lt;sub&gt;2&lt;/sub&gt;, x&lt;sub&gt;3&lt;/sub&gt;, x&lt;sub&gt;4&lt;/sub&gt;''. By defining ψ as the angle of rotation around the ''z''-axis, the Lorentz transformation assumes a form (with ''c''=1):&lt;ref group=R name=mink1&gt;Minkowski (1907/08), p. 59&lt;/ref&gt;

:&lt;math&gt;\begin{align}x'_{1} &amp; =x_{1}\\
x'_{2} &amp; =x_{2}\\
x'_{3} &amp; =x_{3}\cos i\psi+x_{4}\sin i\psi\\
x'_{4} &amp; =-x_{3}\sin i\psi+x_{4}\cos i\psi\\
\cos i\psi &amp; =\frac{1}{\sqrt{1-q^{2}}}
\end{align}
&lt;/math&gt;

Even though Minkowski used the imaginary number iψ, he for once&lt;ref group=R name=mink1 /&gt; directly used the [[w:tangens hyperbolicus]] in the equation for velocity

:&lt;math&gt;-i\tan i\psi=\frac{e^{\psi}-e^{-\psi}}{e^{\psi}+e^{-\psi}}=q&lt;/math&gt; with &lt;math&gt;\psi=\frac{1}{2}\ln\frac{1+q}{1-q}&lt;/math&gt;.

Minkowski's expression can also by written as ψ=atanh(q) and was later called [[w:rapidity]]. He also wrote the Lorentz transformation in matrix form:&lt;ref group=R&gt;Minkowski (1907/08), pp. 65–66, 81–82&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}+x_{4}^{\prime2}\\
\left(x_{1}^{\prime}=x',\ x_{2}^{\prime}=y',\ x_{3}^{\prime}=z',\ x_{4}^{\prime}=it'\right)\\
-x^{2}-y^{2}-z^{2}+t^{2}=-x^{\prime2}-y^{\prime2}-z^{\prime2}+t^{\prime2}\\
\hline x_{h}=\alpha_{h1}x_{1}^{\prime}+\alpha_{h2}x_{2}^{\prime}+\alpha_{h3}x_{3}^{\prime}+\alpha_{h4}x_{4}^{\prime}\\
\mathrm{A}=\mathrm{\left|\begin{matrix}\alpha_{11}, &amp; \alpha_{12}, &amp; \alpha_{13}, &amp; \alpha_{14}\\
\alpha_{21}, &amp; \alpha_{22}, &amp; \alpha_{23}, &amp; \alpha_{24}\\
\alpha_{31}, &amp; \alpha_{32}, &amp; \alpha_{33}, &amp; \alpha_{34}\\
\alpha_{41}, &amp; \alpha_{42}, &amp; \alpha_{43}, &amp; \alpha_{44}
\end{matrix}\right|,\ \begin{align}\bar{\mathrm{A}}\mathrm{A} &amp; =1\\
\left(\det \mathrm{A}\right)^{2} &amp; =1\\
\det \mathrm{A} &amp; =1\\
\alpha_{44} &amp; &gt;0
\end{align}
}
\end{matrix}&lt;/math&gt;

As a graphical representation of the Lorentz transformation he introduced the [[w:Minkowski diagram]], which became a standard tool in textbooks and research articles on relativity:&lt;ref group=R&gt;Minkowski (1908/09), p. 77&lt;/ref&gt;

[[File:Minkowski1.png|center|thumb|400px|Original spacetime diagram by Minkowski in 1908.]]

==={{anchor|Frank}} Frank (1909) – Hyperbolic functions===
Hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity in agreement with ({{equationNote|4b}}):&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

==={{Anchor|Sommerfeld}} Sommerfeld (1909) – Spherical trigonometry===

Using an imaginary rapidity such as Minkowski, [[w:Arnold Sommerfeld]] (1909) formulated a transformation equivalent to Lorentz boost ({{equationNote|4b}}), and the relativistc velocity addition ({{equationNote|4c}}) in terms of trigonometric functions and the [[w:spherical law of cosines]]:&lt;ref group=R&gt;Sommerfeld (1909), p. 826ff.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{array}{lrl}
x'= &amp; x\ \cos\varphi+l\ \sin\varphi, &amp; y'=y\\
l'= &amp; -x\ \sin\varphi+l\ \cos\varphi, &amp; z'=z
\end{array}\right\} \\
\left(\operatorname{tg}\varphi=i\beta,\ \cos\varphi=\frac{1}{\sqrt{1-\beta^{2}}},\ \sin\varphi=\frac{i\beta}{\sqrt{1-\beta^{2}}}\right)\\
\hline \beta=\frac{1}{i}\operatorname{tg}\left(\varphi_{1}+\varphi_{2}\right)=\frac{1}{i}\frac{\operatorname{tg}\varphi_{1}+\operatorname{tg}\varphi_{2}}{1-\operatorname{tg}\varphi_{1}\operatorname{tg}\varphi_{2}}=\frac{\beta_{1}+\beta_{2}}{1+\beta_{1}\beta_{2}}\\
\cos\varphi=\cos\varphi_{1}\cos\varphi_{2}-\sin\varphi_{1}\sin\varphi_{2}\cos\alpha\\
v^{2}=\frac{v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos\alpha-\frac{1}{c^{2}}v_{1}^{2}v_{2}^{2}\sin^{2}\alpha}{\left(1+\frac{1}{c^{2}}v_{1}v_{2}\cos\alpha\right)^{2}}
\end{matrix}&lt;/math&gt;

=== {{anchor|Herglotz1}} Herglotz (1909/10) – Möbius transformation ===

Following [[../Lorentz transformation (Möbius)#Klein2|E:Klein (1889–1897)]] as well as [[../Lorentz transformation (Möbius)#Fricke|E:Fricke &amp; Klein (1897)]] concerning the Cayley absolute, hyperbolic motion and its transformation, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic. The general case (on the left) equivalent to Lorentz-Möbius transformation [[../Lorentz transformation (Möbius)#math_6a|E:'''(6a)''']] and the hyperbolic case (on the right) equivalent to Lorentz transformation [[../Lorentz transformation (hyperbolic)#math_3d|E:'''(3d)''']] or squeeze mapping [[../Lorentz transformation (squeeze)#math_9d|E:'''(9d)''']] are as follows:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;

:&lt;math&gt;\left.\begin{matrix}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\\
z_{1}=x,\ z_{2}=y,\ z_{3}=z,\ z_{4}=t\\
Z=\frac{z_{1}+iz_{2}}{z_{4}-z_{3}}=\frac{x+iy}{t-z},\ Z'=\frac{x'+iy'}{t'-z'}\\
Z=\frac{\alpha Z'+\beta}{\gamma Z'+\delta}
\end{matrix}\right|\begin{matrix}Z=Z'e^{\vartheta}\\
\begin{align}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
\end{align}
\end{matrix}&lt;/math&gt;

==={{anchor|Varicak}} Varićak (1910) – Hyperbolic functions===

Following [[#Sommerfeld|Sommerfeld (1909)]], hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

and showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

Subsequently, other authors such as [[w:E. T. Whittaker]] (1910) or [[w:Alfred Robb]] (1911, who coined the name rapidity) used similar expressions, which are still used in modern textbooks.&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

=== {{anchor|Ignatowski}} Ignatowski (1910) ===

While earlier derivations and formulations of the Lorentz transformation relied from the outset on optics, electrodynamics, or the invariance of the speed of light, [[w:Vladimir Ignatowski]] (1910) showed that it is possible to use the principle of relativity (and related [[w:Group theory|group theoretical]] principles) alone, in order to derive the following transformation between two inertial frames:&lt;ref group=R&gt;Ignatowski (1910), pp. 973–974&lt;/ref&gt;&lt;ref group=R&gt;Ignatowski (1910/11ab)&lt;/ref&gt;

:&lt;math&gt;\begin{align}dx' &amp; =p\ dx-pq\ dt\\
dt' &amp; =-pqn\ dx+p\ dt\\
p &amp; =\frac{1}{\sqrt{1-q^{2}n}}
\end{align}
&lt;/math&gt;

The variable ''n'' can be seen as a space-time constant whose value has to be determined by experiment or taken from a known physical law such as electrodynamics. For that purpose, Ignatowski used the above-mentioned Heaviside ellipsoid representing a contraction of electrostatic fields by ''x''/γ in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation when ''n=1/c''&lt;sup&gt;2&lt;/sup&gt;, resulting in ''p''=γ and the Lorentz transformation ({{equationNote|4b}}). With ''n''=0, no length changes arise and the Galilean transformation follows. Ignatowski's method was further developed and improved by [[w:Philipp Frank]] and [[w:Hermann Rothe]] (1911, 1912),&lt;ref group=R&gt;Frank &amp; Rothe (1911), pp. 825ff; (1912), p. 750ff.&lt;/ref&gt; with various authors developing similar methods in subsequent years.&lt;ref name=baccetti&gt;Baccetti (2011), see references 1–25 therein.&lt;/ref&gt;

==={{anchor|Noether}} Noether (1910), Klein (1910) – Quaternions===

[[w:Felix Klein]] (1908) described [[../Lorentz transformation (Quaternions)#Cayley3|E:Cayley's (1854)]] 4D quaternion multiplications as &quot;Drehstreckungen&quot; (orthogonal substitutions in terms of rotations leaving invariant a quadratic form up to a factor), and pointed out that the modern principle of relativity as provided by Minkowski is essentially only the consequent application of such Drehstreckungen, even though he didn't provide details.&lt;ref group=R&gt;Klein (1908), p. 165&lt;/ref&gt;

In an appendix to Klein's and Sommerfeld's &quot;Theory of the top&quot; (1910), [[w:Fritz Noether]] showed how to formulate hyperbolic rotations using biquaternions with &lt;math&gt;\omega=\sqrt{-1}&lt;/math&gt;, which he also related to the speed of light by setting ω&lt;sup&gt;2&lt;/sup&gt;=-''c''&lt;sup&gt;2&lt;/sup&gt;. He concluded that this is the principal ingredient for a rational representation of the group of Lorentz transformations:&lt;ref group=R&gt;Noether (1910), pp. 939–943&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}V=\frac{Q_{1}vQ_{2}}{T_{1}T_{2}}\\
\hline X^{2}+Y^{2}+Z^{2}+\omega^{2}S^{2}=x^{2}+y^{2}+z^{2}+\omega^{2}s^{2}\\
\hline \begin{align}V &amp; =Xi+Yj+Zk+\omega S\\
v &amp; =xi+yj+zk+\omega s\\
Q_{1} &amp; =(+Ai+Bj+Ck+D)+\omega(A'i+B'j+C'k+D')\\
Q_{2} &amp; =(-Ai-Bj-Ck+D)+\omega(A'i+B'j+C'k-D')\\
T_{1}T_{2} &amp; =T_{1}^{2}=T_{2}^{2}=A^{2}+B^{2}+C^{2}+D^{2}+\omega^{2}\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Besides citing quaternion related standard works such as [[#Cayley3|Cayley (1854)]], Noether referred to the entries in Klein's encyclopedia by [[w:Eduard Study]] (1899) and the French version by [[w:Élie Cartan]] (1908).&lt;ref&gt;Cartan &amp; Study (1908), sections 35–36&lt;/ref&gt; Cartan's version contains a description of Study's [[w:dual number]]s, Clifford's biquaternions (including the choice &lt;math&gt;\omega=\sqrt{-1}&lt;/math&gt; for hyperbolic geometry), and Clifford algebra, with references to [[../Lorentz transformation (Quaternions)#Stephanos|Stephanos (1883)]], [[../Lorentz transformation (Quaternions)#Buchheim|Buchheim (1884/85)]], [[../Lorentz transformation (Quaternions)#Vahlen|Vahlen (1901/02)]] and others.

Citing Noether, Klein himself published in August 1910 the following quaternion substitutions forming the group of Lorentz transformations:&lt;ref group=R&gt;Klein (1910), p. 300&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align} &amp; \left(i_{1}x'+i_{2}y'+i_{3}z'+ict'\right)\\
 &amp; \quad-\left(i_{1}x_{0}+i_{2}y_{0}+i_{3}z_{0}+ict_{0}\right)
\end{align}
=\frac{\left[\begin{align} &amp; \left(i_{1}(A+iA')+i_{2}(B+iB')+i_{3}(C+iC')+i_{4}(D+iD')\right)\\
 &amp; \quad\cdot\left(i_{1}x+i_{2}y+i_{3}z+ict\right)\\
 &amp; \quad\quad\cdot\left(i_{1}(A-iA')+i_{2}(B-iB')+i_{3}(C-iC')-(D-iD')\right)
\end{align}
\right]}{\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)-\left(A^{2}+B^{2}+C^{2}+D^{2}\right)}\\
\hline \text{where}\\
AA'+BB'+CC'+DD'=0\\
A^{2}+B^{2}+C^{2}+D^{2}&gt;A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}
\end{matrix}&lt;/math&gt;

or in March 1911&lt;ref group=R&gt;Klein (1911), pp. 602ff.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}g'=\frac{pg\pi}{M}\\
\hline \begin{align}g &amp; =\sqrt{-1}ct+ix+jy+kz\\
g' &amp; =\sqrt{-1}ct'+ix'+jy'+kz'\\
p &amp; =(D+\sqrt{-1}D')+i(A+\sqrt{-1}A')+j(B+\sqrt{-1}B')+k(C+\sqrt{-1}C')\\
\pi &amp; =(D-\sqrt{-1}D')-i(A-\sqrt{-1}A')-j(B-\sqrt{-1}B')-k(C-\sqrt{-1}C')\\
M &amp; =\left(A^{2}+B^{2}+C^{2}+D^{2}\right)-\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)\\
 &amp; AA'+BB'+CC'+DD'=0\\
 &amp; A^{2}+B^{2}+C^{2}+D^{2}&gt;A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}
\end{align}
\end{matrix}&lt;/math&gt;

==={{anchor|Conway}} Conway (1911), Silberstein (1911) – Quaternions===

[[w:Arthur W. Conway]] in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms of velocity λ:&lt;ref group=R&gt;Conway (1911), p. 8&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}\mathtt{D} &amp; =\mathbf{a}^{-1}\mathtt{D}'\mathbf{a}^{-1}\\
\mathtt{\sigma} &amp; =\mathbf{a}\mathtt{\sigma}'\mathbf{a}^{-1}
\end{align}
\\
e=\mathbf{a}^{-1}e'\mathbf{a}^{-1}\\
\hline a=\left(1-hc^{-1}\lambda\right)^{\frac{1}{2}}\left(1+c^{-2}\lambda^{2}\right)^{-\frac{1}{4}}
\end{matrix}&lt;/math&gt;

Also [[w:Ludwik Silberstein]] in November 1911&lt;ref group=R&gt;Silberstein (1911/12), p. 793&lt;/ref&gt; as well as in 1914,&lt;ref&gt;Silberstein (1914), p. 156&lt;/ref&gt; formulated the Lorentz transformation in terms of velocity ''v'':

:&lt;math&gt;\begin{matrix}q'=QqQ\\
\hline \begin{align}q &amp; =\mathbf{r}+l=xi+yj+zk+\iota ct\\
q &amp; '=\mathbf{r}'+l'=x'i+y'j+z'k+\iota ct'\\
Q &amp; =\frac{1}{\sqrt{2}}\left(\sqrt{1+\gamma}+\mathrm{u}\sqrt{1-\gamma}\right)\\
 &amp; =\cos\alpha+\mathrm{u}\sin\alpha=e^{\alpha\mathrm{u}}\\
 &amp; \left\{ \gamma=\left(1-v^{2}/c^{2}\right)^{-1/2},\ 2\alpha=\operatorname{arctg}\ \left(\iota\frac{v}{c}\right)\right\} 
\end{align}
\end{matrix}&lt;/math&gt;

Silberstein cites [[../Lorentz transformation (Quaternions)#Cayley3|E:Cayley (1854, 1855)]] and Study's encyclopedia entry (in the extended French version of Cartan in 1908), as well as the appendix of Klein's and Sommerfeld's book.

==={{anchor|Herglotz2}} Ignatowski (1910/11), Herglotz (1911), and others – Vector transformation===

{{See|w:Lorentz transformation#Vector transformations}}

[[w:Vladimir Ignatowski]] (1910, published 1911) defined the vectorial velocity addition ({{equationNote|4g}}) as well as general Lorentz boost ({{equationNote|4f}}) as&lt;ref group=R&gt;Ignatowski (1910/11a), p. 23; (1910/11b), p. 22&lt;/ref&gt; 

:&lt;math&gt;\begin{matrix}\begin{matrix}\mathfrak{v} =\frac{\mathfrak{v}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{v}'+pq\mathfrak{c}_{0}}{p\left(1+nq\mathfrak{c}_{0}\mathfrak{v}'\right)} &amp; \left|\begin{align}\mathfrak{A}' &amp; =\mathfrak{A}+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}-pqb\mathfrak{c}_{0}\\
b' &amp; =pb-pqn\mathfrak{A}\mathfrak{c}_{0}\\
\\
\mathfrak{A} &amp; =\mathfrak{A}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}'+pqb'\mathfrak{c}_{0}\\
b &amp; =pb'+pqn\mathfrak{A}'\mathfrak{c}_{0}
\end{align}
\right.\end{matrix}\\
\left[\mathfrak{v}=\mathbf{u},\ \mathfrak{A}=\mathbf{x},\ b=t,\ \mathfrak{c}_{0}=\frac{\mathbf{v}}{v},\ p=\gamma,\ n=\frac{1}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

An equivalent transformation was given by [[w:Gustav Herglotz]] (1911)&lt;ref group=R&gt;Herglotz (1911), p. 497&lt;/ref&gt; using '''v'''=''(v&lt;sub&gt;x&lt;/sub&gt;, v&lt;sub&gt;y&lt;/sub&gt;, v&lt;sub&gt;z&lt;/sub&gt;)'' and '''r'''=''(x, y, z)'':

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{0} &amp; =x+\alpha u(ux+vy+wz)-\beta ut\\
y^{0} &amp; =y+\alpha v(ux+vy+wz)-\beta vt\\
z^{0} &amp; =z+\alpha w(ux+vy+wz)-\beta wt\\
t^{0} &amp; =-\beta(ux+vy+wz)+\beta t\\
 &amp; \alpha=\frac{1}{\sqrt{1-s^{2}}\left(1+\sqrt{1-s^{2}}\right)},\ \beta=\frac{1}{\sqrt{1-s^{2}}}
\end{align}
\right| &amp; \begin{align}x' &amp; =x+\alpha v_{x}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{x}t\\
y' &amp; =y+\alpha v_{y}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{y}t\\
z' &amp; =z+\alpha v_{z}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{z}t\\
t' &amp; =-\gamma\left(v_{x}x+v_{y}y+v_{z}z\right)+\gamma t\\
 &amp; \alpha=\frac{\gamma^{2}}{\gamma+1},\ \gamma=\frac{1}{\sqrt{1-v^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

Or by [[w:Ludwik Silberstein]] (1911 on the left, 1914 on the right):&lt;ref group=R&gt;Silberstein (1911/12), p. 792; (1914), p. 123&lt;/ref&gt;

:&lt;math&gt;\begin{array}{c|c}
\begin{align}\mathbf{r}' &amp; =\mathbf{r}+(\gamma-1)(\mathbf{ru})\mathbf{u}+i\beta\gamma lu\\
l' &amp; =\gamma\left[l-i\beta(\mathbf{ru})\right]
\end{align}
 &amp; \begin{align}\mathbf{r}' &amp; =\mathbf{r}+\left[\frac{\gamma-1}{v^{2}}(\mathbf{vr})-\gamma t\right]\mathbf{v}\\
t' &amp; =\gamma\left[t-\frac{1}{c^{2}}(\mathbf{vr})\right]
\end{align}
\end{array}&lt;/math&gt;

[[w:Erwin Madelung]] (1922) provided the matrix form&lt;ref&gt;Madelung (1921), p. 207&lt;/ref&gt;

:&lt;math&gt;\begin{array}{c|c|c|c|c}
 &amp; x &amp; y &amp; z &amp; t\\
\hline x' &amp; 1-\frac{v_{x}^{2}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{x}v_{y}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{x}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; \frac{-v_{x}}{\sqrt{1-\beta^{2}}}\\
y' &amp; -\frac{v_{x}v_{y}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; 1-\frac{v_{y}^{2}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{y}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; \frac{-v_{y}}{\sqrt{1-\beta^{2}}}\\
z' &amp; -\frac{v_{x}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{y}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; 1-\frac{v_{z}^{2}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; \frac{-v_{z}}{\sqrt{1-\beta^{2}}}\\
t' &amp; \frac{-v_{x}}{c^{2}\sqrt{1-\beta^{2}}} &amp; \frac{-v_{y}}{c^{2}\sqrt{1-\beta^{2}}} &amp; \frac{-v_{z}}{c^{2}\sqrt{1-\beta^{2}}} &amp; \frac{1}{\sqrt{1-\beta^{2}}}
\end{array}&lt;/math&gt;

These formulas were called &quot;general Lorentz transformation without rotation&quot; by [[w:Christian Møller]] (1952),&lt;ref&gt;Møller (1952/55), pp. 41–43&lt;/ref&gt; who in addition gave an even more general Lorentz transformation in which the Cartesian axes have different orientations, using a [[w:Three-dimensional rotation operator|rotation operator]] &lt;math&gt;\mathfrak{D}&lt;/math&gt;. In this case, '''v′'''=''(v′&lt;sub&gt;x&lt;/sub&gt;, v′&lt;sub&gt;y&lt;/sub&gt;, v′&lt;sub&gt;z&lt;/sub&gt;)'' is not equal to -'''v'''=''(-v&lt;sub&gt;x&lt;/sub&gt;, -v&lt;sub&gt;y&lt;/sub&gt;, -v&lt;sub&gt;z&lt;/sub&gt;)'', but the relation &lt;math&gt;\mathbf{v}'=-\mathfrak{D}\mathbf{v}&lt;/math&gt; holds instead, with the result

:&lt;math&gt;\begin{array}{c}
\begin{align}\mathbf{x}' &amp; =\mathfrak{D}^{-1}\mathbf{x}-\mathbf{v}'\left\{ \left(\gamma-1\right)(\mathbf{x\cdot v})/v^{2}-\gamma t\right\} \\
t' &amp; =\gamma\left(t-(\mathbf{v}\cdot\mathbf{x})/c^{2}\right)
\end{align}
\end{array}&lt;/math&gt;

==={{anchor|Borel}} Borel (1913–14) – Cayley–Hermite parameter===

Borel (1913) started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and [[../Lorentz transformation (Cayley-Hermite)#Cayley|E:Cayley's (1846)]] parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions:&lt;ref group=R&gt;Borel (1913/14), p. 39&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}-z^{2}-1=0\\
\hline {\scriptstyle \begin{align}\delta a &amp; =\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta b &amp; =2(\lambda\mu+\nu\rho), &amp; \delta c &amp; =-2(\lambda\nu+\mu\rho),\\
\delta a' &amp; =2(\lambda\mu-\nu\rho), &amp; \delta b' &amp; =-\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta c' &amp; =2(\lambda\rho-\mu\nu),\\
\delta a'' &amp; =2(\lambda\nu-\mu\rho), &amp; \delta b'' &amp; =2(\lambda\rho+\mu\nu), &amp; \delta c'' &amp; =-\left(\lambda^{2}+\mu^{2}+\nu^{2}+\rho^{2}\right),
\end{align}
}\\
\left(\delta=\lambda^{2}+\mu^{2}-\rho^{2}-\nu^{2}\right)\\
\lambda=\nu=0\rightarrow\text{Hyperbolic rotation}
\end{matrix}&lt;/math&gt;

In four dimensions:&lt;ref group=R&gt;Borel (1913/14), p. 41&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}-\left(t_{1}-t_{2}\right)^{2}\\
\hline {\scriptstyle \begin{align} &amp; \left(\mu^{2}+\nu^{2}-\alpha^{2}\right)\cos\varphi+\left(\lambda^{2}-\beta^{2}-\gamma^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi-\gamma\operatorname{sh}{\theta}\\
 &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi+\gamma\operatorname{sh}{\theta} &amp;  &amp; \left(\mu^{2}+\nu^{2}-\beta^{2}\right)\cos\varphi+\left(\mu^{2}-\alpha^{2}-\gamma^{2}\right)\operatorname{ch}{\theta}\\
 &amp; -(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi-\beta\operatorname{sh}{\theta} &amp;  &amp; -(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\lambda\sin\varphi+\alpha\operatorname{sh}{\theta}\\
 &amp; (\gamma\mu-\beta\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta} &amp;  &amp; -(\alpha\nu-\lambda\gamma)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
\\
 &amp; \quad-(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi+\beta\operatorname{sh}{\theta} &amp;  &amp; \quad(\beta\nu-\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta}\\
 &amp; \quad-(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})-\lambda\sin\varphi-\alpha\operatorname{sh}{\theta} &amp;  &amp; \quad(\lambda\gamma-\alpha\nu)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
 &amp; \quad\left(\lambda^{2}+\mu^{2}-\gamma^{2}\right)\cos\varphi+\left(\nu^{2}-\alpha^{2}-\beta^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; \quad(\alpha\mu-\beta\lambda)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta}\\
 &amp; \quad(\beta\gamma-\alpha\mu)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta} &amp;  &amp; \quad-\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\cos\varphi+\left(\lambda^{2}+\mu^{2}+\nu^{2}\right)\operatorname{ch}{\theta}
\end{align}
}\\
\left(\alpha^{2}+\beta^{2}+\gamma^{2}-\lambda^{2}-\mu^{2}-\nu^{2}=-1\right)
\end{matrix}&lt;/math&gt;

==={{anchor|Gruner}} Gruner (1921) – Trigonometric Lorentz boosts===

In order to simplify the graphical representation of Minkowski space, [[w:Paul Gruner]] (1921) (with the aid of Josef Sauter) developed what is now called [[w:Loedel diagram]]s, using the following relations:&lt;ref group=R&gt;Gruner (1921a),&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}v=\alpha\cdot c;\quad\beta=\frac{1}{\sqrt{1-\alpha^{2}}}\\
\sin\varphi=\alpha;\quad\beta=\frac{1}{\cos\varphi};\quad\alpha\beta=\tan\varphi\\
\hline x'=\frac{x}{\cos\varphi}-t\cdot\tan\varphi,\quad t'=\frac{t}{\cos\varphi}-x\cdot\tan\varphi
\end{matrix}&lt;/math&gt;

In another paper Gruner used the alternative relations:&lt;ref group=R&gt;Gruner (1921b)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\alpha=\frac{v}{c};\ \beta=\frac{1}{\sqrt{1-\alpha^{2}}};\\
\cos\theta=\alpha=\frac{v}{c};\ \sin\theta=\frac{1}{\beta};\ \cot\theta=\alpha\cdot\beta\\
\hline x'=\frac{x}{\sin\theta}-t\cdot\cot\theta,\quad t'=\frac{t}{\sin\theta}-x\cdot\cot\theta
\end{matrix}&lt;/math&gt;

==References==
===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|abra1905}}
*{{#section:History of Topics in Special Relativity/relsource|bor14}}
*{{#section:History of Topics in Special Relativity/relsource|buch08}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04a}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04b}}
*{{#section:History of Topics in Special Relativity/relsource|con11qua}}
*{{#section:History of Topics in Special Relativity/relsource|einst05elek}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|fra11}}
*{{#section:History of Topics in Special Relativity/relsource|fra12}}
*{{#section:History of Topics in Special Relativity/relsource|gans05}}
*{{#section:History of Topics in Special Relativity/relsource|grun21a}}
*{{#section:History of Topics in Special Relativity/relsource|grun21b}}
*{{#section:History of Topics in Special Relativity/relsource|heav89}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|herg11ela}}
*{{#section:History of Topics in Special Relativity/relsource|igna10}}
*{{#section:History of Topics in Special Relativity/relsource|ignat10prin2}}
*{{#section:History of Topics in Special Relativity/relsource|klein08}}
*{{#section:History of Topics in Special Relativity/relsource|klein10b}}
*{{#section:History of Topics in Special Relativity/relsource|klein10}}
*{{#section:History of Topics in Special Relativity/relsource|klein11}}
*{{#section:History of Topics in Special Relativity/relsource|lar97}}
*{{#section:History of Topics in Special Relativity/relsource|lar29}}
*{{#section:History of Topics in Special Relativity/relsource|lar00}}
*{{#section:History of Topics in Special Relativity/relsource|lar04a}}
*{{#section:History of Topics in Special Relativity/relsource|lar04b}}
*{{#section:History of Topics in Special Relativity/relsource|lor92elek}}
*{{#section:History of Topics in Special Relativity/relsource|lor92b}}
*{{#section:History of Topics in Special Relativity/relsource|lor95}}
*{{#section:History of Topics in Special Relativity/relsource|lor99}}
*{{#section:History of Topics in Special Relativity/relsource|lor04}}
*{{#section:History of Topics in Special Relativity/relsource|lor16}}
*{{#section:History of Topics in Special Relativity/relsource|mink07a}}
*{{#section:History of Topics in Special Relativity/relsource|mink07b}}
*{{#section:History of Topics in Special Relativity/relsource|mink08}}
*{{#section:History of Topics in Special Relativity/relsource|plum10}}
*{{#section:History of Topics in Special Relativity/relsource|poi00}}
*{{#section:History of Topics in Special Relativity/relsource|poi04}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05a}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05b}}
*{{#section:History of Topics in Special Relativity/relsource|sea97}}
*{{#section:History of Topics in Special Relativity/relsource|silber11quat}}
*{{#section:History of Topics in Special Relativity/relsource|silber12quat}}
*{{#section:History of Topics in Special Relativity/relsource|silber14quat}}
*{{#section:History of Topics in Special Relativity/relsource|som09}}
*{{#section:History of Topics in Special Relativity/relsource|thom89}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}
*{{#section:History of Topics in Special Relativity/relsource|voi87}}
*{{#section:History of Topics in Special Relativity/relsource|wien04}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L4}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
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      <text bytes="64204" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via velocity==
===Boosts===
In the [[w:theory of relativity]], Lorentz transformations exhibit the symmetry of [[w:Minkowski spacetime]] by using a constant ''c'' as the [[w:speed of light]], and a parameter ''v'' as the relative [[w:velocity]] between two [[w:inertial reference frames]]. The corresponding formulas are identical to [[../Lorentz transformation (hyperbolic)|E:Lorentz transformations via hyperbolic functions]] introduced long before relativity was developed. In particular, the hyperbolic angle &lt;math&gt;\eta&lt;/math&gt; can be interpreted as the velocity related [[w:rapidity]] &lt;math&gt;\tanh\eta=\beta=v/c&lt;/math&gt;, so that &lt;math&gt;\gamma=\cosh\eta&lt;/math&gt; is the [[w:Lorentz factor]], &lt;math&gt;\beta\gamma=\sinh\eta&lt;/math&gt; the [[w:proper velocity]], &lt;math&gt;u'=c\tanh q&lt;/math&gt; the velocity of another object, &lt;math&gt;u=c\tanh(q+\eta)&lt;/math&gt; the [[w:velocity-addition formula]], thus transformation [[../Lorentz transformation (hyperbolic)#math_3b|E:'''(3b)''']] becomes:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}\gamma-x_{1}\beta\gamma\\
x_{1}^{\prime} &amp; =-x_{0}\beta\gamma+x_{1}\gamma\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\gamma+x_{1}^{\prime}\beta\gamma\\
x_{1} &amp; =x_{0}^{\prime}\beta\gamma+x_{1}^{\prime}\gamma\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\left|{\scriptstyle \begin{align}\beta^{2}\gamma^{2}-\gamma^{2} &amp; =-1 &amp; (a)\\
\gamma^{2}-\beta^{2}\gamma^{2} &amp; =1 &amp; (b)\\
\frac{\beta\gamma}{\gamma} &amp; =\beta &amp; (c)\\
\frac{1}{\sqrt{1-\beta^{2}}} &amp; =\gamma &amp; (d)\\
\frac{\beta}{\sqrt{1-\beta^{2}}} &amp; =\beta\gamma &amp; (e)\\
\frac{u'+v}{1+\frac{u'v}{c^{2}}} &amp; =u &amp; (f)
\end{align}
}\right.
\end{matrix}&lt;/math&gt;|{{equationRef|4a}}}}

Or in four dimensions and by setting &lt;math&gt;x_{0}=ct,\ x_{1}=x,\ x_{2}=y&lt;/math&gt; and adding an unchanged ''z'' the familiar form follows

{{NumBlk|:|&lt;math&gt;\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}+z^{2}=-c^{2}t^{\prime2}+x^{\prime2}+y^{\prime2}+z^{\prime2}\\
\hline \left.\begin{align}t' &amp; =\gamma\left(t-x\frac{v}{c^{2}}\right)\\
x' &amp; =\gamma(x-vt)\\
y' &amp; =y\\
z' &amp; =z
\end{align}
\right|\begin{align}t &amp; =\gamma\left(t'+x\frac{v}{c^{2}}\right)\\
x &amp; =\gamma(x'+vt')\\
y &amp; =y'\\
z &amp; =z'
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|4b}}}}

Analogous transformations have been introduced by [[#Voigt|Voigt (1887)]] in terms of an incompessible medium, and by [[#Lorentz1|Lorentz (1892, 1895)]] who analyzed [[w:Maxwell's equations]], they were completed by [[#Larmor|Larmor (1897, 1900)]] and [[#Lorentz2|Lorentz (1899, 1904)]], and brought into their modern form by [[#Poincare3|Poincaré (1905)]] who gave the transformation the name of Lorentz.&lt;ref&gt;Miller (1981), chapter 1&lt;/ref&gt; Eventually, [[#Einstein|Einstein (1905)]] showed in his development of [[w:special relativity]] that the transformations follow from the [[w:principle of relativity]] and constant light speed alone by modifying the traditional concepts of space and time, without requiring a [[w:Lorentz ether theory|mechanical aether]] in contradistinction to Lorentz and Poincaré.&lt;ref&gt;Miller (1981), chapter 4–7&lt;/ref&gt; [[#Minkowski|Minkowski (1907–1908)]] used them to argue that space and time are inseparably connected as [[w:spacetime]]. [[#Minkowski|Minkowski (1907–1908)]], [[#Frank|Frank (1909)]]  and [[#Varicak|Varićak (1910)]] showed the relation to imaginary and hyperbolic functions. Important contributions to the mathematical understanding of the Lorentz transformation were also made by other authors such as [[#Herglotz1|Herglotz (1909/10)]], [[#Ignatowski|Ignatowski (1910)]], [[#Noether|Noether (1910) and Klein (1910)]], [[#Borel|Borel (1913–14)]].

===Velocity addition and aberration===
In exact analogy to Beltrami coordinates in equation [[../Lorentz transformation (hyperbolic)#math_3e|E:'''(3e)''']], one can substitute &lt;math&gt;\left[\tfrac{u_{x}}{c},\ \tfrac{u_{y}}{c},\ 1\right]=\left[\tfrac{x}{ct},\ \tfrac{y}{ct},\ \tfrac{ct}{ct}\right]&lt;/math&gt; in ({{equationNote|4a}}), producing the Lorentz transformation of velocities (or [[w:velocity addition formula]]) :

{{NumBlk|:|&lt;math&gt;\begin{align}u_{x}^{\prime} &amp; =\frac{-c^{2}\sinh\eta+u_{x}c\cosh\eta}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{x}-c\tanh\eta}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}-v}{1-\frac{v}{c^{2}}u{}_{x}}\\
u_{y}^{\prime} &amp; =\frac{cu_{y}}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\tanh^{2}\eta}}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u{}_{x}}\\
\\
u_{x} &amp; =\frac{c^{2}\sinh\eta+u_{x}^{\prime}c\cosh\eta}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+c\tanh\eta}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+v}{1+\frac{v}{c^{2}}u_{x}^{\prime}}\\
u_{y} &amp; =\frac{cy'}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u_{x}^{\prime}}
\end{align}
&lt;/math&gt;|{{equationRef|4c}}}}

or using trigonometric and hyperbolic identities as in equation [[../Lorentz transformation (hyperbolic)#math_3f|E:'''(3f)''']], it becomes the hyperbolic law of cosines:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{matrix}u^{2}=u_{x}^{2}+u_{y}^{2}\\
u'^{2}=u_{x}^{\prime2}+u_{y}^{\prime2}
\end{matrix}\left|\begin{matrix}u_{x}=u\cos\alpha\\
u_{y}=u\sin\alpha\\
\\
u_{x}^{\prime}=u'\cos\alpha'\\
u_{y}^{\prime}=u'\sin\alpha'
\end{matrix}\right|\begin{align}u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-\frac{v}{c^{2}}u\cos\alpha}\\
u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{u'\cos\alpha'+v}, &amp; \tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{u\cos\alpha-v}
\end{align}
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(\frac{vu'\sin\alpha'}{c}\right){}^{2}}}{1+\frac{v}{c^{2}}u'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(\frac{vu\sin\alpha}{c}\right){}^{2}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\frac{u^{\prime2}}{c^{2}}}}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{1}{\sqrt{1-\frac{u^{2}}{c^{2}}}}-\frac{v/c}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{u/c}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\cos\alpha\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha
\end{matrix}&lt;/math&gt;|{{equationRef|4d}}}}

and by further setting ''u=u′=c'' one gets the well known [[../Lorentz transformation (hyperbolic)#math_3g|E:Kepler formulas '''(3g)''']], which  express the relativistic [[w:aberration of light]]:&lt;ref&gt;Pauli (1921), pp. 562; 565–566&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\cos\alpha=\frac{\cos\alpha'+\frac{v}{c}}{1+\frac{v}{c}\cos\alpha'},\ \sin\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c}\cos\alpha'},\ \tan\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha'+\frac{v}{c}},\ \tan\frac{\alpha}{2}=\sqrt{\frac{c-v}{c+v}}\tan\frac{\alpha'}{2}\\
\cos\alpha'=\frac{\cos\alpha-\frac{v}{c}}{1-\frac{v}{c}\cos\alpha},\ \sin\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c}\cos\alpha},\ \tan\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha-\frac{v}{c}},\ \tan\frac{\alpha'}{2}=\sqrt{\frac{c+v}{c-v}}\tan\frac{\alpha}{2}
\end{matrix}&lt;/math&gt;|{{equationRef|4e}}}}

Formulas ({{equationNote|4c}}, {{equationNote|4d}}) were given by [[#Einstein|Einstein (1905)]] and [[#Poincare3|Poincaré (1905/06)]], while the relations to the spherical and hyperbolic law of cosines were given by [[#Sommerfeld|Sommerfeld (1909)]] and [[#Varicak|Varićak (1910)]]. The aberration formula for cos(α) was given by [[#Einstein|Einstein (1905)]].&lt;ref group=R name=plum&gt;Plummer (1910), pp. 258-259: After deriving the relativistic expressions for the aberration angles φ' and φ, Plummer remarked on p. 259: ''Another geometrical representation is obtained by assimilating φ' to the eccentric and φ to the true anomaly in an ellipse whose eccentricity is v/U = sin β.''&lt;/ref&gt;&lt;ref name=robin&gt;Robinson (1990), chapter 3-4, analyzed the relation between &quot;Kepler's formula&quot; and the &quot;physical velocity addition formula&quot; in special relativity.&lt;/ref&gt;

===Vectorial Lorentz transformation===
Also Lorentz boosts for arbitrary directions in line with [[../Lorentz transformation (general)#math_1a|E:general Lorentz transformation '''(1a)''']] can be given as:&lt;ref&gt;Møller (1952/55), Chapter II, § 18&lt;/ref&gt;

:&lt;math&gt;\mathbf{x}'=\begin{bmatrix}\gamma &amp; -\gamma\beta n_{x} &amp; -\gamma\beta n_{y} &amp; -\gamma\beta n_{z}\\
-\gamma\beta n_{x} &amp; 1+(\gamma-1)n_{x}^{2} &amp; (\gamma-1)n_{x}n_{y} &amp; (\gamma-1)n_{x}n_{z}\\
-\gamma\beta n_{y} &amp; (\gamma-1)n_{y}n_{x} &amp; 1+(\gamma-1)n_{y}^{2} &amp; (\gamma-1)n_{y}n_{z}\\
-\gamma\beta n_{z} &amp; (\gamma-1)n_{z}n_{x} &amp; (\gamma-1)n_{z}n_{y} &amp; 1+(\gamma-1)n_{z}^{2}
\end{bmatrix}\cdot\mathbf{x},\quad\left[\mathbf{n}=\frac{\mathbf{v}}{v}\right]&lt;/math&gt;

or in vector notation

{{NumBlk|:|&lt;math&gt;\begin{align}t' &amp; =\gamma\left(t-\frac{v\mathbf{n}\cdot\mathbf{r}}{c^{2}}\right)\\
\mathbf{r}' &amp; =\mathbf{r}+(\gamma-1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n}-\gamma tv\mathbf{n}
\end{align}
&lt;/math&gt;|{{equationRef|4f}}}}

The corresponding vectorial velocity addition formula in line with [[../Lorentz transformation (general)#math_1b|E:general Lorentz transformation '''(1b)''']] is given by

{{NumBlk|:|&lt;math&gt;\mathbf{u}'=\frac{1}{1+\frac{\mathbf{v}\cdot\mathbf{u}}{c^{2}}}\left[\frac{\mathbf{u}}{\gamma_{\mathbf{v}}}+\mathbf{v}+\frac{1}{c^{2}}\frac{\gamma_{\mathbf{v}}}{\gamma_{\mathbf{v}}+1}(\mathbf{u}\cdot\mathbf{v})\mathbf{v}\right]&lt;/math&gt;|{{equationRef|4g}}}}

Such transformations were formulated by [[#Herglotz2|Ignatowski (1910), Herglotz (1911)]] and others.

==Historical notation==
==={{anchor|Voigt}} Voigt (1887) ===

[[w:Woldemar Voigt]] (1887)&lt;ref group=R&gt;Voigt (1887), p. 45&lt;/ref&gt; developed a transformation in connection with the [[w:Doppler effect]] and an incompressible medium, being in modern notation:&lt;ref&gt;Miller (1981), 114–115&lt;/ref&gt;&lt;ref name=pais&gt;Pais (1982), Kap. 6b&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\xi_{1} &amp; =x_{1}-\varkappa t\\
\eta_{1} &amp; =y_{1}q\\
\zeta_{1} &amp; =z_{1}q\\
\tau &amp; =t-\frac{\varkappa x_{1}}{\omega^{2}}\\
q &amp; =\sqrt{1-\frac{\varkappa^{2}}{\omega^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =\frac{y}{\gamma}\\
z^{\prime} &amp; =\frac{z}{\gamma}\\
t^{\prime} &amp; =t-\frac{vx}{c^{2}}\\
\frac{1}{\gamma} &amp; =\sqrt{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

If the right-hand sides of his equations are multiplied by γ they are the modern Lorentz transformation ({{equationNote|4b}}). In Voigt's theory the speed of light is invariant, but his transformations mix up a relativistic boost together with a rescaling of space-time. Optical phenomena in free space are [[w:Scale invariance|scale]], [[w:Conformal map|conformal]] (using the factor λ discussed [[#Lorsph|above]]), and [[w:Lorentz covariance|Lorentz invariant]], so the combination is invariant too.&lt;ref name=pais /&gt; For instance, Lorentz transformations can be extended by using &lt;math&gt;l=\sqrt{\lambda}&lt;/math&gt;:&lt;ref group=R&gt;Lorentz (1915/16), p. 197&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\quad y^{\prime}=ly,\quad z^{\prime}=lz,\quad t^{\prime}=\gamma l\left(t-x\frac{v}{c^{2}}\right)&lt;/math&gt;.

''l''=1/γ gives the Voigt transformation, ''l''=1 the Lorentz transformation. But scale transformations are not a symmetry of all the laws of nature, only of electromagnetism, so these transformations cannot be used to formulate a [[w:principle of relativity]] in general. It was demonstrated by Poincaré and Einstein that one has to set ''l''=1 in order to make the above transformation symmetric and to form a group as required by the relativity principle, therefore the Lorentz transformation is the only viable choice.

Voigt sent his 1887 paper to Lorentz in 1908,&lt;ref&gt;Voigt's transformations and the beginning of the relativistic revolution, Ricardo Heras, arXiv:1411.2559 [https://arxiv.org/abs/1411.2559]&lt;/ref&gt; and that was acknowledged in 1909: {{Quote|In a paper &quot;Über das Doppler'sche Princip&quot;, published in 1887 (Gött. Nachrichten, p. 41) and which to my regret has escaped my notice all these years, Voigt has applied to equations of the form (7) (§ 3 of this book) [namely &lt;math&gt;\Delta\Psi-\tfrac{1}{c^{2}}\tfrac{\partial^{2}\Psi}{\partial t^{2}}=0&lt;/math&gt;] a transformation equivalent to the formulae (287) and (288) [namely &lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\ y^{\prime}=ly,\ z^{\prime}=lz,\ t^{\prime}=\gamma l\left(t-\tfrac{v}{c^{2}}x\right)&lt;/math&gt;]. The idea of the transformations used above (and in § 44) might therefore have been borrowed from Voigt and the proof that it does not alter the form of the equations for the ''free'' ether is contained in his paper.&lt;ref group=R&gt;Lorentz (1915/16), p. 198&lt;/ref&gt;}}

Also [[w:Hermann Minkowski]] said in 1908 that the transformations which play the main role in the principle of relativity were first examined by Voigt in 1887. Voigt responded in the same paper by saying that his theory was based on an elastic theory of light, not an electromagnetic one. However, he concluded that some results were actually the same.&lt;ref group=R&gt;Bucherer (1908), p. 762&lt;/ref&gt;

==={{anchor|Heaviside}} Heaviside (1888), Thomson (1889), Searle (1896)===

In 1888, [[w:Oliver Heaviside]]&lt;ref group=R&gt;Heaviside (1888), p. 324&lt;/ref&gt; investigated the properties of [[w:Relativistic electromagnetism|charges in motion]] according to Maxwell's electrodynamics. He calculated, among other things, anisotropies in the electric field of moving bodies represented by this formula:&lt;ref&gt;Brown (2003)&lt;/ref&gt;

:&lt;math&gt;\mathrm{E}=\left(\frac{q\mathrm{r}}{r^{2}}\right)\left(1-\frac{v^{2}\sin^{2}\theta}{c^{2}}\right)^{-3/2}&lt;/math&gt;.

Consequently, [[w:Joseph John Thomson]] (1889)&lt;ref group=R&gt;Thomson (1889), p. 12&lt;/ref&gt; found a way to substantially simplify calculations concerning moving charges by using the following mathematical transformation (like other authors such as Lorentz or Larmor, also Thomson implicitly used the [[w:Galilean transformation]] ''z-vt'' in his equation&lt;ref name=mil /&gt;):

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}z &amp; =\left\{ 1-\frac{\omega^{2}}{v^{2}}\right\} ^{\frac{1}{2}}z'\end{align}
\right| &amp; \begin{align}z^{\ast}=z-vt &amp; =\frac{z'}{\gamma}\end{align}
\end{matrix}&lt;/math&gt;

Thereby, [[w:inhomogeneous electromagnetic wave equation]]s are transformed into a [[w:Poisson equation]].&lt;ref name=mil&gt;Miller (1981), 98–99&lt;/ref&gt; Eventually, [[w:George Frederick Charles Searle]]&lt;ref group=R&gt;Searle (1886), p. 333&lt;/ref&gt; noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called &quot;Heaviside-Ellipsoid&quot; of [[w:axial ratio]]

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align} &amp; \sqrt{\alpha}:1:1\\
\alpha= &amp; 1-\frac{u^{2}}{v^{2}}
\end{align}
\right| &amp; \begin{align} &amp; \frac{1}{\gamma}:1:1\\
\frac{1}{\gamma^{2}} &amp; =1-\frac{v^{2}}{c^{2}}
\end{align}
\end{matrix}&lt;/math&gt;&lt;ref name=mil /&gt;

=== {{anchor|Lorentz1}} Lorentz (1892, 1895) ===

In order to explain the [[w:aberration of light]] and the result of the [[w:Fizeau experiment]] in accordance with [[w:Maxwell's equations]], Lorentz in 1892 developed a model (&quot;[[w:Lorentz ether theory]]&quot;) in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. In order to calculate the optics of moving bodies, Lorentz introduced the following quantities to transform from the aether system into a moving system (it's unknown whether he was influenced by Voigt, Heaviside, and Thomson)&lt;ref group=R&gt;Lorentz (1892a), p. 141&lt;/ref&gt;&lt;ref name=milf&gt;Miller (1982), 1.4 &amp; 1.5&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\mathfrak{x} &amp; =\frac{V}{\sqrt{V^{2}-p^{2}}}x\\
t' &amp; =t-\frac{\varepsilon}{V}\mathfrak{x}\\
\varepsilon &amp; =\frac{p}{\sqrt{V^{2}-p^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)\\
\gamma\frac{v}{c} &amp; =\frac{v}{\sqrt{c^{2}-v^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

where ''x&lt;sup&gt;*&lt;/sup&gt;'' is the [[w:Galilean transformation]] ''x-vt''. Except the additional γ in the time transformation, this is the complete Lorentz transformation ({{equationNote|4b}}).&lt;ref name=milf /&gt; While ''t'' is the &quot;true&quot; time for observers resting in the aether, ''t′'' is an auxiliary variable only for calculating processes for moving systems. It is also important that Lorentz and later also Larmor formulated this transformation in two steps. At first an implicit Galilean transformation, and later the expansion into the &quot;fictitious&quot; electromagnetic system with the aid of the Lorentz transformation. In order to explain the negative result of the [[w:Michelson–Morley experiment]], he (1892b)&lt;ref group=R&gt;Lorentz (1892b), p. 141&lt;/ref&gt; introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced [[w:length contraction]] in his theory (without proof as he admitted). The same hypothesis was already made by [[w:George FitzGerald]] in 1889 based on Heaviside's work. While length contraction was a real physical effect for Lorentz, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation.

In 1895, Lorentz further elaborated on his theory and introduced the &quot;theorem of corresponding states&quot;. This theorem states that a moving observer (relative to the ether) in his &quot;fictitious&quot; field makes the same observations as a resting observers in his &quot;real&quot; field for velocities to first order in ''v/c''. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:&lt;ref group=R&gt;Lorentz (1895), p. 37&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =x^{\prime}\sqrt{1-\frac{\mathfrak{p}^{2}}{V^{2}}}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{x^{\prime}}{\gamma}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\end{matrix}&lt;/math&gt;

For solving optical problems Lorentz used the following transformation, in which the modified time variable was called &quot;local time&quot; ({{lang-de|Ortszeit}}) by him:&lt;ref group=R&gt;Lorentz (1895), p. 49 for local time and p. 56 for spatial coordinates.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\mathrm{x}-\mathfrak{p}_{x}t\\
y &amp; =\mathrm{y}-\mathfrak{p}_{y}t\\
z &amp; =\mathrm{z}-\mathfrak{p}_{z}t\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}}x-\frac{\mathfrak{p}_{y}}{V^{2}}y-\frac{\mathfrak{p}_{z}}{V^{2}}z
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-v_{x}t\\
y^{\prime} &amp; =y-v_{y}t\\
z^{\prime} &amp; =z-v_{z}t\\
t^{\prime} &amp; =t-\frac{v_{x}}{c^{2}}x'-\frac{v_{y}}{c^{2}}y'-\frac{v_{z}}{c^{2}}z'
\end{align}
\end{matrix}&lt;/math&gt;

With this concept Lorentz could explain the [[w:Doppler effect]], the [[w:aberration of light]], and the [[w:Fizeau experiment]].&lt;ref&gt;Janssen (1995), 3.1&lt;/ref&gt;

=== {{anchor|Larmor}} Larmor (1897, 1900) ===

In 1897, Larmor extended the work of Lorentz and derived the following transformation&lt;ref group=R&gt;Larmor (1897), p. 229&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =x\varepsilon^{\frac{1}{2}}\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-vx/c^{2}\\
dt_{1} &amp; =dt^{\prime}\varepsilon^{-\frac{1}{2}}\\
\varepsilon &amp; =\left(1-v^{2}/c^{2}\right)^{-1}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-\frac{vx^{\ast}}{c^{2}}=t-\frac{v(x-vt)}{c^{2}}\\
dt_{1} &amp; =\frac{dt^{\prime}}{\gamma}\\
\gamma^{2} &amp; =\frac{1}{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

Larmor noted that if it is assumed that the constitution of molecules is electrical then the FitzGerald–Lorentz contraction is a consequence of this transformation, explaining the [[w:Michelson–Morley experiment]]. It's notable that Larmor was the first who recognized that some sort of [[w:time dilation]] is a consequence of this transformation as well, because &quot;individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 1/γ&quot;.&lt;ref&gt;Darrigol (2000), Chap. 8.5&lt;/ref&gt;&lt;ref&gt;Macrossan (1986)&lt;/ref&gt; Larmor wrote his electrodynamical equations and transformations neglecting terms of higher order than ''(v/c)''&lt;sup&gt;2&lt;/sup&gt; – when his 1897 paper was reprinted in 1929, Larmor added the following comment in which he described how they can be made valid to all orders of ''v/c'':&lt;ref group=R&gt;Larmor (1897/1929), p. 39&lt;/ref&gt;

{{Quote|Nothing need be neglected: the transformation is ''exact'' if ''v/c''&lt;sup&gt;2&lt;/sup&gt; is replaced by ''εv/c''&lt;sup&gt;2&lt;/sup&gt; in the equations and also in the change following from ''t'' to ''t′'', as is worked out in ''Aether and Matter'' (1900), p. 168, and as Lorentz found it to be in 1904, thereby stimulating the modern schemes of intrinsic relational relativity.}}

In line with that comment, in his book Aether and Matter published in 1900, Larmor used a modified local time ''t″=t′-εvx′/c&lt;sup&gt;2&lt;/sup&gt;'' instead of the 1897 expression ''t′=t-vx/c&lt;sup&gt;2&lt;/sup&gt;'' by replacing ''v/c''&lt;sup&gt;2&lt;/sup&gt; with ''εv/c''&lt;sup&gt;2&lt;/sup&gt;, so that ''t″'' is now identical to the one given by Lorentz in 1892, which he combined with a Galilean transformation for the ''x′, y′, z′, t′'' coordinates:&lt;ref group=R&gt;Larmor (1900), p. 168&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime} &amp; =t^{\prime}-\varepsilon vx^{\prime}/c^{2}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime}=t^{\prime}-\frac{\gamma^{2}vx^{\prime}}{c^{2}} &amp; =\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor ''(v/c)''&lt;sup&gt;2&lt;/sup&gt;, and so he sought the transformations which were &quot;accurate to second order&quot; (as he put it). Thus he wrote the final transformations (where ''x′=x-vt'' and ''t″'' as given above) as:&lt;ref group=R&gt;Larmor (1900), p. 174&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =\varepsilon^{\frac{1}{2}}x^{\prime}\\
y_{1} &amp; =y^{\prime}\\
z_{1} &amp; =z^{\prime}\\
dt_{1} &amp; =\varepsilon^{-\frac{1}{2}}dt^{\prime\prime}=\varepsilon^{-\frac{1}{2}}\left(dt^{\prime}-\frac{v}{c^{2}}\varepsilon dx^{\prime}\right)\\
t_{1} &amp; =\varepsilon^{-\frac{1}{2}}t^{\prime}-\frac{v}{c^{2}}\varepsilon^{\frac{1}{2}}x^{\prime}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\prime}=\gamma(x-vt)\\
y_{1} &amp; =y'=y\\
z_{1} &amp; =z'=z\\
dt_{1} &amp; =\frac{dt^{\prime\prime}}{\gamma}=\frac{1}{\gamma}\left(dt^{\prime}-\frac{\gamma^{2}vdx^{\prime}}{c^{2}}\right)=\gamma\left(dt-\frac{vdx}{c^{2}}\right)\\
t_{1} &amp; =\frac{t^{\prime}}{\gamma}-\frac{\gamma vx^{\prime}}{c^{2}}=\gamma\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

by which he arrived at the complete Lorentz transformation ({{equationNote|4b}}). Larmor showed that Maxwell's equations were invariant under this two-step transformation, &quot;to second order in ''v/c''&quot; – it was later shown by Lorentz (1904) and Poincaré (1905) that they are indeed invariant under this transformation to all orders in ''v/c''.

Larmor gave credit to Lorentz in two papers published in 1904, in which he used the term &quot;Lorentz transformation&quot; for Lorentz's first order transformations of coordinates and field configurations:

{{Quote|p. 583: [..] Lorentz's transformation for passing from the field of activity of a stationary electrodynamic material system to that of one moving with uniform velocity of translation through the aether.&lt;br /&gt; p. 585: [..] the Lorentz transformation has shown us what is not so immediately obvious [..]&lt;ref group=R&gt;Larmor (1904a), p. 583, 585&lt;/ref&gt; &lt;br /&gt; p. 622: [..] the transformation first developed by Lorentz: namely, each point in space is to have its own origin from which time is measured, its &quot;local time&quot; in Lorentz's phraseology, and then the values of the electric and magnetic vectors [..] at all points in the aether between the molecules in the system at rest, are the same as those of the vectors [..] at the corresponding points in the convected system at the same local times.&lt;ref group=R&gt;Larmor (1904b), p. 622&lt;/ref&gt;}}

=== {{anchor|Lorentz2}} Lorentz (1899, 1904) ===

Also Lorentz extended his theorem of corresponding states in 1899. First he wrote a transformation equivalent to the one from 1892 (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1899), p. 429&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}x\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}-\mathfrak{p}_{x}^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Then he introduced a factor ε of which he said he has no means of determining it, and modified his transformation as follows (where the above value of ''t′'' has to be inserted):&lt;ref group=R&gt;Lorentz (1899), p. 439&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\frac{\varepsilon}{k}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon x^{\prime\prime}\\
t^{\prime} &amp; =k\varepsilon t^{\prime\prime}\\
k &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{\varepsilon}{\gamma}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon z^{\prime\prime}\\
t^{\prime}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right) &amp; =\gamma\varepsilon t^{\prime\prime}\\
\gamma &amp; =\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
\end{align}
\end{matrix}&lt;/math&gt;

This is equivalent to the complete Lorentz transformation ({{equationNote|4b}}) when solved for ''x″'' and ''t″'' and with ε=1. Like Larmor, Lorentz noticed in 1899&lt;ref group=R&gt;Lorentz (1899), p. 442&lt;/ref&gt; also some sort of time dilation effect in relation to the frequency of oscillating electrons ''&quot;that in ''S'' the time of vibrations be ''kε'' times as great as in ''S&lt;sub&gt;0&lt;/sub&gt;''&quot;'', where ''S&lt;sub&gt;0&lt;/sub&gt;'' is the aether frame.&lt;ref&gt;Janssen (1995), Kap. 3.3&lt;/ref&gt;

In 1904 he rewrote the equations in the following form by setting ''l''=1/ε (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1904), p. 812&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =klx\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =\frac{l}{k}t-kl\frac{w}{c^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma lx^{\ast}=\gamma l(x-vt)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t^{\prime} &amp; =\frac{lt}{\gamma}-\frac{\gamma lvx^{\ast}}{c^{2}}=\gamma l\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Under the assumption that ''l=1'' when ''v''=0, he demonstrated that ''l=1'' must be the case at all velocities, therefore length contraction can only arise in the line of motion. So by setting the factor ''l'' to unity, Lorentz's transformations now assumed the same form as Larmor's and are now completed. Unlike Larmor, who restricted himself to show the covariance of Maxwell's equations to second order, Lorentz tried to widen its covariance to all orders in ''v/c''. He also derived the correct formulas for the velocity dependence of [[w:electromagnetic mass]], and concluded that the transformation formulas must apply to all forces of nature, not only electrical ones.&lt;ref group=R&gt;Lorentz (1904), p. 826&lt;/ref&gt; However, he didn't achieve full covariance of the transformation equations for charge density and velocity.&lt;ref&gt;Miller (1981), Chap. 1.12.2&lt;/ref&gt; When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:&lt;ref&gt;Janssen (1995), Chap. 3.5.6&lt;/ref&gt;

{{Quote|One will notice that in this work the transformation equations of Einstein’s Relativity Theory have not quite been attained. [..] On this circumstance depends the clumsiness of many of the further considerations in this work.}}

Lorentz's 1904 transformation was cited and used by [[w:Alfred Bucherer]] in July 1904:&lt;ref group=R&gt;Bucherer, p. 129; Definition of s on p. 32&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\sqrt{s}x,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{\sqrt{s}}-\sqrt{s}\frac{u}{v^{2}}x,\quad s=1-\frac{u^{2}}{v^{2}}&lt;/math&gt;

or by [[w:Wilhelm Wien]] in July 1904:&lt;ref group=R&gt;Wien (1904), p. 394&lt;/ref&gt;

:&lt;math&gt;x=kx',\quad y=y',\quad z=z',\quad t'=kt-\frac{v}{kc^{2}}x&lt;/math&gt;

or by [[w:Emil Cohn]] in November 1904 (setting the speed of light to unity):&lt;ref group=R&gt;Cohn (1904a), pp. 1296-1297&lt;/ref&gt;

:&lt;math&gt;x=\frac{x_{0}}{k},\quad y=y_{0},\quad z=z_{0},\quad t=kt_{0},\quad t_{1}=t_{0}-w\cdot r_{0},\quad k^{2}=\frac{1}{1-w^{2}}&lt;/math&gt;

or by [[w:Richard Gans]] in February 1905:&lt;ref group=R&gt;Gans (1905), p. 169&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=kx,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{k}-\frac{kwx}{c^{2}},\quad k^{2}=\frac{c^{2}}{c^{2}-w^{2}}&lt;/math&gt;

=== {{anchor|Poincare3}} Poincaré (1900, 1905) ===

==== Local time ====

Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time. However, [[w:Henri Poincaré]] in 1900 commented on the origin of Lorentz's &quot;wonderful invention&quot; of local time.&lt;ref&gt;Darrigol (2005), Kap. 4&lt;/ref&gt; He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed &lt;math&gt;c&lt;/math&gt; in both directions, which lead to what is nowadays called [[w:relativity of simultaneity]], although Poincaré's calculation does not involve length contraction or time dilation.&lt;ref group=R&gt;Poincaré (1900), pp. 272–273&lt;/ref&gt; In order to synchronise the clocks here on Earth (the ''x*, t''* frame) a light signal from one clock (at the origin) is sent to another (at ''x''*), and is sent back. It's supposed that the Earth is moving with speed ''v'' in the ''x''-direction (= ''x''*-direction) in some rest system (''x, t'') (''i.e.'' the [[w:luminiferous aether]] system for Lorentz and Larmor). The time of flight outwards is

:&lt;math&gt;\delta t_{a}=\frac{x^{\ast}}{\left(c-v\right)}&lt;/math&gt;

and the time of flight back is

:&lt;math&gt;\delta t_{b}=\frac{x^{\ast}}{\left(c+v\right)}&lt;/math&gt;.

The elapsed time on the clock when the signal is returned is ''δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;'' and the time ''t*=(δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;)/2'' is ascribed to the moment when the light signal reached the distant clock. In the rest frame the time ''t=δt&lt;sub&gt;a&lt;/sub&gt;'' is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus

:&lt;math&gt;t^{\ast}=t-\frac{\gamma^{2}vx^{*}}{c^{2}}&lt;/math&gt;

identical to Lorentz (1892). By dropping the factor γ&lt;sup&gt;2&lt;/sup&gt; under the assumption that &lt;math&gt;\tfrac{v^{2}}{c^{2}}\ll1&lt;/math&gt;, Poincaré gave the result ''t*=t-vx*/c&lt;sup&gt;2&lt;/sup&gt;'', which is the form used by Lorentz in 1895.

Similar physical interpretations of local time were later given by [[w:Emil Cohn]] (1904)&lt;ref group=R&gt;Cohn (1904b), p. 1408&lt;/ref&gt; and [[w:Max Abraham]] (1905).&lt;ref group=R&gt;Abraham (1905), § 42&lt;/ref&gt;

==== Lorentz transformation ====

On June 5, 1905 (published June 9) Poincaré formulated transformation equations which are algebraically equivalent to those of Larmor and Lorentz and gave them the modern form ({{equationNote|4b}}):&lt;ref group=R&gt;Poincaré (1905), p. 1505&lt;/ref&gt;

:&lt;math&gt;\begin{align}x^{\prime} &amp; =kl(x+\varepsilon t)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =kl(t+\varepsilon x)\\
k &amp; =\frac{1}{\sqrt{1-\varepsilon^{2}}}
\end{align}
&lt;/math&gt;.

Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name &quot;Lorentz transformation&quot;.&lt;ref&gt;Pais (1982), Chap. 6c&lt;/ref&gt;&lt;ref&gt;Katzir (2005), 280–288&lt;/ref&gt; Poincaré set the speed of light to unity, pointed out the group characteristics of the transformation by setting ''l''=1, and modified/corrected Lorentz's derivation of the equations of electrodynamics in some details in order to fully satisfy the principle of relativity, ''i.e.'' making them fully Lorentz covariant.&lt;ref&gt;Miller (1981), Chap. 1.14&lt;/ref&gt;

In July 1905 (published in January 1906)&lt;ref group=R&gt;Poincaré (1905/06), pp. 129ff&lt;/ref&gt; Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of the [[w:principle of least action]]; he demonstrated in more detail the group characteristics of the transformation, which he called [[w:Lorentz group]], and he showed that the combination ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;+z&lt;sup&gt;2&lt;/sup&gt;-t&lt;sup&gt;2&lt;/sup&gt;'' is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing &lt;math&gt;ct\sqrt{-1}&lt;/math&gt; as a fourth imaginary coordinate, and he used an early form of [[w:four-vector]]s. He also formulated the velocity addition formula ({{equationNote|4c}}), which he had already derived in unpublished letters to Lorentz from May 1905:&lt;ref group=R&gt;Poincaré (1905/06), p. 144&lt;/ref&gt;

:&lt;math&gt;\xi'=\frac{\xi+\varepsilon}{1+\xi\varepsilon},\ \eta'=\frac{\eta}{k(1+\xi\varepsilon)}&lt;/math&gt;.

==={{anchor|Einstein}} Einstein (1905) – Special relativity===

On June 30, 1905 (published September 1905) Einstein published what is now called [[w:special relativity]] and gave a new derivation of the transformation, which was based only on the principle on relativity and the principle of the constancy of the speed of light. While Lorentz considered &quot;local time&quot; to be a mathematical stipulation device for explaining the Michelson-Morley experiment, Einstein showed that the coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. For quantities of first order in ''v/c'' this was also done by Poincaré in 1900, while Einstein derived the complete transformation by this method. Unlike Lorentz and Poincaré who still distinguished between real time in the aether and apparent time for moving observers, Einstein showed that the transformations concern the nature of space and time.&lt;ref&gt;Miller (1981), Chap. 6&lt;/ref&gt;&lt;ref&gt;Pais (1982), Kap. 7&lt;/ref&gt;&lt;ref&gt;Darrigol (2005), Chap. 6&lt;/ref&gt;

The notation for this transformation is equivalent to Poincaré's of 1905 and ({{equationNote|4b}}), except that Einstein didn't set the speed of light to unity:&lt;ref group=R&gt;Einstein (1905), p. 902&lt;/ref&gt;

:&lt;math&gt;\begin{align}\tau &amp; =\beta\left(t-\frac{v}{V^{2}}x\right)\\
\xi &amp; =\beta(x-vt)\\
\eta &amp; =y\\
\zeta &amp; =z\\
\beta &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}
\end{align}
&lt;/math&gt;

Einstein also defined the velocity addition formula ({{equationNote|4c}}, {{equationNote|4d}}):&lt;ref group=R&gt;Einstein (1905), § 5 and § 9&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=\frac{w_{\xi}+v}{1+\frac{vw_{\xi}}{V^{2}}}t,\ y=\frac{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}{1+\frac{vw_{\xi}}{V^{2}}}w_{\eta}t\\
U^{2}=\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2},\ w^{2}=w_{\xi}^{2}+w_{\eta}^{2},\ \alpha=\operatorname{arctg}\frac{w_{y}}{w_{x}}\\
U=\frac{\sqrt{\left(v^{2}+w^{2}+2vw\cos\alpha\right)-\left(\frac{vw\sin\alpha}{V}\right)^{2}}}{1+\frac{vw\cos\alpha}{V^{2}}}
\end{matrix}\left|\begin{matrix}\frac{u_{x}-v}{1-\frac{u_{x}v}{V^{2}}}=u_{\xi}\\
\frac{u_{y}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\eta}\\
\frac{u_{z}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\zeta}
\end{matrix}\right.&lt;/math&gt;

and the light aberration formula ({{equationNote|4e}}):&lt;ref group=R&gt;Einstein (1905), § 7&lt;/ref&gt;

:&lt;math&gt;\cos\varphi'=\frac{\cos\varphi-\frac{v}{V}}{1-\frac{v}{V}\cos\varphi}&lt;/math&gt;

=== {{anchor|Minkowski}} Minkowski (1907–1908) – Spacetime ===

The work on the principle of relativity by Lorentz, Einstein, [[w:Max Planck|Planck]], together with Poincaré's four-dimensional approach, were further elaborated and combined with the [[w:hyperboloid model]] by [[w:Hermann Minkowski]] in 1907 and 1908.&lt;ref group=R&gt;Minkowski (1907/15), pp. 927ff&lt;/ref&gt;&lt;ref group=R&gt;Minkowski (1907/08), pp. 53ff&lt;/ref&gt; Minkowski particularly reformulated electrodynamics in a four-dimensional way ([[w:Minkowski spacetime]]).&lt;ref&gt;Walter (1999a), (1999b), (2018)&lt;/ref&gt; For instance, he wrote ''x, y, z, it'' in the form ''x&lt;sub&gt;1&lt;/sub&gt;, x&lt;sub&gt;2&lt;/sub&gt;, x&lt;sub&gt;3&lt;/sub&gt;, x&lt;sub&gt;4&lt;/sub&gt;''. By defining ψ as the angle of rotation around the ''z''-axis, the Lorentz transformation assumes a form (with ''c''=1):&lt;ref group=R name=mink1&gt;Minkowski (1907/08), p. 59&lt;/ref&gt;

:&lt;math&gt;\begin{align}x'_{1} &amp; =x_{1}\\
x'_{2} &amp; =x_{2}\\
x'_{3} &amp; =x_{3}\cos i\psi+x_{4}\sin i\psi\\
x'_{4} &amp; =-x_{3}\sin i\psi+x_{4}\cos i\psi\\
\cos i\psi &amp; =\frac{1}{\sqrt{1-q^{2}}}
\end{align}
&lt;/math&gt;

Even though Minkowski used the imaginary number iψ, he for once&lt;ref group=R name=mink1 /&gt; directly used the [[w:tangens hyperbolicus]] in the equation for velocity

:&lt;math&gt;-i\tan i\psi=\frac{e^{\psi}-e^{-\psi}}{e^{\psi}+e^{-\psi}}=q&lt;/math&gt; with &lt;math&gt;\psi=\frac{1}{2}\ln\frac{1+q}{1-q}&lt;/math&gt;.

Minkowski's expression can also by written as ψ=atanh(q) and was later called [[w:rapidity]]. He also wrote the Lorentz transformation in matrix form:&lt;ref group=R&gt;Minkowski (1907/08), pp. 65–66, 81–82&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}+x_{4}^{\prime2}\\
\left(x_{1}^{\prime}=x',\ x_{2}^{\prime}=y',\ x_{3}^{\prime}=z',\ x_{4}^{\prime}=it'\right)\\
-x^{2}-y^{2}-z^{2}+t^{2}=-x^{\prime2}-y^{\prime2}-z^{\prime2}+t^{\prime2}\\
\hline x_{h}=\alpha_{h1}x_{1}^{\prime}+\alpha_{h2}x_{2}^{\prime}+\alpha_{h3}x_{3}^{\prime}+\alpha_{h4}x_{4}^{\prime}\\
\mathrm{A}=\mathrm{\left|\begin{matrix}\alpha_{11}, &amp; \alpha_{12}, &amp; \alpha_{13}, &amp; \alpha_{14}\\
\alpha_{21}, &amp; \alpha_{22}, &amp; \alpha_{23}, &amp; \alpha_{24}\\
\alpha_{31}, &amp; \alpha_{32}, &amp; \alpha_{33}, &amp; \alpha_{34}\\
\alpha_{41}, &amp; \alpha_{42}, &amp; \alpha_{43}, &amp; \alpha_{44}
\end{matrix}\right|,\ \begin{align}\bar{\mathrm{A}}\mathrm{A} &amp; =1\\
\left(\det \mathrm{A}\right)^{2} &amp; =1\\
\det \mathrm{A} &amp; =1\\
\alpha_{44} &amp; &gt;0
\end{align}
}
\end{matrix}&lt;/math&gt;

As a graphical representation of the Lorentz transformation he introduced the [[w:Minkowski diagram]], which became a standard tool in textbooks and research articles on relativity:&lt;ref group=R&gt;Minkowski (1908/09), p. 77&lt;/ref&gt;

[[File:Minkowski1.png|center|thumb|400px|Original spacetime diagram by Minkowski in 1908.]]

==={{anchor|Frank}} Frank (1909) – Hyperbolic functions===
Hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity in agreement with ({{equationNote|4b}}):&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

==={{Anchor|Sommerfeld}} Sommerfeld (1909) – Spherical trigonometry===

Using an imaginary rapidity such as Minkowski, [[w:Arnold Sommerfeld]] (1909) formulated a transformation equivalent to Lorentz boost ({{equationNote|4b}}), and the relativistc velocity addition ({{equationNote|4c}}) in terms of trigonometric functions and the [[w:spherical law of cosines]]:&lt;ref group=R&gt;Sommerfeld (1909), p. 826ff.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{array}{lrl}
x'= &amp; x\ \cos\varphi+l\ \sin\varphi, &amp; y'=y\\
l'= &amp; -x\ \sin\varphi+l\ \cos\varphi, &amp; z'=z
\end{array}\right\} \\
\left(\operatorname{tg}\varphi=i\beta,\ \cos\varphi=\frac{1}{\sqrt{1-\beta^{2}}},\ \sin\varphi=\frac{i\beta}{\sqrt{1-\beta^{2}}}\right)\\
\hline \beta=\frac{1}{i}\operatorname{tg}\left(\varphi_{1}+\varphi_{2}\right)=\frac{1}{i}\frac{\operatorname{tg}\varphi_{1}+\operatorname{tg}\varphi_{2}}{1-\operatorname{tg}\varphi_{1}\operatorname{tg}\varphi_{2}}=\frac{\beta_{1}+\beta_{2}}{1+\beta_{1}\beta_{2}}\\
\cos\varphi=\cos\varphi_{1}\cos\varphi_{2}-\sin\varphi_{1}\sin\varphi_{2}\cos\alpha\\
v^{2}=\frac{v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos\alpha-\frac{1}{c^{2}}v_{1}^{2}v_{2}^{2}\sin^{2}\alpha}{\left(1+\frac{1}{c^{2}}v_{1}v_{2}\cos\alpha\right)^{2}}
\end{matrix}&lt;/math&gt;

=== {{anchor|Herglotz1}} Herglotz (1909/10) – Möbius transformation ===

Following [[../Lorentz transformation (Möbius)#Klein2|E:Klein (1889–1897)]] as well as [[../Lorentz transformation (Möbius)#Fricke|E:Fricke &amp; Klein (1897)]] concerning the Cayley absolute, hyperbolic motion and its transformation, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic. The general case (on the left) equivalent to Lorentz-Möbius transformation [[../Lorentz transformation (Möbius)#math_6a|E:'''(6a)''']] and the hyperbolic case (on the right) equivalent to Lorentz transformation [[../Lorentz transformation (hyperbolic)#math_3d|E:'''(3d)''']] or squeeze mapping [[../Lorentz transformation (squeeze)#math_9d|E:'''(9d)''']] are as follows:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;

:&lt;math&gt;\left.\begin{matrix}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\\
z_{1}=x,\ z_{2}=y,\ z_{3}=z,\ z_{4}=t\\
Z=\frac{z_{1}+iz_{2}}{z_{4}-z_{3}}=\frac{x+iy}{t-z},\ Z'=\frac{x'+iy'}{t'-z'}\\
Z=\frac{\alpha Z'+\beta}{\gamma Z'+\delta}
\end{matrix}\right|\begin{matrix}Z=Z'e^{\vartheta}\\
\begin{align}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
\end{align}
\end{matrix}&lt;/math&gt;

==={{anchor|Varicak}} Varićak (1910) – Hyperbolic functions===

Following [[#Sommerfeld|Sommerfeld (1909)]], hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

and showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

Subsequently, other authors such as [[w:E. T. Whittaker]] (1910) or [[w:Alfred Robb]] (1911, who coined the name rapidity) used similar expressions, which are still used in modern textbooks.&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

=== {{anchor|Ignatowski}} Ignatowski (1910) ===

While earlier derivations and formulations of the Lorentz transformation relied from the outset on optics, electrodynamics, or the invariance of the speed of light, [[w:Vladimir Ignatowski]] (1910) showed that it is possible to use the principle of relativity (and related [[w:Group theory|group theoretical]] principles) alone, in order to derive the following transformation between two inertial frames:&lt;ref group=R&gt;Ignatowski (1910), pp. 973–974&lt;/ref&gt;&lt;ref group=R&gt;Ignatowski (1910/11ab)&lt;/ref&gt;

:&lt;math&gt;\begin{align}dx' &amp; =p\ dx-pq\ dt\\
dt' &amp; =-pqn\ dx+p\ dt\\
p &amp; =\frac{1}{\sqrt{1-q^{2}n}}
\end{align}
&lt;/math&gt;

The variable ''n'' can be seen as a space-time constant whose value has to be determined by experiment or taken from a known physical law such as electrodynamics. For that purpose, Ignatowski used the above-mentioned Heaviside ellipsoid representing a contraction of electrostatic fields by ''x''/γ in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation when ''n=1/c''&lt;sup&gt;2&lt;/sup&gt;, resulting in ''p''=γ and the Lorentz transformation ({{equationNote|4b}}). With ''n''=0, no length changes arise and the Galilean transformation follows. Ignatowski's method was further developed and improved by [[w:Philipp Frank]] and [[w:Hermann Rothe]] (1911, 1912),&lt;ref group=R&gt;Frank &amp; Rothe (1911), pp. 825ff; (1912), p. 750ff.&lt;/ref&gt; with various authors developing similar methods in subsequent years.&lt;ref name=baccetti&gt;Baccetti (2011), see references 1–25 therein.&lt;/ref&gt;

==={{anchor|Noether}} Noether (1910), Klein (1910) – Quaternions===

[[w:Felix Klein]] (1908) described [[../Lorentz transformation (Quaternions)#Cayley3|E:Cayley's (1854)]] 4D quaternion multiplications as &quot;Drehstreckungen&quot; (orthogonal substitutions in terms of rotations leaving invariant a quadratic form up to a factor), and pointed out that the modern principle of relativity as provided by Minkowski is essentially only the consequent application of such Drehstreckungen, even though he didn't provide details.&lt;ref group=R&gt;Klein (1908), p. 165&lt;/ref&gt;

In an appendix to Klein's and Sommerfeld's &quot;Theory of the top&quot; (1910), [[w:Fritz Noether]] showed how to formulate hyperbolic rotations using biquaternions with &lt;math&gt;\omega=\sqrt{-1}&lt;/math&gt;, which he also related to the speed of light by setting ω&lt;sup&gt;2&lt;/sup&gt;=-''c''&lt;sup&gt;2&lt;/sup&gt;. He concluded that this is the principal ingredient for a rational representation of the group of Lorentz transformations:&lt;ref group=R&gt;Noether (1910), pp. 939–943&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}V=\frac{Q_{1}vQ_{2}}{T_{1}T_{2}}\\
\hline X^{2}+Y^{2}+Z^{2}+\omega^{2}S^{2}=x^{2}+y^{2}+z^{2}+\omega^{2}s^{2}\\
\hline \begin{align}V &amp; =Xi+Yj+Zk+\omega S\\
v &amp; =xi+yj+zk+\omega s\\
Q_{1} &amp; =(+Ai+Bj+Ck+D)+\omega(A'i+B'j+C'k+D')\\
Q_{2} &amp; =(-Ai-Bj-Ck+D)+\omega(A'i+B'j+C'k-D')\\
T_{1}T_{2} &amp; =T_{1}^{2}=T_{2}^{2}=A^{2}+B^{2}+C^{2}+D^{2}+\omega^{2}\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Besides citing quaternion related standard works such as [[#Cayley3|Cayley (1854)]], Noether referred to the entries in Klein's encyclopedia by [[w:Eduard Study]] (1899) and the French version by [[w:Élie Cartan]] (1908).&lt;ref&gt;Cartan &amp; Study (1908), sections 35–36&lt;/ref&gt; Cartan's version contains a description of Study's [[w:dual number]]s, Clifford's biquaternions (including the choice &lt;math&gt;\omega=\sqrt{-1}&lt;/math&gt; for hyperbolic geometry), and Clifford algebra, with references to [[../Lorentz transformation (Quaternions)#Stephanos|Stephanos (1883)]], [[../Lorentz transformation (Quaternions)#Buchheim|Buchheim (1884/85)]], [[../Lorentz transformation (Quaternions)#Vahlen|Vahlen (1901/02)]] and others.

Citing Noether, Klein himself published in August 1910 the following quaternion substitutions forming the group of Lorentz transformations:&lt;ref group=R&gt;Klein (1910), p. 300&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align} &amp; \left(i_{1}x'+i_{2}y'+i_{3}z'+ict'\right)\\
 &amp; \quad-\left(i_{1}x_{0}+i_{2}y_{0}+i_{3}z_{0}+ict_{0}\right)
\end{align}
=\frac{\left[\begin{align} &amp; \left(i_{1}(A+iA')+i_{2}(B+iB')+i_{3}(C+iC')+i_{4}(D+iD')\right)\\
 &amp; \quad\cdot\left(i_{1}x+i_{2}y+i_{3}z+ict\right)\\
 &amp; \quad\quad\cdot\left(i_{1}(A-iA')+i_{2}(B-iB')+i_{3}(C-iC')-(D-iD')\right)
\end{align}
\right]}{\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)-\left(A^{2}+B^{2}+C^{2}+D^{2}\right)}\\
\hline \text{where}\\
AA'+BB'+CC'+DD'=0\\
A^{2}+B^{2}+C^{2}+D^{2}&gt;A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}
\end{matrix}&lt;/math&gt;

or in March 1911&lt;ref group=R&gt;Klein (1911), pp. 602ff.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}g'=\frac{pg\pi}{M}\\
\hline \begin{align}g &amp; =\sqrt{-1}ct+ix+jy+kz\\
g' &amp; =\sqrt{-1}ct'+ix'+jy'+kz'\\
p &amp; =(D+\sqrt{-1}D')+i(A+\sqrt{-1}A')+j(B+\sqrt{-1}B')+k(C+\sqrt{-1}C')\\
\pi &amp; =(D-\sqrt{-1}D')-i(A-\sqrt{-1}A')-j(B-\sqrt{-1}B')-k(C-\sqrt{-1}C')\\
M &amp; =\left(A^{2}+B^{2}+C^{2}+D^{2}\right)-\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)\\
 &amp; AA'+BB'+CC'+DD'=0\\
 &amp; A^{2}+B^{2}+C^{2}+D^{2}&gt;A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}
\end{align}
\end{matrix}&lt;/math&gt;

==={{anchor|Conway}} Conway (1911), Silberstein (1911) – Quaternions===

[[w:Arthur W. Conway]] in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms of velocity λ:&lt;ref group=R&gt;Conway (1911), p. 8&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}\mathtt{D} &amp; =\mathbf{a}^{-1}\mathtt{D}'\mathbf{a}^{-1}\\
\mathtt{\sigma} &amp; =\mathbf{a}\mathtt{\sigma}'\mathbf{a}^{-1}
\end{align}
\\
e=\mathbf{a}^{-1}e'\mathbf{a}^{-1}\\
\hline a=\left(1-hc^{-1}\lambda\right)^{\frac{1}{2}}\left(1+c^{-2}\lambda^{2}\right)^{-\frac{1}{4}}
\end{matrix}&lt;/math&gt;

Also [[w:Ludwik Silberstein]] in November 1911&lt;ref group=R&gt;Silberstein (1911/12), p. 793&lt;/ref&gt; as well as in 1914,&lt;ref&gt;Silberstein (1914), p. 156&lt;/ref&gt; formulated the Lorentz transformation in terms of velocity ''v'':

:&lt;math&gt;\begin{matrix}q'=QqQ\\
\hline \begin{align}q &amp; =\mathbf{r}+l=xi+yj+zk+\iota ct\\
q &amp; '=\mathbf{r}'+l'=x'i+y'j+z'k+\iota ct'\\
Q &amp; =\frac{1}{\sqrt{2}}\left(\sqrt{1+\gamma}+\mathrm{u}\sqrt{1-\gamma}\right)\\
 &amp; =\cos\alpha+\mathrm{u}\sin\alpha=e^{\alpha\mathrm{u}}\\
 &amp; \left\{ \gamma=\left(1-v^{2}/c^{2}\right)^{-1/2},\ 2\alpha=\operatorname{arctg}\ \left(\iota\frac{v}{c}\right)\right\} 
\end{align}
\end{matrix}&lt;/math&gt;

Silberstein cites [[../Lorentz transformation (Quaternions)#Cayley3|E:Cayley (1854, 1855)]] and Study's encyclopedia entry (in the extended French version of Cartan in 1908), as well as the appendix of Klein's and Sommerfeld's book.

==={{anchor|Herglotz2}} Ignatowski (1910/11), Herglotz (1911), and others – Vector transformation===

{{See|w:Lorentz transformation#Vector transformations}}

[[w:Vladimir Ignatowski]] (1910, published 1911) defined the vectorial velocity addition ({{equationNote|4g}}) as well as general Lorentz boost ({{equationNote|4f}}) as&lt;ref group=R&gt;Ignatowski (1910/11a), p. 23; (1910/11b), p. 22&lt;/ref&gt; 

:&lt;math&gt;\begin{matrix}\begin{matrix}\mathfrak{v} =\frac{\mathfrak{v}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{v}'+pq\mathfrak{c}_{0}}{p\left(1+nq\mathfrak{c}_{0}\mathfrak{v}'\right)} &amp; \left|\begin{align}\mathfrak{A}' &amp; =\mathfrak{A}+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}-pqb\mathfrak{c}_{0}\\
b' &amp; =pb-pqn\mathfrak{A}\mathfrak{c}_{0}\\
\\
\mathfrak{A} &amp; =\mathfrak{A}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}'+pqb'\mathfrak{c}_{0}\\
b &amp; =pb'+pqn\mathfrak{A}'\mathfrak{c}_{0}
\end{align}
\right.\end{matrix}\\
\left[\mathfrak{v}=\mathbf{u},\ \mathfrak{A}=\mathbf{x},\ b=t,\ \mathfrak{c}_{0}=\frac{\mathbf{v}}{v},\ p=\gamma,\ n=\frac{1}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

An equivalent transformation was given by [[w:Gustav Herglotz]] (1911)&lt;ref group=R&gt;Herglotz (1911), p. 497&lt;/ref&gt; using '''v'''=''(v&lt;sub&gt;x&lt;/sub&gt;, v&lt;sub&gt;y&lt;/sub&gt;, v&lt;sub&gt;z&lt;/sub&gt;)'' and '''r'''=''(x, y, z)'':

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{0} &amp; =x+\alpha u(ux+vy+wz)-\beta ut\\
y^{0} &amp; =y+\alpha v(ux+vy+wz)-\beta vt\\
z^{0} &amp; =z+\alpha w(ux+vy+wz)-\beta wt\\
t^{0} &amp; =-\beta(ux+vy+wz)+\beta t\\
 &amp; \alpha=\frac{1}{\sqrt{1-s^{2}}\left(1+\sqrt{1-s^{2}}\right)},\ \beta=\frac{1}{\sqrt{1-s^{2}}}
\end{align}
\right| &amp; \begin{align}x' &amp; =x+\alpha v_{x}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{x}t\\
y' &amp; =y+\alpha v_{y}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{y}t\\
z' &amp; =z+\alpha v_{z}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{z}t\\
t' &amp; =-\gamma\left(v_{x}x+v_{y}y+v_{z}z\right)+\gamma t\\
 &amp; \alpha=\frac{\gamma^{2}}{\gamma+1},\ \gamma=\frac{1}{\sqrt{1-v^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

Or by [[w:Ludwik Silberstein]] (1911 on the left, 1914 on the right):&lt;ref group=R&gt;Silberstein (1911/12), p. 792; (1914), p. 123&lt;/ref&gt;

:&lt;math&gt;\begin{array}{c|c}
\begin{align}\mathbf{r}' &amp; =\mathbf{r}+(\gamma-1)(\mathbf{ru})\mathbf{u}+i\beta\gamma lu\\
l' &amp; =\gamma\left[l-i\beta(\mathbf{ru})\right]
\end{align}
 &amp; \begin{align}\mathbf{r}' &amp; =\mathbf{r}+\left[\frac{\gamma-1}{v^{2}}(\mathbf{vr})-\gamma t\right]\mathbf{v}\\
t' &amp; =\gamma\left[t-\frac{1}{c^{2}}(\mathbf{vr})\right]
\end{align}
\end{array}&lt;/math&gt;

[[w:Erwin Madelung]] (1922) provided the matrix form&lt;ref&gt;Madelung (1921), p. 207&lt;/ref&gt;

:&lt;math&gt;\begin{array}{c|c|c|c|c}
 &amp; x &amp; y &amp; z &amp; t\\
\hline x' &amp; 1-\frac{v_{x}^{2}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{x}v_{y}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{x}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; \frac{-v_{x}}{\sqrt{1-\beta^{2}}}\\
y' &amp; -\frac{v_{x}v_{y}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; 1-\frac{v_{y}^{2}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{y}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; \frac{-v_{y}}{\sqrt{1-\beta^{2}}}\\
z' &amp; -\frac{v_{x}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; -\frac{v_{y}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; 1-\frac{v_{z}^{2}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) &amp; \frac{-v_{z}}{\sqrt{1-\beta^{2}}}\\
t' &amp; \frac{-v_{x}}{c^{2}\sqrt{1-\beta^{2}}} &amp; \frac{-v_{y}}{c^{2}\sqrt{1-\beta^{2}}} &amp; \frac{-v_{z}}{c^{2}\sqrt{1-\beta^{2}}} &amp; \frac{1}{\sqrt{1-\beta^{2}}}
\end{array}&lt;/math&gt;

These formulas were called &quot;general Lorentz transformation without rotation&quot; by [[w:Christian Møller]] (1952),&lt;ref&gt;Møller (1952/55), pp. 41–43&lt;/ref&gt; who in addition gave an even more general Lorentz transformation in which the Cartesian axes have different orientations, using a [[w:Three-dimensional rotation operator|rotation operator]] &lt;math&gt;\mathfrak{D}&lt;/math&gt;. In this case, '''v′'''=''(v′&lt;sub&gt;x&lt;/sub&gt;, v′&lt;sub&gt;y&lt;/sub&gt;, v′&lt;sub&gt;z&lt;/sub&gt;)'' is not equal to -'''v'''=''(-v&lt;sub&gt;x&lt;/sub&gt;, -v&lt;sub&gt;y&lt;/sub&gt;, -v&lt;sub&gt;z&lt;/sub&gt;)'', but the relation &lt;math&gt;\mathbf{v}'=-\mathfrak{D}\mathbf{v}&lt;/math&gt; holds instead, with the result

:&lt;math&gt;\begin{array}{c}
\begin{align}\mathbf{x}' &amp; =\mathfrak{D}^{-1}\mathbf{x}-\mathbf{v}'\left\{ \left(\gamma-1\right)(\mathbf{x\cdot v})/v^{2}-\gamma t\right\} \\
t' &amp; =\gamma\left(t-(\mathbf{v}\cdot\mathbf{x})/c^{2}\right)
\end{align}
\end{array}&lt;/math&gt;

==={{anchor|Borel}} Borel (1913–14) – Cayley–Hermite parameter===

Borel (1913) started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and [[../Lorentz transformation (Cayley-Hermite)#Cayley|E:Cayley's (1846)]] parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions:&lt;ref group=R&gt;Borel (1913/14), p. 39&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}-z^{2}-1=0\\
\hline {\scriptstyle \begin{align}\delta a &amp; =\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta b &amp; =2(\lambda\mu+\nu\rho), &amp; \delta c &amp; =-2(\lambda\nu+\mu\rho),\\
\delta a' &amp; =2(\lambda\mu-\nu\rho), &amp; \delta b' &amp; =-\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta c' &amp; =2(\lambda\rho-\mu\nu),\\
\delta a'' &amp; =2(\lambda\nu-\mu\rho), &amp; \delta b'' &amp; =2(\lambda\rho+\mu\nu), &amp; \delta c'' &amp; =-\left(\lambda^{2}+\mu^{2}+\nu^{2}+\rho^{2}\right),
\end{align}
}\\
\left(\delta=\lambda^{2}+\mu^{2}-\rho^{2}-\nu^{2}\right)\\
\lambda=\nu=0\rightarrow\text{Hyperbolic rotation}
\end{matrix}&lt;/math&gt;

In four dimensions:&lt;ref group=R&gt;Borel (1913/14), p. 41&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}-\left(t_{1}-t_{2}\right)^{2}\\
\hline {\scriptstyle \begin{align} &amp; \left(\mu^{2}+\nu^{2}-\alpha^{2}\right)\cos\varphi+\left(\lambda^{2}-\beta^{2}-\gamma^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi-\gamma\operatorname{sh}{\theta}\\
 &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi+\gamma\operatorname{sh}{\theta} &amp;  &amp; \left(\mu^{2}+\nu^{2}-\beta^{2}\right)\cos\varphi+\left(\mu^{2}-\alpha^{2}-\gamma^{2}\right)\operatorname{ch}{\theta}\\
 &amp; -(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi-\beta\operatorname{sh}{\theta} &amp;  &amp; -(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\lambda\sin\varphi+\alpha\operatorname{sh}{\theta}\\
 &amp; (\gamma\mu-\beta\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta} &amp;  &amp; -(\alpha\nu-\lambda\gamma)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
\\
 &amp; \quad-(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi+\beta\operatorname{sh}{\theta} &amp;  &amp; \quad(\beta\nu-\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta}\\
 &amp; \quad-(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})-\lambda\sin\varphi-\alpha\operatorname{sh}{\theta} &amp;  &amp; \quad(\lambda\gamma-\alpha\nu)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
 &amp; \quad\left(\lambda^{2}+\mu^{2}-\gamma^{2}\right)\cos\varphi+\left(\nu^{2}-\alpha^{2}-\beta^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; \quad(\alpha\mu-\beta\lambda)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta}\\
 &amp; \quad(\beta\gamma-\alpha\mu)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta} &amp;  &amp; \quad-\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\cos\varphi+\left(\lambda^{2}+\mu^{2}+\nu^{2}\right)\operatorname{ch}{\theta}
\end{align}
}\\
\left(\alpha^{2}+\beta^{2}+\gamma^{2}-\lambda^{2}-\mu^{2}-\nu^{2}=-1\right)
\end{matrix}&lt;/math&gt;

==={{anchor|Gruner}} Gruner (1921) – Trigonometric Lorentz boosts===

In order to simplify the graphical representation of Minkowski space, [[w:Paul Gruner]] (1921) (with the aid of Josef Sauter) developed what is now called [[w:Loedel diagram]]s, using the following relations:&lt;ref group=R&gt;Gruner (1921a),&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}v=\alpha\cdot c;\quad\beta=\frac{1}{\sqrt{1-\alpha^{2}}}\\
\sin\varphi=\alpha;\quad\beta=\frac{1}{\cos\varphi};\quad\alpha\beta=\tan\varphi\\
\hline x'=\frac{x}{\cos\varphi}-t\cdot\tan\varphi,\quad t'=\frac{t}{\cos\varphi}-x\cdot\tan\varphi
\end{matrix}&lt;/math&gt;

In another paper Gruner used the alternative relations:&lt;ref group=R&gt;Gruner (1921b)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\alpha=\frac{v}{c};\ \beta=\frac{1}{\sqrt{1-\alpha^{2}}};\\
\cos\theta=\alpha=\frac{v}{c};\ \sin\theta=\frac{1}{\beta};\ \cot\theta=\alpha\cdot\beta\\
\hline x'=\frac{x}{\sin\theta}-t\cdot\cot\theta,\quad t'=\frac{t}{\sin\theta}-x\cdot\cot\theta
\end{matrix}&lt;/math&gt;

==References==
===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|abra1905}}
*{{#section:History of Topics in Special Relativity/relsource|bor14}}
*{{#section:History of Topics in Special Relativity/relsource|buch08}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04a}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04b}}
*{{#section:History of Topics in Special Relativity/relsource|con11qua}}
*{{#section:History of Topics in Special Relativity/relsource|einst05elek}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|fra11}}
*{{#section:History of Topics in Special Relativity/relsource|fra12}}
*{{#section:History of Topics in Special Relativity/relsource|gans05}}
*{{#section:History of Topics in Special Relativity/relsource|grun21a}}
*{{#section:History of Topics in Special Relativity/relsource|grun21b}}
*{{#section:History of Topics in Special Relativity/relsource|heav89}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|herg11ela}}
*{{#section:History of Topics in Special Relativity/relsource|igna10}}
*{{#section:History of Topics in Special Relativity/relsource|ignat10prin2}}
*{{#section:History of Topics in Special Relativity/relsource|klein08}}
*{{#section:History of Topics in Special Relativity/relsource|klein10b}}
*{{#section:History of Topics in Special Relativity/relsource|klein10}}
*{{#section:History of Topics in Special Relativity/relsource|klein11}}
*{{#section:History of Topics in Special Relativity/relsource|lar97}}
*{{#section:History of Topics in Special Relativity/relsource|lar29}}
*{{#section:History of Topics in Special Relativity/relsource|lar00}}
*{{#section:History of Topics in Special Relativity/relsource|lar04a}}
*{{#section:History of Topics in Special Relativity/relsource|lar04b}}
*{{#section:History of Topics in Special Relativity/relsource|lor92elek}}
*{{#section:History of Topics in Special Relativity/relsource|lor92b}}
*{{#section:History of Topics in Special Relativity/relsource|lor95}}
*{{#section:History of Topics in Special Relativity/relsource|lor99}}
*{{#section:History of Topics in Special Relativity/relsource|lor04}}
*{{#section:History of Topics in Special Relativity/relsource|lor16}}
*{{#section:History of Topics in Special Relativity/relsource|mink07a}}
*{{#section:History of Topics in Special Relativity/relsource|mink07b}}
*{{#section:History of Topics in Special Relativity/relsource|mink08}}
*{{#section:History of Topics in Special Relativity/relsource|plum10}}
*{{#section:History of Topics in Special Relativity/relsource|poi00}}
*{{#section:History of Topics in Special Relativity/relsource|poi04}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05a}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05b}}
*{{#section:History of Topics in Special Relativity/relsource|sea97}}
*{{#section:History of Topics in Special Relativity/relsource|silber11quat}}
*{{#section:History of Topics in Special Relativity/relsource|silber12quat}}
*{{#section:History of Topics in Special Relativity/relsource|silber14quat}}
*{{#section:History of Topics in Special Relativity/relsource|som09}}
*{{#section:History of Topics in Special Relativity/relsource|thom89}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}
*{{#section:History of Topics in Special Relativity/relsource|voi87}}
*{{#section:History of Topics in Special Relativity/relsource|wien04}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L4}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
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  <page>
    <title>History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)</title>
    <ns>0</ns>
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      <id>2411867</id>
      <parentid>2392451</parentid>
      <timestamp>2022-08-04T17:25:08Z</timestamp>
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      <text bytes="26975" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via Cayley–Hermite transformation ==

The [[../Lorentz transformation (general)#math_Q1|E:general transformation '''(Q1)''']] of any quadratic form into itself can also be given using ''arbitrary'' parameters based on the [[w:Cayley transform]] ('''I'''-'''T''')&lt;sup&gt;−1&lt;/sup&gt;·('''I'''+'''T'''), where '''I''' is the [[w:identity matrix]], '''T''' an arbitrary [[w:antisymmetric matrix]], and by adding '''A''' as symmetric matrix defining the quadratic form (there is no primed '''A' ''' because the coefficients are assumed to be the same on both sides):&lt;ref&gt;Hawkins (2013), pp. 210–214&lt;/ref&gt;&lt;ref&gt;Meyer (1899), p. 329&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}q=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}=q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}\cdot\mathbf{x}'\\
\hline \\
\mathbf{x}=(\mathbf{I}-\mathbf{T}\cdot\mathbf{A})^{-1}\cdot(\mathbf{I}+\mathbf{T}\cdot\mathbf{A})\cdot\mathbf{x}'\\
\text{or}\\
\mathbf{x}=\mathbf{A}^{-1}\cdot(\mathbf{A}-\mathbf{T})\cdot(\mathbf{A}+\mathbf{T})^{-1}\cdot\mathbf{A}\cdot\mathbf{x}'
\end{matrix}&lt;/math&gt;|{{equationRef|Q2}}}}

After [[#Cayley|Cayley (1846)]] introduced transformations related to sums of positive squares, [[#Hermite|Hermite (1853/54, 1854)]] derived transformations for arbitrary quadratic forms, whose result was reformulated in terms of matrices by [[#Cayley|Cayley (1855a, 1855b)]]. For instance, the choice '''A'''=diag(1,1,1) gives an orthogonal transformation which can be used to describe spatial rotations corresponding to the [[w:Euler–Rodrigues formula|w:Euler-Rodrigues parameter]]s ''[a,b,c,d]'' discovered by [[#Euler|Euler (1771) and Rodrigues (1840)]], which can be interpreted as the coefficients of [[w:quaternion]]s. Setting ''d=1'', the equations have the form:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(1,1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a &amp; -b\\
-a &amp; 0 &amp; c\\
b &amp; -c &amp; 0
\end{vmatrix}}\\
\hline x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[\begin{matrix}1-a^{2}-b^{2}+c^{2} &amp; 2(bc-a) &amp; 2(ac+b)\\
2(bc+a) &amp; 1-a^{2}+b^{2}-c^{2} &amp; 2(ab-c)\\
2(ac-b) &amp; 2(ab+c) &amp; 1+a^{2}-b^{2}-c^{2}
\end{matrix}\right]\cdot\mathbf{x}\\
\left(\kappa=1+a^{2}+b^{2}+c^{2}\right)
\end{matrix}&lt;/math&gt;|{{equationRef|Q3}}}}

Also the Lorentz interval and the general Lorentz transformation in any dimension can be produced by the Cayley–Hermite formalism.&lt;ref group=R&gt;Borel (1914), pp. 39–41&lt;/ref&gt;&lt;ref group=R&gt;Brill (1925)&lt;/ref&gt;&lt;ref&gt;Klein (1928), Chapter III, § 2B&lt;/ref&gt;&lt;ref&gt;Lorente (2003), section 3.3&lt;/ref&gt; For instance, the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']] with ''n''=1 follows from ({{equationNote|Q2}}) with:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(-1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a\\
-a &amp; 0
\end{vmatrix}}\\
\hline -x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{1-a^{2}}\left[\begin{matrix}1+a^{2} &amp; -2a\\
-2a &amp; 1+a^{2}
\end{matrix}\right]\cdot\mathbf{x}
\end{matrix}\Rightarrow\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}x_{0} &amp; =x_{0}^{\prime}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}}+x_{1}^{\prime}\frac{2\beta_{0}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{x_{0}^{\prime}\left(1+\beta_{0}^{2}\right)+x_{1}^{\prime}2\beta_{0}}{1-\beta_{0}^{2}}\\
x_{1} &amp; =x_{0}^{\prime}\frac{2\beta_{0}}{1-\beta_{0}^{2}}+x_{1}^{\prime}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{x_{0}^{\prime}2\beta_{0}+x_{1}^{\prime}\left(1+\beta_{0}^{2}\right)}{1-\beta_{0}^{2}}\\
\\
x_{0}^{\prime} &amp; =x_{0}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}}-x_{1}\frac{2\beta_{0}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{x_{0}\left(1+\beta_{0}^{2}\right)-x_{1}2\beta_{0}}{1-\beta_{0}^{2}}\\
x_{1}^{\prime} &amp; =-x_{0}\frac{2\beta_{0}}{1-\beta_{0}^{2}}+x_{1}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{-x_{0}2\beta_{0}+x_{1}\left(1+\beta_{0}^{2}\right)}{1-\beta_{0}^{2}}
\end{align}
\right|{\scriptstyle \begin{align}\frac{2\beta_{0}}{1+\beta_{0}^{2}} &amp; =\beta\\
\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} &amp; =\gamma\\
\frac{2\beta_{0}}{1-\beta_{0}^{2}} &amp; =\beta\gamma
\end{align}
}
\end{matrix}&lt;/math&gt;|{{equationRef|5a}}}}

This becomes [[../Lorentz transformation (velocity)#math_4a|E:Lorentz boost '''(4a)''']] by setting &lt;math&gt;\tfrac{2a}{1+a^{2}}=\tfrac{v}{c}&lt;/math&gt;, which is equivalent to the relation &lt;math&gt;\tfrac{2\beta_{0}}{1+\beta_{0}^{2}}=\tfrac{v}{c}&lt;/math&gt; known from [[w:Loedel diagram]]s, thus ({{equationNote|5a}}) can be interpreted as a Lorentz boost from the viewpoint of a &quot;median frame&quot; in which two other inertial frames are moving with equal speed &lt;math&gt;\beta_0&lt;/math&gt; in opposite directions.

Furthermore, Lorentz transformation [[../Lorentz transformation (general)#math_1a|E:'''(1a)''']]) with ''n''=2 is given by:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(-1,1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a &amp; -b\\
-a &amp; 0 &amp; c\\
b &amp; -c &amp; 0
\end{vmatrix}}\\
\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[\begin{matrix}1+a^{2}+b^{2}+c^{2} &amp; -2(bc-a) &amp; -2(ac+b)\\
2(bc+a) &amp; 1+a^{2}-b^{2}-c^{2} &amp; 2(ab-c)\\
2(ac-b) &amp; -2(ab-c) &amp; 1-a^{2}+b^{2}-c^{2}
\end{matrix}\right]\cdot\mathbf{x}\\
\left(\kappa=1-a^{2}-b^{2}+c^{2}\right)
\end{matrix}&lt;/math&gt;|{{equationRef|5b}}}}

or using ''n''=3:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(-1,1,1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a &amp; -b &amp; c\\
-a &amp; 0 &amp; d &amp; e\\
b &amp; -d &amp; 0 &amp; f\\
-c &amp; -e &amp; -f &amp; 0
\end{vmatrix}}\\
\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[{\scriptstyle \begin{align} &amp; 1+a^{2}+b^{2}+c^{2}+ &amp;  &amp; 2(-bd+a+ec+pf) &amp;  &amp; 2(-ad-b+fc-pe) &amp;  &amp; 2(pd+fb-ea+c)\\
 &amp; \quad d^{2}+e^{2}+f^{2}+p^{2} &amp;  &amp; 1+a^{2}-b^{2}-c^{2} &amp;  &amp; 2(-d-ab+pc-fe) &amp;  &amp; 2(fd+pb+ca-e)\\
 &amp; 2(bd+a-ec+pf) &amp;  &amp; \quad-d^{2}-e^{2}+f^{2}+p^{2} &amp;  &amp; 1-a^{2}+b^{2}-c^{2} &amp;  &amp; 2(-ed-cb+pa-f)\\
 &amp; 2(ad-b-fc-pe) &amp;  &amp; 2(d-ab-pc-fe) &amp;  &amp; \quad-d^{2}+e^{2}-f^{2}+p^{2} &amp;  &amp; 1-a^{2}-b^{2}+-c^{2}\\
 &amp; 2(pd-fb+ea+c) &amp;  &amp; 2(fd-pb+ca+e) &amp;  &amp; 2(-ed-cb-pa+f) &amp;  &amp; \quad+d^{2}-e^{2}-f^{2}+p^{2}
\end{align}
}\right]\cdot\mathbf{x}\\
\left(\begin{align}\kappa &amp; =1-a^{2}-b^{2}-c^{2}+d^{2}+e^{2}+f^{2}-p^{2}\\
p &amp; =af+be+cd
\end{align}
\right)
\end{matrix}&lt;/math&gt;|{{equationRef|5c}}}}

The transformation of a binary quadratic form of which Lorentz transformation ({{equationNote|5a}}) is a special case was given by [[#Hermite|Hermite (1854)]], equations containing Lorentz transformations ({{equationNote|5a}}, {{equationNote|5b}}, {{equationNote|5c}}) as special cases were given by [[#Cayley|Cayley (1855)]], Lorentz transformation ({{equationNote|5a}}) was given (up to a sign change) by [[#Laguerre|Laguerre (1882)]], [[#Darboux2|Darboux (1887)]], [[#Smith|Smith (1900)]] in relation to Laguerre geometry, and Lorentz transformation ({{equationNote|5b}}) was given by [[#Bachmann|Bachmann (1869)]]. In relativity, equations similar to ({{equationNote|5b}}, {{equationNote|5c}}) were first employed by [[#Borel|Borel (1913)]] to represent Lorentz transformations.

==Historical notation==
==={{anchor|Euler}} Euler (1771) – Euler-Rodrigues parameter===

Euler (1771) demonstrated the invariance of quadratic forms in terms of sum of squares under a linear substitution and its coefficients, now known as [[w:orthogonal transformation]]. The transformation in three dimensions was given as

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}+Z^{2}=x^{2}+y^{2}+z^{2}\\
\hline \begin{align}X &amp; =Ax+By+Cz\\
Y &amp; =Dx+Ey+Fz\\
Z &amp; =Gx+Hy+Iz
\end{align}
\begin{matrix}\left|{\scriptstyle \begin{align}1 &amp; =AA+DD+GG\\
1 &amp; =BB+EE+HH\\
1 &amp; =CC+FF+II\\
0 &amp; =AB+DE+GH\\
0 &amp; =AG+DF+GI\\
0 &amp; =BC+EF+HI
\end{align}
}\right.\end{matrix}\end{matrix}&lt;/math&gt;

in which the coefficiens ''A,B,C,D,E,F,G,H,I'' were related by Euler to four arbitrary parameter ''p,q,r,s'', which where rediscovered by [[w:Olinde Rodrigues]] (1840) who related them to rotation angles&lt;ref group=M&gt;Rodrigues (1840), p. 405&lt;/ref&gt;:&lt;ref group=M&gt;Euler (1771), p. 101&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}A &amp; =\frac{pp+qq-rr-ss}{u} &amp; B &amp; =\frac{2pq+2ps}{u} &amp; C &amp; =\frac{2qs-2pr}{u}\\
D &amp; =\frac{2qr-2ps}{u} &amp; E &amp; =\frac{pp-qq+rr-ss}{u} &amp; F &amp; =\frac{2pq+2rs}{u}\\
G &amp; =\frac{2qs+2pr}{u} &amp; H &amp; =\frac{2rs-2pq}{u} &amp; I &amp; =\frac{pp-qq-rr+ss}{u}
\end{align}
\\
(u=pp+qq+rr+ss)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These are now called (improper) [[w:Euler–Rodrigues formula|Euler–Rodrigues parameters]] in line with equation ({{equationNote|Q3}}).&lt;/p&gt;

==={{anchor|Cayley}} Cayley (1846–1884)===
===={{anchor|Cayley}} Euler–Rodrigues parameter and Cayley–Hermite transformation====

The [[w:Euler–Rodrigues formula|Euler–Rodrigues parameters]] discovered by [[#Euler|Euler (1871) and Rodrigues (1840)]] leaving invariant &lt;math&gt;x_{0}^{2}+x_{1}^{2}+x_{2}^{2}&lt;/math&gt; were extended to &lt;math&gt;x_{0}^{2}+\dots+x_{n}^{2}&lt;/math&gt; by [[w:Arthur Cayley]] (1846) as a byproduct of what is now called the [[w:Cayley transform]] using the method of skew–symmetric coefficients.&lt;ref group=M name=cayl1&gt;Cayley (1846)&lt;/ref&gt; Following Cayley's methods, a general transformation for quadratic forms into themselves in three (1853) and arbitrary (1854) dimensions was provided by [[#Hermite|Hermite (1853, 1854)]]. Hermite's formula was simplified and brought into matrix form equivalent to ({{equationNote|Q2}}) by Cayley (1855a)&lt;ref group=M&gt;Cayley (1855a), p. 288&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle (\mathrm{x,y,z}\dots)=\left(\left|\begin{matrix}a, &amp; h, &amp; g &amp; \dots\\
h, &amp; b, &amp; f &amp; \dots\\
g, &amp; f, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|^{-1}\left|\begin{matrix}a, &amp; h-\nu, &amp; g+\mu &amp; \dots\\
h+\nu, &amp; b, &amp; f-\lambda &amp; \dots\\
g-\mu, &amp; f+\lambda, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|\left|\begin{matrix}a, &amp; h+\nu, &amp; g-\mu &amp; \dots\\
h-\nu, &amp; b, &amp; f+\lambda &amp; \dots\\
g+\mu, &amp; f-\lambda, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|^{-1}\left|\begin{matrix}a, &amp; h, &amp; g &amp; \dots\\
h, &amp; b, &amp; f &amp; \dots\\
g, &amp; f, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|\right)\ ^{\frown}(x,y,z\dots)}&lt;/math&gt;

which he abbreviated in 1858, where &lt;math&gt;\Upsilon&lt;/math&gt; is any skew-symmetric matrix:&lt;ref group=M name=cayl1858&gt;Cayley (1858), p. 39&lt;/ref&gt;&lt;ref&gt;Hawkins (2013), p. 214&lt;/ref&gt;

:&lt;math&gt;(\mathrm{x,y,z})=\left(\Omega^{-1}(\Omega-\Upsilon)(\Omega+\Upsilon)^{-1}\Omega\right)(x,y,z)&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The Cayley–Hermite transformation becomes equivalent to the Lorentz transformation ({{equationNote|5a}}) by setting Ω=diag(-1,1) and  ({{equationNote|5b}}) by setting Ω=diag(-1,1,1) and ({{equationNote|5c}}) by setting Ω=diag(-1,1,1,1).&lt;/p&gt;

Using the parameters of (1855a), Cayley in a subsequent paper (1855b) particularly discussed several special cases, such as:&lt;ref group=M&gt;Cayley (1855b), p. 210&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}=ax^{2}+by^{2}\\
\hline (\mathrm{x,y})=\frac{1}{ab+\nu^{2}}\cdot\left[\begin{matrix}ab-\nu^{2}, &amp; -2\nu b\\
2\nu a, &amp; ab-\nu^{2}
\end{matrix}\right](x,y)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to the Lorentz transformation ({{equationNote|5a}}) in 1+1 dimensions by setting ''(a,b)''=(-1,1) and Lorentz boost [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] by additionally setting &lt;math&gt;\tfrac{2\nu}{1+\nu^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

or:&lt;ref group=M&gt;Cayley (1855b), p. 211&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}+c\mathrm{z}^{2}=ax^{2}+by^{2}+cz^{2}\\
\hline (\mathrm{x,y,z})=\frac{1}{abc+a\lambda^{2}+b\mu^{2}+c\nu^{2}}\times\left[\begin{matrix}abc+a\lambda^{2}-b\mu^{2}-c\nu^{2}, &amp; 2(\lambda\mu-c\nu)b &amp; 2(\nu\lambda+b\mu)c\\
2(\lambda\mu+c\nu)a, &amp; abc-a\lambda^{2}+b\mu^{2}-c\nu^{2} &amp; 2(\mu\nu-a\lambda)c\\
2(\nu\lambda-b\mu)a &amp; 2(\mu\nu+a\lambda)b &amp; abc-a\lambda^{2}-b\mu^{2}-c\nu^{2}
\end{matrix}\right](x,y,z)
\end{matrix}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to the Lorentz transformation ({{equationNote|5b}}) by setting ''(a,b,c)''=(-1,1,1).&lt;/p&gt;

or:&lt;ref group=M&gt;Cayley (1855b), pp. 212–213&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}+c\mathrm{z}^{2}+d\mathrm{w}^{2}=ax^{2}+by^{2}+cz^{2}+dw^{2}\\
\hline (\mathrm{x,y,z,w})=\frac{1}{k}\cdot\left[\begin{align} &amp; abcd-bc\rho^{2}+ca\sigma^{2}+ab\tau^{2}+ad\lambda^{2} &amp;  &amp; 2b\left(-cd\nu-\tau\phi+d\lambda\mu-c\rho\sigma\right),\\
 &amp; \quad-bd\mu^{2}-cd\nu^{2}-\phi^{2}, &amp;  &amp; abcd+bc\rho^{2}-ca\sigma^{2}+ab\tau^{2}-ad\lambda^{2}\\
 &amp; 2a\left(cd\nu+\tau\phi+d\lambda\mu-c\rho\sigma\right), &amp;  &amp; \quad+bd\mu^{2}-cd\nu^{2}-\phi^{2},\\
 &amp; 2a\left(-bd\mu-\sigma\phi+d\lambda\nu-b\rho\tau\right), &amp;  &amp; 2b\left(ad\lambda+\rho\phi+d\mu\nu-a\sigma\tau\right),\\
 &amp; 2a\left(bc\rho+\lambda\phi+c\nu\sigma-b\mu\tau\right), &amp;  &amp; 2b\left(ac\sigma+\mu\phi-c\nu\rho+a\lambda\tau\right),\\
\\
 &amp; \quad2c\left(bd\mu+\sigma\phi+d\lambda\nu-b\rho\tau\right), &amp;  &amp; \quad2d\left(-bc\rho-\lambda\phi+c\nu\sigma-b\mu\tau\right)\\
 &amp; \quad2c\left(-ad\lambda-\rho\phi+d\mu\nu-a\sigma\tau\right), &amp;  &amp; \quad2d\left(-ac\sigma-\mu\phi-c\nu\rho+a\lambda\tau\right)\\
 &amp; \quad abcd+bc\rho^{2}+ca\sigma^{2}-ab\tau^{2}-ad\lambda^{2} &amp;  &amp; \quad2d\left(-ab\tau-\nu\phi+b\mu\rho-a\lambda\sigma\right)\\
 &amp; \quad\quad-bd\mu^{2}+cd\nu^{2}-\phi^{2}, &amp;  &amp; \quad abcd-bc\rho^{2}-ca\sigma^{2}-ab\tau^{2}+ad\lambda^{2}\\
 &amp; \quad2c\left(ab\tau+\nu\phi+b\mu\rho-a\lambda\sigma\right), &amp;  &amp; \quad\quad+bd\mu^{2}+cd\nu^{2}-\phi^{2},
\end{align}
\right]\cdot(x,y,z,w)\\
\left(\begin{align}k &amp; =abcd+bc\rho^{2}+ca\sigma^{2}+ab\tau^{2}+ad\lambda^{2}+bd\mu^{2}+cd\nu^{2}+\phi^{2}\\
\phi &amp; =\lambda\rho+\mu\sigma+\nu\tau
\end{align}
\right)
\end{matrix}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to the Lorentz transformation ({{equationNote|5c}}) by setting ''(a,b,c,d)''=(-1,1,1,1).&lt;/p&gt;

==={{anchor|Hermite}} Hermite (1853, 1854) – Cayley–Hermite transformation===

[[w:Charles Hermite]] (1853) extended the number theoretical work of [[../Lorentz transformation (general)#Gauss2|E:Gauss (1801)]] and others (including himself) by additionally analyzing ''indefinite'' ternary quadratic forms that can be transformed into the Lorentz interval ±''(x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;)'', and by using [[#Cayley|Cayley's (1846)]] method of skew–symmetric coefficients he derived transformations leaving invariant almost all types of ternary quadratic forms.&lt;ref group=M name=herm&gt;Hermite (1853/54a), p. 307ff.&lt;/ref&gt; This was generalized by him in 1854 to ''n'' dimensions:&lt;ref group=M&gt;Hermite (1854b), p. 64&lt;/ref&gt;&lt;ref&gt;Hawkins (2013), p. 212&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}f\left(X_{1},X_{2},\dots\right)=f\left(x_{1},x_{2},\dots\right)\\
\hline X_{r}=2\xi_{r}-x_{r}=\xi_{r}-\frac{1}{2}\sum_{s=1}^{n}\lambda_{r,s}\frac{df}{d\xi_{s}}\\
\left(\lambda_{r,s}=-\lambda_{s,r},\ \lambda_{r,r}=0\right)
\end{matrix}&lt;/math&gt;

This result was subsequently expressed in matrix form by [[#Cayley|Cayley (1855)]], while [[w:Ferdinand Georg Frobenius]] (1877) added some modifications in order to include some special cases of quadratic forms that cannot be dealt with by the Cayley–Hermite transformation.&lt;ref group=M&gt;Frobenius (1877)&lt;/ref&gt;&lt;ref&gt;Hawkins (2013), pp. 219ff&lt;/ref&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to equation ({{equationNote|Q2}}), and becomes the Lorentz transformation by setting the coefficients of the quadratic form ''f'' to diag(-1,1,...1).&lt;/p&gt;

For instance, the special case of the transformation of a binary quadratic form into itself was given by Hermite as follows:&lt;ref group=M&gt;Hermite (1854b), pp. 64–65&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}f=ax^{2}+2bxy+cy^{2}\\
\hline \begin{align}X &amp; =\frac{\left(1-2\lambda b+\lambda^{2}D\right)x-2\lambda cy}{1-\lambda^{2}D} &amp;  &amp; =x(t-bu)-cuy\\
Y &amp; =\frac{2\lambda ax+\left(1+2\lambda b+\lambda^{2}D\right)y}{1-\lambda^{2}D} &amp;  &amp; =xau+(t+bu)y
\end{align}
\\
\left(b^{2}-ac=D,\ t=\frac{1+\lambda^{2}D}{1-\lambda^{2}D},\ u=\frac{2\lambda}{1-\lambda^{2}D},\ t^{2}-Du^{2}=1\right)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to Lorentz boost ({{equationNote|5a}}) by setting ''(a,b,c)''=(-1,0,1) and Lorentz boost of velocity by additionally setting &lt;math&gt;\tfrac{2\lambda}{1+\lambda^{2}}=\tfrac{v}{c}&lt;/math&gt; which produces ''t''=γ and ''u''=βγ.&lt;/p&gt;

=== {{anchor|Bachmann}} Bachmann (1869) – Cayley–Hermite transformation ===

[[w:Paul Gustav Heinrich Bachmann]] (1869) adapted [[#Hermite|Hermite's (1853/54)]] transformation of ternary quadratic forms to the case of integer transformations. He particularly analyzed the Lorentz interval and its transformation, and also alluded to the analogue result of [[#Gauss3|Gauss (1800)]] in terms of Cayley–Klein parameters, while Bachmann formulated his result in terms of the Cayley–Hermite transformation:&lt;ref group=M&gt;Bachmann (1869), p. 303&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+x^{\prime2}-x^{\prime\prime2}\\
\hline \begin{align}\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X &amp; =\left(p^{2}-q^{2}+q^{\prime2}-q^{\prime\prime2}\right)x-2(pq''+qq')x'-2(pq'+qq'')x''\\
\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X' &amp; =2(pq''-qq')x+\left(p^{2}+q^{2}-q^{\prime2}-q^{\prime\prime2}\right)x'+2(pq-q'q'')x''\\
\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X'' &amp; =-2(pq'-qq')x+2(pq+q'q'')x'+\left(p^{2}+q^{2}+q^{\prime2}+q^{\prime\prime2}\right)x''
\end{align}
\end{matrix}&lt;/math&gt;

He described this transformation in 1898 in the first part of his &quot;arithmetics of quadratic forms&quot; as well.&lt;ref&gt;Bachmann (1898), pp. 101–102&lt;/ref&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|5b}}), producing the relation &lt;math&gt;X^{2}+X^{\prime2}-X^{\prime\prime2}=x^{2}+x^{\prime2}-x^{\prime\prime2}&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Laguerre}} Laguerre (1882) – Laguerre inversion===

After previous work by [[w:Albert Ribaucour]] (1870),&lt;ref group=M&gt;Ribaucour (1870)&lt;/ref&gt; a transformation which transforms oriented spheres into oriented spheres, oriented planes into oriented planes, and oriented lines into oriented lines, was explicitly formulated by [[w:Edmond Laguerre]] (1882) as &quot;[[w:Spherical wave transformation#Transformation by reciprocal directions|transformation by reciprocal directions]]&quot; which was later called &quot;Laguerre inversion/transformation&quot;. It can be seen as a special case of the conformal group in terms of [[../Lorentz transformation (conformal)#Lie|E:Lie's transformations of oriented spheres]]. In two dimensions the transformation or oriented lines has the form (''R'' being the radius):&lt;ref group=M name=laguerre&gt;Laguerre (1882), pp. 550–551.&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}D' &amp; =\frac{D\left(1+\alpha^{2}\right)-2\alpha R}{1-\alpha^{2}}\\
R' &amp; =\frac{2\alpha D-R\left(1+\alpha^{2}\right)}{1-\alpha^{2}}
\end{align}
\right|\begin{align}D^{2}-D^{\prime2} &amp; =R^{2}-R^{\prime2}\\
D-D' &amp; =\alpha(R-R')\\
D+D' &amp; =\frac{1}{\alpha}(R+R')
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change) to Lorentz transformation ({{equationNote|5a}}) in terms of Cayley–Hermite parameters (even though Laguerre didn't use the Cayley-Hermite transformation ({{equationNote|Q2}})). The Lorentz boost of velocity follows with &lt;math&gt;\tfrac{2\alpha}{1+\alpha^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Darboux2}} Darboux (1887) – Laguerre inversion===

Following [[#Laguerre|Laguerre (1882)]], [[w:Gaston Darboux]] (1887) presented the Laguerre inversions in four dimensions using coordinates ''x,y,z,R'':&lt;ref group=M name=darboux&gt;Darboux (1887)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{\prime2}+y^{\prime2}+z^{\prime2}-R^{\prime2}=x^{2}+y^{2}+z^{2}-R^{2}\\
\hline \begin{align}x' &amp; =x, &amp; z' &amp; =\frac{1+k^{2}}{1-k^{2}}z-\frac{2kR}{1-k^{2}},\\
y' &amp; =y, &amp; R' &amp; =\frac{2kz}{1-k^{2}}-\frac{1+k^{2}}{1-k^{2}}R,
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change for ''R'') to Lorentz transformation ({{equationNote|5a}}) in terms of Cayley–Hermite parameters (even though Darboux didn't use the Cayley-Hermite transformation ({{equationNote|Q2}})). The Lorentz boost of velocity follows with &lt;math&gt;\tfrac{2k}{1+k^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

Darboux rewrote these equations as follows:

:&lt;math&gt;\begin{matrix}x^{\prime2}+y^{\prime2}+z^{\prime2}-R^{\prime2}=x^{2}+y^{2}+z^{2}-R^{2}\\
\hline \begin{align}z'+R' &amp; =\frac{1+k}{1-k}(z-R)\\
z'-R' &amp; =\frac{1-k}{1+k}(z+R)
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change for ''R'') to a squeeze mapping.&lt;/p&gt;

==={{anchor|Smith}} Smith (1900) – Laguerre inversion===

[[w:Percey F. Smith]] (1900) followed [[#Laguerre|Laguerre (1882)]] and [[#Darboux2|Darboux (1887)]] and defined the Laguerre inversion as follows:&lt;ref group=M&gt;Smith (1900), p. 159&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{\prime2}-p^{2}=R^{\prime2}-R^{2}\\
\hline p'=\frac{\kappa^{2}+1}{\kappa^{2}-1}p-\frac{2\kappa}{\kappa^{2}-1}R,\quad R'=\frac{2\kappa}{\kappa^{2}-1}p-\frac{\kappa^{2}+1}{\kappa^{2}-1}R
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change) to Lorentz transformation ({{equationNote|5a}}) in terms of Cayley–Hermite parameters (even though Smith didn't use the Cayley-Hermite transformation ({{equationNote|Q2}})). The Lorentz boost of velocity follows with &lt;math&gt;\tfrac{2\kappa}{1+\kappa^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Borel}} Borel (1913–14) – Cayley–Hermite parameter===

Borel (1913) started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and [[#Cayley|Cayley's (1846)]] parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions equivalent to ({{equationNote|5b}}):&lt;ref group=R&gt;Borel (1913/14), p. 39&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}-z^{2}-1=0\\
\hline {\scriptstyle \begin{align}\delta a &amp; =\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta b &amp; =2(\lambda\mu+\nu\rho), &amp; \delta c &amp; =-2(\lambda\nu+\mu\rho),\\
\delta a' &amp; =2(\lambda\mu-\nu\rho), &amp; \delta b' &amp; =-\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta c' &amp; =2(\lambda\rho-\mu\nu),\\
\delta a'' &amp; =2(\lambda\nu-\mu\rho), &amp; \delta b'' &amp; =2(\lambda\rho+\mu\nu), &amp; \delta c'' &amp; =-\left(\lambda^{2}+\mu^{2}+\nu^{2}+\rho^{2}\right),
\end{align}
}\\
\left(\delta=\lambda^{2}+\mu^{2}-\rho^{2}-\nu^{2}\right)\\
\lambda=\nu=0\rightarrow\text{Hyperbolic rotation}
\end{matrix}&lt;/math&gt;

In four dimensions equivalent to ({{equationNote|5c}}):&lt;ref group=R&gt;Borel (1913/14), p. 41&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}-\left(t_{1}-t_{2}\right)^{2}\\
\hline {\scriptstyle \begin{align} &amp; \left(\mu^{2}+\nu^{2}-\alpha^{2}\right)\cos\varphi+\left(\lambda^{2}-\beta^{2}-\gamma^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi-\gamma\operatorname{sh}{\theta}\\
 &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi+\gamma\operatorname{sh}{\theta} &amp;  &amp; \left(\mu^{2}+\nu^{2}-\beta^{2}\right)\cos\varphi+\left(\mu^{2}-\alpha^{2}-\gamma^{2}\right)\operatorname{ch}{\theta}\\
 &amp; -(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi-\beta\operatorname{sh}{\theta} &amp;  &amp; -(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\lambda\sin\varphi+\alpha\operatorname{sh}{\theta}\\
 &amp; (\gamma\mu-\beta\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta} &amp;  &amp; -(\alpha\nu-\lambda\gamma)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
\\
 &amp; \quad-(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi+\beta\operatorname{sh}{\theta} &amp;  &amp; \quad(\beta\nu-\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta}\\
 &amp; \quad-(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})-\lambda\sin\varphi-\alpha\operatorname{sh}{\theta} &amp;  &amp; \quad(\lambda\gamma-\alpha\nu)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
 &amp; \quad\left(\lambda^{2}+\mu^{2}-\gamma^{2}\right)\cos\varphi+\left(\nu^{2}-\alpha^{2}-\beta^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; \quad(\alpha\mu-\beta\lambda)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta}\\
 &amp; \quad(\beta\gamma-\alpha\mu)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta} &amp;  &amp; \quad-\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\cos\varphi+\left(\lambda^{2}+\mu^{2}+\nu^{2}\right)\operatorname{ch}{\theta}
\end{align}
}\\
\left(\alpha^{2}+\beta^{2}+\gamma^{2}-\lambda^{2}-\mu^{2}-\nu^{2}=-1\right)
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bach69}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay46gau}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay55quad}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay55gau}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay58quad}} 
*{{#section:History of Topics in Special Relativity/mathsource|dar87cou}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul71}} 
*{{#section:History of Topics in Special Relativity/mathsource|fro77}} 
*{{#section:History of Topics in Special Relativity/mathsource|herm53}} 
*{{#section:History of Topics in Special Relativity/mathsource|herm54}} 
*{{#section:History of Topics in Special Relativity/mathsource|lagu81}} 
*{{#section:History of Topics in Special Relativity/mathsource|lagu82}} 
*{{#section:History of Topics in Special Relativity/mathsource|rib70}} 
*{{#section:History of Topics in Special Relativity/mathsource|rod40}} 
*{{#section:History of Topics in Special Relativity/mathsource|smi00}}

===Historical relativity sources===
{{reflist|3|group=R}}
{{#section:History of Topics in Special Relativity/relsource|bor14}}
{{#section:History of Topics in Special Relativity/relsource|brill25}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L6}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
      <sha1>i8pa63o0zsd9asb7b2tm1ilo2sigf0b</sha1>
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      <text bytes="26920" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via Cayley–Hermite transformation ==

The [[../Lorentz transformation (general)#math_Q1|E:general transformation '''(Q1)''']] of any quadratic form into itself can also be given using ''arbitrary'' parameters based on the [[w:Cayley transform]] ('''I'''-'''T''')&lt;sup&gt;−1&lt;/sup&gt;·('''I'''+'''T'''), where '''I''' is the [[w:identity matrix]], '''T''' an arbitrary [[w:antisymmetric matrix]], and by adding '''A''' as symmetric matrix defining the quadratic form (there is no primed '''A' ''' because the coefficients are assumed to be the same on both sides):&lt;ref&gt;Hawkins (2013), pp. 210–214&lt;/ref&gt;&lt;ref&gt;Meyer (1899), p. 329&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}q=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}=q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}\cdot\mathbf{x}'\\
\hline \\
\mathbf{x}=(\mathbf{I}-\mathbf{T}\cdot\mathbf{A})^{-1}\cdot(\mathbf{I}+\mathbf{T}\cdot\mathbf{A})\cdot\mathbf{x}'\\
\text{or}\\
\mathbf{x}=\mathbf{A}^{-1}\cdot(\mathbf{A}-\mathbf{T})\cdot(\mathbf{A}+\mathbf{T})^{-1}\cdot\mathbf{A}\cdot\mathbf{x}'
\end{matrix}&lt;/math&gt;|{{equationRef|Q2}}}}

After [[#Cayley|Cayley (1846)]] introduced transformations related to sums of positive squares, [[#Hermite|Hermite (1853/54, 1854)]] derived transformations for arbitrary quadratic forms, whose result was reformulated in terms of matrices by [[#Cayley|Cayley (1855a, 1855b)]]. For instance, the choice '''A'''=diag(1,1,1) gives an orthogonal transformation which can be used to describe spatial rotations corresponding to the [[w:Euler–Rodrigues formula|w:Euler-Rodrigues parameter]]s ''[a,b,c,d]'' discovered by [[#Euler|Euler (1771) and Rodrigues (1840)]], which can be interpreted as the coefficients of [[w:quaternion]]s. Setting ''d=1'', the equations have the form:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(1,1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a &amp; -b\\
-a &amp; 0 &amp; c\\
b &amp; -c &amp; 0
\end{vmatrix}}\\
\hline x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[\begin{matrix}1-a^{2}-b^{2}+c^{2} &amp; 2(bc-a) &amp; 2(ac+b)\\
2(bc+a) &amp; 1-a^{2}+b^{2}-c^{2} &amp; 2(ab-c)\\
2(ac-b) &amp; 2(ab+c) &amp; 1+a^{2}-b^{2}-c^{2}
\end{matrix}\right]\cdot\mathbf{x}\\
\left(\kappa=1+a^{2}+b^{2}+c^{2}\right)
\end{matrix}&lt;/math&gt;|{{equationRef|Q3}}}}

Also the Lorentz interval and the general Lorentz transformation in any dimension can be produced by the Cayley–Hermite formalism.&lt;ref group=R&gt;Borel (1914), pp. 39–41&lt;/ref&gt;&lt;ref group=R&gt;Brill (1925)&lt;/ref&gt;&lt;ref&gt;Klein (1928), Chapter III, § 2B&lt;/ref&gt;&lt;ref&gt;Lorente (2003), section 3.3&lt;/ref&gt; For instance, the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']] with ''n''=1 follows from ({{equationNote|Q2}}) with:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(-1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a\\
-a &amp; 0
\end{vmatrix}}\\
\hline -x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{1-a^{2}}\left[\begin{matrix}1+a^{2} &amp; -2a\\
-2a &amp; 1+a^{2}
\end{matrix}\right]\cdot\mathbf{x}
\end{matrix}\Rightarrow\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}x_{0} &amp; =x_{0}^{\prime}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}}+x_{1}^{\prime}\frac{2\beta_{0}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{x_{0}^{\prime}\left(1+\beta_{0}^{2}\right)+x_{1}^{\prime}2\beta_{0}}{1-\beta_{0}^{2}}\\
x_{1} &amp; =x_{0}^{\prime}\frac{2\beta_{0}}{1-\beta_{0}^{2}}+x_{1}^{\prime}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{x_{0}^{\prime}2\beta_{0}+x_{1}^{\prime}\left(1+\beta_{0}^{2}\right)}{1-\beta_{0}^{2}}\\
\\
x_{0}^{\prime} &amp; =x_{0}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}}-x_{1}\frac{2\beta_{0}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{x_{0}\left(1+\beta_{0}^{2}\right)-x_{1}2\beta_{0}}{1-\beta_{0}^{2}}\\
x_{1}^{\prime} &amp; =-x_{0}\frac{2\beta_{0}}{1-\beta_{0}^{2}}+x_{1}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{-x_{0}2\beta_{0}+x_{1}\left(1+\beta_{0}^{2}\right)}{1-\beta_{0}^{2}}
\end{align}
\right|{\scriptstyle \begin{align}\frac{2\beta_{0}}{1+\beta_{0}^{2}} &amp; =\beta\\
\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} &amp; =\gamma\\
\frac{2\beta_{0}}{1-\beta_{0}^{2}} &amp; =\beta\gamma
\end{align}
}
\end{matrix}&lt;/math&gt;|{{equationRef|5a}}}}

This becomes [[../Lorentz transformation (velocity)#math_4a|E:Lorentz boost '''(4a)''']] by setting &lt;math&gt;\tfrac{2a}{1+a^{2}}=\tfrac{v}{c}&lt;/math&gt;, which is equivalent to the relation &lt;math&gt;\tfrac{2\beta_{0}}{1+\beta_{0}^{2}}=\tfrac{v}{c}&lt;/math&gt; known from [[w:Loedel diagram]]s, thus ({{equationNote|5a}}) can be interpreted as a Lorentz boost from the viewpoint of a &quot;median frame&quot; in which two other inertial frames are moving with equal speed &lt;math&gt;\beta_0&lt;/math&gt; in opposite directions.

Furthermore, Lorentz transformation [[../Lorentz transformation (general)#math_1a|E:'''(1a)''']]) with ''n''=2 is given by:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(-1,1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a &amp; -b\\
-a &amp; 0 &amp; c\\
b &amp; -c &amp; 0
\end{vmatrix}}\\
\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[\begin{matrix}1+a^{2}+b^{2}+c^{2} &amp; -2(bc-a) &amp; -2(ac+b)\\
2(bc+a) &amp; 1+a^{2}-b^{2}-c^{2} &amp; 2(ab-c)\\
2(ac-b) &amp; -2(ab-c) &amp; 1-a^{2}+b^{2}-c^{2}
\end{matrix}\right]\cdot\mathbf{x}\\
\left(\kappa=1-a^{2}-b^{2}+c^{2}\right)
\end{matrix}&lt;/math&gt;|{{equationRef|5b}}}}

or using ''n''=3:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(-1,1,1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a &amp; -b &amp; c\\
-a &amp; 0 &amp; d &amp; e\\
b &amp; -d &amp; 0 &amp; f\\
-c &amp; -e &amp; -f &amp; 0
\end{vmatrix}}\\
\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[{\scriptstyle \begin{align} &amp; 1+a^{2}+b^{2}+c^{2}+ &amp;  &amp; 2(-bd+a+ec+pf) &amp;  &amp; 2(-ad-b+fc-pe) &amp;  &amp; 2(pd+fb-ea+c)\\
 &amp; \quad d^{2}+e^{2}+f^{2}+p^{2} &amp;  &amp; 1+a^{2}-b^{2}-c^{2} &amp;  &amp; 2(-d-ab+pc-fe) &amp;  &amp; 2(fd+pb+ca-e)\\
 &amp; 2(bd+a-ec+pf) &amp;  &amp; \quad-d^{2}-e^{2}+f^{2}+p^{2} &amp;  &amp; 1-a^{2}+b^{2}-c^{2} &amp;  &amp; 2(-ed-cb+pa-f)\\
 &amp; 2(ad-b-fc-pe) &amp;  &amp; 2(d-ab-pc-fe) &amp;  &amp; \quad-d^{2}+e^{2}-f^{2}+p^{2} &amp;  &amp; 1-a^{2}-b^{2}+-c^{2}\\
 &amp; 2(pd-fb+ea+c) &amp;  &amp; 2(fd-pb+ca+e) &amp;  &amp; 2(-ed-cb-pa+f) &amp;  &amp; \quad+d^{2}-e^{2}-f^{2}+p^{2}
\end{align}
}\right]\cdot\mathbf{x}\\
\left(\begin{align}\kappa &amp; =1-a^{2}-b^{2}-c^{2}+d^{2}+e^{2}+f^{2}-p^{2}\\
p &amp; =af+be+cd
\end{align}
\right)
\end{matrix}&lt;/math&gt;|{{equationRef|5c}}}}

The transformation of a binary quadratic form of which Lorentz transformation ({{equationNote|5a}}) is a special case was given by [[#Hermite|Hermite (1854)]], equations containing Lorentz transformations ({{equationNote|5a}}, {{equationNote|5b}}, {{equationNote|5c}}) as special cases were given by [[#Cayley|Cayley (1855)]], Lorentz transformation ({{equationNote|5a}}) was given (up to a sign change) by [[#Laguerre|Laguerre (1882)]], [[#Darboux2|Darboux (1887)]], [[#Smith|Smith (1900)]] in relation to Laguerre geometry, and Lorentz transformation ({{equationNote|5b}}) was given by [[#Bachmann|Bachmann (1869)]]. In relativity, equations similar to ({{equationNote|5b}}, {{equationNote|5c}}) were first employed by [[#Borel|Borel (1913)]] to represent Lorentz transformations.

==Historical notation==
==={{anchor|Euler}} Euler (1771) – Euler-Rodrigues parameter===

Euler (1771) demonstrated the invariance of quadratic forms in terms of sum of squares under a linear substitution and its coefficients, now known as [[w:orthogonal transformation]]. The transformation in three dimensions was given as

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}+Z^{2}=x^{2}+y^{2}+z^{2}\\
\hline \begin{align}X &amp; =Ax+By+Cz\\
Y &amp; =Dx+Ey+Fz\\
Z &amp; =Gx+Hy+Iz
\end{align}
\begin{matrix}\left|{\scriptstyle \begin{align}1 &amp; =AA+DD+GG\\
1 &amp; =BB+EE+HH\\
1 &amp; =CC+FF+II\\
0 &amp; =AB+DE+GH\\
0 &amp; =AG+DF+GI\\
0 &amp; =BC+EF+HI
\end{align}
}\right.\end{matrix}\end{matrix}&lt;/math&gt;

in which the coefficiens ''A,B,C,D,E,F,G,H,I'' were related by Euler to four arbitrary parameter ''p,q,r,s'', which where rediscovered by [[w:Olinde Rodrigues]] (1840) who related them to rotation angles&lt;ref group=M&gt;Rodrigues (1840), p. 405&lt;/ref&gt;:&lt;ref group=M&gt;Euler (1771), p. 101&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}A &amp; =\frac{pp+qq-rr-ss}{u} &amp; B &amp; =\frac{2pq+2ps}{u} &amp; C &amp; =\frac{2qs-2pr}{u}\\
D &amp; =\frac{2qr-2ps}{u} &amp; E &amp; =\frac{pp-qq+rr-ss}{u} &amp; F &amp; =\frac{2pq+2rs}{u}\\
G &amp; =\frac{2qs+2pr}{u} &amp; H &amp; =\frac{2rs-2pq}{u} &amp; I &amp; =\frac{pp-qq-rr+ss}{u}
\end{align}
\\
(u=pp+qq+rr+ss)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These are now called (improper) [[w:Euler–Rodrigues formula|Euler–Rodrigues parameters]] in line with equation ({{equationNote|Q3}}).&lt;/p&gt;

==={{anchor|Cayley}} Cayley (1846–1884) – Cayley–Hermite transformation===
The [[w:Euler–Rodrigues formula|Euler–Rodrigues parameters]] discovered by [[#Euler|Euler (1871) and Rodrigues (1840)]] leaving invariant &lt;math&gt;x_{0}^{2}+x_{1}^{2}+x_{2}^{2}&lt;/math&gt; were extended to &lt;math&gt;x_{0}^{2}+\dots+x_{n}^{2}&lt;/math&gt; by [[w:Arthur Cayley]] (1846) as a byproduct of what is now called the [[w:Cayley transform]] using the method of skew–symmetric coefficients.&lt;ref group=M name=cayl1&gt;Cayley (1846)&lt;/ref&gt; Following Cayley's methods, a general transformation for quadratic forms into themselves in three (1853) and arbitrary (1854) dimensions was provided by [[#Hermite|Hermite (1853, 1854)]]. Hermite's formula was simplified and brought into matrix form equivalent to ({{equationNote|Q2}}) by Cayley (1855a)&lt;ref group=M&gt;Cayley (1855a), p. 288&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle (\mathrm{x,y,z}\dots)=\left(\left|\begin{matrix}a, &amp; h, &amp; g &amp; \dots\\
h, &amp; b, &amp; f &amp; \dots\\
g, &amp; f, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|^{-1}\left|\begin{matrix}a, &amp; h-\nu, &amp; g+\mu &amp; \dots\\
h+\nu, &amp; b, &amp; f-\lambda &amp; \dots\\
g-\mu, &amp; f+\lambda, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|\left|\begin{matrix}a, &amp; h+\nu, &amp; g-\mu &amp; \dots\\
h-\nu, &amp; b, &amp; f+\lambda &amp; \dots\\
g+\mu, &amp; f-\lambda, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|^{-1}\left|\begin{matrix}a, &amp; h, &amp; g &amp; \dots\\
h, &amp; b, &amp; f &amp; \dots\\
g, &amp; f, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|\right)\ ^{\frown}(x,y,z\dots)}&lt;/math&gt;

which he abbreviated in 1858, where &lt;math&gt;\Upsilon&lt;/math&gt; is any skew-symmetric matrix:&lt;ref group=M name=cayl1858&gt;Cayley (1858), p. 39&lt;/ref&gt;&lt;ref&gt;Hawkins (2013), p. 214&lt;/ref&gt;

:&lt;math&gt;(\mathrm{x,y,z})=\left(\Omega^{-1}(\Omega-\Upsilon)(\Omega+\Upsilon)^{-1}\Omega\right)(x,y,z)&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The Cayley–Hermite transformation becomes equivalent to the Lorentz transformation ({{equationNote|5a}}) by setting Ω=diag(-1,1) and  ({{equationNote|5b}}) by setting Ω=diag(-1,1,1) and ({{equationNote|5c}}) by setting Ω=diag(-1,1,1,1).&lt;/p&gt;

Using the parameters of (1855a), Cayley in a subsequent paper (1855b) particularly discussed several special cases, such as:&lt;ref group=M&gt;Cayley (1855b), p. 210&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}=ax^{2}+by^{2}\\
\hline (\mathrm{x,y})=\frac{1}{ab+\nu^{2}}\cdot\left[\begin{matrix}ab-\nu^{2}, &amp; -2\nu b\\
2\nu a, &amp; ab-\nu^{2}
\end{matrix}\right](x,y)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to the Lorentz transformation ({{equationNote|5a}}) in 1+1 dimensions by setting ''(a,b)''=(-1,1) and Lorentz boost [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] by additionally setting &lt;math&gt;\tfrac{2\nu}{1+\nu^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

or:&lt;ref group=M&gt;Cayley (1855b), p. 211&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}+c\mathrm{z}^{2}=ax^{2}+by^{2}+cz^{2}\\
\hline (\mathrm{x,y,z})=\frac{1}{abc+a\lambda^{2}+b\mu^{2}+c\nu^{2}}\times\left[\begin{matrix}abc+a\lambda^{2}-b\mu^{2}-c\nu^{2}, &amp; 2(\lambda\mu-c\nu)b &amp; 2(\nu\lambda+b\mu)c\\
2(\lambda\mu+c\nu)a, &amp; abc-a\lambda^{2}+b\mu^{2}-c\nu^{2} &amp; 2(\mu\nu-a\lambda)c\\
2(\nu\lambda-b\mu)a &amp; 2(\mu\nu+a\lambda)b &amp; abc-a\lambda^{2}-b\mu^{2}-c\nu^{2}
\end{matrix}\right](x,y,z)
\end{matrix}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to the Lorentz transformation ({{equationNote|5b}}) by setting ''(a,b,c)''=(-1,1,1).&lt;/p&gt;

or:&lt;ref group=M&gt;Cayley (1855b), pp. 212–213&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}+c\mathrm{z}^{2}+d\mathrm{w}^{2}=ax^{2}+by^{2}+cz^{2}+dw^{2}\\
\hline (\mathrm{x,y,z,w})=\frac{1}{k}\cdot\left[\begin{align} &amp; abcd-bc\rho^{2}+ca\sigma^{2}+ab\tau^{2}+ad\lambda^{2} &amp;  &amp; 2b\left(-cd\nu-\tau\phi+d\lambda\mu-c\rho\sigma\right),\\
 &amp; \quad-bd\mu^{2}-cd\nu^{2}-\phi^{2}, &amp;  &amp; abcd+bc\rho^{2}-ca\sigma^{2}+ab\tau^{2}-ad\lambda^{2}\\
 &amp; 2a\left(cd\nu+\tau\phi+d\lambda\mu-c\rho\sigma\right), &amp;  &amp; \quad+bd\mu^{2}-cd\nu^{2}-\phi^{2},\\
 &amp; 2a\left(-bd\mu-\sigma\phi+d\lambda\nu-b\rho\tau\right), &amp;  &amp; 2b\left(ad\lambda+\rho\phi+d\mu\nu-a\sigma\tau\right),\\
 &amp; 2a\left(bc\rho+\lambda\phi+c\nu\sigma-b\mu\tau\right), &amp;  &amp; 2b\left(ac\sigma+\mu\phi-c\nu\rho+a\lambda\tau\right),\\
\\
 &amp; \quad2c\left(bd\mu+\sigma\phi+d\lambda\nu-b\rho\tau\right), &amp;  &amp; \quad2d\left(-bc\rho-\lambda\phi+c\nu\sigma-b\mu\tau\right)\\
 &amp; \quad2c\left(-ad\lambda-\rho\phi+d\mu\nu-a\sigma\tau\right), &amp;  &amp; \quad2d\left(-ac\sigma-\mu\phi-c\nu\rho+a\lambda\tau\right)\\
 &amp; \quad abcd+bc\rho^{2}+ca\sigma^{2}-ab\tau^{2}-ad\lambda^{2} &amp;  &amp; \quad2d\left(-ab\tau-\nu\phi+b\mu\rho-a\lambda\sigma\right)\\
 &amp; \quad\quad-bd\mu^{2}+cd\nu^{2}-\phi^{2}, &amp;  &amp; \quad abcd-bc\rho^{2}-ca\sigma^{2}-ab\tau^{2}+ad\lambda^{2}\\
 &amp; \quad2c\left(ab\tau+\nu\phi+b\mu\rho-a\lambda\sigma\right), &amp;  &amp; \quad\quad+bd\mu^{2}+cd\nu^{2}-\phi^{2},
\end{align}
\right]\cdot(x,y,z,w)\\
\left(\begin{align}k &amp; =abcd+bc\rho^{2}+ca\sigma^{2}+ab\tau^{2}+ad\lambda^{2}+bd\mu^{2}+cd\nu^{2}+\phi^{2}\\
\phi &amp; =\lambda\rho+\mu\sigma+\nu\tau
\end{align}
\right)
\end{matrix}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to the Lorentz transformation ({{equationNote|5c}}) by setting ''(a,b,c,d)''=(-1,1,1,1).&lt;/p&gt;

==={{anchor|Hermite}} Hermite (1853, 1854) – Cayley–Hermite transformation===

[[w:Charles Hermite]] (1853) extended the number theoretical work of [[../Lorentz transformation (general)#Gauss2|E:Gauss (1801)]] and others (including himself) by additionally analyzing ''indefinite'' ternary quadratic forms that can be transformed into the Lorentz interval ±''(x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;)'', and by using [[#Cayley|Cayley's (1846)]] method of skew–symmetric coefficients he derived transformations leaving invariant almost all types of ternary quadratic forms.&lt;ref group=M name=herm&gt;Hermite (1853/54a), p. 307ff.&lt;/ref&gt; This was generalized by him in 1854 to ''n'' dimensions:&lt;ref group=M&gt;Hermite (1854b), p. 64&lt;/ref&gt;&lt;ref&gt;Hawkins (2013), p. 212&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}f\left(X_{1},X_{2},\dots\right)=f\left(x_{1},x_{2},\dots\right)\\
\hline X_{r}=2\xi_{r}-x_{r}=\xi_{r}-\frac{1}{2}\sum_{s=1}^{n}\lambda_{r,s}\frac{df}{d\xi_{s}}\\
\left(\lambda_{r,s}=-\lambda_{s,r},\ \lambda_{r,r}=0\right)
\end{matrix}&lt;/math&gt;

This result was subsequently expressed in matrix form by [[#Cayley|Cayley (1855)]], while [[w:Ferdinand Georg Frobenius]] (1877) added some modifications in order to include some special cases of quadratic forms that cannot be dealt with by the Cayley–Hermite transformation.&lt;ref group=M&gt;Frobenius (1877)&lt;/ref&gt;&lt;ref&gt;Hawkins (2013), pp. 219ff&lt;/ref&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to equation ({{equationNote|Q2}}), and becomes the Lorentz transformation by setting the coefficients of the quadratic form ''f'' to diag(-1,1,...1).&lt;/p&gt;

For instance, the special case of the transformation of a binary quadratic form into itself was given by Hermite as follows:&lt;ref group=M&gt;Hermite (1854b), pp. 64–65&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}f=ax^{2}+2bxy+cy^{2}\\
\hline \begin{align}X &amp; =\frac{\left(1-2\lambda b+\lambda^{2}D\right)x-2\lambda cy}{1-\lambda^{2}D} &amp;  &amp; =x(t-bu)-cuy\\
Y &amp; =\frac{2\lambda ax+\left(1+2\lambda b+\lambda^{2}D\right)y}{1-\lambda^{2}D} &amp;  &amp; =xau+(t+bu)y
\end{align}
\\
\left(b^{2}-ac=D,\ t=\frac{1+\lambda^{2}D}{1-\lambda^{2}D},\ u=\frac{2\lambda}{1-\lambda^{2}D},\ t^{2}-Du^{2}=1\right)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to Lorentz boost ({{equationNote|5a}}) by setting ''(a,b,c)''=(-1,0,1) and Lorentz boost of velocity by additionally setting &lt;math&gt;\tfrac{2\lambda}{1+\lambda^{2}}=\tfrac{v}{c}&lt;/math&gt; which produces ''t''=γ and ''u''=βγ.&lt;/p&gt;

=== {{anchor|Bachmann}} Bachmann (1869) – Cayley–Hermite transformation ===

[[w:Paul Gustav Heinrich Bachmann]] (1869) adapted [[#Hermite|Hermite's (1853/54)]] transformation of ternary quadratic forms to the case of integer transformations. He particularly analyzed the Lorentz interval and its transformation, and also alluded to the analogue result of [[#Gauss3|Gauss (1800)]] in terms of Cayley–Klein parameters, while Bachmann formulated his result in terms of the Cayley–Hermite transformation:&lt;ref group=M&gt;Bachmann (1869), p. 303&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+x^{\prime2}-x^{\prime\prime2}\\
\hline \begin{align}\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X &amp; =\left(p^{2}-q^{2}+q^{\prime2}-q^{\prime\prime2}\right)x-2(pq''+qq')x'-2(pq'+qq'')x''\\
\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X' &amp; =2(pq''-qq')x+\left(p^{2}+q^{2}-q^{\prime2}-q^{\prime\prime2}\right)x'+2(pq-q'q'')x''\\
\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X'' &amp; =-2(pq'-qq')x+2(pq+q'q'')x'+\left(p^{2}+q^{2}+q^{\prime2}+q^{\prime\prime2}\right)x''
\end{align}
\end{matrix}&lt;/math&gt;

He described this transformation in 1898 in the first part of his &quot;arithmetics of quadratic forms&quot; as well.&lt;ref&gt;Bachmann (1898), pp. 101–102&lt;/ref&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|5b}}), producing the relation &lt;math&gt;X^{2}+X^{\prime2}-X^{\prime\prime2}=x^{2}+x^{\prime2}-x^{\prime\prime2}&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Laguerre}} Laguerre (1882) – Laguerre inversion===

After previous work by [[w:Albert Ribaucour]] (1870),&lt;ref group=M&gt;Ribaucour (1870)&lt;/ref&gt; a transformation which transforms oriented spheres into oriented spheres, oriented planes into oriented planes, and oriented lines into oriented lines, was explicitly formulated by [[w:Edmond Laguerre]] (1882) as &quot;[[w:Spherical wave transformation#Transformation by reciprocal directions|transformation by reciprocal directions]]&quot; which was later called &quot;Laguerre inversion/transformation&quot;. It can be seen as a special case of the conformal group in terms of [[../Lorentz transformation (conformal)#Lie|E:Lie's transformations of oriented spheres]]. In two dimensions the transformation or oriented lines has the form (''R'' being the radius):&lt;ref group=M name=laguerre&gt;Laguerre (1882), pp. 550–551.&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}D' &amp; =\frac{D\left(1+\alpha^{2}\right)-2\alpha R}{1-\alpha^{2}}\\
R' &amp; =\frac{2\alpha D-R\left(1+\alpha^{2}\right)}{1-\alpha^{2}}
\end{align}
\right|\begin{align}D^{2}-D^{\prime2} &amp; =R^{2}-R^{\prime2}\\
D-D' &amp; =\alpha(R-R')\\
D+D' &amp; =\frac{1}{\alpha}(R+R')
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change) to Lorentz transformation ({{equationNote|5a}}) in terms of Cayley–Hermite parameters (even though Laguerre didn't use the Cayley-Hermite transformation ({{equationNote|Q2}})). The Lorentz boost of velocity follows with &lt;math&gt;\tfrac{2\alpha}{1+\alpha^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Darboux2}} Darboux (1887) – Laguerre inversion===

Following [[#Laguerre|Laguerre (1882)]], [[w:Gaston Darboux]] (1887) presented the Laguerre inversions in four dimensions using coordinates ''x,y,z,R'':&lt;ref group=M name=darboux&gt;Darboux (1887)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{\prime2}+y^{\prime2}+z^{\prime2}-R^{\prime2}=x^{2}+y^{2}+z^{2}-R^{2}\\
\hline \begin{align}x' &amp; =x, &amp; z' &amp; =\frac{1+k^{2}}{1-k^{2}}z-\frac{2kR}{1-k^{2}},\\
y' &amp; =y, &amp; R' &amp; =\frac{2kz}{1-k^{2}}-\frac{1+k^{2}}{1-k^{2}}R,
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change for ''R'') to Lorentz transformation ({{equationNote|5a}}) in terms of Cayley–Hermite parameters (even though Darboux didn't use the Cayley-Hermite transformation ({{equationNote|Q2}})). The Lorentz boost of velocity follows with &lt;math&gt;\tfrac{2k}{1+k^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

Darboux rewrote these equations as follows:

:&lt;math&gt;\begin{matrix}x^{\prime2}+y^{\prime2}+z^{\prime2}-R^{\prime2}=x^{2}+y^{2}+z^{2}-R^{2}\\
\hline \begin{align}z'+R' &amp; =\frac{1+k}{1-k}(z-R)\\
z'-R' &amp; =\frac{1-k}{1+k}(z+R)
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change for ''R'') to a squeeze mapping.&lt;/p&gt;

==={{anchor|Smith}} Smith (1900) – Laguerre inversion===

[[w:Percey F. Smith]] (1900) followed [[#Laguerre|Laguerre (1882)]] and [[#Darboux2|Darboux (1887)]] and defined the Laguerre inversion as follows:&lt;ref group=M&gt;Smith (1900), p. 159&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{\prime2}-p^{2}=R^{\prime2}-R^{2}\\
\hline p'=\frac{\kappa^{2}+1}{\kappa^{2}-1}p-\frac{2\kappa}{\kappa^{2}-1}R,\quad R'=\frac{2\kappa}{\kappa^{2}-1}p-\frac{\kappa^{2}+1}{\kappa^{2}-1}R
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change) to Lorentz transformation ({{equationNote|5a}}) in terms of Cayley–Hermite parameters (even though Smith didn't use the Cayley-Hermite transformation ({{equationNote|Q2}})). The Lorentz boost of velocity follows with &lt;math&gt;\tfrac{2\kappa}{1+\kappa^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Borel}} Borel (1913–14) – Cayley–Hermite parameter===

Borel (1913) started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and [[#Cayley|Cayley's (1846)]] parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions equivalent to ({{equationNote|5b}}):&lt;ref group=R&gt;Borel (1913/14), p. 39&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}-z^{2}-1=0\\
\hline {\scriptstyle \begin{align}\delta a &amp; =\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta b &amp; =2(\lambda\mu+\nu\rho), &amp; \delta c &amp; =-2(\lambda\nu+\mu\rho),\\
\delta a' &amp; =2(\lambda\mu-\nu\rho), &amp; \delta b' &amp; =-\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta c' &amp; =2(\lambda\rho-\mu\nu),\\
\delta a'' &amp; =2(\lambda\nu-\mu\rho), &amp; \delta b'' &amp; =2(\lambda\rho+\mu\nu), &amp; \delta c'' &amp; =-\left(\lambda^{2}+\mu^{2}+\nu^{2}+\rho^{2}\right),
\end{align}
}\\
\left(\delta=\lambda^{2}+\mu^{2}-\rho^{2}-\nu^{2}\right)\\
\lambda=\nu=0\rightarrow\text{Hyperbolic rotation}
\end{matrix}&lt;/math&gt;

In four dimensions equivalent to ({{equationNote|5c}}):&lt;ref group=R&gt;Borel (1913/14), p. 41&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}-\left(t_{1}-t_{2}\right)^{2}\\
\hline {\scriptstyle \begin{align} &amp; \left(\mu^{2}+\nu^{2}-\alpha^{2}\right)\cos\varphi+\left(\lambda^{2}-\beta^{2}-\gamma^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi-\gamma\operatorname{sh}{\theta}\\
 &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi+\gamma\operatorname{sh}{\theta} &amp;  &amp; \left(\mu^{2}+\nu^{2}-\beta^{2}\right)\cos\varphi+\left(\mu^{2}-\alpha^{2}-\gamma^{2}\right)\operatorname{ch}{\theta}\\
 &amp; -(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi-\beta\operatorname{sh}{\theta} &amp;  &amp; -(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\lambda\sin\varphi+\alpha\operatorname{sh}{\theta}\\
 &amp; (\gamma\mu-\beta\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta} &amp;  &amp; -(\alpha\nu-\lambda\gamma)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
\\
 &amp; \quad-(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi+\beta\operatorname{sh}{\theta} &amp;  &amp; \quad(\beta\nu-\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta}\\
 &amp; \quad-(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})-\lambda\sin\varphi-\alpha\operatorname{sh}{\theta} &amp;  &amp; \quad(\lambda\gamma-\alpha\nu)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
 &amp; \quad\left(\lambda^{2}+\mu^{2}-\gamma^{2}\right)\cos\varphi+\left(\nu^{2}-\alpha^{2}-\beta^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; \quad(\alpha\mu-\beta\lambda)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta}\\
 &amp; \quad(\beta\gamma-\alpha\mu)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta} &amp;  &amp; \quad-\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\cos\varphi+\left(\lambda^{2}+\mu^{2}+\nu^{2}\right)\operatorname{ch}{\theta}
\end{align}
}\\
\left(\alpha^{2}+\beta^{2}+\gamma^{2}-\lambda^{2}-\mu^{2}-\nu^{2}=-1\right)
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bach69}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay46gau}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay55quad}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay55gau}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay58quad}} 
*{{#section:History of Topics in Special Relativity/mathsource|dar87cou}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul71}} 
*{{#section:History of Topics in Special Relativity/mathsource|fro77}} 
*{{#section:History of Topics in Special Relativity/mathsource|herm53}} 
*{{#section:History of Topics in Special Relativity/mathsource|herm54}} 
*{{#section:History of Topics in Special Relativity/mathsource|lagu81}} 
*{{#section:History of Topics in Special Relativity/mathsource|lagu82}} 
*{{#section:History of Topics in Special Relativity/mathsource|rib70}} 
*{{#section:History of Topics in Special Relativity/mathsource|rod40}} 
*{{#section:History of Topics in Special Relativity/mathsource|smi00}}

===Historical relativity sources===
{{reflist|3|group=R}}
{{#section:History of Topics in Special Relativity/relsource|bor14}}
{{#section:History of Topics in Special Relativity/relsource|brill25}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L6}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
      <sha1>sf8mg0d72z6zulx195zzvpzh7hrpplu</sha1>
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    <revision>
      <id>2411870</id>
      <parentid>2411868</parentid>
      <timestamp>2022-08-04T17:26:17Z</timestamp>
      <contributor>
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        <id>52339</id>
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      <comment>/* {{anchor|Cayley}} Cayley (1846–1884) – Cayley–Hermite transformation */</comment>
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      <text bytes="26920" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via Cayley–Hermite transformation ==

The [[../Lorentz transformation (general)#math_Q1|E:general transformation '''(Q1)''']] of any quadratic form into itself can also be given using ''arbitrary'' parameters based on the [[w:Cayley transform]] ('''I'''-'''T''')&lt;sup&gt;−1&lt;/sup&gt;·('''I'''+'''T'''), where '''I''' is the [[w:identity matrix]], '''T''' an arbitrary [[w:antisymmetric matrix]], and by adding '''A''' as symmetric matrix defining the quadratic form (there is no primed '''A' ''' because the coefficients are assumed to be the same on both sides):&lt;ref&gt;Hawkins (2013), pp. 210–214&lt;/ref&gt;&lt;ref&gt;Meyer (1899), p. 329&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}q=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}=q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}\cdot\mathbf{x}'\\
\hline \\
\mathbf{x}=(\mathbf{I}-\mathbf{T}\cdot\mathbf{A})^{-1}\cdot(\mathbf{I}+\mathbf{T}\cdot\mathbf{A})\cdot\mathbf{x}'\\
\text{or}\\
\mathbf{x}=\mathbf{A}^{-1}\cdot(\mathbf{A}-\mathbf{T})\cdot(\mathbf{A}+\mathbf{T})^{-1}\cdot\mathbf{A}\cdot\mathbf{x}'
\end{matrix}&lt;/math&gt;|{{equationRef|Q2}}}}

After [[#Cayley|Cayley (1846)]] introduced transformations related to sums of positive squares, [[#Hermite|Hermite (1853/54, 1854)]] derived transformations for arbitrary quadratic forms, whose result was reformulated in terms of matrices by [[#Cayley|Cayley (1855a, 1855b)]]. For instance, the choice '''A'''=diag(1,1,1) gives an orthogonal transformation which can be used to describe spatial rotations corresponding to the [[w:Euler–Rodrigues formula|w:Euler-Rodrigues parameter]]s ''[a,b,c,d]'' discovered by [[#Euler|Euler (1771) and Rodrigues (1840)]], which can be interpreted as the coefficients of [[w:quaternion]]s. Setting ''d=1'', the equations have the form:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(1,1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a &amp; -b\\
-a &amp; 0 &amp; c\\
b &amp; -c &amp; 0
\end{vmatrix}}\\
\hline x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[\begin{matrix}1-a^{2}-b^{2}+c^{2} &amp; 2(bc-a) &amp; 2(ac+b)\\
2(bc+a) &amp; 1-a^{2}+b^{2}-c^{2} &amp; 2(ab-c)\\
2(ac-b) &amp; 2(ab+c) &amp; 1+a^{2}-b^{2}-c^{2}
\end{matrix}\right]\cdot\mathbf{x}\\
\left(\kappa=1+a^{2}+b^{2}+c^{2}\right)
\end{matrix}&lt;/math&gt;|{{equationRef|Q3}}}}

Also the Lorentz interval and the general Lorentz transformation in any dimension can be produced by the Cayley–Hermite formalism.&lt;ref group=R&gt;Borel (1914), pp. 39–41&lt;/ref&gt;&lt;ref group=R&gt;Brill (1925)&lt;/ref&gt;&lt;ref&gt;Klein (1928), Chapter III, § 2B&lt;/ref&gt;&lt;ref&gt;Lorente (2003), section 3.3&lt;/ref&gt; For instance, the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']] with ''n''=1 follows from ({{equationNote|Q2}}) with:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(-1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a\\
-a &amp; 0
\end{vmatrix}}\\
\hline -x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{1-a^{2}}\left[\begin{matrix}1+a^{2} &amp; -2a\\
-2a &amp; 1+a^{2}
\end{matrix}\right]\cdot\mathbf{x}
\end{matrix}\Rightarrow\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}x_{0} &amp; =x_{0}^{\prime}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}}+x_{1}^{\prime}\frac{2\beta_{0}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{x_{0}^{\prime}\left(1+\beta_{0}^{2}\right)+x_{1}^{\prime}2\beta_{0}}{1-\beta_{0}^{2}}\\
x_{1} &amp; =x_{0}^{\prime}\frac{2\beta_{0}}{1-\beta_{0}^{2}}+x_{1}^{\prime}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{x_{0}^{\prime}2\beta_{0}+x_{1}^{\prime}\left(1+\beta_{0}^{2}\right)}{1-\beta_{0}^{2}}\\
\\
x_{0}^{\prime} &amp; =x_{0}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}}-x_{1}\frac{2\beta_{0}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{x_{0}\left(1+\beta_{0}^{2}\right)-x_{1}2\beta_{0}}{1-\beta_{0}^{2}}\\
x_{1}^{\prime} &amp; =-x_{0}\frac{2\beta_{0}}{1-\beta_{0}^{2}}+x_{1}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{-x_{0}2\beta_{0}+x_{1}\left(1+\beta_{0}^{2}\right)}{1-\beta_{0}^{2}}
\end{align}
\right|{\scriptstyle \begin{align}\frac{2\beta_{0}}{1+\beta_{0}^{2}} &amp; =\beta\\
\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} &amp; =\gamma\\
\frac{2\beta_{0}}{1-\beta_{0}^{2}} &amp; =\beta\gamma
\end{align}
}
\end{matrix}&lt;/math&gt;|{{equationRef|5a}}}}

This becomes [[../Lorentz transformation (velocity)#math_4a|E:Lorentz boost '''(4a)''']] by setting &lt;math&gt;\tfrac{2a}{1+a^{2}}=\tfrac{v}{c}&lt;/math&gt;, which is equivalent to the relation &lt;math&gt;\tfrac{2\beta_{0}}{1+\beta_{0}^{2}}=\tfrac{v}{c}&lt;/math&gt; known from [[w:Loedel diagram]]s, thus ({{equationNote|5a}}) can be interpreted as a Lorentz boost from the viewpoint of a &quot;median frame&quot; in which two other inertial frames are moving with equal speed &lt;math&gt;\beta_0&lt;/math&gt; in opposite directions.

Furthermore, Lorentz transformation [[../Lorentz transformation (general)#math_1a|E:'''(1a)''']]) with ''n''=2 is given by:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(-1,1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a &amp; -b\\
-a &amp; 0 &amp; c\\
b &amp; -c &amp; 0
\end{vmatrix}}\\
\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[\begin{matrix}1+a^{2}+b^{2}+c^{2} &amp; -2(bc-a) &amp; -2(ac+b)\\
2(bc+a) &amp; 1+a^{2}-b^{2}-c^{2} &amp; 2(ab-c)\\
2(ac-b) &amp; -2(ab-c) &amp; 1-a^{2}+b^{2}-c^{2}
\end{matrix}\right]\cdot\mathbf{x}\\
\left(\kappa=1-a^{2}-b^{2}+c^{2}\right)
\end{matrix}&lt;/math&gt;|{{equationRef|5b}}}}

or using ''n''=3:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(-1,1,1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a &amp; -b &amp; c\\
-a &amp; 0 &amp; d &amp; e\\
b &amp; -d &amp; 0 &amp; f\\
-c &amp; -e &amp; -f &amp; 0
\end{vmatrix}}\\
\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[{\scriptstyle \begin{align} &amp; 1+a^{2}+b^{2}+c^{2}+ &amp;  &amp; 2(-bd+a+ec+pf) &amp;  &amp; 2(-ad-b+fc-pe) &amp;  &amp; 2(pd+fb-ea+c)\\
 &amp; \quad d^{2}+e^{2}+f^{2}+p^{2} &amp;  &amp; 1+a^{2}-b^{2}-c^{2} &amp;  &amp; 2(-d-ab+pc-fe) &amp;  &amp; 2(fd+pb+ca-e)\\
 &amp; 2(bd+a-ec+pf) &amp;  &amp; \quad-d^{2}-e^{2}+f^{2}+p^{2} &amp;  &amp; 1-a^{2}+b^{2}-c^{2} &amp;  &amp; 2(-ed-cb+pa-f)\\
 &amp; 2(ad-b-fc-pe) &amp;  &amp; 2(d-ab-pc-fe) &amp;  &amp; \quad-d^{2}+e^{2}-f^{2}+p^{2} &amp;  &amp; 1-a^{2}-b^{2}+-c^{2}\\
 &amp; 2(pd-fb+ea+c) &amp;  &amp; 2(fd-pb+ca+e) &amp;  &amp; 2(-ed-cb-pa+f) &amp;  &amp; \quad+d^{2}-e^{2}-f^{2}+p^{2}
\end{align}
}\right]\cdot\mathbf{x}\\
\left(\begin{align}\kappa &amp; =1-a^{2}-b^{2}-c^{2}+d^{2}+e^{2}+f^{2}-p^{2}\\
p &amp; =af+be+cd
\end{align}
\right)
\end{matrix}&lt;/math&gt;|{{equationRef|5c}}}}

The transformation of a binary quadratic form of which Lorentz transformation ({{equationNote|5a}}) is a special case was given by [[#Hermite|Hermite (1854)]], equations containing Lorentz transformations ({{equationNote|5a}}, {{equationNote|5b}}, {{equationNote|5c}}) as special cases were given by [[#Cayley|Cayley (1855)]], Lorentz transformation ({{equationNote|5a}}) was given (up to a sign change) by [[#Laguerre|Laguerre (1882)]], [[#Darboux2|Darboux (1887)]], [[#Smith|Smith (1900)]] in relation to Laguerre geometry, and Lorentz transformation ({{equationNote|5b}}) was given by [[#Bachmann|Bachmann (1869)]]. In relativity, equations similar to ({{equationNote|5b}}, {{equationNote|5c}}) were first employed by [[#Borel|Borel (1913)]] to represent Lorentz transformations.

==Historical notation==
==={{anchor|Euler}} Euler (1771) – Euler-Rodrigues parameter===

Euler (1771) demonstrated the invariance of quadratic forms in terms of sum of squares under a linear substitution and its coefficients, now known as [[w:orthogonal transformation]]. The transformation in three dimensions was given as

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}+Z^{2}=x^{2}+y^{2}+z^{2}\\
\hline \begin{align}X &amp; =Ax+By+Cz\\
Y &amp; =Dx+Ey+Fz\\
Z &amp; =Gx+Hy+Iz
\end{align}
\begin{matrix}\left|{\scriptstyle \begin{align}1 &amp; =AA+DD+GG\\
1 &amp; =BB+EE+HH\\
1 &amp; =CC+FF+II\\
0 &amp; =AB+DE+GH\\
0 &amp; =AG+DF+GI\\
0 &amp; =BC+EF+HI
\end{align}
}\right.\end{matrix}\end{matrix}&lt;/math&gt;

in which the coefficiens ''A,B,C,D,E,F,G,H,I'' were related by Euler to four arbitrary parameter ''p,q,r,s'', which where rediscovered by [[w:Olinde Rodrigues]] (1840) who related them to rotation angles&lt;ref group=M&gt;Rodrigues (1840), p. 405&lt;/ref&gt;:&lt;ref group=M&gt;Euler (1771), p. 101&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}A &amp; =\frac{pp+qq-rr-ss}{u} &amp; B &amp; =\frac{2pq+2ps}{u} &amp; C &amp; =\frac{2qs-2pr}{u}\\
D &amp; =\frac{2qr-2ps}{u} &amp; E &amp; =\frac{pp-qq+rr-ss}{u} &amp; F &amp; =\frac{2pq+2rs}{u}\\
G &amp; =\frac{2qs+2pr}{u} &amp; H &amp; =\frac{2rs-2pq}{u} &amp; I &amp; =\frac{pp-qq-rr+ss}{u}
\end{align}
\\
(u=pp+qq+rr+ss)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These are now called (improper) [[w:Euler–Rodrigues formula|Euler–Rodrigues parameters]] in line with equation ({{equationNote|Q3}}).&lt;/p&gt;

==={{anchor|Cayley}} Cayley (1846–1855) – Cayley–Hermite transformation===
The [[w:Euler–Rodrigues formula|Euler–Rodrigues parameters]] discovered by [[#Euler|Euler (1871) and Rodrigues (1840)]] leaving invariant &lt;math&gt;x_{0}^{2}+x_{1}^{2}+x_{2}^{2}&lt;/math&gt; were extended to &lt;math&gt;x_{0}^{2}+\dots+x_{n}^{2}&lt;/math&gt; by [[w:Arthur Cayley]] (1846) as a byproduct of what is now called the [[w:Cayley transform]] using the method of skew–symmetric coefficients.&lt;ref group=M name=cayl1&gt;Cayley (1846)&lt;/ref&gt; Following Cayley's methods, a general transformation for quadratic forms into themselves in three (1853) and arbitrary (1854) dimensions was provided by [[#Hermite|Hermite (1853, 1854)]]. Hermite's formula was simplified and brought into matrix form equivalent to ({{equationNote|Q2}}) by Cayley (1855a)&lt;ref group=M&gt;Cayley (1855a), p. 288&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle (\mathrm{x,y,z}\dots)=\left(\left|\begin{matrix}a, &amp; h, &amp; g &amp; \dots\\
h, &amp; b, &amp; f &amp; \dots\\
g, &amp; f, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|^{-1}\left|\begin{matrix}a, &amp; h-\nu, &amp; g+\mu &amp; \dots\\
h+\nu, &amp; b, &amp; f-\lambda &amp; \dots\\
g-\mu, &amp; f+\lambda, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|\left|\begin{matrix}a, &amp; h+\nu, &amp; g-\mu &amp; \dots\\
h-\nu, &amp; b, &amp; f+\lambda &amp; \dots\\
g+\mu, &amp; f-\lambda, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|^{-1}\left|\begin{matrix}a, &amp; h, &amp; g &amp; \dots\\
h, &amp; b, &amp; f &amp; \dots\\
g, &amp; f, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|\right)\ ^{\frown}(x,y,z\dots)}&lt;/math&gt;

which he abbreviated in 1858, where &lt;math&gt;\Upsilon&lt;/math&gt; is any skew-symmetric matrix:&lt;ref group=M name=cayl1858&gt;Cayley (1858), p. 39&lt;/ref&gt;&lt;ref&gt;Hawkins (2013), p. 214&lt;/ref&gt;

:&lt;math&gt;(\mathrm{x,y,z})=\left(\Omega^{-1}(\Omega-\Upsilon)(\Omega+\Upsilon)^{-1}\Omega\right)(x,y,z)&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The Cayley–Hermite transformation becomes equivalent to the Lorentz transformation ({{equationNote|5a}}) by setting Ω=diag(-1,1) and  ({{equationNote|5b}}) by setting Ω=diag(-1,1,1) and ({{equationNote|5c}}) by setting Ω=diag(-1,1,1,1).&lt;/p&gt;

Using the parameters of (1855a), Cayley in a subsequent paper (1855b) particularly discussed several special cases, such as:&lt;ref group=M&gt;Cayley (1855b), p. 210&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}=ax^{2}+by^{2}\\
\hline (\mathrm{x,y})=\frac{1}{ab+\nu^{2}}\cdot\left[\begin{matrix}ab-\nu^{2}, &amp; -2\nu b\\
2\nu a, &amp; ab-\nu^{2}
\end{matrix}\right](x,y)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to the Lorentz transformation ({{equationNote|5a}}) in 1+1 dimensions by setting ''(a,b)''=(-1,1) and Lorentz boost [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] by additionally setting &lt;math&gt;\tfrac{2\nu}{1+\nu^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

or:&lt;ref group=M&gt;Cayley (1855b), p. 211&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}+c\mathrm{z}^{2}=ax^{2}+by^{2}+cz^{2}\\
\hline (\mathrm{x,y,z})=\frac{1}{abc+a\lambda^{2}+b\mu^{2}+c\nu^{2}}\times\left[\begin{matrix}abc+a\lambda^{2}-b\mu^{2}-c\nu^{2}, &amp; 2(\lambda\mu-c\nu)b &amp; 2(\nu\lambda+b\mu)c\\
2(\lambda\mu+c\nu)a, &amp; abc-a\lambda^{2}+b\mu^{2}-c\nu^{2} &amp; 2(\mu\nu-a\lambda)c\\
2(\nu\lambda-b\mu)a &amp; 2(\mu\nu+a\lambda)b &amp; abc-a\lambda^{2}-b\mu^{2}-c\nu^{2}
\end{matrix}\right](x,y,z)
\end{matrix}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to the Lorentz transformation ({{equationNote|5b}}) by setting ''(a,b,c)''=(-1,1,1).&lt;/p&gt;

or:&lt;ref group=M&gt;Cayley (1855b), pp. 212–213&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}+c\mathrm{z}^{2}+d\mathrm{w}^{2}=ax^{2}+by^{2}+cz^{2}+dw^{2}\\
\hline (\mathrm{x,y,z,w})=\frac{1}{k}\cdot\left[\begin{align} &amp; abcd-bc\rho^{2}+ca\sigma^{2}+ab\tau^{2}+ad\lambda^{2} &amp;  &amp; 2b\left(-cd\nu-\tau\phi+d\lambda\mu-c\rho\sigma\right),\\
 &amp; \quad-bd\mu^{2}-cd\nu^{2}-\phi^{2}, &amp;  &amp; abcd+bc\rho^{2}-ca\sigma^{2}+ab\tau^{2}-ad\lambda^{2}\\
 &amp; 2a\left(cd\nu+\tau\phi+d\lambda\mu-c\rho\sigma\right), &amp;  &amp; \quad+bd\mu^{2}-cd\nu^{2}-\phi^{2},\\
 &amp; 2a\left(-bd\mu-\sigma\phi+d\lambda\nu-b\rho\tau\right), &amp;  &amp; 2b\left(ad\lambda+\rho\phi+d\mu\nu-a\sigma\tau\right),\\
 &amp; 2a\left(bc\rho+\lambda\phi+c\nu\sigma-b\mu\tau\right), &amp;  &amp; 2b\left(ac\sigma+\mu\phi-c\nu\rho+a\lambda\tau\right),\\
\\
 &amp; \quad2c\left(bd\mu+\sigma\phi+d\lambda\nu-b\rho\tau\right), &amp;  &amp; \quad2d\left(-bc\rho-\lambda\phi+c\nu\sigma-b\mu\tau\right)\\
 &amp; \quad2c\left(-ad\lambda-\rho\phi+d\mu\nu-a\sigma\tau\right), &amp;  &amp; \quad2d\left(-ac\sigma-\mu\phi-c\nu\rho+a\lambda\tau\right)\\
 &amp; \quad abcd+bc\rho^{2}+ca\sigma^{2}-ab\tau^{2}-ad\lambda^{2} &amp;  &amp; \quad2d\left(-ab\tau-\nu\phi+b\mu\rho-a\lambda\sigma\right)\\
 &amp; \quad\quad-bd\mu^{2}+cd\nu^{2}-\phi^{2}, &amp;  &amp; \quad abcd-bc\rho^{2}-ca\sigma^{2}-ab\tau^{2}+ad\lambda^{2}\\
 &amp; \quad2c\left(ab\tau+\nu\phi+b\mu\rho-a\lambda\sigma\right), &amp;  &amp; \quad\quad+bd\mu^{2}+cd\nu^{2}-\phi^{2},
\end{align}
\right]\cdot(x,y,z,w)\\
\left(\begin{align}k &amp; =abcd+bc\rho^{2}+ca\sigma^{2}+ab\tau^{2}+ad\lambda^{2}+bd\mu^{2}+cd\nu^{2}+\phi^{2}\\
\phi &amp; =\lambda\rho+\mu\sigma+\nu\tau
\end{align}
\right)
\end{matrix}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to the Lorentz transformation ({{equationNote|5c}}) by setting ''(a,b,c,d)''=(-1,1,1,1).&lt;/p&gt;

==={{anchor|Hermite}} Hermite (1853, 1854) – Cayley–Hermite transformation===

[[w:Charles Hermite]] (1853) extended the number theoretical work of [[../Lorentz transformation (general)#Gauss2|E:Gauss (1801)]] and others (including himself) by additionally analyzing ''indefinite'' ternary quadratic forms that can be transformed into the Lorentz interval ±''(x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;)'', and by using [[#Cayley|Cayley's (1846)]] method of skew–symmetric coefficients he derived transformations leaving invariant almost all types of ternary quadratic forms.&lt;ref group=M name=herm&gt;Hermite (1853/54a), p. 307ff.&lt;/ref&gt; This was generalized by him in 1854 to ''n'' dimensions:&lt;ref group=M&gt;Hermite (1854b), p. 64&lt;/ref&gt;&lt;ref&gt;Hawkins (2013), p. 212&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}f\left(X_{1},X_{2},\dots\right)=f\left(x_{1},x_{2},\dots\right)\\
\hline X_{r}=2\xi_{r}-x_{r}=\xi_{r}-\frac{1}{2}\sum_{s=1}^{n}\lambda_{r,s}\frac{df}{d\xi_{s}}\\
\left(\lambda_{r,s}=-\lambda_{s,r},\ \lambda_{r,r}=0\right)
\end{matrix}&lt;/math&gt;

This result was subsequently expressed in matrix form by [[#Cayley|Cayley (1855)]], while [[w:Ferdinand Georg Frobenius]] (1877) added some modifications in order to include some special cases of quadratic forms that cannot be dealt with by the Cayley–Hermite transformation.&lt;ref group=M&gt;Frobenius (1877)&lt;/ref&gt;&lt;ref&gt;Hawkins (2013), pp. 219ff&lt;/ref&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to equation ({{equationNote|Q2}}), and becomes the Lorentz transformation by setting the coefficients of the quadratic form ''f'' to diag(-1,1,...1).&lt;/p&gt;

For instance, the special case of the transformation of a binary quadratic form into itself was given by Hermite as follows:&lt;ref group=M&gt;Hermite (1854b), pp. 64–65&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}f=ax^{2}+2bxy+cy^{2}\\
\hline \begin{align}X &amp; =\frac{\left(1-2\lambda b+\lambda^{2}D\right)x-2\lambda cy}{1-\lambda^{2}D} &amp;  &amp; =x(t-bu)-cuy\\
Y &amp; =\frac{2\lambda ax+\left(1+2\lambda b+\lambda^{2}D\right)y}{1-\lambda^{2}D} &amp;  &amp; =xau+(t+bu)y
\end{align}
\\
\left(b^{2}-ac=D,\ t=\frac{1+\lambda^{2}D}{1-\lambda^{2}D},\ u=\frac{2\lambda}{1-\lambda^{2}D},\ t^{2}-Du^{2}=1\right)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to Lorentz boost ({{equationNote|5a}}) by setting ''(a,b,c)''=(-1,0,1) and Lorentz boost of velocity by additionally setting &lt;math&gt;\tfrac{2\lambda}{1+\lambda^{2}}=\tfrac{v}{c}&lt;/math&gt; which produces ''t''=γ and ''u''=βγ.&lt;/p&gt;

=== {{anchor|Bachmann}} Bachmann (1869) – Cayley–Hermite transformation ===

[[w:Paul Gustav Heinrich Bachmann]] (1869) adapted [[#Hermite|Hermite's (1853/54)]] transformation of ternary quadratic forms to the case of integer transformations. He particularly analyzed the Lorentz interval and its transformation, and also alluded to the analogue result of [[#Gauss3|Gauss (1800)]] in terms of Cayley–Klein parameters, while Bachmann formulated his result in terms of the Cayley–Hermite transformation:&lt;ref group=M&gt;Bachmann (1869), p. 303&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+x^{\prime2}-x^{\prime\prime2}\\
\hline \begin{align}\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X &amp; =\left(p^{2}-q^{2}+q^{\prime2}-q^{\prime\prime2}\right)x-2(pq''+qq')x'-2(pq'+qq'')x''\\
\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X' &amp; =2(pq''-qq')x+\left(p^{2}+q^{2}-q^{\prime2}-q^{\prime\prime2}\right)x'+2(pq-q'q'')x''\\
\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X'' &amp; =-2(pq'-qq')x+2(pq+q'q'')x'+\left(p^{2}+q^{2}+q^{\prime2}+q^{\prime\prime2}\right)x''
\end{align}
\end{matrix}&lt;/math&gt;

He described this transformation in 1898 in the first part of his &quot;arithmetics of quadratic forms&quot; as well.&lt;ref&gt;Bachmann (1898), pp. 101–102&lt;/ref&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|5b}}), producing the relation &lt;math&gt;X^{2}+X^{\prime2}-X^{\prime\prime2}=x^{2}+x^{\prime2}-x^{\prime\prime2}&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Laguerre}} Laguerre (1882) – Laguerre inversion===

After previous work by [[w:Albert Ribaucour]] (1870),&lt;ref group=M&gt;Ribaucour (1870)&lt;/ref&gt; a transformation which transforms oriented spheres into oriented spheres, oriented planes into oriented planes, and oriented lines into oriented lines, was explicitly formulated by [[w:Edmond Laguerre]] (1882) as &quot;[[w:Spherical wave transformation#Transformation by reciprocal directions|transformation by reciprocal directions]]&quot; which was later called &quot;Laguerre inversion/transformation&quot;. It can be seen as a special case of the conformal group in terms of [[../Lorentz transformation (conformal)#Lie|E:Lie's transformations of oriented spheres]]. In two dimensions the transformation or oriented lines has the form (''R'' being the radius):&lt;ref group=M name=laguerre&gt;Laguerre (1882), pp. 550–551.&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}D' &amp; =\frac{D\left(1+\alpha^{2}\right)-2\alpha R}{1-\alpha^{2}}\\
R' &amp; =\frac{2\alpha D-R\left(1+\alpha^{2}\right)}{1-\alpha^{2}}
\end{align}
\right|\begin{align}D^{2}-D^{\prime2} &amp; =R^{2}-R^{\prime2}\\
D-D' &amp; =\alpha(R-R')\\
D+D' &amp; =\frac{1}{\alpha}(R+R')
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change) to Lorentz transformation ({{equationNote|5a}}) in terms of Cayley–Hermite parameters (even though Laguerre didn't use the Cayley-Hermite transformation ({{equationNote|Q2}})). The Lorentz boost of velocity follows with &lt;math&gt;\tfrac{2\alpha}{1+\alpha^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Darboux2}} Darboux (1887) – Laguerre inversion===

Following [[#Laguerre|Laguerre (1882)]], [[w:Gaston Darboux]] (1887) presented the Laguerre inversions in four dimensions using coordinates ''x,y,z,R'':&lt;ref group=M name=darboux&gt;Darboux (1887)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{\prime2}+y^{\prime2}+z^{\prime2}-R^{\prime2}=x^{2}+y^{2}+z^{2}-R^{2}\\
\hline \begin{align}x' &amp; =x, &amp; z' &amp; =\frac{1+k^{2}}{1-k^{2}}z-\frac{2kR}{1-k^{2}},\\
y' &amp; =y, &amp; R' &amp; =\frac{2kz}{1-k^{2}}-\frac{1+k^{2}}{1-k^{2}}R,
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change for ''R'') to Lorentz transformation ({{equationNote|5a}}) in terms of Cayley–Hermite parameters (even though Darboux didn't use the Cayley-Hermite transformation ({{equationNote|Q2}})). The Lorentz boost of velocity follows with &lt;math&gt;\tfrac{2k}{1+k^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

Darboux rewrote these equations as follows:

:&lt;math&gt;\begin{matrix}x^{\prime2}+y^{\prime2}+z^{\prime2}-R^{\prime2}=x^{2}+y^{2}+z^{2}-R^{2}\\
\hline \begin{align}z'+R' &amp; =\frac{1+k}{1-k}(z-R)\\
z'-R' &amp; =\frac{1-k}{1+k}(z+R)
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change for ''R'') to a squeeze mapping.&lt;/p&gt;

==={{anchor|Smith}} Smith (1900) – Laguerre inversion===

[[w:Percey F. Smith]] (1900) followed [[#Laguerre|Laguerre (1882)]] and [[#Darboux2|Darboux (1887)]] and defined the Laguerre inversion as follows:&lt;ref group=M&gt;Smith (1900), p. 159&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{\prime2}-p^{2}=R^{\prime2}-R^{2}\\
\hline p'=\frac{\kappa^{2}+1}{\kappa^{2}-1}p-\frac{2\kappa}{\kappa^{2}-1}R,\quad R'=\frac{2\kappa}{\kappa^{2}-1}p-\frac{\kappa^{2}+1}{\kappa^{2}-1}R
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change) to Lorentz transformation ({{equationNote|5a}}) in terms of Cayley–Hermite parameters (even though Smith didn't use the Cayley-Hermite transformation ({{equationNote|Q2}})). The Lorentz boost of velocity follows with &lt;math&gt;\tfrac{2\kappa}{1+\kappa^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Borel}} Borel (1913–14) – Cayley–Hermite parameter===

Borel (1913) started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and [[#Cayley|Cayley's (1846)]] parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions equivalent to ({{equationNote|5b}}):&lt;ref group=R&gt;Borel (1913/14), p. 39&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}-z^{2}-1=0\\
\hline {\scriptstyle \begin{align}\delta a &amp; =\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta b &amp; =2(\lambda\mu+\nu\rho), &amp; \delta c &amp; =-2(\lambda\nu+\mu\rho),\\
\delta a' &amp; =2(\lambda\mu-\nu\rho), &amp; \delta b' &amp; =-\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta c' &amp; =2(\lambda\rho-\mu\nu),\\
\delta a'' &amp; =2(\lambda\nu-\mu\rho), &amp; \delta b'' &amp; =2(\lambda\rho+\mu\nu), &amp; \delta c'' &amp; =-\left(\lambda^{2}+\mu^{2}+\nu^{2}+\rho^{2}\right),
\end{align}
}\\
\left(\delta=\lambda^{2}+\mu^{2}-\rho^{2}-\nu^{2}\right)\\
\lambda=\nu=0\rightarrow\text{Hyperbolic rotation}
\end{matrix}&lt;/math&gt;

In four dimensions equivalent to ({{equationNote|5c}}):&lt;ref group=R&gt;Borel (1913/14), p. 41&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}-\left(t_{1}-t_{2}\right)^{2}\\
\hline {\scriptstyle \begin{align} &amp; \left(\mu^{2}+\nu^{2}-\alpha^{2}\right)\cos\varphi+\left(\lambda^{2}-\beta^{2}-\gamma^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi-\gamma\operatorname{sh}{\theta}\\
 &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi+\gamma\operatorname{sh}{\theta} &amp;  &amp; \left(\mu^{2}+\nu^{2}-\beta^{2}\right)\cos\varphi+\left(\mu^{2}-\alpha^{2}-\gamma^{2}\right)\operatorname{ch}{\theta}\\
 &amp; -(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi-\beta\operatorname{sh}{\theta} &amp;  &amp; -(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\lambda\sin\varphi+\alpha\operatorname{sh}{\theta}\\
 &amp; (\gamma\mu-\beta\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta} &amp;  &amp; -(\alpha\nu-\lambda\gamma)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
\\
 &amp; \quad-(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi+\beta\operatorname{sh}{\theta} &amp;  &amp; \quad(\beta\nu-\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta}\\
 &amp; \quad-(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})-\lambda\sin\varphi-\alpha\operatorname{sh}{\theta} &amp;  &amp; \quad(\lambda\gamma-\alpha\nu)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
 &amp; \quad\left(\lambda^{2}+\mu^{2}-\gamma^{2}\right)\cos\varphi+\left(\nu^{2}-\alpha^{2}-\beta^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; \quad(\alpha\mu-\beta\lambda)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta}\\
 &amp; \quad(\beta\gamma-\alpha\mu)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta} &amp;  &amp; \quad-\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\cos\varphi+\left(\lambda^{2}+\mu^{2}+\nu^{2}\right)\operatorname{ch}{\theta}
\end{align}
}\\
\left(\alpha^{2}+\beta^{2}+\gamma^{2}-\lambda^{2}-\mu^{2}-\nu^{2}=-1\right)
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bach69}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay46gau}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay55quad}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay55gau}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay58quad}} 
*{{#section:History of Topics in Special Relativity/mathsource|dar87cou}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul71}} 
*{{#section:History of Topics in Special Relativity/mathsource|fro77}} 
*{{#section:History of Topics in Special Relativity/mathsource|herm53}} 
*{{#section:History of Topics in Special Relativity/mathsource|herm54}} 
*{{#section:History of Topics in Special Relativity/mathsource|lagu81}} 
*{{#section:History of Topics in Special Relativity/mathsource|lagu82}} 
*{{#section:History of Topics in Special Relativity/mathsource|rib70}} 
*{{#section:History of Topics in Special Relativity/mathsource|rod40}} 
*{{#section:History of Topics in Special Relativity/mathsource|smi00}}

===Historical relativity sources===
{{reflist|3|group=R}}
{{#section:History of Topics in Special Relativity/relsource|bor14}}
{{#section:History of Topics in Special Relativity/relsource|brill25}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L6}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
      <sha1>fcov03w6eih6a4xuk20y5of7ij0xbqm</sha1>
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        <id>52339</id>
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      <comment>/* {{anchor|Borel}} Borel (1913–14) – Cayley–Hermite parameter */</comment>
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      <text bytes="26933" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via Cayley–Hermite transformation ==

The [[../Lorentz transformation (general)#math_Q1|E:general transformation '''(Q1)''']] of any quadratic form into itself can also be given using ''arbitrary'' parameters based on the [[w:Cayley transform]] ('''I'''-'''T''')&lt;sup&gt;−1&lt;/sup&gt;·('''I'''+'''T'''), where '''I''' is the [[w:identity matrix]], '''T''' an arbitrary [[w:antisymmetric matrix]], and by adding '''A''' as symmetric matrix defining the quadratic form (there is no primed '''A' ''' because the coefficients are assumed to be the same on both sides):&lt;ref&gt;Hawkins (2013), pp. 210–214&lt;/ref&gt;&lt;ref&gt;Meyer (1899), p. 329&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}q=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}=q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}\cdot\mathbf{x}'\\
\hline \\
\mathbf{x}=(\mathbf{I}-\mathbf{T}\cdot\mathbf{A})^{-1}\cdot(\mathbf{I}+\mathbf{T}\cdot\mathbf{A})\cdot\mathbf{x}'\\
\text{or}\\
\mathbf{x}=\mathbf{A}^{-1}\cdot(\mathbf{A}-\mathbf{T})\cdot(\mathbf{A}+\mathbf{T})^{-1}\cdot\mathbf{A}\cdot\mathbf{x}'
\end{matrix}&lt;/math&gt;|{{equationRef|Q2}}}}

After [[#Cayley|Cayley (1846)]] introduced transformations related to sums of positive squares, [[#Hermite|Hermite (1853/54, 1854)]] derived transformations for arbitrary quadratic forms, whose result was reformulated in terms of matrices by [[#Cayley|Cayley (1855a, 1855b)]]. For instance, the choice '''A'''=diag(1,1,1) gives an orthogonal transformation which can be used to describe spatial rotations corresponding to the [[w:Euler–Rodrigues formula|w:Euler-Rodrigues parameter]]s ''[a,b,c,d]'' discovered by [[#Euler|Euler (1771) and Rodrigues (1840)]], which can be interpreted as the coefficients of [[w:quaternion]]s. Setting ''d=1'', the equations have the form:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(1,1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a &amp; -b\\
-a &amp; 0 &amp; c\\
b &amp; -c &amp; 0
\end{vmatrix}}\\
\hline x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[\begin{matrix}1-a^{2}-b^{2}+c^{2} &amp; 2(bc-a) &amp; 2(ac+b)\\
2(bc+a) &amp; 1-a^{2}+b^{2}-c^{2} &amp; 2(ab-c)\\
2(ac-b) &amp; 2(ab+c) &amp; 1+a^{2}-b^{2}-c^{2}
\end{matrix}\right]\cdot\mathbf{x}\\
\left(\kappa=1+a^{2}+b^{2}+c^{2}\right)
\end{matrix}&lt;/math&gt;|{{equationRef|Q3}}}}

Also the Lorentz interval and the general Lorentz transformation in any dimension can be produced by the Cayley–Hermite formalism.&lt;ref group=R&gt;Borel (1914), pp. 39–41&lt;/ref&gt;&lt;ref group=R&gt;Brill (1925)&lt;/ref&gt;&lt;ref&gt;Klein (1928), Chapter III, § 2B&lt;/ref&gt;&lt;ref&gt;Lorente (2003), section 3.3&lt;/ref&gt; For instance, the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']] with ''n''=1 follows from ({{equationNote|Q2}}) with:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(-1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a\\
-a &amp; 0
\end{vmatrix}}\\
\hline -x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{1-a^{2}}\left[\begin{matrix}1+a^{2} &amp; -2a\\
-2a &amp; 1+a^{2}
\end{matrix}\right]\cdot\mathbf{x}
\end{matrix}\Rightarrow\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}x_{0} &amp; =x_{0}^{\prime}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}}+x_{1}^{\prime}\frac{2\beta_{0}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{x_{0}^{\prime}\left(1+\beta_{0}^{2}\right)+x_{1}^{\prime}2\beta_{0}}{1-\beta_{0}^{2}}\\
x_{1} &amp; =x_{0}^{\prime}\frac{2\beta_{0}}{1-\beta_{0}^{2}}+x_{1}^{\prime}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{x_{0}^{\prime}2\beta_{0}+x_{1}^{\prime}\left(1+\beta_{0}^{2}\right)}{1-\beta_{0}^{2}}\\
\\
x_{0}^{\prime} &amp; =x_{0}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}}-x_{1}\frac{2\beta_{0}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{x_{0}\left(1+\beta_{0}^{2}\right)-x_{1}2\beta_{0}}{1-\beta_{0}^{2}}\\
x_{1}^{\prime} &amp; =-x_{0}\frac{2\beta_{0}}{1-\beta_{0}^{2}}+x_{1}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{-x_{0}2\beta_{0}+x_{1}\left(1+\beta_{0}^{2}\right)}{1-\beta_{0}^{2}}
\end{align}
\right|{\scriptstyle \begin{align}\frac{2\beta_{0}}{1+\beta_{0}^{2}} &amp; =\beta\\
\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} &amp; =\gamma\\
\frac{2\beta_{0}}{1-\beta_{0}^{2}} &amp; =\beta\gamma
\end{align}
}
\end{matrix}&lt;/math&gt;|{{equationRef|5a}}}}

This becomes [[../Lorentz transformation (velocity)#math_4a|E:Lorentz boost '''(4a)''']] by setting &lt;math&gt;\tfrac{2a}{1+a^{2}}=\tfrac{v}{c}&lt;/math&gt;, which is equivalent to the relation &lt;math&gt;\tfrac{2\beta_{0}}{1+\beta_{0}^{2}}=\tfrac{v}{c}&lt;/math&gt; known from [[w:Loedel diagram]]s, thus ({{equationNote|5a}}) can be interpreted as a Lorentz boost from the viewpoint of a &quot;median frame&quot; in which two other inertial frames are moving with equal speed &lt;math&gt;\beta_0&lt;/math&gt; in opposite directions.

Furthermore, Lorentz transformation [[../Lorentz transformation (general)#math_1a|E:'''(1a)''']]) with ''n''=2 is given by:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(-1,1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a &amp; -b\\
-a &amp; 0 &amp; c\\
b &amp; -c &amp; 0
\end{vmatrix}}\\
\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[\begin{matrix}1+a^{2}+b^{2}+c^{2} &amp; -2(bc-a) &amp; -2(ac+b)\\
2(bc+a) &amp; 1+a^{2}-b^{2}-c^{2} &amp; 2(ab-c)\\
2(ac-b) &amp; -2(ab-c) &amp; 1-a^{2}+b^{2}-c^{2}
\end{matrix}\right]\cdot\mathbf{x}\\
\left(\kappa=1-a^{2}-b^{2}+c^{2}\right)
\end{matrix}&lt;/math&gt;|{{equationRef|5b}}}}

or using ''n''=3:

{{NumBlk|:|&lt;math&gt;\begin{matrix}\mathbf{A}=\operatorname{diag}(-1,1,1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a &amp; -b &amp; c\\
-a &amp; 0 &amp; d &amp; e\\
b &amp; -d &amp; 0 &amp; f\\
-c &amp; -e &amp; -f &amp; 0
\end{vmatrix}}\\
\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[{\scriptstyle \begin{align} &amp; 1+a^{2}+b^{2}+c^{2}+ &amp;  &amp; 2(-bd+a+ec+pf) &amp;  &amp; 2(-ad-b+fc-pe) &amp;  &amp; 2(pd+fb-ea+c)\\
 &amp; \quad d^{2}+e^{2}+f^{2}+p^{2} &amp;  &amp; 1+a^{2}-b^{2}-c^{2} &amp;  &amp; 2(-d-ab+pc-fe) &amp;  &amp; 2(fd+pb+ca-e)\\
 &amp; 2(bd+a-ec+pf) &amp;  &amp; \quad-d^{2}-e^{2}+f^{2}+p^{2} &amp;  &amp; 1-a^{2}+b^{2}-c^{2} &amp;  &amp; 2(-ed-cb+pa-f)\\
 &amp; 2(ad-b-fc-pe) &amp;  &amp; 2(d-ab-pc-fe) &amp;  &amp; \quad-d^{2}+e^{2}-f^{2}+p^{2} &amp;  &amp; 1-a^{2}-b^{2}+-c^{2}\\
 &amp; 2(pd-fb+ea+c) &amp;  &amp; 2(fd-pb+ca+e) &amp;  &amp; 2(-ed-cb-pa+f) &amp;  &amp; \quad+d^{2}-e^{2}-f^{2}+p^{2}
\end{align}
}\right]\cdot\mathbf{x}\\
\left(\begin{align}\kappa &amp; =1-a^{2}-b^{2}-c^{2}+d^{2}+e^{2}+f^{2}-p^{2}\\
p &amp; =af+be+cd
\end{align}
\right)
\end{matrix}&lt;/math&gt;|{{equationRef|5c}}}}

The transformation of a binary quadratic form of which Lorentz transformation ({{equationNote|5a}}) is a special case was given by [[#Hermite|Hermite (1854)]], equations containing Lorentz transformations ({{equationNote|5a}}, {{equationNote|5b}}, {{equationNote|5c}}) as special cases were given by [[#Cayley|Cayley (1855)]], Lorentz transformation ({{equationNote|5a}}) was given (up to a sign change) by [[#Laguerre|Laguerre (1882)]], [[#Darboux2|Darboux (1887)]], [[#Smith|Smith (1900)]] in relation to Laguerre geometry, and Lorentz transformation ({{equationNote|5b}}) was given by [[#Bachmann|Bachmann (1869)]]. In relativity, equations similar to ({{equationNote|5b}}, {{equationNote|5c}}) were first employed by [[#Borel|Borel (1913)]] to represent Lorentz transformations.

==Historical notation==
==={{anchor|Euler}} Euler (1771) – Euler-Rodrigues parameter===

Euler (1771) demonstrated the invariance of quadratic forms in terms of sum of squares under a linear substitution and its coefficients, now known as [[w:orthogonal transformation]]. The transformation in three dimensions was given as

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}+Z^{2}=x^{2}+y^{2}+z^{2}\\
\hline \begin{align}X &amp; =Ax+By+Cz\\
Y &amp; =Dx+Ey+Fz\\
Z &amp; =Gx+Hy+Iz
\end{align}
\begin{matrix}\left|{\scriptstyle \begin{align}1 &amp; =AA+DD+GG\\
1 &amp; =BB+EE+HH\\
1 &amp; =CC+FF+II\\
0 &amp; =AB+DE+GH\\
0 &amp; =AG+DF+GI\\
0 &amp; =BC+EF+HI
\end{align}
}\right.\end{matrix}\end{matrix}&lt;/math&gt;

in which the coefficiens ''A,B,C,D,E,F,G,H,I'' were related by Euler to four arbitrary parameter ''p,q,r,s'', which where rediscovered by [[w:Olinde Rodrigues]] (1840) who related them to rotation angles&lt;ref group=M&gt;Rodrigues (1840), p. 405&lt;/ref&gt;:&lt;ref group=M&gt;Euler (1771), p. 101&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}A &amp; =\frac{pp+qq-rr-ss}{u} &amp; B &amp; =\frac{2pq+2ps}{u} &amp; C &amp; =\frac{2qs-2pr}{u}\\
D &amp; =\frac{2qr-2ps}{u} &amp; E &amp; =\frac{pp-qq+rr-ss}{u} &amp; F &amp; =\frac{2pq+2rs}{u}\\
G &amp; =\frac{2qs+2pr}{u} &amp; H &amp; =\frac{2rs-2pq}{u} &amp; I &amp; =\frac{pp-qq-rr+ss}{u}
\end{align}
\\
(u=pp+qq+rr+ss)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;These are now called (improper) [[w:Euler–Rodrigues formula|Euler–Rodrigues parameters]] in line with equation ({{equationNote|Q3}}).&lt;/p&gt;

==={{anchor|Cayley}} Cayley (1846–1855) – Cayley–Hermite transformation===
The [[w:Euler–Rodrigues formula|Euler–Rodrigues parameters]] discovered by [[#Euler|Euler (1871) and Rodrigues (1840)]] leaving invariant &lt;math&gt;x_{0}^{2}+x_{1}^{2}+x_{2}^{2}&lt;/math&gt; were extended to &lt;math&gt;x_{0}^{2}+\dots+x_{n}^{2}&lt;/math&gt; by [[w:Arthur Cayley]] (1846) as a byproduct of what is now called the [[w:Cayley transform]] using the method of skew–symmetric coefficients.&lt;ref group=M name=cayl1&gt;Cayley (1846)&lt;/ref&gt; Following Cayley's methods, a general transformation for quadratic forms into themselves in three (1853) and arbitrary (1854) dimensions was provided by [[#Hermite|Hermite (1853, 1854)]]. Hermite's formula was simplified and brought into matrix form equivalent to ({{equationNote|Q2}}) by Cayley (1855a)&lt;ref group=M&gt;Cayley (1855a), p. 288&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle (\mathrm{x,y,z}\dots)=\left(\left|\begin{matrix}a, &amp; h, &amp; g &amp; \dots\\
h, &amp; b, &amp; f &amp; \dots\\
g, &amp; f, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|^{-1}\left|\begin{matrix}a, &amp; h-\nu, &amp; g+\mu &amp; \dots\\
h+\nu, &amp; b, &amp; f-\lambda &amp; \dots\\
g-\mu, &amp; f+\lambda, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|\left|\begin{matrix}a, &amp; h+\nu, &amp; g-\mu &amp; \dots\\
h-\nu, &amp; b, &amp; f+\lambda &amp; \dots\\
g+\mu, &amp; f-\lambda, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|^{-1}\left|\begin{matrix}a, &amp; h, &amp; g &amp; \dots\\
h, &amp; b, &amp; f &amp; \dots\\
g, &amp; f, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|\right)\ ^{\frown}(x,y,z\dots)}&lt;/math&gt;

which he abbreviated in 1858, where &lt;math&gt;\Upsilon&lt;/math&gt; is any skew-symmetric matrix:&lt;ref group=M name=cayl1858&gt;Cayley (1858), p. 39&lt;/ref&gt;&lt;ref&gt;Hawkins (2013), p. 214&lt;/ref&gt;

:&lt;math&gt;(\mathrm{x,y,z})=\left(\Omega^{-1}(\Omega-\Upsilon)(\Omega+\Upsilon)^{-1}\Omega\right)(x,y,z)&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;The Cayley–Hermite transformation becomes equivalent to the Lorentz transformation ({{equationNote|5a}}) by setting Ω=diag(-1,1) and  ({{equationNote|5b}}) by setting Ω=diag(-1,1,1) and ({{equationNote|5c}}) by setting Ω=diag(-1,1,1,1).&lt;/p&gt;

Using the parameters of (1855a), Cayley in a subsequent paper (1855b) particularly discussed several special cases, such as:&lt;ref group=M&gt;Cayley (1855b), p. 210&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}=ax^{2}+by^{2}\\
\hline (\mathrm{x,y})=\frac{1}{ab+\nu^{2}}\cdot\left[\begin{matrix}ab-\nu^{2}, &amp; -2\nu b\\
2\nu a, &amp; ab-\nu^{2}
\end{matrix}\right](x,y)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to the Lorentz transformation ({{equationNote|5a}}) in 1+1 dimensions by setting ''(a,b)''=(-1,1) and Lorentz boost [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] by additionally setting &lt;math&gt;\tfrac{2\nu}{1+\nu^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

or:&lt;ref group=M&gt;Cayley (1855b), p. 211&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}+c\mathrm{z}^{2}=ax^{2}+by^{2}+cz^{2}\\
\hline (\mathrm{x,y,z})=\frac{1}{abc+a\lambda^{2}+b\mu^{2}+c\nu^{2}}\times\left[\begin{matrix}abc+a\lambda^{2}-b\mu^{2}-c\nu^{2}, &amp; 2(\lambda\mu-c\nu)b &amp; 2(\nu\lambda+b\mu)c\\
2(\lambda\mu+c\nu)a, &amp; abc-a\lambda^{2}+b\mu^{2}-c\nu^{2} &amp; 2(\mu\nu-a\lambda)c\\
2(\nu\lambda-b\mu)a &amp; 2(\mu\nu+a\lambda)b &amp; abc-a\lambda^{2}-b\mu^{2}-c\nu^{2}
\end{matrix}\right](x,y,z)
\end{matrix}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to the Lorentz transformation ({{equationNote|5b}}) by setting ''(a,b,c)''=(-1,1,1).&lt;/p&gt;

or:&lt;ref group=M&gt;Cayley (1855b), pp. 212–213&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}+c\mathrm{z}^{2}+d\mathrm{w}^{2}=ax^{2}+by^{2}+cz^{2}+dw^{2}\\
\hline (\mathrm{x,y,z,w})=\frac{1}{k}\cdot\left[\begin{align} &amp; abcd-bc\rho^{2}+ca\sigma^{2}+ab\tau^{2}+ad\lambda^{2} &amp;  &amp; 2b\left(-cd\nu-\tau\phi+d\lambda\mu-c\rho\sigma\right),\\
 &amp; \quad-bd\mu^{2}-cd\nu^{2}-\phi^{2}, &amp;  &amp; abcd+bc\rho^{2}-ca\sigma^{2}+ab\tau^{2}-ad\lambda^{2}\\
 &amp; 2a\left(cd\nu+\tau\phi+d\lambda\mu-c\rho\sigma\right), &amp;  &amp; \quad+bd\mu^{2}-cd\nu^{2}-\phi^{2},\\
 &amp; 2a\left(-bd\mu-\sigma\phi+d\lambda\nu-b\rho\tau\right), &amp;  &amp; 2b\left(ad\lambda+\rho\phi+d\mu\nu-a\sigma\tau\right),\\
 &amp; 2a\left(bc\rho+\lambda\phi+c\nu\sigma-b\mu\tau\right), &amp;  &amp; 2b\left(ac\sigma+\mu\phi-c\nu\rho+a\lambda\tau\right),\\
\\
 &amp; \quad2c\left(bd\mu+\sigma\phi+d\lambda\nu-b\rho\tau\right), &amp;  &amp; \quad2d\left(-bc\rho-\lambda\phi+c\nu\sigma-b\mu\tau\right)\\
 &amp; \quad2c\left(-ad\lambda-\rho\phi+d\mu\nu-a\sigma\tau\right), &amp;  &amp; \quad2d\left(-ac\sigma-\mu\phi-c\nu\rho+a\lambda\tau\right)\\
 &amp; \quad abcd+bc\rho^{2}+ca\sigma^{2}-ab\tau^{2}-ad\lambda^{2} &amp;  &amp; \quad2d\left(-ab\tau-\nu\phi+b\mu\rho-a\lambda\sigma\right)\\
 &amp; \quad\quad-bd\mu^{2}+cd\nu^{2}-\phi^{2}, &amp;  &amp; \quad abcd-bc\rho^{2}-ca\sigma^{2}-ab\tau^{2}+ad\lambda^{2}\\
 &amp; \quad2c\left(ab\tau+\nu\phi+b\mu\rho-a\lambda\sigma\right), &amp;  &amp; \quad\quad+bd\mu^{2}+cd\nu^{2}-\phi^{2},
\end{align}
\right]\cdot(x,y,z,w)\\
\left(\begin{align}k &amp; =abcd+bc\rho^{2}+ca\sigma^{2}+ab\tau^{2}+ad\lambda^{2}+bd\mu^{2}+cd\nu^{2}+\phi^{2}\\
\phi &amp; =\lambda\rho+\mu\sigma+\nu\tau
\end{align}
\right)
\end{matrix}}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to the Lorentz transformation ({{equationNote|5c}}) by setting ''(a,b,c,d)''=(-1,1,1,1).&lt;/p&gt;

==={{anchor|Hermite}} Hermite (1853, 1854) – Cayley–Hermite transformation===

[[w:Charles Hermite]] (1853) extended the number theoretical work of [[../Lorentz transformation (general)#Gauss2|E:Gauss (1801)]] and others (including himself) by additionally analyzing ''indefinite'' ternary quadratic forms that can be transformed into the Lorentz interval ±''(x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;)'', and by using [[#Cayley|Cayley's (1846)]] method of skew–symmetric coefficients he derived transformations leaving invariant almost all types of ternary quadratic forms.&lt;ref group=M name=herm&gt;Hermite (1853/54a), p. 307ff.&lt;/ref&gt; This was generalized by him in 1854 to ''n'' dimensions:&lt;ref group=M&gt;Hermite (1854b), p. 64&lt;/ref&gt;&lt;ref&gt;Hawkins (2013), p. 212&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}f\left(X_{1},X_{2},\dots\right)=f\left(x_{1},x_{2},\dots\right)\\
\hline X_{r}=2\xi_{r}-x_{r}=\xi_{r}-\frac{1}{2}\sum_{s=1}^{n}\lambda_{r,s}\frac{df}{d\xi_{s}}\\
\left(\lambda_{r,s}=-\lambda_{s,r},\ \lambda_{r,r}=0\right)
\end{matrix}&lt;/math&gt;

This result was subsequently expressed in matrix form by [[#Cayley|Cayley (1855)]], while [[w:Ferdinand Georg Frobenius]] (1877) added some modifications in order to include some special cases of quadratic forms that cannot be dealt with by the Cayley–Hermite transformation.&lt;ref group=M&gt;Frobenius (1877)&lt;/ref&gt;&lt;ref&gt;Hawkins (2013), pp. 219ff&lt;/ref&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to equation ({{equationNote|Q2}}), and becomes the Lorentz transformation by setting the coefficients of the quadratic form ''f'' to diag(-1,1,...1).&lt;/p&gt;

For instance, the special case of the transformation of a binary quadratic form into itself was given by Hermite as follows:&lt;ref group=M&gt;Hermite (1854b), pp. 64–65&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}f=ax^{2}+2bxy+cy^{2}\\
\hline \begin{align}X &amp; =\frac{\left(1-2\lambda b+\lambda^{2}D\right)x-2\lambda cy}{1-\lambda^{2}D} &amp;  &amp; =x(t-bu)-cuy\\
Y &amp; =\frac{2\lambda ax+\left(1+2\lambda b+\lambda^{2}D\right)y}{1-\lambda^{2}D} &amp;  &amp; =xau+(t+bu)y
\end{align}
\\
\left(b^{2}-ac=D,\ t=\frac{1+\lambda^{2}D}{1-\lambda^{2}D},\ u=\frac{2\lambda}{1-\lambda^{2}D},\ t^{2}-Du^{2}=1\right)
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This becomes equivalent to Lorentz boost ({{equationNote|5a}}) by setting ''(a,b,c)''=(-1,0,1) and Lorentz boost of velocity by additionally setting &lt;math&gt;\tfrac{2\lambda}{1+\lambda^{2}}=\tfrac{v}{c}&lt;/math&gt; which produces ''t''=γ and ''u''=βγ.&lt;/p&gt;

=== {{anchor|Bachmann}} Bachmann (1869) – Cayley–Hermite transformation ===

[[w:Paul Gustav Heinrich Bachmann]] (1869) adapted [[#Hermite|Hermite's (1853/54)]] transformation of ternary quadratic forms to the case of integer transformations. He particularly analyzed the Lorentz interval and its transformation, and also alluded to the analogue result of [[#Gauss3|Gauss (1800)]] in terms of Cayley–Klein parameters, while Bachmann formulated his result in terms of the Cayley–Hermite transformation:&lt;ref group=M&gt;Bachmann (1869), p. 303&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+x^{\prime2}-x^{\prime\prime2}\\
\hline \begin{align}\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X &amp; =\left(p^{2}-q^{2}+q^{\prime2}-q^{\prime\prime2}\right)x-2(pq''+qq')x'-2(pq'+qq'')x''\\
\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X' &amp; =2(pq''-qq')x+\left(p^{2}+q^{2}-q^{\prime2}-q^{\prime\prime2}\right)x'+2(pq-q'q'')x''\\
\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X'' &amp; =-2(pq'-qq')x+2(pq+q'q'')x'+\left(p^{2}+q^{2}+q^{\prime2}+q^{\prime\prime2}\right)x''
\end{align}
\end{matrix}&lt;/math&gt;

He described this transformation in 1898 in the first part of his &quot;arithmetics of quadratic forms&quot; as well.&lt;ref&gt;Bachmann (1898), pp. 101–102&lt;/ref&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent to Lorentz transformation ({{equationNote|5b}}), producing the relation &lt;math&gt;X^{2}+X^{\prime2}-X^{\prime\prime2}=x^{2}+x^{\prime2}-x^{\prime\prime2}&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Laguerre}} Laguerre (1882) – Laguerre inversion===

After previous work by [[w:Albert Ribaucour]] (1870),&lt;ref group=M&gt;Ribaucour (1870)&lt;/ref&gt; a transformation which transforms oriented spheres into oriented spheres, oriented planes into oriented planes, and oriented lines into oriented lines, was explicitly formulated by [[w:Edmond Laguerre]] (1882) as &quot;[[w:Spherical wave transformation#Transformation by reciprocal directions|transformation by reciprocal directions]]&quot; which was later called &quot;Laguerre inversion/transformation&quot;. It can be seen as a special case of the conformal group in terms of [[../Lorentz transformation (conformal)#Lie|E:Lie's transformations of oriented spheres]]. In two dimensions the transformation or oriented lines has the form (''R'' being the radius):&lt;ref group=M name=laguerre&gt;Laguerre (1882), pp. 550–551.&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}D' &amp; =\frac{D\left(1+\alpha^{2}\right)-2\alpha R}{1-\alpha^{2}}\\
R' &amp; =\frac{2\alpha D-R\left(1+\alpha^{2}\right)}{1-\alpha^{2}}
\end{align}
\right|\begin{align}D^{2}-D^{\prime2} &amp; =R^{2}-R^{\prime2}\\
D-D' &amp; =\alpha(R-R')\\
D+D' &amp; =\frac{1}{\alpha}(R+R')
\end{align}
&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change) to Lorentz transformation ({{equationNote|5a}}) in terms of Cayley–Hermite parameters (even though Laguerre didn't use the Cayley-Hermite transformation ({{equationNote|Q2}})). The Lorentz boost of velocity follows with &lt;math&gt;\tfrac{2\alpha}{1+\alpha^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Darboux2}} Darboux (1887) – Laguerre inversion===

Following [[#Laguerre|Laguerre (1882)]], [[w:Gaston Darboux]] (1887) presented the Laguerre inversions in four dimensions using coordinates ''x,y,z,R'':&lt;ref group=M name=darboux&gt;Darboux (1887)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{\prime2}+y^{\prime2}+z^{\prime2}-R^{\prime2}=x^{2}+y^{2}+z^{2}-R^{2}\\
\hline \begin{align}x' &amp; =x, &amp; z' &amp; =\frac{1+k^{2}}{1-k^{2}}z-\frac{2kR}{1-k^{2}},\\
y' &amp; =y, &amp; R' &amp; =\frac{2kz}{1-k^{2}}-\frac{1+k^{2}}{1-k^{2}}R,
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change for ''R'') to Lorentz transformation ({{equationNote|5a}}) in terms of Cayley–Hermite parameters (even though Darboux didn't use the Cayley-Hermite transformation ({{equationNote|Q2}})). The Lorentz boost of velocity follows with &lt;math&gt;\tfrac{2k}{1+k^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

Darboux rewrote these equations as follows:

:&lt;math&gt;\begin{matrix}x^{\prime2}+y^{\prime2}+z^{\prime2}-R^{\prime2}=x^{2}+y^{2}+z^{2}-R^{2}\\
\hline \begin{align}z'+R' &amp; =\frac{1+k}{1-k}(z-R)\\
z'-R' &amp; =\frac{1-k}{1+k}(z+R)
\end{align}
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change for ''R'') to a squeeze mapping.&lt;/p&gt;

==={{anchor|Smith}} Smith (1900) – Laguerre inversion===

[[w:Percey F. Smith]] (1900) followed [[#Laguerre|Laguerre (1882)]] and [[#Darboux2|Darboux (1887)]] and defined the Laguerre inversion as follows:&lt;ref group=M&gt;Smith (1900), p. 159&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{\prime2}-p^{2}=R^{\prime2}-R^{2}\\
\hline p'=\frac{\kappa^{2}+1}{\kappa^{2}-1}p-\frac{2\kappa}{\kappa^{2}-1}R,\quad R'=\frac{2\kappa}{\kappa^{2}-1}p-\frac{\kappa^{2}+1}{\kappa^{2}-1}R
\end{matrix}&lt;/math&gt;

&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;This is equivalent (up to a sign change) to Lorentz transformation ({{equationNote|5a}}) in terms of Cayley–Hermite parameters (even though Smith didn't use the Cayley-Hermite transformation ({{equationNote|Q2}})). The Lorentz boost of velocity follows with &lt;math&gt;\tfrac{2\kappa}{1+\kappa^{2}}=\tfrac{v}{c}&lt;/math&gt;.&lt;/p&gt;

==={{anchor|Borel}} Borel (1913–14) – Cayley–Hermite parameter===

[[w:Émile Borel]] (1913) started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and [[#Cayley|Cayley's (1846)]] parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions equivalent to ({{equationNote|5b}}):&lt;ref group=R&gt;Borel (1913/14), p. 39&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}-z^{2}-1=0\\
\hline {\scriptstyle \begin{align}\delta a &amp; =\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta b &amp; =2(\lambda\mu+\nu\rho), &amp; \delta c &amp; =-2(\lambda\nu+\mu\rho),\\
\delta a' &amp; =2(\lambda\mu-\nu\rho), &amp; \delta b' &amp; =-\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta c' &amp; =2(\lambda\rho-\mu\nu),\\
\delta a'' &amp; =2(\lambda\nu-\mu\rho), &amp; \delta b'' &amp; =2(\lambda\rho+\mu\nu), &amp; \delta c'' &amp; =-\left(\lambda^{2}+\mu^{2}+\nu^{2}+\rho^{2}\right),
\end{align}
}\\
\left(\delta=\lambda^{2}+\mu^{2}-\rho^{2}-\nu^{2}\right)\\
\lambda=\nu=0\rightarrow\text{Hyperbolic rotation}
\end{matrix}&lt;/math&gt;

In four dimensions equivalent to ({{equationNote|5c}}):&lt;ref group=R&gt;Borel (1913/14), p. 41&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}-\left(t_{1}-t_{2}\right)^{2}\\
\hline {\scriptstyle \begin{align} &amp; \left(\mu^{2}+\nu^{2}-\alpha^{2}\right)\cos\varphi+\left(\lambda^{2}-\beta^{2}-\gamma^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi-\gamma\operatorname{sh}{\theta}\\
 &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi+\gamma\operatorname{sh}{\theta} &amp;  &amp; \left(\mu^{2}+\nu^{2}-\beta^{2}\right)\cos\varphi+\left(\mu^{2}-\alpha^{2}-\gamma^{2}\right)\operatorname{ch}{\theta}\\
 &amp; -(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi-\beta\operatorname{sh}{\theta} &amp;  &amp; -(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\lambda\sin\varphi+\alpha\operatorname{sh}{\theta}\\
 &amp; (\gamma\mu-\beta\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta} &amp;  &amp; -(\alpha\nu-\lambda\gamma)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
\\
 &amp; \quad-(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi+\beta\operatorname{sh}{\theta} &amp;  &amp; \quad(\beta\nu-\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta}\\
 &amp; \quad-(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})-\lambda\sin\varphi-\alpha\operatorname{sh}{\theta} &amp;  &amp; \quad(\lambda\gamma-\alpha\nu)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
 &amp; \quad\left(\lambda^{2}+\mu^{2}-\gamma^{2}\right)\cos\varphi+\left(\nu^{2}-\alpha^{2}-\beta^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; \quad(\alpha\mu-\beta\lambda)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta}\\
 &amp; \quad(\beta\gamma-\alpha\mu)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta} &amp;  &amp; \quad-\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\cos\varphi+\left(\lambda^{2}+\mu^{2}+\nu^{2}\right)\operatorname{ch}{\theta}
\end{align}
}\\
\left(\alpha^{2}+\beta^{2}+\gamma^{2}-\lambda^{2}-\mu^{2}-\nu^{2}=-1\right)
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bach69}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay46gau}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay55quad}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay55gau}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay58quad}} 
*{{#section:History of Topics in Special Relativity/mathsource|dar87cou}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul71}} 
*{{#section:History of Topics in Special Relativity/mathsource|fro77}} 
*{{#section:History of Topics in Special Relativity/mathsource|herm53}} 
*{{#section:History of Topics in Special Relativity/mathsource|herm54}} 
*{{#section:History of Topics in Special Relativity/mathsource|lagu81}} 
*{{#section:History of Topics in Special Relativity/mathsource|lagu82}} 
*{{#section:History of Topics in Special Relativity/mathsource|rib70}} 
*{{#section:History of Topics in Special Relativity/mathsource|rod40}} 
*{{#section:History of Topics in Special Relativity/mathsource|smi00}}

===Historical relativity sources===
{{reflist|3|group=R}}
{{#section:History of Topics in Special Relativity/relsource|bor14}}
{{#section:History of Topics in Special Relativity/relsource|brill25}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L6}}

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
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[[History of Topics in Special Relativity/Lorentz transformation (introduction)|Introduction]]

[[History of Topics in Special Relativity/Lorentz transformation (general)|Most general Lorentz transformations]]

[[History of Topics in Special Relativity/Lorentz transformation (imaginary)|LT via imaginary orthogonal transformation]]

[[History of Topics in Special Relativity/Lorentz transformation (hyperbolic)|LT via hyperbolic functions]]

[[History of Topics in Special Relativity/Lorentz transformation (velocity)|LT via velocity]]

[[History of Topics in Special Relativity/Lorentz transformation (conformal)|LT via sphere transformation]]

[[History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)|LT via Cayley–Hermite transformation]]

[[History of Topics in Special Relativity/Lorentz transformation (Möbius)|LT via Cayley–Klein parameters, Möbius and spin transformations]]

[[History of Topics in Special Relativity/Lorentz transformation (Quaternions)|LT via quaternions and hyperbolic numbers]]

[[History of Topics in Special Relativity/Lorentz transformation (trigonometric)|LT via trigonometric functions]]

[[History of Topics in Special Relativity/Lorentz transformation (squeeze)|LT via squeeze mappings]]}}
|}</text>
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  </page>
  <page>
    <title>Talk:WikiJournal Preprints/A history of coronaviruses</title>
    <ns>1</ns>
    <id>268005</id>
    <revision>
      <id>2412064</id>
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      <timestamp>2022-08-05T07:12:37Z</timestamp>
      <contributor>
        <username>Rwatson1955</username>
        <id>2831054</id>
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      <text bytes="13056" xml:space="preserve">{{#section-h:{{ARTICLEPAGENAMEE}}}}

== plagiarism check ==
'''{{Pass}}''' Report from [https://tools.wmflabs.org/copyvios/?lang=en&amp;project=wikiversity&amp;action=search&amp;use_engine=1&amp;turnitin=1 WMF copyvios tool]: 0% Plagiarism, 100% Unique. Flagged terms, like “Department of Life Sciences, Pachhunga University College”, “Severe acute respiratory syndrome-related coronavirus”, “Middle East respiratory syndrome coronavirus (MERS-CoV)” and “International Committee on Taxonomy of Viruses” were appropriate. '''--'''[[User:علاء |&lt;span style=&quot;color:black;font-family:Script MT Bold;font-size:16px;&quot;&gt;Alaa&lt;/span&gt; ]] [[User_talk:علاء |:)..!]] 13:10, 18 September 2020 (UTC)

== Peer review 1 ==

{{review|Q=Q109450388|reviewer=Christopher Burrell|affiliation=University of Adelaide|link=|date=7 November 2021|pdf=|text=This article contains extensive reviewing of the early literature and an interesting presentation some of the earlier discoveries. However it has a number of serious problems that require thorough correction and editing before the article can be published, otherwise it is likely to add to confusion in the field. I recommend that the author be congratulated on the scope of the article, but that it is reviewed by an authority in the field of coronaviruses

1. Care in distinguishing the name of the disease from the name of the virus. eg COVID-19 is the correct term for the disease, SARS-CoV-2 is the correct name of the virus. The earlier sections should keep this distinction.

2. Balance in breadth and depth. The article is selective in choice of examples. There are 46 coronavirus species and the article should explain the basis for selection. There is little on SARS-CoV-2, but the author may have taken the view that this example is too recent to qualify in a historical article

3. The sections headed “Feline infectious peritonitis virus” and “porcine epidemic diarrhea virus” are confusing. Feline coronavirus (FCoV) can mutate into feline infectious peritonitis virus (FIPV) and the lin k with pigs is obscure.

4. The final section on evolutionary history is too simplistic, and makes no mention of the uncertainty and assumptions associated with these estimates.

5. There are errors in use of terms for naming taxa versus terms for naming viruses themselves. Species names should be in italic script, subspecies and isolates in Roman, eg Avian coronavirus is the species and “Infectious bronchitis” and “Infectious bronchitis Beaudette” are the different virus isolates. “Severe acute respiratory syndrome-related coronavirus”, is the name of the species

6. Section headed Tyrrell and Almeida. The immunological procedure is “to react virus with antibody to form a complex”, not “fuse”.
{{Response|
;14 December 2021
I appreciate the insightful comments and suggestions. I hope I address them well:

1. Names of the diseases and viruses are properly written.

2. There are 45 species of coronaviruses (note that under the family ''Coronaviridae'', one group ''Letovirinae'' with 1 species is not a coronavirus). History is reflected by landmark events, as such only those that have notable stories are discussed. Undue weight cannot be given to SARS-CoV-2 either.

3. FIPV and FCoV and PTGV are elaborated.

4. Evolutionary history is expanded, yet avoiding systematic analysis as it can be a digression of an encyclopaedic purpose.

5. Names are distinguished; however, for historical context, reader should note that at the time the name was accepted, italics are used (e.g., ''Bovine coronavirus'') but written in regular fonts after demotion to a strain/sub-species (e.g., Bovine coronavirus).

6. Wording corrected.}}
}}

==Peer review 2==
{{review
|Q=Q57438616
|reviewer =Stanley Perlman
|affiliation=University of Iowa
|date = 12 December 2021
|text =
1. Discovery of mouse coronaviruses. These viruses are now all called mouse hepatitis virus, with JHM and A59 representing different strains. JHM is in fact primarily if not solely a pathogen of the central nervous system, while the A59 strains causes hepatitis and enteritis. Other strains of MHV are enteric. Some of these strains cause an endemic infection of mouse vivarium, which are extremely difficult to clear.

2. Severe acute respiratory syndrome-“The infection was believed to have been contained in China, but an infected individual carried it to Hong Kong on 21 February and spread it in the hospital“. The virus was spread within a hotel in Hong Kong as well as in hospitals.

3. “ In October, it was found that the infection was acquired from the masked palm civets (Paguma larvata)”. Chinese ferret badgers and raccoon dogs were also considered possible intermediate hosts.

4. “The origin of the virus is not known. Malayan pangolins (Manis javanica) which are available in the live-animal market in Wuhan city has been studied as a probable source as its virus is closely related to the SARS-CoV-2”-The first sentence is correct but the second one is an overstatement. There are several postulated intermediate hosts, including raccoon dogs and minks but none are proven. The evidence is weak or non-existent for a specific intermediate host.

5. MERS-coronavirus-“In 2013, a study revealed that the virus was 100% genetically identical to the coronavirus of the Egyptian tomb bat (Taphozous perforatus coronavirus HKU4) from Bisha, Saudi Arabia, indicating its origin.” This is no longer believed to be true and the finding likely represented a lab contamination. The sentence should be removed.

6. Feline infectious peritonitis virus-“A new disease that caused inflammation of the abdomen (peritonitis) in cats was reported in 1966,[121] the virus was identified in 1968,[122] and was named by ICTV as Feline infectious peritonitis virus (FIPV) in 1991”. This is true but it is now known that ‘feline coronavirus’ is much more common and causes diarrhea. It is often persistent and during persistence, it mutates to gain ability to infect macrphages, causes FIPV. This section could be modified to include this information.

7. Evolutionary History: “Some experimental evidences indicate that animal coronaviruses switch hosts from one mammal to another, such as dog coronavirus (Canine respiratory coronavirus), which originated from a cattle virus (Bovine coronavirus).[146] Zoonotic coronaviruses emerged recently. For instance, there is an evidence that the human coronavirus OC43 (HCoV-OC43) came from cattle around 1890, which makes it also likely the first zoonotic coronavirus.[147] SARS-CoV was transmitted from bats in 1998 (4.08 years prior to the outbreak in 2003),[148] and diverged from bat coronavirus in around 1962. SARS-CoV-2 evolved from bat coronavirus in around 1948.” This paragraph should be modified. It is not known whether canine respiratory coronavirus originated from a cattle virus. The putative 1998 cross species transmission of SARS-CoV is speculative. Finally SARS-CoV-2 may have diverged from other bat CoV in 1948, but it remained a bat virus

{{Response|
;18 December 2021

1. JHM and A59 are explained and elaborated. Additional information is also made mentioning rat viruses. By the way, they are officially called ''Murine coronavirus''.

2. The location is added.

3. Possible intermediate hosts are now described.

4. The sentence is modified with additional explanation on other possible hosts.

5. The sentence remains, as I found no evidence of lab contamination (kindly provide if any). Recent literature are consistent with the position that MERS-CoV is bat derived. Also note that Wikipedia has a policy that it is not needed to make [[w:WP:RIGHTGREATWRONGS|right great wrongs]] as long as reliable sources hold.

6. ''Alphacoronavirus 1'' is revised and updated to incorporate the suggested points.

7. The last paragraph is revised and modified, with additional information.}}

}}

== Editorial comment ==

{{editor comment
|Q=Q38382414
|date=2021-12-15
|Additional note from an approached virologist who was unable to provide a detailed review but requested for a comment to be passed on:

Please check that the use of virus taxonomy nomenclature is sufficiently (briefly) explained when first used in the article and checked for consistency throughout.

{{Response|
;18 December 2021
Copy edits are made to make distinction in nomenclature. However, as the text indicates, the taxonomy is always confusing. As explained above, for historical context, reader should note that at the time the name was accepted, italics are used (e.g., ''Bovine coronavirus'') but written in regular fonts before acceptance or after demotion to a strain/sub-species (e.g., Bovine coronavirus).}}
}}
==Editorial note==
The editorial board have agreed that this should be accepted and I have added to the technical editor task sheet [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 10:14, 21 June 2022 (UTC)
==Editorial note==
Reviewer #2 made some final suggestions on 26 June 2022:

1. There are now nine HCoV. Two recently identified ones are: Hu-PDCoV, which is a porcine delta coronavirus that crossed species and was identified in Haiti and Hu-CCoV, which was a recombinant  canine CoV that was isolated from children in Malaysia with pneumonia

2. HKU4 is believed to be found in Tylonycteris pachypus. This is described in reference 128. As I mentioned in my original review, the Egyptian tomb bat is not the source for the virus and the virus isolated from this bat (100% identity with MERS-CoV) is considered a lab contaminant. another issue is that MERS-CoV is an endemic  camel virus at present and occasionally crosses species to infect humans. It likely originated in bats, but the human infection only results from camels. In Saudi Arabia, the camel and human viruses are nearly identical.

3. The numbers of COVID-19 cases and deaths should be updated.

4. Last paragraph: &quot;Although no details are yet available, but it is generally believed that MERS-CoV originated from bat coronavirus (collectively namedSevere acute respiratory syndrome–related coronavirus)[127] specifically suggested to have evolved from the common ancestor of BtCoV-HKU4 and BtCoV-HKU5.[170][171] . Reference 127 is the early manuscript that misidentified  MERS-CoV as being identical to a virus from at the Egyptian tomb bat. The nomenclature and information have been corrected over the past 9 years. MERS-CoV is no longer referred to as a Severe acute respiratory syndrome-related CoV.

I have halted acceptance until the above comments are replied to.

[[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 09:22, 27 June 2022 (UTC)

:Here are my responses:
:1. It is interesting to learn new species of HCoV. However, the article does not emphasise on HCoV. As a matter of fact, Hu-PDCoV and Hu-CCoV are not yet recognised by ICTV. The number of coronavirus species is updated.
:2. It would be awfully wrong to say “the human infection [of Mers-CoV] only results from camels.” Human-to-human infection is the major transmission, while camel-to-human infection is regarded as less common. The reference is supported and still cited as legitimate by others (see below).
:I reiterate my previous response to this comment: “The sentence remains, as I found no evidence of lab contamination (kindly provide if any). Recent literature [see https://doi.org/10.1002/rmv.2150, https://doi.org/10.14202/vetworld.2021.190-199, [[doi:10.1016/B978-0-12-801238-3.11634-4|10.1016/B978-0-12-801238-3.11634-4]], etc.] are consistent with the position that MERS-CoV is bat derived [still upholding [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3837665/ Memish ''et al.'', 2013]]. Also note that Wikipedia has a policy that it is not needed to make [[w:WP:RIGHTGREATWRONGS|right great wrongs]] as long as reliable sources hold.”
:3. COVID-19 statistics are updated.
:4. There must be a misreading by this comment: “MERS-CoV is no longer referred to as a Severe acute respiratory syndrome-related CoV.” Nowhere does the manuscript states that MERS-CoV is called Severe acute respiratory syndrome-related CoV. As to the said reference, the justification is as above point 2. I nonetheless changed the wordings for clarity. [[User:Chhandama|Chhandama]] ([[User talk:Chhandama|discuss]] • [[Special:Contributions/Chhandama|contribs]]) 11:09, 29 June 2022 (UTC)
::{{re|Rwatson1955}} The author has responded to reviewer 2's comments (see above). Please take a look. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 22:16, 11 July 2022 (UTC)

==Editorial note==
I have reinstated this on the technical editor sheet for acceptance and pdf generation [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 07:12, 5 August 2022 (UTC)</text>
      <sha1>b6wzafmjnfvxsp7b883zhvlky466144</sha1>
    </revision>
  </page>
  <page>
    <title>Theology</title>
    <ns>0</ns>
    <id>271058</id>
    <revision>
      <id>2411913</id>
      <parentid>2248826</parentid>
      <timestamp>2022-08-04T20:29:58Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* See Also */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1211" xml:space="preserve">[[File:P religion world.svg|right|220px|World religions]]

'''Theology''' is the systematic study of the nature of the divine and, more broadly, of religious belief. It is taught as an academic discipline, typically in universities and seminaries. It occupies itself with the unique content of analyzing the supernatural, but also deals with religious epistemology, asks and seeks to answer the question of revelation. Revelation pertains to the acceptance of God, gods, or deities, as not only transcendent or above the natural world, but also willing and able to interact with the natural world and, in particular, to reveal themselves to humankind. While theology has turned into a secular field, religious adherents still consider theology to be a discipline that helps them live and understand concepts such as life and love and that helps them lead lives of obedience to the deities they follow or worship.&lt;ref&gt;[[Wikipedia: Theology]]&lt;/ref&gt;

== Resources ==

== Theology Categories ==
&lt;categorytree mode=&quot;categories&quot; showcount=&quot;on&quot;&gt;Theology&lt;/categorytree&gt;

== See Also ==
* [[School:Theology]]
* [[School:Religious Studies]]
* [[Wikipedia: Theology]]

== References ==
{{Reflist}}

[[Category:Theology| ]]</text>
      <sha1>iwtblwfcmucn7vzws5p3p2i5ts50glt</sha1>
    </revision>
  </page>
  <page>
    <title>WikiJournal User Group/Technical editors/tasks</title>
    <ns>0</ns>
    <id>271287</id>
    <revision>
      <id>2412065</id>
      <parentid>2411004</parentid>
      <timestamp>2022-08-05T07:14:47Z</timestamp>
      <contributor>
        <username>Rwatson1955</username>
        <id>2831054</id>
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      <minor />
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9497" xml:space="preserve">&lt;noinclude&gt;{{WikiJ top menu}}__NOTOC__
Tasks for the technical editors can be added at the bottom of the table ([https://en.wikiversity.org/wiki/WikiJournal_User_Group/Technical_editors/tasks?veaction=edit activate editing mode], click bottom row, click chevron that appears on the left, select 'insert below')

Tech eds can claim tasks by adding their name to the right ([[WikiJournal User Group/Editorial guidelines/Technical editor summary|process guidelines)]]

{| class=&quot;wikitable sortable&quot;
|+
!item
!task
!person
!time taken
!completion
!comments
|-
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|editorial process training
|Logan Smith
|120
|2021-01-11
|Complete
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|editorial process training
|Joshua Langfus
|120
|2021-01-11
|Complete
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|editorial process training
|Wilson Jacobs
|120
|2021-01-11
|Complete
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|editorial process training
|Emma Choplin
|120
|2021-01-11
|Complete
|-
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|editorial process training
|Jenna Harmon
|120
|2021-01-14
|Complete
|-
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|editorial process training
|Cody Naccarato
|120
|2021-01-14
|Complete
|-
|[[WikiJournal Preprints/The effect of local millet drink (Kunu) on the testis and epididymis of adult male wistar rats|The effect of local millet drink (Kunu) on the testis and epididymis of adult male wistar rats]]
|create/link author wikidata items
|Wilson Jacobs
|90
|2021-01-26
|Complete
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|[[WikiJournal of Science/Virtual colony count|Virtual colony count]]
|format and upload PDF
|Wilson Jacobs
|150
|2021-02-13
|Complete
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|Virtual colony count
|upload PDF
|Wilson Jacobs
|300
|2021-02-14
|Complete
|-
|[[WikiJournal of Science/Evolved human male preferences for female body shape|Evolved human male preferences for female body shape]]
|format PDF
|Jenna Harmon
|150
|2021-01-21
|Complete
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|Evolved human male preferences for female body shape
|upload PDF
|Jenna Harmon
|30
|2021-02-05
|Complete
|-
|[[WikiJournal of Science/Arabinogalactan-proteins|Arabinogalactan-proteins]]
|format PDF and upload
|Jenna Harmon
|180
|2021-02-13
|Complete
|-
|[[WikiJournal Preprints/Does the packaging of health information affect the assessment of its reliability? A randomized controlled trial protocol|Does the packaging of health information affect the assessment of its reliability? A randomized controlled trial protocol]]
|peer-review processing
|Logan Smith
|60
|2021-02-01
|Complete
|-
|[[WikiJournal Preprints/Does the packaging of health information affect the assessment of its reliability? A randomized controlled trial protocol|Does the packaging of health information affect the assessment of its reliability? A randomized controlled trial protocol]]
|peer-review processing
|Logan Smith
|30
|2021-02-03
|Complete
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|[[WikiJournal Preprints/Affine symmetric group|Affine symmetric group]]
|peer-review processing
|Logan Smith
|30
|2021-02-04
|Complete
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|[[WikiJournal Preprints/Affine symmetric group|Affine symmetric group]]
|peer-review processing
|Logan Smith
|30
|2021-02-10
|Complete
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|[[WikiJournal Preprints/The Kivu Ebola epidemic|Kivu Ebola epidemic]]
|process newly submitted article
|Wilson Jacobs
|150
|2021-02-09
|Complete 
|-
|Wikidata items of each WikiJMed article ([https://w.wiki/445Z query]; [[wikidata:Q96317242#P50|example]])
|add email address to corresponding author
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|Wikidata items of each WikiJSci article ([https://w.wiki/445a query]; [[wikidata:Q96317242#P50|example]])
|add email address to corresponding author
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|Wikidata items of each WikiJHum article ([https://w.wiki/445b query]; [[wikidata:Q96317242#P50|example]])
|add email address to corresponding author
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|Authors on this list ([https://w.wiki/4fY2 query], [https://author-disambiguator.toolforge.org/work_item_oauth.php disambiguator])
|Create a wikidata item for each author and disambiguate any of their other publications on wikidata
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|-
|[[WikiJournal of Science/Structural Model of Bacteriophage T4|Structural Model of Bacteriophage T4]]
|format PDF and upload
|Jenna Harmon
|130
|2022-01-31
|Complete
|-
|[[WikiJournal of Science/A broad introduction to RNA-Seq|A broad introduction to RNA-Seq]]
|format PDF and upload
|Jenna 
Harmon
|130
|2022-02-17
|Complete
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|[[WikiJournal Preprints/“Collect, acquire, analyze, report, and disseminate statistical data related to the science and engineering enterprise…”: The National Center for Science and Engineering Statistics|“Collect, acquire, analyze, report, and disseminate statistical data related to the science and engineering enterprise…”: The National Center for Science and Engineering Statistics]]
|format PDF and upload
|Jenna Harmon
|120
|2022-02-11
|Complete
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|Wikidata items of each WikiJMed author ([https://w.wiki/4463 query])
|add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all authors
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|Wikidata items of each WikiJSci author ([https://w.wiki/4462 query])
|add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all authors
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|Wikidata items of each WikiJHum author ([https://w.wiki/445q query])
|add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all authors
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|Wikidata items of each WikiJMed reviewer ([https://w.wiki/445v query])
|add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all peer reviewers
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|Wikidata items of each WikiJSci reviewer ([https://w.wiki/445x query])
|add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all peer reviewers
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|Wikidata items of each WikiJHum reviewer ([https://w.wiki/445u query])
|add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all peer reviewers
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|Edit all titles to sentence case ([[Talk:WikiJournal User Group#Article title format - Why no consistency?|discussion]])
|''on hold until consensus''
Pagemove the articles, update wikidata, update the PDFs
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|Update 2021 [[WikiJournal of Medicine/Citation metrics|WikiJMed]] citation metrics 
|Review the number of articles published in the past 2 years (2019-2021) that cite WikiJMed articles to calculate impact factor
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|Backfill [[WikiJournal of Science/Citation metrics|citation metrics]] for WikiJSci articles from 2019-2021
|Similar to the update for WikiJMed citation metrics, this one will also need to calculate impact factors for 2019, 2020 and 2021
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|Calculate citation metrics for WikiJHum articles from 2018-2021
|Similar to the other citation metrics tasks, this one will need to create a page from scratch for WikiJHum and calculate impact factors for 2018, 2019, 2020 and 2021
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|[[WikiJournal Preprints/“It’s all about people skills”: Perspectives on the social license of the forest products industry from rural North America]]
|Using [[:File:Soc license forestry NA Annotated text and reviewers comments - Ian Thomson.pdf|reviewer's annotated PDF]], extract the original text from the file and transfer onto the bare wiki page.
|Emma Chiu
|180
|2022-06-20
|Complete
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|[[WikiJournal of Medicine/History of penicillin|History of penicillin]]
|format PDF and upload
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|[[WikiJournal of Medicine/Phage Therapy|Phage Therapy]]
|format PDF and upload
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|[[WikiJournal of Science/“Collect, acquire, analyze, report, and disseminate statistical data related to the science and engineering enterprise…”: The National Center for Science and Engineering Statistics|“Collect, acquire, analyze, report, and disseminate statistical data related to the science and engineering enterprise…”: The National Center for Science and Engineering Statistics]]
|format PDF and upload
|Jenna Harmon
|95
|2022-05-06
|complete
|-
|[[WikiJournal User Group/Editorial guidelines#Registering article in DOAJ|DOAJ]]
|upload WikiJMed and WikiJSci [[WikiJournal User Group/Editorial guidelines#Registering article in DOAJ|article metadata to DOAJ]]
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|[[WikiJournal Preprints/The Kivu Ebola epidemic|Kivu Ebola epidemic]]
|format PDF and upload 
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|[[WikiJournal Preprints/Parenting_stress|Parenting stress]]
|format PDF and upload
|Natalie Charamut
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|[[WikiJournal Preprints/Leptospirosis|Leptospirosis]]
|format PDF and upload
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|[[WikiJournal Preprints/Melioidosis|Melioidosis]]
|Ready to be accepted and pdf generated please
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|[[WikiJournal Preprints/A history of coronaviruses|A history of coronaviruses]]
|Almost ready - awaiting final references - no action yet	
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|[[WikiJournal Preprints/The effect of local millet drink (Kunu) on the testis and epididymis of adult male wistar rats|The effect of local millet drink (Kunu) on the testis and epididymis of adult male wistar rats]]
|Using [[:File:Kunu and wistar rates after review tracked changes.pdf|updated manuscript's PDF]], copy the text from the file and transfer onto the wiki page
|Peter Agan
|
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|Crossref reference deposit (see [[Talk:WikiJournal User Group#Talk:WikiJournal User Group|discussion]])
|[[WikiJournal User Group/Editorial guidelines#Submitting reference metadata|Submit metadata with references for published articles onto Crossref platform]]
|
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|-
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|editorial process training 1 &amp; 2 
|Ellen Sussman
|120
|2022-07-11
|Complete
|-
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|editorial process training 1 &amp; 2 
|Andrew Neil
|120
|2022-07-11
|Complete
|-
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|editorial process training 1 &amp; 2
|Natalie Charamut
|120
|2022-07-11
|Complete
|-
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|editorial process training 1 &amp; 2 
|Emma Chiu
|120
|2022-07-11
|Complete
|}</text>
      <sha1>db3shgnsdp60hs6qxaxl0xi0k5hrxlw</sha1>
    </revision>
  </page>
  <page>
    <title>Quasi-minimal prime</title>
    <ns>0</ns>
    <id>271693</id>
    <revision>
      <id>2412016</id>
      <parentid>2410930</parentid>
      <timestamp>2022-08-05T03:26:07Z</timestamp>
      <contributor>
        <ip>2402:7500:901:2B90:BD69:8142:3D96:F08F</ip>
      </contributor>
      <comment>/* Condensed table */</comment>
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      <text bytes="1959013" xml:space="preserve">A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime &gt; ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime &gt; 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 &gt; 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 &gt; 10 by changing the order.

(using A−Z to represent digit values 10 to 35)

For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')).

I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E&lt;sub&gt;800873&lt;/sub&gt;, and base 36 has quasi-minimal prime P&lt;sub&gt;81993&lt;/sub&gt;SZ).

Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to:

* Prove that all elements in ''S'', when read as base ''b'' representation, are primes &gt; ''b''.

* Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are &gt; ''b'', are composite.

* Prove that all primes &gt; ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are &gt; ''b'', are composite).

e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to:

* Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes &gt; 10.

* Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are &gt; 10 are composite.

* Prove that all primes &gt; 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers &gt; 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite).

==Condensed table==

{|class=wikitable
|''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form of largest known quasi-minimal prime base ''b''
|-
|2||1||11||2||3
|-
|3||3||111||3||13
|-
|4||5||221||3||41
|-
|5||22||10&lt;sub&gt;93&lt;/sub&gt;13||96||5&lt;sup&gt;95&lt;/sup&gt;+8
|-
|6||11||40041||5||5209
|-
|7||71||3&lt;sub&gt;16&lt;/sub&gt;1||17||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2
|-
|8||75||4&lt;sub&gt;220&lt;/sub&gt;7||221||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||1161||3×9&lt;sup&gt;1160&lt;/sup&gt;+10
|-
|10||77||50&lt;sub&gt;28&lt;/sub&gt;27||31||5×10&lt;sup&gt;30&lt;/sup&gt;+27
|-
|11&lt;sup&gt;*&lt;/sup&gt;||1068||57&lt;sub&gt;62668&lt;/sub&gt;||62669||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10
|-
|12||106||40&lt;sub&gt;39&lt;/sub&gt;77||42||4×12&lt;sup&gt;41&lt;/sup&gt;+91
|-
|13&lt;sup&gt;*&lt;/sup&gt;||3195~3197||80&lt;sub&gt;32017&lt;/sub&gt;111||32021||8×13&lt;sup&gt;32020&lt;/sup&gt;+183
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||19699||5×14&lt;sup&gt;19698&lt;/sup&gt;−1
|-
|15||1284||7&lt;sub&gt;155&lt;/sub&gt;97||157||(15&lt;sup&gt;157&lt;/sup&gt;+59)/2
|-
|16&lt;sup&gt;*&lt;/sup&gt;||2346~2347||4&lt;sub&gt;72785&lt;/sub&gt;DD||72787||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15
|-
|17&lt;sup&gt;*&lt;/sup&gt;||10408~10428||F70&lt;sub&gt;186767&lt;/sub&gt;1||186770||262×17&lt;sup&gt;186768&lt;/sup&gt;+1
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||6271||12×18&lt;sup&gt;6270&lt;/sup&gt;+221
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||6271||16×20&lt;sup&gt;6270&lt;/sup&gt;+13
|-
|21&lt;sup&gt;*&lt;/sup&gt;||13375~13396||CF&lt;sub&gt;479147&lt;/sub&gt;0K||479150||(51×21&lt;sup&gt;479149&lt;/sup&gt;−1243)/4
|-
|22&lt;sup&gt;*&lt;/sup&gt;||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||22003||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||8134||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49
|-
|30&lt;sup&gt;*&lt;/sup&gt;||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||34206||25×30&lt;sup&gt;34205&lt;/sup&gt;−1
|-
|36&lt;sup&gt;*&lt;/sup&gt;||35257~35263||P&lt;sub&gt;81993&lt;/sub&gt;SZ||81995||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7
|}

&lt;sup&gt;*&lt;/sup&gt; Data assumes the primality of the [[w:probable prime|probable prime]]s.

Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes &gt; 10&lt;sup&gt;25000&lt;/sup&gt;), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below.

Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists))

Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists))

==Data for quasi-minimal primes==

===Base 2===

11

===Base 3===

12, 21, 111

===Base 4===

11, 13, 23, 31, 221

===Base 5===

12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013

===Base 6===

11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041

===Base 7===

14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331

===Base 8===

13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447

===Base 9===

12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011

===Base 10===

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027

===Base 11===

12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 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===Base 12===

11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077

===Base 13===

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===Base 14===

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===Base 15===

12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797

===Base 16===

11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD

===Base 17===

12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 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====Additional known quasi-minimal primes (not necessarily the next)====

570&lt;sub&gt;51310&lt;/sub&gt;1, 49&lt;sub&gt;111333&lt;/sub&gt;, 970&lt;sub&gt;166047&lt;/sub&gt;1, F70&lt;sub&gt;186767&lt;/sub&gt;1

===Base 18===

11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, 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===Base 20===

13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, GGJ0J, GGJJJ, GJ00J, GJGJJ, GJJJ7, H00EH, H024B, H04B1, H0E4B, H0F41, H0H11, H0HEH, H4E0B, H6I11, HAAA3, HAAG3, HAGA3, HB44B, HBBBB, HBHHH, HGGA3, HGGG3, HH001, HH061, HH3AH, HH6F1, HHG63, HHIA3, HI011, HIII3, I0029, I02I9, I085H, I0A41, I0F29, I2229, I2929, I2I29, I2I99, I2II9, I33A3, I3EHH, I422B, I424B, I4EEB, I4IEB, I80G1, I9IA9, IA099, IA309, IA841, IA939, IA9A9, IAF09, IAF99, IAI99, IAIA9, IB0A1, IB4IB, IB54B, IBB4B, IBBIB, IBE0H, IBH0H, IC929, ICC29, IF841, IFAG1, IGA81, IHB4B, II0A9, II42B, II44B, IIB5B, III4B, IIIA3, IIIIB, J00G1, J00JB, J0601, J06D1, J06I1, J0861, J08B1, J0B01, J0CGJ, J0E0B, J0GA1, J0GG1, J0I2B, J0J2B, J0JIB, J55CJ, J60A1, J60G1, J6D81, JA777, JA7A7, JAJG1, JAJJ1, JAJJ7, JB00B, JB08B, JC00J, JC0GJ, JEECJ, JGGGJ, JGGJJ, JJ0B1, JJ0JB, JJ55J, JJAG1, JJAJ1, JJC0J, JJE0B, JJEEB, JJGG1, JJGGJ, JJGJJ, JJJ01, JJJEB, JJJJB, 17555J, 175EEJ, 1E0007, 1JE55J, 1JJJ5J, 20005H, 2000HH, 222I29, 22E889, 22EE89, 24222B, 2999I9, 29III9, 2A000H, 2B0I0H, 2I2999, 2I9I29, 2III29, 333AF3, 336IA3, 36A3F3, 36I3A3, 388E8H, 3AF333, 3EHHHH, 3HEHHH, 3HHEHH, 40008B, 40054B, 40405B, 40448B, 415E5J, 44440B, 44452B, 44524B, 44E08B, 45444B, 4AAAE7, 4AEEE7, 4E4447, 4GEEE7, 4I544B, 4IEEEB, 500EFJ, 500FCJ, 5055FJ, 50AFFJ, 50EEFJ, 50EF5J, 50F5FJ, 50FFCJ, 52224B, 54EEEB, 5558AH, 555CAJ, 5585AH, 55F55J, 55FFFJ, 5888AH, 588A5H, 5E555J, 5F055J, 5F05CJ, 5F5F5J, 5FFFCJ, 5JEEEB, 60006D, 6000F1, 6000G3, 6001G1, 600A63, 600G1D, 600G6D, 603A33, 606G6D, 608001, 608F89, 608IF9, 60A333, 60A363, 60AAA3, 60CCCD, 60D04D, 60D0DD, 60F8I9, 60FF89, 60GCG3, 636AF3, 63A333, 63A3F3, 63AFF3, 63IIA3, 64AAAD, 660089, 660A33, 660DDD, 660F89, 662289, 666A33, 666D4D, 66A6A3, 66AF33, 66AF63, 66D0DD, 66FF89, 6A33F3, 6AFF33, 6AG6A3, 6AGA63, 6D000D, 6D004D, 6D0D4D, 6F6689, 7000G7, 7060DD, 70AAAD, 70D0CD, 733C33, 7366C3, 73AAA3, 7600DD, 760CDD, 766333, 76664D, 766FC3, 76CCDD, 76DD4D, 770G07, 771JEJ, 7771EJ, 777481, 7777I1, 777F81, 77C7JJ, 77EC77, 77EEC7, 77EJJJ, 77GCG7, 77JECJ, 7A7E77, 7AAAG3, 7AAF63, 7AEEE7, 7C7GG7, 7CCCAD, 7CCCG7, 7CGJJJ, 7D000D, 7J000D, 7JCGGJ, 80005B, 8000G1, 8000I1, 800GA1, 800I01, 80GGA1, 84405B, 84454B, 888EAH, 88E8AH, 8A4441, 8B8EAH, 8E8IF9, 8I0AG1, 902229, 909A09, 909C29, 909FA9, 90IC29, 9222I9, 92I999, 944EE7, 94AEE7, 977GG7, 97AEE7, 97GGG7, 990I29, 999I29, 99I299, 99IIA9, 9A9447, 9A9AF9, 9A9FF9, 9CII29, 9EEEE7, 9FA9A9, 9I9C29, 9I9I29, 9ICI29, 9IIIA9, A1555J, A1E55J, A333C9, A336A3, A33889, A33F63, A36663, A444E7, A4AAE7, A555FJ, A666A3, A6AAA3, A7A7E7, A7AE77, A944E7, A9A9F9, A9AAA9, AA3389, AA9AF9, AAA099, AAA3C9, AAA7E7, AAA939, AAACG9, AAAFF9, AAAI99, AAAIC9, AAE4E7, AAG9A9, AAGAA3, ADE55J, AF6663, AF9FF9, AGA963, AGG6A3, AH4441, AI000H, AI99C9, AI9AA9, AII099, B0004B, B0054B, B0080B, B00H0H, B04IIB, B05B8B, B0A6I1, B0BB8B, B0BIBB, B0E0IH, B0HA0H, B0I0HH, B0I4IB, B0IIBB, B888AH, BB058B, BB05BB, BB080B, BB0I4B, BBB04B, BBB08B, BBIB8B, BE88AH, BEHA0H, BEHHAH, BHEA0H, BHEHAH, BHHA0H, BIE00H, C007A7, C00GG7, C00JA7, C00JAJ, C00JC7, C0C7A7, C0JAC7, C0JJC7, C3G999, CAJJ77, CC0629, CC0929, CCC007, CCCCC7, CCCCG3, CCCGG7, CCCI29, CCI029, CD4GGJ, CE0009, CEFF69, CEFFF9, CEI609, CII029, CIII29, CJACC7, CJAJC7, CJJJA7, D000D7, D000J7, D004CD, D0D007, DCCDC7, DDD04D, DDDAI1, DEJJJJ, DG000D, DG5E5J, DGGG5J, DJ00CJ, DJEEE7, DJJJ5J, E00097, E00A0H, E00AA9, E02289, E028I9, E029I9, E055AJ, E0774J, E0777J, E07JCJ, E0AF5J, E0BI8B, E0E4CJ, E0EA5J, E0ECJJ, E0FFF9, E0H0HH, E0HA0H, E0IF29, E0J5AJ, E0JJAJ, E29299, E444E7, E4EEE7, E4IIIB, E66289, E66629, E66909, E66F69, E69609, E69999, E7774J, E77C77, E7C7C7, E8B8AH, E90029, E90229, E90909, E90FF9, E92299, E94EE7, E97EE7, E990F9, E99I09, E9F669, E9I029, EA8AF9, EAAAI9, EAIAA9, EAJJJJ, EBIBIB, EC7CC7, EE08F9, EE408B, EE8IF9, EEAF5J, EEE84B, EEEC07, EEEC77, EEEE0B, EEF55J, EEFFFJ, EEJ5AJ, EFE009, EFEF09, EFF669, EFFFF9, EGGGG9, EIF669, F004AJ, F00A41, F00JA1, F04481, F055CJ, F0A15J, F0EI09, F0G01J, F0J0GJ, F0J0JJ, F0JJ5J, F14441, F68F89, F68IF9, F800G1, F8E8I9, F99299, FA0009, FA1JJJ, FAAIA9, FAII09, FAII99, FCC929, FCE229, FCE609, FCEF29, FE68F9, FE8FA9, FF0E09, FF68F9, FF9FA9, FFC4AJ, FFC929, FFCE09, FFE009, FFEE89, FFEF09, FFFA99, FFFFFJ, FFFGJJ, FG1JJJ, FI99A9, FIA999, FIAAA9, FIAII9, FIIA09, G00093, G000EJ, G000G9, G00781, G007G3, G00C09, G07GC3, G09033, G0C903, G0CGG3, G0G903, G0G933, G0GCC3, G0GGC9, G30003, G36003, G5000J, G5E0EJ, G60303, G6AGG3, G6GAA3, G7AAA3, G7G363, G90AA9, G90C63, G9AAA9, GAAAA3, GAAGG3, GC0003, GCCC93, GCCCC3, GCG903, GDEEE7, GE0007, GG0CG3, GGA8G1, GGC6G3, GGCCG3, GGCGC9, GGG0I9, GGG363, GGG6G3, GGG75J, GGG8G1, GGG963, GGGAA3, GGGCG3, GGGCG9, H044EB, H0BIBB, H0E00B, H0F011, H0IIEB, H2444B, H400EB, H44IEB, H4EEEB, HBB4IB, HE000B, HEE40B, HEEE4B, HEHHAH, HF0011, HF4441, HH3HEH, HHGAA3, HHGAG3, HHHAG3, HHHEAH, HI0001, HIBIBB, HIEBBB, I0000H, I000A9, I002HH, I00A09, I00A99, I00AG1, I00AI9, I00H3H, I02999, I03HEH, I09AF9, I0A009, I0E0HH, I0I299, I0IIA9, I4445B, I4I45B, I90I29, I99029, I99A09, I9I029, IA3999, IB000H, IB00EH, IB0HEH, IB5B8B, IF2999, II0929, II92I9, II99A9, II9C29, IIAFF9, IIF299, III5BB, III8BB, J0000B, J00081, J000IB, J0010D, J006A1, J00BI1, J00I8B, J00IA1, J05EEJ, J06G81, J0C0JJ, J0E0CJ, J0EEEB, J0J0EB, J5EAEJ, J5EEAJ, J608I1, JE505J, JEEEEB, JJ0001, JJ0JCJ, JJ0JGJ, JJAJC7, JJJAE7, JJJBI1, JJJI8B, 14GGGG1, 1J5555J, 1JGGGG1, 205555H, 20I000H, 222222B, 2929999, 2BI000H, 3333A33, 3333G09, 333A3A3, 333AAA3, 336AAA3, 336AF33, 33II3A3, 363AF33, 3GIIII9, 400445B, 404454B, 404800B, 415555J, 440045B, 440080B, 442222B, 444004B, 444422B, 444444B, 4444G47, 444GGG7, 444I4IB, 44AAA47, 44I44IB, 44I4IIB, 4AAAAA7, 4EEE4E7, 4I4454B, 5000AFJ, 525555H, 555552H, 555555J, 555585H, 58BBBBB, 5BB8BBB, 5E0E55J, 5EBBBBB, 60003A3, 60008I9, 6000DDD, 6006DDD, 60080A1, 60080I1, 600A3A3, 600D8I1, 600F841, 60CCDDD, 60CGGG3, 60DDD0D, 60DDDDD, 60GGG63, 60GGGC3, 66600A3, 66603A3, 66608I9, 6666089, 66666A3, 6666C29, 6666F89, 666AFF3, 66DDDDD, 6800001, 6AGGGG3, 6CCCC29, 6G6AAA3, 6GAGGG3, 6GCCC63, 6GGG633, 700000D, 70000CD, 7000EC7, 70070C7, 700EEC7, 7070EC7, 70EEEC7, 76666CD, 77333A3, 7733A33, 775EE5J, 777A777, 777CGJJ, 777GE5J, 77A7777, 77JCJJJ, 77JJJCJ, 77JJJJJ, 7A77777, 7A777A7, 7AAAA63, 7AAAAAD, 7C00007, 7E77JCJ, 7EE7JCJ, 7EEC007, 7EEEEC7, 7G5E55J, 7GE555J, 7GJJJJJ, 7J6666D, 9000029, 9000299, 9009I29, 900I2I9, 90FAAA9, 90I0299, 90I0I29, 90I9929, 90II029, 90II299, 90II2I9, 94444E7, 97777E7, 9902I99, 99902I9, 9992929, 9992999, 9999929, 99I0C29, 99I9029, 99I92I9, 99III29, 9AAAF09, 9AFFFF9, 9I0AAA9, 9IAAA09, 9II0I29, 9II9929, 9III029, 9III2I9, 9IIIC29, A5F555J, A777E77, A994447, A9AAF99, AA00089, AA6A6A3, AAA0009, AAA3999, AAA9A99, AAA9AA9, AAAAA47, AAAAA77, AAAAA99, AAAAG99, AAAAIA9, AAAD447, AAAGAC9, AAGAAAD, AAGAAC9, AF00009, AGAAA99, AGAGG63, AGGGA63, AI0C999, AJ777A7, B000B8B, B000HEH, B000I0H, B00I00H, B0B805B, B0HHHEH, B0IHE0H, B0III8B, B8B000B, BB8BBBB, BBB8B0B, BEH0HHH, BHHHEHH, C00000J, C000C07, C066629, C0AJJJJ, C0C0A47, C0G000J, C0JAJJJ, C0JJAJJ, C0JJJJJ, C6CCG33, CC004A7, CC00A47, CC6CG33, CC92229, CCCC7G7, CCCGGG3, CE66609, CEF6669, D00007D, D00007J, D00071J, D000C0J, D000I01, D00700D, D00C007, D00C0C7, D00G00D, D0C00C7, D0CDCC7, D444447, D7JJJJJ, DC0000J, DDD7DDD, DDDCCC7, DJ0000J, DJ000EJ, DJJJJ0J, DJJJJJJ, E000009, E00084B, E000889, E000909, E000HAH, E008AF9, E04000B, E0AA8F9, E0CJJJJ, E0HHHHH, E40400B, E6660F9, E666609, E669969, E699669, E7CCC07, E996669, E99I669, E9FAAA9, EB8BBBB, EBBII8B, EBH0HHH, EE092I9, EE44IIB, EE4E4E7, EEBB8BB, EEC0007, EEE0IBB, EEE0IIB, EEEBB8B, EEEE44B, EEJCJJJ, EF00F09, EF0555J, EFAAAA9, EFF0009, EFF00F9, EFF60F9, EI0AAA9, EI660F9, EIIBB8B, EJ0CJJJ, EJ5055J, EJJ0CJJ, F0E00F9, F0E88I9, F0JJJJ1, F299929, F4G555J, F5555CJ, F5555FJ, F6668I9, F8GGGG1, F929999, F999C29, FA84441, FAAA9A9, FCE00F9, FCEF009, FCFEE29, FE00F09, FEE2889, FEF0009, FFAA9A9, FFCEE29, FFFF4AJ, FIII9A9, FIIIAI9, FJJJJ81, G000009, G0000J7, G000303, G000AA9, G000CC3, G000GG3, G003003, G0090C3, G00CG03, G00D007, G00G0G3, G00GG03, G00GG93, G080001, G0G00I9, G0G09C3, G0G0C93, G0G0GG9, G0GG093, G0GGC63, G0GGGG9, G44GG47, G6AAGA3, G999999, GAA9AA9, GCCCGG9, GCGGCC9, GCGGG03, GEE4447, GEEEE47, GG0C007, GG0GC09, GG0GGG3, GG6AAA3, GG6AGA3, GG6GAG3, GGEEE47, GGG00C9, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 21===

12, 18, 1A, 1G, 1K, 21, 25, 2B, 2H, 2J, 34, 38, 3A, 3G, 3K, 45, 4D, 4H, 4J, 52, 54, 58, 61, 65, 6B, 6D, 72, 74, 7A, 7G, 7K, 85, 8B, 8D, 92, 94, 98, 9A, A1, AD, AH, AJ, B2, B8, BA, BK, C5, CB, CH, CJ, D4, D8, DA, DK, ED, EH, EJ, F2, FG, G1, GB, GD, GH, H2, HA, HG, I1, I5, IB, IJ, J2, JA, JK, K1, KB, KD, KJ, 111, 115, 11H, 135, 13H, 13J, 151, 15B, 15H, 164, 175, 17B, 17D, 17J, 191, 19B, 19D, 19H, 1B1, 1B5, 1BB, 1BJ, 1D5, 1DD, 1DJ, 1E4, 1F1, 1F5, 1FD, 1FH, 1HB, 1HD, 1I4, 1JD, 1JH, 1JJ, 22D, 232, 29G, 29K, 2DG, 2F4, 2FK, 31H, 322, 33D, 351, 355, 35B, 35J, 362, 371, 37B, 37D, 37H, 37J, 39B, 39J, 3B5, 3BD, 3BH, 3D1, 3D5, 3DB, 3DD, 3DH, 3E2, 3FJ, 3HD, 3HH, 3HJ, 3J1, 3JB, 3JJ, 414, 481, 49K, 4B4, 4BG, 4C1, 4EB, 4F4, 4F8, 4FA, 4FK, 50G, 51B, 51D, 51H, 531, 535, 53D, 53J, 551, 56A, 56G, 56K, 575, 57J, 595, 59H, 5B1, 5B5, 5BB, 5CA, 5CG, 5CK, 5F1, 5FB, 5FJ, 5GA, 5GG, 5HH, 5IA, 5J5, 5JD, 5JH, 60H, 632, 66H, 66J, 678, 68J, 69G, 6F8, 6FA, 6GJ, 6H8, 6HK, 6IH, 6J4, 6JG, 6KH, 711, 71B, 71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, GGJ, GJ4, GK5, H0K, H15, H1B, H1J, H31, H3D, H3H, H48, H51, H55, H5J, H6K, H75, H84, H88, H91, H95, H9D, H9H, HBD, HC4, HC8, HDJ, HF5, HFB, HFH, HHD, HHJ, HI4, HI8, HJ5, HJB, HKK, I0D, I2D, I6H, I78, I8H, I9K, IFA, IFK, IID, J08, J1J, J31, J35, J3J, J44, J5H, J68, J6G, J71, J7B, J7D, J7H, J8G, J95, J9D, JBD, JBH, JBJ, JCG, JDB, JDH, JE4, JE8, JEG, JF5, JFD, JFJ, JG4, JGG, JH1, JH5, JHB, JHH, JI4, JJ1, JJ5, K0H, K5G, K6H, K9K, KG5, KH4, KKH, 101B, 103D, 1044, 1055, 10DH, 10FB, 10H1, 10H5, 10J1, 110J, 11CD, 11EB, 11ID, 130D, 13C1, 13EB, 141B, 1444, 1471, 14FB, 1505, 15CD, 167H, 16FJ, 16HH, 176H, 17C1, 17IH, 190J, 1BF4, 1BID, 1C04, 1C3D, 1CD1, 1D6H, 1DE1, 1DEB, 1E1B, 1E71, 1E95, 1EFB, 1F0B, 1F4B, 1FB4, 1H01, 1H05, 1H41, 1H6H, 1H6J, 1HC1, 1HE1, 1HF4, 1IDH, 1IHH, 1JE5, 1JF4, 20CD, 20D2, 20ID, 2278, 22F8, 239D, 23FD, 2478, 2708, 27FD, 293D, 29DD, 2A78, 2C0D, 2CF8, 2F0A, 2F9D, 2FAA, 2FDD, 2GFA, 2IF8, 3015, 301D, 301J, 3031, 3035, 305D, 30F1, 30FB, 30FD, 30HB, 30I2, 3105, 31C1, 329D, 330H, 336J, 33E1, 33EB, 33IH, 35CD, 363H, 390H, 39E5, 39ID, 3B0J, 3B6J, 3BC1, 3BEB, 3CC2, 3CF1, 3CFD, 3D32, 3E3B, 3E91, 3EF1, 3EFB, 3F05, 3FC1, 3FE1, 3FEB, 3H01, 3H0B, 3HC1, 3I3H, 3I9D, 404B, 4078, 40E1, 411B, 4131, 4171, 41FB, 4331, 4391, 43BB, 43FB, 43I2, 4432, 444B, 44E1, 477B, 4931, 4971, 499B, 49CG, 4A32, 4A4B, 4A78, 4A9G, 4B1B, 4B31, 4B3B, 4BFB, 4C78, 4E32, 4F1B, 4FB1, 5011, 509D, 50BD, 50DB, 50DD, 50EK, 50FD, 50H1, 50HJ, 50IK, 50JJ, 5101, 5105, 51E1, 530B, 53EB, 53IH, 550D, 550H, 55AB, 55BG, 55DG, 55E5, 55KH, 563H, 56JJ, 5701, 570H, 57EB, 57ID, 57IH, 590D, 59C1, 59CD, 5A3B, 5A7B, 5AGK, 5AIG, 5AKG, 5B0D, 5B9G, 5BCD, 5BGJ, 5BIH, 5C91, 5CD1, 5D0D, 5D0J, 5D6H, 5D6J, 5DC1, 5DG5, 5DIH, 5DJG, 5E0A, 5E11, 5E3B, 5E71, 5EAK, 5EEA, 5EKK, 5F0H, 5F9K, 5G0K, 5G9J, 5GF5, 5GIK, 5H0B, 5H5K, 5H9K, 5HE5, 5HID, 5I3H, 5I7D, 5IKG, 5J6J, 5JE1, 5KAG, 5KEK, 5KFH, 5KH5, 60FK, 63FH, 67JH, 69JH, 6E9K, 6F0K, 6F64, 6FJH, 6H04, 6HE4, 6HFJ, 6IF4, 6J9H, 6J9J, 7015, 7031, 703B, 7055, 705J, 7088, 709B, 709D, 709H, 70DJ, 70I8, 710J, 716J, 730B, 73C1, 73E5, 7501, 769H, 76FJ, 76JJ, 770D, 77E1, 77E5, 77EB, 7888, 78FH, 78H1, 78JH, 790H, 79CD, 79IH, 7B0J, 7B6H, 7C31, 7C91, 7C9D, 7E35, 7E75, 7E91, 7EF1, 7F01, 7F78, 7FCD, 7FH8, 7FIH, 7FJ8, 7H81, 7HC1, 7HF8, 7J6J, 7J78, 7JC1, 7JF8, 7JH8, 7JJ8, 809K, 80H8, 81H1, 81HH, 81J1, 839H, 83FH, 869K, 871H, 8731, 8771, 8788, 879J, 87F1, 87FJ, 87JH, 888H, 8911, 8931, 896K, 8A78, 8A9K, 8CC1, 8E9K, 8F3H, 8F9H, 8FF1, 8FHH, 8FJH, 8G9K, 8GJ8, 8H3J, 8H7H, 8H8K, 8HEK, 8HF1, 8HFJ, 8HH1, 8HJJ, 8HK8, 8J7J, 8J91, 8JC8, 8JFH, 8JJH, 8K78, 8K8H, 8KH8, 901J, 9035, 903B, 9071, 90CG, 90GG, 90JD, 9101, 91E5, 9301, 93E1, 93EB, 95CD, 95DG, 95FK, 967J, 969H, 96HH, 96HJ, 96JJ, 9705, 97C1, 9905, 995K, 999G, 99CD, 99EB, 99K5, 9B5G, 9BE5, 9BID, 9C0G, 9C11, 9C71, 9C9D, 9CEG, 9CGG, 9D05, 9D5G, 9DDG, 9DE5, 9E35, 9E3B, 9E71, 9EB5, 9EEG, 9EF1, 9EFB, 9F0J, 9F6H, 9F9K, 9FC1, 9FIH, 9FKH, 9G35, 9G95, 9GIK, 9GJJ, 9GKG, 9H0H, 9H5K, 9H6H, 9HKH, 9I6K, 9I9H, 9IDD, 9J6J, 9JEB, 9JGJ, 9JID, 9JJG, 9K35, 9K95, 9KEG, A33B, A3F5, A432, A478, A53B, A5IK, A788, A795, A878, A975, A99B, A9CG, A9GG, AAAB, AB44, AB55, ABFB, AC32, AC78, ACFA, AE32, AE5A, AE9K, AEEB, AEFA, AEFK, AF75, AFCA, AFCK, AFEA, AFFB, AG9G, AGFA, AI32, B064, B091, B09H, B0B1, B0D5, B0DH, B0EG, B0FB, B0FH, B0H5, B0HD, B114, B141, B14B, B1B4, B1E1, B411, B431, B444, B4B1, B59G, B5G5, B5JG, B67H, B67J, B69H, B6FH, B6FJ, B6HH, B6JH, B96H, B9DG, B9G5, BB14, BBB4, BC1D, BC31, BCDD, BD05, BD0H, BD6H, BDC1, BDE5, BDIH, BE0G, BE31, BEBB, BEE4, BEIG, BF0B, BF0H, BF6J, BFEB, BFIH, BG0G, BG55, BG95, BGG4, BGGG, BGIG, BGJ5, BH0B, BHEB, BIEG, BIGG, BIHH, BJ01, BJ0J, BJC1, BJF4, BJID, C041, C081, C09G, C0E1, C0FK, C104, C131, C171, C1C4, C1D1, C2F8, C2FA, C39D, C3F1, C478, C6FK, C7FD, C8C1, C90G, C911, C971, C99D, CAFK, CC78, CC81, CCE1, CCF8, CD02, CD31, CDFD, CE01, CE32, CF0A, CF48, CF64, CF68, CF6A, CF91, CFAK, CFDD, CFEA, CFEK, CIF4, CK32, CK9G, CKFA, D01D, D01H, D07B, D0DB, D0F1, D0FD, D0HD, D0HH, D10D, D10H, D16H, D1EB, D1IH, D23D, D3E1, D56H, D63J, D6CG, D6IG, D70B, D70D, D70J, D99G, D9GJ, D9ID, D9JG, DBC1, DBCD, DBE1, DBJG, DC11, DC9D, DCF1, DCFD, DCGG, DCIG, DD0H, DD6J, DDGJ, DDID, DDJG, DE31, DE75, DFC1, DFID, DFIH, DG22, DG6G, DG75, DG9J, DGCG, DGIG, DH6H, DHE5, DI6G, DICG, DIDH, DIFD, DIFH, DIGG, DJCD, DJDG, DJG5, E032, E05A, E0EB, E11B, E2F8, E31B, E331, E375, E395, E3C2, E3F1, E3F5, E3FB, E3I2, E53B, E555, E591, E59B, E5EG, E6F4, E755, E775, E89K, E915, E935, E93B, E96K, E9FB, EA32, EAEB, EAFK, EB44, EB95, ECFA, ECFK, EE32, EEEB, EEFK, EF08, EF35, EF48, EF55, EF64, EF68, EF75, EF95, EFB1, EFCA, EFCK, EFEA, EFIA, EFKK, EGF4, EIF4, EKF4, EKF8, EKFA, F008, F051, F075, F095, F09H, F0CA, F0CK, F0DD, F0FB, F0H1, F0IA, F0JJ, F105, F10B, F10D, F16J, F1B4, F1EB, F1ID, F1J4, F36H, F468, F47B, F491, F4C8, F501, F50B, F648, F66A, F69H, F705, F778, F864, F8F1, F90D, F99K, F9C1, F9HK, FA6A, FA75, FA7B, FA95, FAEA, FAFB, FB0D, FB14, FBCD, FBE1, FBJ4, FC1D, FC68, FC9D, FCAA, FCCK, FCD1, FCEA, FD6H, FDIH, FE0K, FE1B, FE64, FE95, 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====Additional known quasi-minimal primes (not necessarily the next)====

40&lt;sub&gt;47333&lt;/sub&gt;9G, CF&lt;sub&gt;479147&lt;/sub&gt;0K

===Base 22===

11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, IHH, IIJ, IJ7, IL7, J07, J55, J69, J8J, J99, J9J, JA5, JAH, JE7, JEH, JHH, JI9, JJ5, JJ9, JJH, JK7, K2J, K51, K5D, K75, K81, KA1, KB1, KB7, KBJ, KD1, KEJ, KG5, KIF, KJ5, KKD, KKJ, L0D, L47, L7F, L97, LAF, LD3, LD9, LDD, LEF, LGD, LI7, LJ7, LJJ, LLD, 104H, 10D5, 1205, 12B5, 140H, 1433, 144H, 14AH, 14B3, 16ED, 1AIH, 1B43, 1DD5, 1DDD, 1E6D, 1EGD, 1G05, 1GDD, 1GED, 1GGD, 1HB3, 1HHH, 1IAH, 200L, 2015, 2051, 20A1, 20DJ, 20GD, 20IL, 21B5, 21DD, 220D, 226D, 228D, 22B5, 22G5, 22K5, 22KD, 2555, 2557, 2581, 25C1, 26A1, 26B1, 2725, 2755, 2801, 2861, 288D, 28B1, 28KD, 2AA1, 2B25, 2B51, 2BB1, 2C81, 2D6D, 2DA1, 2DDJ, 2DGD, 2G0J, 2GB5, 2GDD, 2GGJ, 2I0L, 2I6L, 2ICL, 2J05, 2JK5, 2K07, 2K2D, 2K55, 2K6D, 2KB5, 2KI7, 2L2D, 2L8D, 2LK7, 302L, 30G3, 320L, 32IL, 332L, 33G3, 33G9, 36AF, 36EF, 382J, 388J, 39EL, 3AAF, 3BBJ, 3CG9, 3E2J, 3E6F, 3E6L, 3EEF, 3FAF, 3G69, 3GI3, 3GI9, 3IG9, 3LEL, 3LG3, 4025, 404H, 4063, 4075, 40AH, 40B5, 40B7, 40GH, 4225, 4363, 4447, 444H, 447H, 4487, 44B7, 44H7, 4525, 4555, 45B5, 4663, 4777, 47GH, 4807, 4B0H, 4BB5, 4BG3, 4EBH, 4G4H, 4GB3, 4HH3, 4I03, 4I63, 4IGH, 5069, 5077, 50KD, 5255, 52IJ, 5455, 5477, 5507, 5527, 556F, 557F, 5587, 56EF, 56FF, 56GD, 56KD, 5771, 57C1, 57EF, 5807, 580J, 589J, 58E7, 58IJ, 58K7, 5905, 5945, 5957, 5969, 5989, 598J, 5999, 59EJ, 59K7, 5AEF, 5B25, 5B55, 5BA5, 5BI7, 5BIJ, 5BK5, 5BK7, 5C21, 5EAF, 5EE7, 5F09, 5F6F, 5F95, 5FKD, 5G6D, 5G89, 5I09, 5I0F, 5I27, 5I6F, 5I89, 5IB7, 5IBJ, 5IFF, 5J77, 5K07, 5K25, 5K27, 5K6F, 5K87, 5KK7, 6013, 60A1, 60AF, 60EL, 6281, 62B1, 62KD, 63G9, 6403, 6643, 66EL, 66G9, 68KD, 69B3, 6A01, 6A0F, 6AAF, 6AFF, 6B21, 6B8D, 6BBD, 6BK1, 6D03, 6D43, 6D69, 6D6F, 6D93, 6D9F, 6E21, 6E81, 6E8D, 6EBD, 6ECL, 6ELL, 6FD9, 6GBD, 6GG9, 6IEF, 6K21, 6K8D, 6KBD, 6KE1, 6L43, 6LB3, 6LDF, 6LEL, 6LG9, 700F, 7027, 7057, 70EH, 70LF, 710H, 71G5, 7207, 7225, 7255, 727L, 7505, 7507, 755F, 75EF, 75F5, 75I7, 766F, 7681, 76CF, 7771, 7781, 77IL, 7861, 7A0H, 7A4H, 7BB5, 7C6F, 7EEH, 7EHL, 7FG5, 7G45, 7HEL, 7I0L, 7I27, 7IE7, 7IGH, 7K07, 7K61, 7K77, 7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, H0D1, H0HJ, H0LL, H1HH, H20D, H22D, H26D, H2I7, H2K7, H30L, H3KJ, H40H, H447, H4G3, H4H3, H4HH, H66L, H6ED, 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5IIIIIIIIIIIIIIIIIIIIIIIIIIF, D555555555555555555555550555, EEAAAAAAAAAAAAAAAAAAAAAAAAAF, HHHHHHHHHHHHHHHHHHHHHHHHEEBH, K66666666666666666666666666F, LLLLLLLLLLLLLLLLLLLLLLLLEEB7, D5555555555555555555555555A55, GGGGGGGGGGGGGGGGGGGGGGGGGGGG3, GIG0000000000000000000000000H, HH00000000000000000000000001H, K0000000000000000000000005KEF, 5BBBBBBBBBBBBBBBBBBBBBBBBBBBBD, HB0000000000000000000000000001, K000000000000000000000000505EF, L7777777777777777777777777772L, 2000000000000000000000000000CB1, C8CCCCCCCCCCCCCCCCCCCCCCCCCCCC9, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKFF, JE0000000000000000000000000000J, K000000000000000000000000000261, A0000000000000000000000000004I4H, HD000000000000000000000000000001, K000000000000000000000000000EC01, K0FFFFFFFFFFFFFFFFFFFFFFFFFFFFCF, D0002222222222222222222222222222D, FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL2L, I700000000000000000000000000000GH, K00000000000000000000000000000E61, 20000000000000000000000000000000JJ, DD5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL0KE7, 44HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, F0000000000000000000000000000000000000000000000000000000L89, J000000000000000000000000000000000000000000000000000000JJIJ, K222222222222222222222222222222222222222222222222222222222DD, KCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCF, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKCF, D55KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, K000000000000000000000000000000000000000000000000000000000000077, 22222222222222222222222222222222222222222222222222222222222222222D, K000000000000000000000000000000000000000000000000000000000000008IJ, DFA00000000000000000000000000000000000000000000000000000000000000005, CC4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJE0J, 777777777777777777777777777777777777777777777777777777777777777777777777EK7, 6000000000000000000000000000000000000000000000000000000000000000000000000000000043, KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKB5, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEEH, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 50000000000000000000000000000000000000000000000000000000000000000000000000000000002C1, K0000000000000000000000000000000000000000000000000000000000000000000000000000000000055EF, H700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000K1, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0I7, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, A400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, E0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000071, 7LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIL, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLK77, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEI7, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIB3, I7G00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 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77EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEK7, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJKJ, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777772L, BKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK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KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK5

===Base 24===

15, 17, 1D, 1H, 1J, 1N, 25, 2B, 2D, 2J, 2N, 31, 37, 3B, 3H, 41, 45, 47, 4B, 4D, 4H, 57, 5B, 5H, 5J, 65, 67, 6D, 6J, 6N, 75, 7B, 7D, 7N, 81, 85, 87, 8J, 97, 9B, 9D, 9H, 9N, A1, AB, AH, AN, B5, B7, BD, BH, BJ, C5, CJ, CN, D1, D5, DJ, E1, EB, ED, EH, EN, F7, FD, FJ, FN, G5, GD, GH, H1, HB, HD, HN, I1, I7, IB, IH, J1, J5, J7, JB, JN, K7, KB, KJ, KN, L5, LH, LJ, MD, MJ, N5, NB, NH, NJ, 101, 10B, 111, 1F1, 1FB, 1GB, 1LB, 201, 221, 22H, 261, 271, 277, 28H, 2A7, 2C7, 2G7, 2H7, 2L1, 2L7, 2MH, 305, 30D, 30J, 33N, 34N, 35D, 35N, 38D, 395, 3A5, 3AJ, 3CD, 3DD, 3DN, 3E5, 3EJ, 3GJ, 3IJ, 3JJ, 3K5, 3KD, 3ND, 43N, 44N, 49J, 4EJ, 4GJ, 4GN, 4NN, 50N, 535, 54N, 551, 55N, 5C1, 5CD, 5E5, 5K1, 5KD, 5LN, 5M5, 5N1, 601, 60B, 61B, 66H, 68B, 691, 6CH, 6FH, 6GB, 6HH, 6MH, 70H, 70J, 711, 761, 771, 77H, 77J, 78H, 7C7, 7CH, 7FH, 7G7, 7H7, 7HH, 7IJ, 7JJ, 7K1, 7M1, 7M7, 80D, 82H, 83N, 88D, 88H, 8AD, 8CD, 8DB, 8DD, 8DH, 8DN, 8GB, 8KD, 8MB, 8MH, 905, 911, 921, 935, 955, 99J, 9AJ, 9G1, 9JJ, 9K5, 9L1, 9M5, A0J, A3J, A95, AA7, AD7, AE5, AG7, AGJ, AI5, AIJ, AJD, AL7, ALD, B01, B0N, B11, B61, B6B, B8N, B91, BIN, BL1, BLN, BNN, C1B, C21, C27, C2H, C3D, C61, C8H, C91, CA7, CB1, CBB, CC7, CCB, CCD, CDD, CFB, CG1, CGB, CK1, CL1, CMB, CMH, D0B, D3D, D3N, D4N, D6B, D6H, D7H, D8B, D8N, DAD, DCD, DCH, DDH, DDN, DG7, DGB, DID, DMN, DND, E05, E4J, EA7, EEJ, EF5, EGJ, EI5, EJJ, EM5, EM7, F01, F21, F51, F8H, F95, FC1, FF1, FFB, FKH, FM5, G0B, G0N, G11, G3N, G6B, G77, G7J, G8B, G8N, G91, GA7, GBB, GC7, GFB, GG1, GGJ, GGN, GK1, GL1, GLN, GMN, GN1, GNN, H0J, H2H, H3J, H4J, H77, HA5, HA7, HE5, HFH, HIJ, HJJ, HKH, HL7, HMH, I0N, I3D, I3J, I3N, I4N, I5D, I95, IA5, IAJ, IE5, IEJ, IF5, IGJ, IJD, IK5, IKD, J0D, J4J, J8D, JAD, JDH, JEJ, JFH, JHH, JKD, JMH, K35, K6H, KCD, KFH, KH5, KLD, KM1, L01, L0B, L0D, L0N, L61, L6B, L8D, LA7, LC7, LDD, LF1, LG7, LGB, LGN, LID, LK1, LKD, LL1, LLB, LLD, LMN, LNN, M0H, M11, M21, M4N, M71, M91, M95, MA5, MA7, MBN, MC7, MF1, MF5, MFB, MFH, MG7, MI5, MIN, ML1, ML7, MLB, MMH, N01, N21, N4N, N71, N8N, NC1, ND7, NE7, NG1, NID, NK1, NL7, NMN, NN7, 11CB, 11MB, 1291, 12G1, 16C1, 16CB, 16K1, 186B, 18CB, 19K1, 1BK1, 1C8B, 1K91, 1KC1, 1KL1, 1L21, 1LC1, 1LM1, 1M61, 1M8B, 1MG1, 206H, 20CH, 20M7, 21C1, 21M1, 2207, 260H, 26KH, 2991, 2C6H, 2CC1, 2CM1, 2F11, 2FHH, 2MC1, 2MK1, 2MM1, 308N, 30GN, 30IN, 30LN, 30MN, 333J, 33JD, 33LD, 343J, 344J, 35I5, 380N, 393J, 394J, 3A3D, 3FI5, 3IMN, 3J3D, 3JID, 3L3D, 3L8N, 3M0N, 3M55, 3NGN, 404J, 408N, 40LN, 434J, 44AJ, 4ILN, 4JAJ, 4L8N, 5091, 5095, 50F1, 50I5, 51L1, 5211, 5291, 52G1, 53ID, 53MN, 5595, 55AD, 56F1, 588N, 58MN, 58ND, 5961, 5991, 5A5D, 5AAD, 5F91, 5GF1, 5GIN, 5I05, 5I55, 5I8D, 5IDD, 5IDN, 5IIN, 5IMN, 5KI5, 5M61, 5M8N, 5N3N, 602H, 6211, 62F1, 62G1, 66C1, 66FB, 66M1, 66MB, 6B21, 6BM1, 6BMB, 6C6B, 6CF1, 6CLB, 6FG1, 6K21, 6K2H, 6KG1, 6KKH, 6L21, 6LCB, 6LM1, 6MB1, 6MBB, 6MG1, 6MK1, 7001, 7027, 7207, 726H, 739J, 793J, 79C1, 7A4J, 7A9J, 7AE7, 7C01, 7CC1, 7FL1, 7G21, 7G9J, 7GAJ, 7GC1, 7HGJ, 7J2H, 7J6H, 7MKH, 800B, 800H, 804N, 806H, 808N, 80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, NK3D, NK8D, 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5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, 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EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEG7, 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===Base 30===

11, 17, 1B, 1D, 1H, 1N, 1T, 21, 27, 2B, 2D, 2J, 2N, 2T, 37, 3B, 3D, 3H, 3J, 3N, 47, 4B, 4H, 4J, 4T, 51, 57, 5D, 5H, 5N, 5T, 61, 6B, 6D, 6H, 6J, 71, 7D, 7H, 7J, 7N, 7T, 81, 8B, 8H, 8N, 8T, 91, 97, 9B, 9D, 9N, A7, AB, AD, AH, B1, B7, BH, BJ, BN, BT, C7, CD, CJ, CN, CT, D7, DB, DJ, DT, E1, EB, ED, EJ, EN, ET, F7, FB, FD, FH, FT, G7, GB, GJ, GN, GT, HB, HD, I1, I7, IH, IN, IT, J1, J7, JH, JN, JT, K1, K7, KD, KH, KJ, L1, LB, LD, LH, LN, LT, M1, MD, MH, MN, N1, NB, NJ, NT, O7, OD, OJ, ON, P1, P7, PB, PJ, PN, Q7, QH, QT, R1, RB, RD, RH, RJ, RT, SD, SH, SJ, SN, T7, TB, TD, TH, 10J, 15J, 1IJ, 1JJ, 1LJ, 1MJ, 1QJ, 22H, 29H, 2EH, 2GH, 30T, 331, 33T, 36T, 39T, 3A1, 3C1, 3G1, 3KT, 3MT, 3OT, 3S1, 3T1, 40D, 40N, 431, 44D, 46N, 48D, 4AN, 4DD, 4DN, 4F1, 4FN, 4GD, 4ID, 4PD, 4S1, 50J, 58J, 59J, 5CB, 5FJ, 5IB, 5IJ, 5JB, 5MB, 5MJ, 5OB, 5SB, 607, 63T, 687, 6E7, 6KT, 6L7, 6M7, 6MT, 6NN, 6QN, 6R7, 6S7, 6ST, 6TT, 70B, 77B, 787, 7KB, 7M7, 7MB, 7SB, 807, 80D, 80J, 84D, 85J, 877, 88J, 89J, 8DD, 8FJ, 8ID, 8IJ, 8JJ, 8M7, 8MJ, 8PD, 8QD, 8R7, 90H, 93T, 95J, 99H, 99J, 9AJ, 9AT, 9EH, 9GH, 9HH, 9HJ, 9JJ, 9MJ, 9OH, 9OT, 9PH, 9ST, 9TT, A01, A0T, A1J, A31, A6N, A6T, AAJ, AAN, AFN, AKN, AKT, ALJ, AMJ, AMT, AO1, AOT, AQ1, AQN, ARN, AT1, B5B, BBD, BCB, BDD, BID, BOB, BPD, BQB, C31, C9H, CC1, CCB, CCH, CF1, CH1, CIB, CKB, CMB, D01, D0H, D2H, D41, D4D, D6N, D8D, D9H, DDH, DDN, DGH, DH1, DHH, DID, DKN, DNN, DO1, DOH, DQN, DS1, EEH, EHH, EM7, EOH, EPH, ER7, F0N, F31, F5J, F8J, F9J, FLJ, FO1, FQ1, FQN, FS1, G01, G4D, G8D, GF1, GGH, GO1, GOH, GQD, GS1, H07, H0J, H0T, H1J, H2H, H31, H4N, H8J, HA1, HAJ, HAT, HC1, HE7, HEH, HFN, HGH, HH7, HIJ, HJJ, HKN, HL7, HNH, HPH, HQN, HS1, HTJ, HTN, I4D, I5B, I5J, I8D, IAJ, IDD, IGD, IIJ, IKB, IMB, IMJ, IOB, IPD, IQD, J8J, J9J, JAJ, JBD, JCB, JFJ, JIJ, JJB, JJD, JLJ, JPD, JQB, K3T, K4N, KAT, KBB, KCB, KMB, KNN, KOB, KOT, KQN, KST, KTT, L0J, L5J, L67, LAJ, LJJ, LQJ, LR7, M0J, M3T, M5B, M7B, M87, M9J, MAT, MFJ, MIJ, MJJ, MKB, MMJ, MOT, MQJ, N07, N0H, N67, N6N, N87, NAN, NDH, NGD, NHH, NKN, NN7, NNH, NPH, NQD, NQN, NR7, O01, O0B, O0H, OAT, OC1, OCH, OEH, OF1, OH1, OKT, OMB, OOT, OPH, OQ1, OQB, OS1, OST, P3T, P6T, P9H, PCH, PDH, PEH, PGD, PHH, PHT, PID, PMT, PPT, PQD, PST, PTT, Q5B, Q6N, Q9J, QAJ, QBB, QBD, QC1, QDN, QFJ, QFN, QGD, QJB, QKN, QLJ, QMB, QND, QNN, QO1, QQ1, QQN, QSB, R4N, R77, R87, RAN, RKN, RM7, RNN, RR7, RS7, S0T, S41, S6T, S87, SBB, SC1, SM7, SMT, SOB, SQ1, SR7, SS1, STT, T0J, T6T, T9T, TAN, TFN, TKN, TKT, TNN, TO1, TOT, TPT, TQ1, TQN, TTN, 18AJ, 19FJ, 1AFJ, 1FAJ, 20CH, 20PH, 2C0H, 2CHH, 2COH, 2H0H, 2HCH, 2HHH, 2POH, 2PPH, 3001, 34Q1, 3F41, 3FF1, 3QF1, 3SPT, 3SST, 3TTT, 40A1, 4441, 44KN, 44QN, 4AC1, 4C01, 4CA1, 4DG1, 4GA1, 4GC1, 4NND, 4NRN, 4OA1, 4Q4N, 4QA1, 4QRN, 4RQN, 550B, 555B, 55KB, 5A5J, 5AJJ, 5BKB, 5JQJ, 5KKB, 5QQB, 5QQJ, 604N, 606T, 60FN, 60KN, 60OT, 60PT, 660N, 660T, 664N, 66AT, 66FN, 66KN, 66TN, 6A0N, 6A9T, 6AAT, 6FKN, 6FRN, 6K0N, 6KFN, 6KKN, 6O6T, 6PAT, 6RFN, 6RRN, 6T6N, 6TRN, 7067, 70R7, 75BB, 77R7, 7C5B, 7CQB, 7E67, 7IQB, 7O5B, 7OIB, 7QOB, 7R07, 7RE7, 8667, 88L7, 88S7, 8E87, 8EE7, 8EL7, 8J8D, 8LL7, 8LS7, 906T, 908J, 90FJ, 90QJ, 90TJ, 92CH, 966T, 99MT, 99PT, 9I0J, 9I8J, 9LFJ, 9LIJ, 9M6T, 9MKT, 9MMT, 9PKT, 9QQJ, 9TLJ, A04N, A0QJ, A3AT, A3ST, A4A1, A4G1, A5QJ, A8QJ, A90J, A9PT, AA3T, AA41, AAAT, AAF1, AAG1, AAPT, AC41, ACS1, AF0J, AFC1, AFIJ, AG41, AGA1, AGG1, AI8J, AIJJ, AIQJ, AJ5J, AJJJ, AQ8J, AQJJ, AQQJ, AS3T, AT4N, AT5J, ATST, B04D, B0QD, BBIB, BBKB, BI0B, BK0B, BKIB, BKSB, BMIB, BMSB, BQ4D, BQQD, BSMB, C00B, C05B, C0A1, C0BB, C0EH, C0GH, C0Q1, C0QB, C2OH, C2PH, C441, C4Q1, C50B, CB0B, CBSB, CEGH, CG2H, CGHH, CHHH, CHOH, COGH, COO1, COOB, COOH, CQ01, CQ0B, CQQB, CQS1, CS0B, D00N, D0PD, D44N, DA0N, DCEH, DCG1, DCQ1, DDC1, DDF1, DDPD, DDQ1, DECH, DF4N, DFA1, DFRN, DG0D, DGD1, DGQ1, DNEH, DPDD, DQA1, DQPD, DRFN, DRRN, E20H, E667, E767, E8L7, E8S7, ECGH, EG2H, EH67, EH77, ES67, F001, F01J, F0MJ, F0QJ, F44N, F4G1, F4RN, F6RN, FA0J, FAC1, FAF1, FAIJ, FC01, FCG1, FFAJ, FFC1, FFG1, FFIJ, FGA1, FJJJ, FJQJ, FKAN, FR6N, FRFN, G09H, G0EH, G0HH, G20H, G341, G3Q1, G4G1, G92H, GA41, GAC1, GC0H, GCEH, GCG1, GCPH, GD31, GDA1, GDG1, GDPD, GE2H, GE9H, GGG1, GGGD, GHCH, GHHH, GP0D, GP2H, GPPD, GQ31, GQA1, GQG1, H00N, H0CH, H3ST, H4O1, H667, H66N, H677, H69T, H99T, H9CH, H9KT, H9LJ, H9PT, H9QJ, HH6N, HHF1, HHFJ, HHH1, HHHT, HHLJ, HHOH, HHQ1, HHST, HK6T, HL9J, HLMJ, HMKT, HMMT, HNM7, HNRN, HOPT, HQ01, HQ5J, HR0N, HR6N, HS77, HSKT, HT01, I0BB, I0JD, I0JJ, I98J, I9LJ, I9QJ, ICBB, IF0J, IFFJ, IFQJ, IIBD, IJBB, IJID, IL8J, ILFJ, IQ8J, IQJJ, IQQJ, IS0B, ISQB, J00D, J05J, J0BB, J0MJ, J0OB, J4QD, J50B, J5QJ, JBKB, JBMB, JDGD, JDQD, JGDD, JI0D, JIIB, JISB, JKIB, JKKB, JM0B, JOIB, JQ4D, JQDD, JQID, JQJJ, JQMJ, JSIB, JSMB, K06N, K0KB, K0KT, K0MT, K0PT, K0SB, K5QB, K60T, K66T, K6AN, K6FN, K6PT, K6RN, K96T, K99T, KA0N, KF6N, KFKN, KI0B, KK0T, KK6N, KK6T, KKFN, KKKB, KKKT, KKQB, KKTN, KMMT, KQ0B, KQQB, KR0N, KS5B, L0M7, L8E7, L98J, LFFJ, LI8J, LIFJ, LL87, LL9J, LLIJ, LLM7, LM77, LML7, LMM7, M09T, M0E7, M0IB, M0L7, M0M7, M0TT, M55J, M5LJ, M60T, M69T, M707, M767, M777, M7E7, M7S7, M8LJ, M96T, M9KT, MC0B, MCOB, ME07, MIBB, MJ0B, MJBB, MJSB, MLLJ, MLS7, MM07, MM0T, MML7, MMOB, MMR7, MOBB, MOCB, MP0T, MP9T, MPKT, MQ0B, MQIB, MQOB, MS07, MSPT, MSSB, MT8J, MTTT, N00D, N04D, N04N, N0FN, N0ND, N0RN, N2OH, N92H, NCGH, ND0D, ND4N, NDPD, NEGH, NEL7, NF4N, NFRN, NGCH, NGEH, NHM7, NME7, NML7, NMM7, NN0D, NN4D, NN4N, NNDN, NNID, NNND, NNRN, NP4D, NSL7, NSS7, O00T, O03T, O0MT, O0PT, O2HH, O2OH, O341, O4A1, O4G1, O4O1, O5KB, O90T, OA41, OBSB, OC5B, OCOB, OG2H, OG31, OH9H, OH9T, OHHH, OHTT, OICB, OISB, OM0T, OM6T, OMPT, OMTT, OO2H, OO9H, OOA1, OOCB, OOGH, OOKB, OOO1, OSKB, OTMT, P00T, P04D, P08D, P09T, P0KT, P90T, PAAT, PKKT, PO0T, PP0D, PPPD, Q00B, Q0AN, Q0D1, Q0F1, Q0IB, Q0JJ, Q0MJ, Q0OB, Q55J, Q5QJ, Q88D, QA0N, QA41, QA4N, QDF1, QDPD, QF41, QFA1, QG31, QIQJ, QJ0J, QJ8D, QJDD, QJJJ, QJMJ, QJQD, QKKB, QKQB, QOKB, QOOB, QP0D, QPDD, QQ8D, QQID, QQJD, QQPD, QQQB, QS01, QSA1, R00N, R067, R06N, R0FN, R0L7, R60N, RE67, RFRN, RRFN, RRRN, S00B, S3AT, S3ST, S50B, S5QB, S99T, SAAT, SC0B, SE67, SGG1, SICB, SIQB, SK5B, SKIB, SKKT, SKPT, SKQB, SM0B, SMMB, SMSB, SOA1, SOG1, SPOT, SQCB, SQQB, SS0B, SS67, ST31, STG1, T00N, T03T, T041, T04N, T0AT, T0C1, T0MT, T0ST, T1AJ, T3F1, T4A1, T4RN, T5JJ, T66N, T8AJ, T9FJ, T9QJ, TA3T, TAA1, TAG1, TAIJ, TCS1, TFA1, TFAJ, TFF1, TFFJ, TFG1, TGA1, TI9J, TIJJ, TIQJ, TL8J, TL9J, TM0T, TM5J, TM8J, TMAJ, TMLJ, TMST, TSF1, TSG1, TT01, TT5J, TT8J, TTA1, TTC1, TTFJ, TTG1, TTLJ, TTMJ, 30441, 30O41, 30OO1, 34O41, 3AATT, 3ASAT, 3O441, 3Q041, 40G41, 40GQ1, 40O41, 40OG1, 40OO1, 44001, 440C1, 440G1, 4444N, 44CG1, 44GQ1, 4AAA1, 4COG1, 4DAA1, 4GQ41, 4KRRN, 4OGG1, 4Q041, 4QGG1, 5005B, 500BB, 500KB, 500QB, 50K0B, 50Q0B, 50QKB, 555AJ, 555LJ, 555QJ, 55J5J, 5B00B, 5B0BB, 5J55J, 5J5JJ, 5JJ5J, 5JJJJ, 5K00B, 5QJ5J, 5QK0B, 6006N, 6009T, 600AT, 6066N, 60R0N, 666N7, 6696T, 66OPT, 67767, 6900T, 696PT, 69P0T, 6F6AN, 6O9PT, 6OP0T, 6OP9T, 6P99T, 755QB, 77767, 7CBBB, 7IBIB, 7IIBB, 7IICB, 7QIIB, 8888D, 88E67, 8L887, 8SSE7, 9099T, 90KKT, 90KMT, 90MPT, 9690T, 96P9T, 990KT, 9990T, 9999T, 99K6T, 9FFFJ, 9HKMT, 9HMPT, 9ILLJ, 9KKMT, 9KKPT, 9M00T, 9T8QJ, 9TFIJ, A008J, A00NN, A055J, A0FJJ, A0I0J, A0I9J, A0J0J, A44C1, A555J, A9FFJ, A9QIJ, A9T8J, AFFA1, AFFFJ, AFFJJ, AFFQJ, AJ00J, AP99T, ASA9T, ASFF1, ASP9T, ASSAT, AT3TT, ATFQJ, ATQIJ, ATT9J, ATTQJ, B088D, B08GD, B0GGD, B0SKB, B8GGD, BG00D, BIISB, BISIB, BS0IB, BS0SB, BSSKB, BSSSB, C0041, C0G41, C0H0H, C0PPH, C0S01, C4001, C4AG1, C4OG1, C5BBB, CG00H, CGE0H, CGGA1, CGQ41, COSSB, CP20H, CPGPH, CPP2H, CPPOH, CPPPH, CS001, CS55B, CSSQB, D00GD, D0A4N, D0FAN, D0GDD, D0N0D, DAAA1, DAAC1, DAFG1, DDAA1, DDGGD, DFFAN, DFGC1, DGDGD, DGG31, DGGA1, DH0AN, DPPPH, DQDD1, DQGG1, E0L87, E88E7, EC00H, EC02H, EE867, EE887, EEL87, ELE87, F00IJ, F0441, F0AG1, F0F0J, F0F41, F0FF1, F0GG1, F4041, F44A1, F44C1, F4A41, F64KN, F6K6N, FAAA1, FAFQJ, FCAA1, FF041, FF0F1, FF64N, FF6KN, FFA41, FFA4N, FFF4N, FFFF1, FFFFN, FFKRN, FFQMJ, FIJ0J, FJ00J, FKKKN, FNNFN, FQQ0J, FQQMJ, G00DD, G00DH, G00PD, G02PH, G0DDD, G0PDD, GAAA1, GC2HH, GDDDD, GDGDD, GDPPH, GG0PD, GGCA1, GGCQ1, GGDD1, GH4Q1, GHHG1, GII0D, GQ441, H009H, H00G1, H00H1, H04Q1, H0ANN, H0H01, H0H9H, H0HG1, H0HO1, H0O41, H0OHH, H0QG1, H40G1, H4G41, H4GG1, H60AN, HAN0N, HF0G1, HFF41, HFFMJ, HFFQJ, HGQ41, HH3TT, HH401, HH441, HH4G1, HH55J, HH66T, HH6OT, HH96T, HHA0N, HHANN, HHC0H, HHCHH, HHH0N, HHHCH, HHHHN, HHHNN, HHKMT, HHO41, HHP9T, HHPKT, HHTMT, HHTT1, HK9MT, HKKMT, HLLFJ, HM66T, HM7R7, HMM67, HMM77, HMMM7, HMTST, HO9HT, HOO41, HOOOH, HOT3T, HOTT1, HPO9T, HSO3T, HTGG1, I000J, I009J, I00BD, I00QJ, I00SB, I08QJ, I0I0D, I0IQB, I0JIB, I0Q0B, I0SCB, I0SIB, I0SSB, IB00D, IBBSB, IBISB, ICSSB, II0CB, II0ID, II0SB, IIIBB, IIQ0B, IIQIB, IISIB, IJ0SB, IJJQJ, ILLLJ, IQ00J, IQC0B, IQIQB, IQQIB, ISSCB, J00MB, J08GD, J0IID, J0QQD, J55BB, J55JJ, J5J5J, J5JJJ, J88GD, JB0SB, JBB0B, JBBSB, JDDDD, JG0ID, JGG0D, JIIID, JJ0QJ, JJ5JJ, JQ08D, JQ0QJ, JQQ0D, JQQ5J, JS55B, JSK0B, JSS5B, K000T, K006T, K00TN, K0FAN, K505B, K6T0N, K9KMT, K9KPT, K9P0T, KIQIB, KK00N, KK05B, KK0AN, KK0KN, KKM9T, KS0IB, KS0QB, KSQIB, KTR6N, KTRRN, L0087, L08S7, LE087, LEE87, LL8LJ, LMEE7, LMSE7, LMSS7, M00CB, M00PT, M066T, M0CQB, M0K6T, M0MQB, M0MSB, M0QQB, M0SMB, M0SST, M666T, M900T, MBBSB, MEE77, MEL77, MELE7, MES77, MESE7, MESS7, MI00B, MIICB, MIISB, MIQCB, MK9PT, ML7L7, MLEL7, MLLE7, MLME7, MM677, MMBIB, MMBSB, MMCQB, MME77, MMEE7, MMICB, MMISB, MMK6T, MMKKT, MMM9T, MMMMT, MMMTT, MMQQB, MMSCB, MMSKT, MMSMB, MMSS7, MMTST, MOIIB, MOOIB, MOSIB, MQCQB, MR007, MR667, MRL07, MS7L7, MSEL7, MSK9T, MSL77, MSSL7, MT00T, MTMMT, N0DDD, N4NNN, N7LE7, N7S77, NE2CH, NE9CH, NEC2H, NEE77, NFFNN, NFNFN, NII0D, NL777, NL7L7, NLES7, NLLL7, NLS77, NOG9H, NRR0N, O6P9T, O9H6T, O9HPT, O9P9T, OCBBB, OG441, OGAG1, OGGA1, OHH6T, OHOOH, OKIIB, OKK5B, OKKIB, OO5BB, OOBIB, OOIIB, OSCSB, OT3TT, OT441, OTG41, OTGG1, P00PH, P0D0D, P0G2H, P0OGH, P0PGH, PA99T, PGP0H, POOOH, PP20H, PP88D, PPP0H, Q0001, Q000N, Q001J, Q00ID, Q00PD, Q044N, Q04G1, Q0I0D, Q0PPD, Q40G1, Q444N, Q44RN, Q4AA1, Q8QQJ, QAAS1, QAFF1, QAFG1, QASG1, QDGG1, QFGG1, QI00D, QIICB, QIIQB, QIQCB, QOIIB, QPP4D, QQ08J, QQ0CB, QQC0B, QQI0J, QQJ5J, QSFG1, R00E7, R0E07, R0NE7, R6F6N, R6FFN, REE07, RELE7, RFF6N, RL0E7, RLE07, RLEE7, RLLE7, RQR0N, RR0QN, RRQ0N, S03O1, S0AF1, S0AG1, S0O31, S0QIB, S0SIB, S30F1, S30O1, S7QIB, SA3PT, SAAA1, SAFG1, SASST, SCSQB, SF0G1, SFFF1, SI0SB, SISSB, SKSSB, SSCQB, SSCSB, SSMIB, SSPAT, SSSKT, SSSMB, SSSSB, STAF1, STF01, T0001, T0031, T0AF1, T0AS1, T0G31, T0R6N, T0T31, T0TF1, T3AAT, T40G1, T4CG1, T4G41, T5LLJ, T8LLJ, TA441, TA98J, TAFJJ, TAQ5J, TASST, TATAT, TC401, TCGG1, TFQIJ, TFQJJ, TFQMJ, TG441, TGC41, TI8LJ, TLLMJ, TMMMT, TMTMT, TQ8QJ, TSS3T, TT3AT, TT9IJ, TTAAT, TTQIJ, TTS31, TTS3T, TTT9J, TTTST, 20000H, 200OOH, 3440O1, 3TAAST, 404CQ1, 4KKKKN, 4KKKRN, 4QQQQD, 505BBB, 50BB0B, 6000AN, 6444RN, 66666N, 666O9T, 66999T, 669P9T, 66N777, 6A444N, 6FF66N, 6FFF6N, 6R666N, 766767, 77S677, 7IBBBB, 8888E7, 8LLLLJ, 8SSSL7, 9000MT, 90K90T, 90KP0T, 90M90T, 99000T, 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===Base 36===

11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 2UJ, 2WN, 2XN, 30Z, 337, 33B, 33H, 33P, 34H, 34P, 37D, 38Z, 39H, 3AB, 3AP, 3AZ, 3D7, 3DH, 3FD, 3FZ, 3HD, 3HH, 3KZ, 3L7, 3LZ, 3MB, 3O7, 3OZ, 3PD, 3PP, 3Q7, 3RH, 3S7, 3SZ, 3UP, 3UZ, 3XB, 40P, 43B, 43H, 46V, 48B, 48V, 49B, 4AP, 4BB, 4E1, 4FP, 4HB, 4HH, 4HP, 4IH, 4MB, 4OP, 4OV, 4PH, 4RH, 4VB, 4VH, 4WH, 4WP, 4WV, 4X1, 4XH, 55T, 565, 58Z, 5C5, 5E7, 5ET, 5EZ, 5IN, 5KT, 5L7, 5MP, 5NN, 5O5, 5O7, 5OP, 5R7, 5RT, 5RZ, 5ST, 5SZ, 5TN, 5TZ, 5W7, 5XN, 5YN, 61T, 625, 62J, 64V, 661, 66J, 691, 6AV, 6EV, 6IJ, 6J1, 6K5, 6O1, 6OT, 6QJ, 6QT, 6RV, 6SJ, 6T1, 6TT, 6U5, 6UV, 6VV, 6W5, 6Y1, 6YT, 72D, 737, 77D, 791, 7A1, 7AZ, 7D7, 7GD, 7I1, 7IN, 7LN, 7M7, 7MN, 7N7, 7NN, 7RN, 7RZ, 7WN, 7X7, 7Z1, 801, 80V, 82D, 83B, 841, 84H, 877, 881, 887, 88B, 88V, 88Z, 89H, 89V, 8A1, 8AB, 8B7, 8BH, 8D1, 8DH, 8EB, 8EV, 8GD, 8GZ, 8HD, 8IV, 8L7, 8LZ, 8M1, 8MB, 8MD, 8O7, 8OB, 8Q7, 8QD, 8RD, 8S7, 8SH, 8SZ, 8UZ, 8W7, 8WV, 8XV, 8Y1, 90H, 91J, 93B, 93H, 94V, 981, 98H, 99J, 9AH, 9BB, 9C5, 9IH, 9IV, 9JV, 9KH, 9M1, 9OB, 9QJ, 9R1, 9RH, 9SH, 9UJ, 9VB, 9VJ, 9W5, 9X1, 9Y1, 9YB, 9YJ, 9YV, A25, A3P, A3Z, A4H, A61, A81, A8B, AAB, AAH, ABB, AC5, AEZ, AHP, AK1, AKB, AKH, ALZ, AMB, AO5, AOZ, APH, AQP, AR1, ARV, ARZ, AUV, AV5, AVB, AVV, AWV, AX1, AXB, AZ1, B2J, B4B, B4V, B6V, B87, B8H, B9B, BAB, BAH, BBH, BBV, BE7, BEB, BGJ, BHB, BIJ, BJB, BJT, BLJ, BOB, BOT, BQ7, BRV, BS7, BTT, BTV, BVB, BVJ, BVT, BWV, BX7, BXH, BYH, BYV, C25, C2J, C2N, C55, C65, C6J, C95, CCN, CDD, CED, CF5, CFJ, CGD, CIN, CJD, CO5, CRN, CUJ, CXJ, CXN, D37, D3P, D41, D55, D5P, D6T, D77, D81, D9H, DA1, DCD, DD1, DD5, DDH, DDP, DE7, DEP, DF5, DFD, DG7, DHH, DI1, DK1, DK5, DKT, DO1, DOP, DP1, DPD, DQ7, DR7, DRH, DS7, DTT, DU1, DWD, DX7, DYH, E0P, E2D, E3Z, E41, E4P, E71, E81, E87, E8B, EAZ, EDP, EED, EEN, EEV, EFZ, EGT, EI1, EKZ, ELB, EMB, EMN, EN7, EO1, EOT, EPP, EPZ, EQ7, ERN, EST, ETN, ETV, EUP, EUZ, EVT, EWD, EWN, EX1, EYB, EYN, EZD, EZP, F2J, F3B, F3Z, F4P, F65, F8B, F8Z, FCJ, FD5, FEZ, FFB, FFD, FG5, FGD, FGZ, FIJ, FJ5, FJJ, FLZ, FPD, FQD, FSZ, FXB, G07, G0D, G0Z, G3D, G3Z, G55, G6T, G7D, G7Z, G87, G8Z, GA5, GCJ, GD7, GE7, GET, GG5, GG7, GGT, GGZ, GI7, GJD, GKZ, GL7, GLT, GQT, GRT, GS7, GST, GU5, GUZ, GW5, GXJ, GZ7, GZZ, H3D, H3H, H4H, H9B, HAH, HED, HGD, HHP, HIH, HKH, HKP, HLN, HNH, HOB, HOP, HQP, HRD, HRH, HRN, HSH, HWD, HWH, HWP, HYN, I41, I6J, I71, I7N, I87, I8H, I9H, I9J, IA1, IAV, IE1, IFJ, IG7, IHH, II1, IIH, IIV, IJV, IK1, IL7, ILJ, ILN, IM1, INN, IOV, IRH, IUV, IX1, IXH, IXV, IYJ, J2D, J4V, J55, J61, J6J, J8B, J8V, J95, J9J, J9V, JA5, JBB, JD5, JDT, JF5, JFJ, JGJ, JGT, JIV, JJ1, JJD, JJV, JK5, JKT, JLB, JLT, JMN, JN5, JNJ, JQJ, JQN, JQT, JRD, JTB, JVJ, JVN, JVV, JWN, JY1, JYJ, K0B, K25, K3P, K65, K81, K95, K9H, KAH, KC5, KEP, KEZ, KG5, KHP, KK1, KKT, KLB, KLP, KLZ, KM1, KMB, KMP, KOT, KP1, KQP, KR1, KRT, KRZ, KTT, KW5, KX1, KZB, L0N, L0P, L8Z, L9B, LAZ, LBJ, LEN, LET, LFB, LFZ, LG7, LGZ, LIJ, LJJ, LKB, LKP, LLB, LLP, LLT, LMB, LMN, LN7, LO7, LOT, LPZ, LQT, LRN, LTN, LTT, LWP, LX7, LYN, M01, M0P, M2D, M2N, M37, M3B, M41, M61, M77, M87, M8D, M91, ME1, ME7, MEB, MFB, MFP, MGD, MI7, MKB, MM7, MMN, MN7, MNN, MO7, MOB, MOP, MPP, MQP, MS7, MW7, MXN, MYN, N5N, N65, N87, N8D, N8H, N95, NCD, NCJ, NDH, NE7, NFJ, NG5, NG7, NGJ, NJ5, NN7, NND, NO5, NQD, NQJ, NRN, NU5, NWH, NWN, NXH, NXN, NYJ, O0Z, O25, O37, O3B, O3P, O3Z, O41, O4B, O5Z, O61, O71, O7Z, O81, OA5, OAP, OBB, OBV, OC5, OEZ, OF5, OG7, OKP, OKZ, OMB, OO5, OOZ, OP5, OPP, OQT, OR1, OR7, OS7, OST, OTB, OTZ, OU5, OVV, OW5, OX7, OXB, OXV, OYV, OZ5, OZ7, P01, P0D, P0N, P3P, P4H, P4P, P5N, P65, P6V, P8V, PD1, PED, PEZ, PHH, PHP, PI1, PIN, PLN, PLP, PLZ, POP, PP1, PPH, PPV, PQD, PQN, PRV, PV5, PVH, PVV, PWH, PWP, PX1, PXV, PYH, PYN, Q07, Q0P, Q2N, Q37, Q3P, Q6J, Q6T, Q7D, Q8D, Q9J, QCD, QCJ, QD7, QED, QFP, QGT, QI7, QIN, QJN, QLJ, QMD, QMN, QMP, QND, QNJ, QOT, QQJ, QTN, QWN, QYJ, QYT, R0V, R0Z, R37, R3H, R4H, R77, R7D, R7N, R7Z, R81, R8V, R91, RA1, RCD, RCN, RD1, RED, REV, REZ, RGT, RGZ, RHD, RIV, RKH, RKZ, RLN, RMD, RNH, RO7, RQN, RRD, RRT, RRZ, RS7, RSH, RT1, RTV, RU1, RUZ, RVN, RVT, RW7, RWH, RX7, RY1, S0J, S6J, S87, S9H, SAZ, SCJ, SET, SFJ, SG7, SGZ, SI7, SM7, SO7, SOT, SQJ, SQT, SRH, SSH, STT, SW7, SX7, SXH, SYH, T0N, T0Z, T1D, T1T, T6V, T8Z, TBB, TBV, TCN, TD1, TEV, TGT, TKT, TLB, TM1, TO1, TOB, TQN, TR1, TRD, TT1, TTB, TTN, TUZ, TVB, TVN, TVT, TWD, TWV, TXB, TXV, TYV, U2J, U3Z, U61, U8V, U95, UA1, UC5, UEP, UEZ, UFJ, UG5, UJ5, ULZ, UO5, UOP, UOZ, UQP, URV, URZ, UUJ, UW5, UWV, UXJ, UXV, UYJ, V0H, V25, V4V, V55, V6V, V9J, V9V, VBB, VBJ, VCJ, VET, VGJ, VHP, VIH, VIN, VJJ, VJN, VLT, VMP, VNJ, VOB, VP5, VQJ, VQT, VRT, VRV, VSH, VSJ, VST, VTB, VTN, VUP, VW5, VWN, VXH, VXN, VYB, W07, W25, W3D, W3H, W4P, W4V, W57, W6V, W7D, W8H, W95, W9H, WAH, WAV, WCN, WD7, WDD, WDH, WE7, WEN, WF5, WGD, WHH, WK5, WKH, WN7, WNN, WPP, WPV, WQP, WR7, WRD, WRH, WRN, WS7, WU5, WUP, WVN, WWH, WWP, WX7, WXH, WXN, WYN, WYV, X0J, X2J, X2N, X4B, X4H, X4V, X6J, X87, X8B, X91, X9B, XAV, XCN, XEB, XFB, XHB, XHH, XHN, XKB, XLJ, XNH, XO1, XQ7, XS7, XSH, XUJ, XWN, XWV, XX7, XXH, XXV, Y0N, Y1J, Y1T, Y2N, Y3H, Y61, Y8V, Y91, YBV, YCN, YEB, YFJ, YHB, YHN, YIH, YJN, YJT, YLJ, YLN, YMB, YMN, YNH, YOB, YOV, YRH, YTB, YTN, YTT, YTV, YVB, YVH, YWH, YWV, YXB, YY1, YYJ, YYT, YYV, Z01, Z3D, Z3Z, Z71, Z8B, ZB7, ZBB, ZD5, ZDD, ZDP, ZG7, ZGD, ZGZ, ZKB, ZLP, ZM1, ZO5, ZP1, ZPD, ZQD, ZU1, ZX1, ZXB, ZZD, 102J, 106T, 10ET, 10GJ, 10LJ, 10TD, 12FJ, 12LJ, 13ED, 13GD, 13MD, 192J, 1D0T, 1DET, 1E3D, 1EFD, 1F0D, 1FWD, 1G0J, 1GGJ, 1GIJ, 1GUJ, 1I0J, 1ISJ, 1JOT, 1JTT, 1M0D, 1M3D, 1MFD, 1MWD, 1OKT, 1OOT, 1Q3D, 1QDT, 1QKT, 1QTD, 1QWD, 1SLJ, 1TDT, 1TFD, 1TOT, 1TTD, 1TTT, 1UGJ, 1ULJ, 1WFD, 1WWD, 20CJ, 20FJ, 20G5, 20IN, 20JD, 20MD, 20RN, 20WD, 20YN, 2295, 25RN, 260J, 26LJ, 288D, 28DD, 28FD, 296J, 29G5, 29K5, 29LJ, 2C05, 2C0J, 2C0N, 2CJJ, 2CK5, 2CLJ, 2CSJ, 2D25, 2DC5, 2DDD, 2DW5, 2F0J, 2FDD, 2FMD, 2FSJ, 2G65, 2G95, 2GK5, 2I0N, 2JC5, 2JCJ, 2JWD, 2K05, 2L6J, 2L9J, 2N0N, 2NIN, 2NN5, 2NSJ, 2NW5, 2NYN, 2Q0N, 2QNN, 2R0N, 2RFD, 2RWD, 2SJJ, 2SXJ, 2SYJ, 2W05, 2WC5, 2WD5, 2WMD, 2XCJ, 2YCJ, 2YJJ, 2YRN, 2YSJ, 300D, 302D, 304B, 3077, 3087, 308D, 30BH, 30DD, 30FP, 30G7, 30GD, 30MD, 30QP, 30YH, 320D, 333D, 333Z, 33WD, 344B, 34FB, 373Z, 37E7, 37EZ, 37G7, 3807, 38BB, 38E7, 38G7, 38HB, 38I7, 38X7, 390B, 394B, 399B, 39EB, 39FB, 39KB, 3ASH, 3AWH, 3B8B, 3BG7, 3BLB, 3BR7, 3BW7, 3C8D, 3D0P, 3D2D, 3DKP, 3E3D, 3E77, 3EBB, 3EFB, 3EKB, 3EQP, 3ERD, 3ERZ, 3EZZ, 3F4B, 3FEP, 3G2D, 3GCD, 3GED, 3GM7, 3GR7, 3GWD, 3GZD, 3H0B, 3HEP, 3HKB, 3IAH, 3ISH, 3K4B, 3KBH, 3KEB, 3KKB, 3L4B, 3L8B, 3M3D, 3MEP, 3MG7, 3MWP, 3OFP, 3OKB, 3OMP, 3OOB, 3OYB, 3P8H, 3PXH, 3QDP, 3QQD, 3QRD, 3QWD, 3R3D, 3R3Z, 3R8D, 3RG7, 3RM7, 3RQD, 3RR7, 3RZ7, 3SIH, 3W0H, 3WED, 3WM7, 3WQD, 3WW7, 3WYH, 3X8H, 3XE7, 3XG7, 3XKH, 3Y0B, 3Y4B, 3Y8H, 3YAH, 3YSH, 3YYB, 3Z4B, 3ZFP, 404V, 4061, 40EB, 40I1, 40IV, 40KH, 40M1, 40R1, 40XB, 40XV, 43QP, 4441, 444H, 444P, 4481, 448H, 4491, 449H, 44EB, 44EV, 44K1, 44KB, 44KP, 44M1, 44R1, 44RV, 44SH, 44U1, 44VP, 44YB, 46I1, 46K1, 46M1, 480H, 488H, 48AH, 48I1, 48K1, 48U1, 4991, 49EV, 49I1, 49O1, 49UV, 4A0H, 4A41, 4A4B, 4A8H, 4A91, 4ALB, 4ASH, 4BXV, 4E4B, 4EEP, 4EFB, 4ELP, 4EUV, 4EXB, 4FOB, 4I01, 4I0V, 4I4V, 4I61, 4I81, 4I91, 4IEV, 4IY1, 4K0H, 4K91, 4KI1, 4KKH, 4KKP, 4KUP, 4L3P, 4L4P, 4LAB, 4LMP, 4MA1, 4MK1, 4MMP, 4MP1, 4MU1, 4O0B, 4OFB, 4OKB, 4OM1, 4OU1, 4OYB, 4P41, 4P81, 4PO1, 4PYV, 4QLP, 4QQP, 4RAV, 4RK1, 4RRV, 4S8H, 4U3P, 4U91, 4U9V, 4UIV, 4UUP, 4VAV, 4VLP, 4VVV, 4X0B, 4X0V, 4XAB, 4XBV, 4XVV, 4XYV, 4Y0V, 4Y41, 4Y4H, 4Y4V, 4Y81, 4YAV, 4YEV, 4YIV, 4YK1, 4YKB, 4YM1, 4YR1, 4YUV, 4YYB, 5025, 503Z, 5095, 50G5, 50PN, 50QN, 50WP, 50Z7, 5205, 52RN, 53FP, 53M7, 53PZ, 53X7, 53ZZ, 54PP, 550N, 553P, 554P, 5557, 555N, 557Z, 5595, 55EN, 55G5, 55GZ, 55LZ, 55OZ, 55QN, 55Z7, 572N, 577Z, 5787, 57CN, 57FZ, 57QN, 57S7, 57ZZ, 58I7, 58X7, 5955, 5A95, 5AEP, 5CMN, 5CWN, 5E2N, 5EAP, 5ECN, 5EKP, 5EQN, 5F95, 5FLP, 5FQP, 5G05, 5GK5, 5GQ7, 5I37, 5II7, 5IM7, 5L3P, 5LCN, 5LFP, 5LGT, 5LKZ, 5LUP, 5LZZ, 5M07, 5N55, 5NQ7, 5NX7, 5OAZ, 5OFZ, 5OLT, 5OOT, 5P25, 5P95, 5PG5, 5PGZ, 5PPN, 5PQP, 5PW5, 5Q0N, 5Q87, 5QCN, 5QKP, 5QN7, 5QPN, 5QPP, 5QQT, 5QTT, 5QWP, 5QX7, 5S37, 5SS7, 5T6T, 5TQT, 5U4P, 5UGZ, 5W0P, 5W2N, 5W5P, 5WLN, 5WP5, 5WQN, 5XM7, 5Y0T, 5YOT, 5Z25, 5Z4P, 5Z57, 5ZG5, 5ZLZ, 5ZQ7, 5ZWP, 6005, 600V, 601J, 6055, 606V, 60A1, 60C5, 60GT, 60JJ, 60LJ, 60M1, 60R1, 60VT, 60YV, 610J, 61FJ, 6401, 64K1, 6505, 665T, 6665, 66A5, 66F5, 66G5, 66GT, 66KT, 66V5, 66XV, 68OV, 68YV, 690J, 69G5, 69O5, 69XJ, 69XV, 6A55, 6AA1, 6AA5, 6AF5, 6C9J, 6CA5, 6CGJ, 6CYJ, 6EK1, 6ER1, 6FLJ, 6FXJ, 6G0J, 6G65, 6GGJ, 6GJ5, 6GLJ, 6I81, 6I9V, 6J65, 6J6V, 6JET, 6JST, 6JWV, 6K01, 6KET, 6KST, 6L0J, 6L6T, 6LFJ, 6LRT, 6LXJ, 6MA1, 6MM1, 6MX1, 6O8V, 6OV5, 6R0T, 6RET, 6RKT, 6S6T, 6T8V, 6U01, 6U9J, 6UE1, 6UGJ, 6UJJ, 6ULJ, 6UU1, 6V0T, 6VA5, 6VF5, 6VG5, 6VKT, 6VXJ, 6WWV, 6WXV, 6X41, 6X6V, 6XIV, 6XJV, 6XOV, 6Y0V, 6YCJ, 6YXV, 7001, 702N, 7041, 707N, 7081, 70GZ, 70I7, 70KZ, 70ND, 70O7, 70OZ, 70R7, 70U1, 70UZ, 70WD, 70YN, 71QD, 71WD, 720N, 72CN, 72YN, 733Z, 73ED, 73ZD, 7401, 7441, 74M1, 74O1, 74R1, 7641, 76K1, 76M1, 76R1, 76U1, 76X1, 773Z, 7741, 7771, 7781, 77FZ, 77G7, 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8KRH, 8KU1, 8KYH, 8KZ1, 8L4B, 8LBB, 8MG7, 8MR7, 8O4V, 8OI1, 8OK1, 8OOV, 8OX1, 8RAV, 8RHH, 8RUV, 8RX1, 8U4V, 8UK1, 8V4B, 8V9B, 8VHH, 8VRH, 8VVV, 8VYV, 8W0H, 8WFD, 8X07, 8XAH, 8XR1, 8XU1, 8XYB, 8Y6V, 8Y8H, 8YBB, 8YFB, 8YHH, 8YKB, 8YLB, 8YUV, 8YVV, 8YYH, 8Z1D, 8Z4B, 8Z8D, 8Z91, 8ZCD, 8ZEZ, 8ZFD, 8ZFZ, 8ZK1, 8ZO1, 8ZX7, 90A5, 90BJ, 90BV, 90EB, 90EV, 90GJ, 90IJ, 90J5, 90JB, 90LB, 90O1, 90OV, 90XB, 90XV, 9225, 92K5, 9441, 94BH, 94EB, 94KB, 94XB, 9505, 9601, 960V, 96A1, 96J5, 96K1, 96V5, 96WV, 96XV, 984B, 98AV, 98BV, 98FB, 994H, 998B, 998V, 999B, 99A1, 99A5, 99E1, 99G5, 99O1, 99O5, 99WH, 9A05, 9A55, 9A95, 9AA5, 9ABV, 9AE1, 9AEV, 9AI1, 9AO1, 9AOV, 9B0J, 9B0V, 9B8V, 9B9H, 9BCJ, 9BFJ, 9BSJ, 9BVV, 9CCJ, 9CLJ, 9CSJ, 9E01, 9E4B, 9EA1, 9EAB, 9EAV, 9EK1, 9F05, 9F25, 9FA5, 9FBJ, 9FFJ, 9FJB, 9FSJ, 9FXJ, 9G25, 9G2J, 9GIJ, 9GSJ, 9HFB, 9HLB, 9IIJ, 9IJ1, 9J01, 9J0B, 9J0J, 9J9B, 9JCJ, 9JE1, 9K05, 9K41, 9K55, 9K61, 9K8B, 9KA5, 9KI1, 9KK5, 9KU5, 9L2J, 9LCJ, 9O05, 9O0V, 9O95, 9OI1, 9OO1, 9OOV, 9OWV, 9RRV, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ

====Additional known quasi-minimal primes (not necessarily the next)====

P&lt;sub&gt;81993&lt;/sub&gt;SZ

==Unsolved families==

Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')) by covering congruence, algebraic factorization, or combine of them.

{|class=wikitable
|base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family
|-
|13||9{5}||(113×13&lt;sup&gt;''n''&lt;/sup&gt;−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|13||A{3}A||(41×13&lt;sup&gt;''n''+1&lt;/sup&gt;+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|16||{3}AF||(16&lt;sup&gt;''n''+2&lt;/sup&gt;+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1{7}||(23×17&lt;sup&gt;''n''&lt;/sup&gt;−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1F{0}7||32×17&lt;sup&gt;''n''+1&lt;/sup&gt;+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||4{7}A||(71×17&lt;sup&gt;''n''+1&lt;/sup&gt;+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||51{0}D||86×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||70F{0}D||2038×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||8{B}9||(139×17&lt;sup&gt;''n''+1&lt;/sup&gt;−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||9{5}9||(149×17&lt;sup&gt;''n''+1&lt;/sup&gt;+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||95{F}||(2543×17&lt;sup&gt;''n''&lt;/sup&gt;−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||A{D}F||(173×17&lt;sup&gt;''n''+1&lt;/sup&gt;+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}B3||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+190||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}DB||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+232||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2BE||(11×17&lt;sup&gt;''n''+3&lt;/sup&gt;−41579)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2E||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;−2411)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}E9||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+773)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}EE||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+853)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||7{K}Z||(53×36&lt;sup&gt;''n''+1&lt;/sup&gt;+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||B{0}EUV||11×36&lt;sup&gt;''n''+3&lt;/sup&gt;+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||HM{0}N||634×36&lt;sup&gt;''n''+1&lt;/sup&gt;+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||N{0}YYN||23×36&lt;sup&gt;''n''+3&lt;/sup&gt;+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||O{L}Z||(123×36&lt;sup&gt;''n''+1&lt;/sup&gt;+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||S{0}8H||28×36&lt;sup&gt;''n''+2&lt;/sup&gt;+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|}

For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E

For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D}

=== Base 17 ===

All searched to length 29000

* 1{7}
* 1F{0}7
* 4{7}A
* 51{0}D
* 70F{0}D
* 8{B}9
* 9{5}9
* 95{F}
* A{D}F
* B{0}B3
* B{0}DB
* {B}2BE
* {B}2E
* {B}E9
* {B}EE
* D0G{D}
* E9{B}
* F1{9}
* FD0{D}
* G{7}F

=== Base 21 ===

All searched to length 20000

* 2{7}9D
* 2F{C}A
* 4{3}B
* 5{0}DJ
* {5}FEK
* {7}ID
* 99{0}99H
* {9}0D
* {9}D
* B0{H}6H
* B3{0}EB
* B9{0}E5
* B{D}B
* B{H}6H
* DH{D}
* F{9}D
* {F}35
* G{0}FK
* H{0}7771
* H{D}
* {J}6J

=== Base 36 ===

All searched to length 20000

* 7{K}Z
* B{0}EUV
* HM{0}N
* N{0}YYN
* O{L}Z
* S{0}8H

==Primality certificates for the proven primes &gt; 10&lt;sup&gt;299&lt;/sup&gt;==

See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]]

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime
|-
|9||149||76&lt;sub&gt;329&lt;/sub&gt;2||(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642]
|-
|9||150||27&lt;sub&gt;686&lt;/sub&gt;07||(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486]
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||3×9&lt;sup&gt;1160&lt;/sup&gt;+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423]
|-
|11||1065||A&lt;sub&gt;713&lt;/sub&gt;58||11&lt;sup&gt;715&lt;/sup&gt;−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487]
|-
|11||1066||7&lt;sub&gt;759&lt;/sub&gt;44||(7×11&lt;sup&gt;761&lt;/sup&gt;−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840]
|-
|11||1067||557&lt;sub&gt;1011&lt;/sub&gt;||(607×11&lt;sup&gt;1011&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522]
|-
|13||3165||50&lt;sub&gt;270&lt;/sub&gt;44||5×13&lt;sup&gt;272&lt;/sup&gt;+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005]
|-
|13||3166||9&lt;sub&gt;271&lt;/sub&gt;095||(3×13&lt;sup&gt;274&lt;/sup&gt;−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654]
|-
|13||3167||10&lt;sub&gt;286&lt;/sub&gt;7771||13&lt;sup&gt;290&lt;/sup&gt;+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633]
|-
|13||3168||9&lt;sub&gt;308&lt;/sub&gt;1||(3×13&lt;sup&gt;309&lt;/sup&gt;−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=308&amp;c0=-&amp;EN= 13&lt;sup&gt;308&lt;/sup&gt;−1]
|-
|13||3169||B&lt;sub&gt;341&lt;/sub&gt;C4||(11×13&lt;sup&gt;343&lt;/sup&gt;+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618]
|-
|13||3170||8B&lt;sub&gt;343&lt;/sub&gt;||(107×13&lt;sup&gt;343&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736]
|-
|13||3171||710&lt;sub&gt;371&lt;/sub&gt;111||92×13&lt;sup&gt;374&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609]
|-
|13||3172||75&lt;sub&gt;375&lt;/sub&gt;7||(89×13&lt;sup&gt;376&lt;/sup&gt;+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596]
|-
|13||3173||9B0&lt;sub&gt;391&lt;/sub&gt;9||128×13&lt;sup&gt;392&lt;/sup&gt;+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790]
|-
|13||3174||7B0B&lt;sub&gt;397&lt;/sub&gt;||(15923×13&lt;sup&gt;397&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574]
|-
|13||3175||10&lt;sub&gt;414&lt;/sub&gt;93||13&lt;sup&gt;416&lt;/sup&gt;+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240]
|-
|13||3176||81010&lt;sub&gt;415&lt;/sub&gt;1||17746×13&lt;sup&gt;416&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3177||8110&lt;sub&gt;435&lt;/sub&gt;1||1366×13&lt;sup&gt;436&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3178||B7&lt;sub&gt;486&lt;/sub&gt;||(139×13&lt;sup&gt;486&lt;/sup&gt;−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892]
|-
|13||3179||B&lt;sub&gt;563&lt;/sub&gt;C||(11×13&lt;sup&gt;564&lt;/sup&gt;+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=564&amp;c0=-&amp;EN= 13&lt;sup&gt;564&lt;/sup&gt;−1]
|-
|13||3180||1B&lt;sub&gt;576&lt;/sub&gt;||(23×13&lt;sup&gt;576&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=576&amp;c0=-&amp;EN= 13&lt;sup&gt;576&lt;/sup&gt;−1]
|-
|13||3181||80&lt;sub&gt;693&lt;/sub&gt;87||8×13&lt;sup&gt;695&lt;/sup&gt;+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532]
|-
|13||3182||CC5&lt;sub&gt;713&lt;/sub&gt;||(2021×13&lt;sup&gt;713&lt;/sup&gt;−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353]
|-
|13||3183||B&lt;sub&gt;834&lt;/sub&gt;74||(11×13&lt;sup&gt;836&lt;/sup&gt;−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871]
|-
|13||3184||9&lt;sub&gt;968&lt;/sub&gt;B||(3×13&lt;sup&gt;969&lt;/sup&gt;+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244]
|-
|13||3185||10&lt;sub&gt;1295&lt;/sub&gt;181||13&lt;sup&gt;1298&lt;/sup&gt;+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013]
|-
|13||3186||9&lt;sub&gt;1362&lt;/sub&gt;5||(3×13&lt;sup&gt;1363&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776]
|-
|13||3187||7&lt;sub&gt;1504&lt;/sub&gt;1||(7×13&lt;sup&gt;1505&lt;/sup&gt;−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755]
|-
|13||3188||930&lt;sub&gt;1551&lt;/sub&gt;1||120×13&lt;sup&gt;1552&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3189||720&lt;sub&gt;2297&lt;/sub&gt;2||93×13&lt;sup&gt;2298&lt;/sup&gt;+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910]
|-
|13||3190||1770&lt;sub&gt;2703&lt;/sub&gt;17||267×13&lt;sup&gt;2705&lt;/sup&gt;+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825]
|-
|13||3191||390&lt;sub&gt;6266&lt;/sub&gt;1||48×13&lt;sup&gt;6267&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3192||B0&lt;sub&gt;6540&lt;/sub&gt;BBA||11×13&lt;sup&gt;6543&lt;/sup&gt;+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906]
|-
|13||3193||C&lt;sub&gt;10631&lt;/sub&gt;92||13&lt;sup&gt;10633&lt;/sup&gt;−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750]
|-
|14||649||34D&lt;sub&gt;708&lt;/sub&gt;||47×14&lt;sup&gt;708&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||5×14&lt;sup&gt;19698&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2328||880&lt;sub&gt;246&lt;/sub&gt;7||136×16&lt;sup&gt;247&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is &lt; 10&lt;sup&gt;299&lt;/sup&gt;
|-
|16||2329||D4&lt;sub&gt;263&lt;/sub&gt;D||(199×16&lt;sup&gt;264&lt;/sup&gt;+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238]
|-
|16||2330||E0&lt;sub&gt;261&lt;/sub&gt;4DD||14×16&lt;sup&gt;264&lt;/sup&gt;+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352]
|-
|16||2331||8C0&lt;sub&gt;290&lt;/sub&gt;ED||140×16&lt;sup&gt;292&lt;/sup&gt;+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307]
|-
|16||2332||DA&lt;sub&gt;305&lt;/sub&gt;5||(41×16&lt;sup&gt;306&lt;/sup&gt;−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284]
|-
|16||2333||CE80&lt;sub&gt;422&lt;/sub&gt;D||3304×16&lt;sup&gt;423&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257]
|-
|16||2334||5F&lt;sub&gt;544&lt;/sub&gt;6F||6×16&lt;sup&gt;546&lt;/sup&gt;−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967]
|-
|16||2335||88F&lt;sub&gt;545&lt;/sub&gt;||137×16&lt;sup&gt;545&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2336||BE0&lt;sub&gt;792&lt;/sub&gt;BB||190×16&lt;sup&gt;794&lt;/sup&gt;+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938]
|-
|16||2337||D9&lt;sub&gt;1052&lt;/sub&gt;||(68×16&lt;sup&gt;1052&lt;/sup&gt;−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020]
|-
|16||2338||FAF&lt;sub&gt;1062&lt;/sub&gt;45||251×16&lt;sup&gt;1064&lt;/sup&gt;−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610]
|-
|16||2339||F8&lt;sub&gt;1517&lt;/sub&gt;F||(233×16&lt;sup&gt;1518&lt;/sup&gt;+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824]
|-
|16||2340||20&lt;sub&gt;1713&lt;/sub&gt;321||2×16&lt;sup&gt;1716&lt;/sup&gt;+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735]
|-
|16||2341||300F&lt;sub&gt;1960&lt;/sub&gt;AF||769×16&lt;sup&gt;1962&lt;/sup&gt;−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750]
|-
|16||2342||90&lt;sub&gt;3542&lt;/sub&gt;91||9×16&lt;sup&gt;3544&lt;/sup&gt;+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191]
|-
|16||2343||5BC&lt;sub&gt;3700&lt;/sub&gt;D||(459×16&lt;sup&gt;3701&lt;/sup&gt;+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322]
|-
|16||2344||D0B&lt;sub&gt;17804&lt;/sub&gt;||(3131×16&lt;sup&gt;17804&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511]
|-
|18||547||80&lt;sub&gt;298&lt;/sub&gt;B||8×18&lt;sup&gt;299&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is an algebraic factor of ''N''+1, factordb entry of 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is [http://factordb.com/index.php?id=1100000000900149167]
|-
|18||548||H&lt;sub&gt;766&lt;/sub&gt;FH||18&lt;sup&gt;768&lt;/sup&gt;−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490]
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||12×18&lt;sup&gt;6270&lt;/sup&gt;+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437]
|-
|20||3301||H&lt;sub&gt;247&lt;/sub&gt;A0H||(17×20&lt;sup&gt;250&lt;/sup&gt;−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619]
|-
|20||3302||7&lt;sub&gt;249&lt;/sub&gt;A7||(7×20&lt;sup&gt;251&lt;/sup&gt;+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602]
|-
|20||3303||J7&lt;sub&gt;270&lt;/sub&gt;||(368×20&lt;sup&gt;270&lt;/sup&gt;−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462]
|-
|20||3304||J&lt;sub&gt;330&lt;/sub&gt;CCC7||20&lt;sup&gt;334&lt;/sup&gt;−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572]
|-
|20||3305||40&lt;sub&gt;387&lt;/sub&gt;404B||4×20&lt;sup&gt;391&lt;/sup&gt;+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563]
|-
|20||3306||EC0&lt;sub&gt;429&lt;/sub&gt;7||292×20&lt;sup&gt;430&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702]
|-
|20||3307||G&lt;sub&gt;447&lt;/sub&gt;99||(16×20&lt;sup&gt;449&lt;/sup&gt;−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753]
|-
|20||3308||3A&lt;sub&gt;527&lt;/sub&gt;3||(67×20&lt;sup&gt;528&lt;/sup&gt;−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531]
|-
|20||3309||E&lt;sub&gt;566&lt;/sub&gt;C7||(14×20&lt;sup&gt;568&lt;/sup&gt;−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516]
|-
|20||3310||JCJ&lt;sub&gt;629&lt;/sub&gt;||393×20&lt;sup&gt;629&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|20||3311||J&lt;sub&gt;655&lt;/sub&gt;05J||20&lt;sup&gt;658&lt;/sup&gt;−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052]
|-
|20||3312||50&lt;sub&gt;1163&lt;/sub&gt;AJ||5×20&lt;sup&gt;1165&lt;/sup&gt;+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412]
|-
|20||3313||CD&lt;sub&gt;2449&lt;/sub&gt;||(241×20&lt;sup&gt;2449&lt;/sup&gt;−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915]
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||16×20&lt;sup&gt;6270&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457]
|-
|22||7984||I7G0&lt;sub&gt;254&lt;/sub&gt;H||8882×22&lt;sup&gt;255&lt;/sup&gt;+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788]
|-
|22||7985||D0&lt;sub&gt;255&lt;/sub&gt;5EEF||13×22&lt;sup&gt;259&lt;/sup&gt;+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932]
|-
|22||7986||IK&lt;sub&gt;322&lt;/sub&gt;F||(398×22&lt;sup&gt;323&lt;/sup&gt;−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145]
|-
|22||7987||C0&lt;sub&gt;340&lt;/sub&gt;G9||12×22&lt;sup&gt;342&lt;/sup&gt;+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159]
|-
|22||7988||77E&lt;sub&gt;348&lt;/sub&gt;K7||(485×22&lt;sup&gt;350&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779]
|-
|22||7989||J&lt;sub&gt;379&lt;/sub&gt;KJ||(19×22&lt;sup&gt;381&lt;/sup&gt;+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027]
|-
|22||7990||J&lt;sub&gt;388&lt;/sub&gt;EJ||(19×22&lt;sup&gt;390&lt;/sup&gt;−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729]
|-
|22||7991||DJ&lt;sub&gt;400&lt;/sub&gt;||(292×22&lt;sup&gt;400&lt;/sup&gt;−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110]
|-
|22||7992||E&lt;sub&gt;404&lt;/sub&gt;K7||(2×22&lt;sup&gt;406&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298]
|-
|22||7993||66F&lt;sub&gt;453&lt;/sub&gt;B3||(971×22&lt;sup&gt;455&lt;/sup&gt;−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809]
|-
|22||7994||L0&lt;sub&gt;454&lt;/sub&gt;B63||21×22&lt;sup&gt;457&lt;/sup&gt;+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331]
|-
|22||7995||L&lt;sub&gt;483&lt;/sub&gt;G3||22&lt;sup&gt;485&lt;/sup&gt;−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730]
|-
|22||7996||E60&lt;sub&gt;496&lt;/sub&gt;L||314×22&lt;sup&gt;497&lt;/sup&gt;+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239]
|-
|22||7997||I&lt;sub&gt;626&lt;/sub&gt;AF||(6×22&lt;sup&gt;628&lt;/sup&gt;−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334]
|-
|22||7998||K0&lt;sub&gt;760&lt;/sub&gt;EC1||20×22&lt;sup&gt;763&lt;/sup&gt;+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415]
|-
|22||7999||J0&lt;sub&gt;767&lt;/sub&gt;IGGJ||19×22&lt;sup&gt;771&lt;/sup&gt;+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567]
|-
|22||8000||7&lt;sub&gt;959&lt;/sub&gt;K7||(22&lt;sup&gt;961&lt;/sup&gt;+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817]
|-
|22||8001||L&lt;sub&gt;2385&lt;/sub&gt;KE7||22&lt;sup&gt;2388&lt;/sup&gt;−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774]
|-
|22||8002||7&lt;sub&gt;3815&lt;/sub&gt;2L||(22&lt;sup&gt;3817&lt;/sup&gt;−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839]
|-
|24||3400||I0&lt;sub&gt;241&lt;/sub&gt;I5||18×24&lt;sup&gt;243&lt;/sup&gt;+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037]
|-
|24||3401||D0&lt;sub&gt;259&lt;/sub&gt;KKD||13×24&lt;sup&gt;262&lt;/sup&gt;+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725]
|-
|24||3402||C7&lt;sub&gt;298&lt;/sub&gt;||(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235]
|-
|24||3403||20&lt;sub&gt;313&lt;/sub&gt;7||2×24&lt;sup&gt;314&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241]
|-
|24||3404||BC0&lt;sub&gt;331&lt;/sub&gt;B||276×24&lt;sup&gt;332&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842]
|-
|24||3405||N&lt;sub&gt;2644&lt;/sub&gt;LLN||24&lt;sup&gt;2647&lt;/sup&gt;−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089]
|-
|24||3406||D&lt;sub&gt;2698&lt;/sub&gt;LD||(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876]
|-
|24||3407||A0&lt;sub&gt;2951&lt;/sub&gt;8ID||10×24&lt;sup&gt;2954&lt;/sup&gt;+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654]
|-
|24||3408||88N&lt;sub&gt;5951&lt;/sub&gt;||201×24&lt;sup&gt;5951&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606]
|-
|30||2613||AN&lt;sub&gt;206&lt;/sub&gt;||(313×30&lt;sup&gt;206&lt;/sup&gt;−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073]
|-
|30||2614||M&lt;sub&gt;241&lt;/sub&gt;QB||(22×30&lt;sup&gt;243&lt;/sup&gt;+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295]
|-
|30||2615||M0&lt;sub&gt;547&lt;/sub&gt;SS7||22×30&lt;sup&gt;550&lt;/sup&gt;+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988]
|-
|30||2616||C0&lt;sub&gt;1022&lt;/sub&gt;1||12×30&lt;sup&gt;1023&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|30||2617||5&lt;sub&gt;4882&lt;/sub&gt;J||(5×30&lt;sup&gt;4883&lt;/sup&gt;+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423]
|-
|30||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||25×30&lt;sup&gt;34205&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|}

==Unproven PRPs==

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this PRP
|-
|11||1068||57&lt;sub&gt;62668&lt;/sub&gt;||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000003573679860]
|-
|13||3194||C5&lt;sub&gt;23755&lt;/sub&gt;C||(149×13&lt;sup&gt;23756&lt;/sup&gt;+79)/12||[http://factordb.com/index.php?id=1100000003590647776]
|-
|13||3195||80&lt;sub&gt;32017&lt;/sub&gt;111||8×13&lt;sup&gt;32020&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000000490878060]
|-
|16||2345||DB&lt;sub&gt;32234&lt;/sub&gt;||(206×16&lt;sup&gt;32234&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000002383583629]
|-
|16||2346||4&lt;sub&gt;72785&lt;/sub&gt;DD||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15||[http://factordb.com/index.php?id=1100000003615909841]
|-
|22||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21||[http://factordb.com/index.php?id=1100000003594696838]
|-
|30||2618||I0&lt;sub&gt;24608&lt;/sub&gt;D||18×30&lt;sup&gt;24609&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003593967511]
|}

All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10&lt;sup&gt;16&lt;/sup&gt;. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]])

==Proof==

===Base 2===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.

===Base 3===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (2,1), (2,2)

* Case (1,1):
** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes)
*** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime.

===Base 4===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (2,1), (2,3), (3,1), (3,3)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.

===Base 5===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4)

* Case (1,1):
** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes)
*** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes)
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes)
***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb])
***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes)
**** The smallest prime of the form 1{4}4 is 14444.
**** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes)
*** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes)
**** For the 3{0,3}1 family, we can separate this family to four families:
***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01.
****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime.
***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031.
***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001.
***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31.
****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime.
**** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.
* Case (3,4):
** 34 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes)
**** The smallest prime of the form 4{4}1 is 44441.
**** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime.
* Case (4,2):
** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime.

===Base 6===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes)
*** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes)
**** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 4{0}41 is 40041
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes)
*** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.

===Base 7===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

* Case (1,1):
** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes)
*** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {}
**** If there are &gt;=3 number of 1 and 5 in the {}:
***** If there is 111 in the {}, then we have the prime 11111
***** If there is 115 in the {}, then the prime 115 is a subsequence
***** If there is 151 in the {}, then the prime 115 is a subsequence
***** If there is 155 in the {}, then the prime 155 is a subsequence
***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime
***** If there is 515 in the {}, then the prime 515 is a subsequence
***** If there is 551 in the {}, then the prime 551 is a subsequence
***** If there is 555 in the {}, then the prime 551 is a subsequence
**** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1
***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1
****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1
******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21
******** The smallest prime of the form 11{0}21 is 1100021
******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime
******** The smallest prime of the form 1001{0}21 is 100121
********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families
***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1
****** Since 1051 is prime, we only need to consider the family 15{0}1
******* The smallest prime of the form 15{0}1 is 150001
* Case (1,2):
** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes)
*** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {}
**** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime.
**** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime.
**** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2
***** Since 1022 and 1202 are primes, we only need to consider the number 122
****** 122 is not prime.
**** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime.
* Case (1,3):
** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** 14 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime.
* Case (1,6):
** 16 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime.
* Case (2,2):
** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,6):
** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes)
*** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes)
**** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's)
***** For the 3{0,1}1 family:
****** If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
****** If there are no 1's between (3,1), then the form will be 3{0}1
******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime.
****** If there are no 0's between (3,1), then the form will be 3{1}1
******* The smallest prime of the form 3{1}1 is 31111
****** If there are exactly 1 0's between (3,1), then there must be &lt;3 1's between (3,1), or 31111 will be a subsequence.
******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime.
******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011
******** Neither 3101 nor 3011 is prime.
******* If there are no 1's between (3,1), then the number must be 301
******** 301 is not prime.
***** For the 3{3}3{0,1}1 family:
****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence.
****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1
******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence.
******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime.
******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime.
******* If there are no 1's between (33,1), then the form will be 33{0}1
******** The smallest prime of the form 33{0}1 is 33001
****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1
******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence.
******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301
******** The smallest prime of the form 3{3}301 is 33333301
******* If there are no 0's between (3{3}3,1), then the form is 3{3}31
******** The smallest prime of the form 3{3}31 is 33333333333333331
**** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1
***** Numbers containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1,3}3{0,1,3}1
******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes)
******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes)
********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes)
********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence.
********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime.
********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1
*********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime.
********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime.
***** Numbers not containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1}1
******* If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1
******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime.
******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1
******** If there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111
********* None of 31151, 31511, 35111 are primes.
******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 351
********* 351 is not prime.
******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111
********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes.
********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime.
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1
********* Neither 3051 nor 3501 is prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** If there are &gt;=2 5's in {}, then 553 will be a subsequence.
*** If there are no 5's in {}, then the family will be 3{0,3,6}3
**** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
*** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3
**** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes)
***** The smallest prime of the form 3{0}53 is 300053
* Case (3,4):
** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes)
*** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes)
**** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes)
**** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes)
* Case (3,5):
** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes)
*** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence.
*** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence.
*** If there are at least two 1's in {}, then 115 will be a subsequence.
*** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5
**** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime.
*** If there is no 1's in {}, then the family will be 3{0,4,5}5
**** If there are at least to 4's in {}, then 344 and 445 will be subsequences.
**** If there is no 4's in {}, then the family will be 3{0,5}5
***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5
****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime.
****** The smallest prime of the form 3{5}5 is 35555
**** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5
***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes)
****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime.
* Case (3,6):
** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,2):
** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes)
*** If there is no 5's in {}, then the family will be 4{0,2,4,6}4
**** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4
**** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes)
***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence.
***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4
****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes)
***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4
****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4
******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes)
* Case (4,5):
** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** If there are at least two 5's in {}, then 4555 will be a subsequence.
*** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there is no 5's in {}, then the family will be 4{0}5
**** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime.
* Case (4,6):
** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (5,1):
** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 3's in {}, then 533 will be a subsequence.
*** If there is no 3's in {}, then the family will be 5{0,1}1
**** Since 5011 is prime, we only need to consider the family 5{1}{0}1
***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes)
****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime.
****** The smallest prime of the form 51{0}1 is 5100000001
****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime.
****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime.
*** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1
**** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence.
***** Thus we only need to consider the family 5{0}3{0,1}1
****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime.
****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime.
****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence.
****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1
******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0)
******** The smallest prime of the form 5311{0}1 is 531101
* Case (5,2):
** 52 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 1's in {}, then 113 will be a subsequence.
*** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
*** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime.
* Case (5,4):
** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes)
*** If there are no 5's in {}, then the family will be 5{0,3}4
**** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime.
*** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence.
*** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there are at least two 5's in {}, then 5554 will be a subsequence.
* Case (5,5):
** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes)
*** If there are no 4's in {}, then the family will be 5{0,5}5
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
*** If there are no 5's in {}, then the family will be 5{0,4}5
**** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime.
*** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence.
* Case (5,6):
** 56 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,2):
** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (6,3):
** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,4):
** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are at least three 3's in {}, then 3334 will be a subsequence.
*** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4
**** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence.
**** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence.
**** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence.
**** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence.
**** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence.
**** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence.
**** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence.
**** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence.
**** Thus the number can only be 634
***** 634 is not prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,6):
** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}6
**** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime.
*** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6
**** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime.
*** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence.
*** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence.

===Base 8===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7)

* Case (1,1):
** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes)
**** 171 is not prime.
**** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 &gt; 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 &gt; 1, this factorization is nontrivial), thus cannot be prime.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes)
*** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime)
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes)
*** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes)
*** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes)
**** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime.
**** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes)
***** The smallest prime of the form 33{4}1 is 3344441
***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime.
**** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes)
***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime.
***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes)
****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes.
* Case (3,3):
** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes)
*** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes)
**** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime)
**** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461
***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes)
****** The smallest prime of the form 4{4}41 is 444444441
****** The smallest prime of the form 4{4}641 is 444641
***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes)
****** The smallest prime of the form 4{4}411 is 444444441
****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes)
***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461
****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime)
**** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes)
***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime)
***** The smallest prime of the form 4{4}641 is 444641
* Case (4,3):
** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes)
*** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes)
**** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime.
**** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime.
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes)
*** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes)
**** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb])
**** The smallest prime of the form 4{4}77 is 4444477
**** The smallest prime of the form 4{7}7 is 47777
**** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime)
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes)
*** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes)
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
**** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes)
***** 500525 is not prime.
***** The smallest prime of the form 5{5}25 is 555555555555525
***** The smallest prime of the form 5{5}025 is 55555025
***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes)
***** The smallest prime of the form 5{5}0525 is 5550525
***** The smallest prime of the form 5{5}00525 is 5500525
***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes)
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence)
**** The form is 6{0,1,7}4{0,1,7}1
***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1
****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411
******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41
******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41
********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.
********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime.
********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime.
******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411
******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime)
*** Numbers not containing 4:
**** The form is 6{0,1,7}1
***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1
****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime.
****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1
******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1
******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.
******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime)
* Case (6,3):
** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,7):
** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them
**** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17
***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime.
**** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77
***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime.
**** If there is neither 1 nor 7 between them, then the form is 6{0,4}7
***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes)
****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime.
****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime.
* Case (7,1):
** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes)
*** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes)
**** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes)
***** The smallest prime of the form 7{7}1 is 7777777777771
***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime)
***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime.
***** The smallest prime of the form 7{7}4444441 is 77774444441
****** Since this prime has just 4 7's, we only need to consider the families with &lt;=3 7's
******* The smallest prime of the form 7{4}1 is 744444441
******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime.
******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes)
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.

===Base 10===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,9):
** 19 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 2{0}21 is 20021
**** The smallest prime of the form 22{0}1 is 22000001
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes)
*** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes)
**** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime.
**** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime.
* Case (2,9):
** 29 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,9):
** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** 47 is prime, and thus the only minimal prime in this family.
* Case (4,9):
** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime.
* Case (5,1):
** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes)
**** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes)
***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 5{5}1 is 555555555551
**** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581
***** 581 is not prime.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,7):
** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027
**** The smallest prime of the form 52{0}7 is 5200007
* Case (5,9):
** 59 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,3):
** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,9):
** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49
***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49
****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes)
******* The smallest prime of the form 6{6}49 is 666649
******** Since this prime has just 4 6's, we only need to consider the families with &lt;=3 6's
********* The smallest prime of the form 6{0}49 is 60000049
********* The smallest prime of the form 66{0}49 is 66000049
********* The smallest prime of the form 666{0}49 is 66600049
* Case (7,1):
** 71 is prime, and thus the only minimal prime in this family.
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,9):
** 79 is prime, and thus the only minimal prime in this family.
* Case (8,1):
** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes)
**** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes)
***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime)
***** The smallest prime of the form 80{5}1 is 80555551
* Case (8,3):
** 83 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime.
* Case (8,9):
** 89 is prime, and thus the only minimal prime in this family.
* Case (9,1):
** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes)
**** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021
***** None of 921, 9201, 9021 are primes.
**** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801
***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
* Case (9,3):
** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (9,7):
** 97 is prime, and thus the only minimal prime in this family.
* Case (9,9):
** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9
***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes)
****** The smallest prime of the form 94{6}9 is 946669

===Base 12===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,B):
** 1B is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes)
*** The smallest prime of the form 2{0}1 is 2001
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (2,B):
** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes)
**** None of 20B, 22B, 29B, 209B, 220B are primes.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,B):
** 3B is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes)
**** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 44{A}1 is 44AAA1
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes)
**** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077
* Case (4,B):
** 4B is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes)
*** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime.
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (5,B):
** 5B is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,5):
** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes)
**** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 6{0}A5 is 600A5
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,B):
** 6B is prime, and thus the only minimal prime in this family.
* Case (7,1):
** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes)
*** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1
**** None of 741, 7B1, 74B1 are primes.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,B):
** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes)
**** The smallest prime of the form 7{9}B is 7999B
**** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes)
**** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime)
* Case (8,1):
** 81 is prime, and thus the only minimal prime in this family.
* Case (8,5):
** 85 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** 87 is prime, and thus the only minimal prime in this family.
* Case (8,B):
** 8B is prime, and thus the only minimal prime in this family.
* Case (9,1):
** 91 is prime, and thus the only minimal prime in this family.
* Case (9,5):
** 95 is prime, and thus the only minimal prime in this family.
* Case (9,7):
** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes)
**** None of 947, 977, 997, 9477, 9977 are primes.
* Case (9,B):
** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes)
**** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes)
***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime)
***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime)
**** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes)
***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes)
****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime.
****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime)
****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime.
**** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B
***** None of 99B, 999B, 979B, 9799B, 9979B are primes.
* Case (A,1):
** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes)
**** For the A{A}{0}1 family:
***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form AA{0}1 is AA000001
***** The smallest prime of the form AAA{0}1 is AAA0001
***** The smallest prime of the form AAAA{0}1 is AAAA1
****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc.
**** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime.
* Case (A,5):
** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes)
*** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes)
**** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form A{0}65 is A00065
* Case (A,7):
** A7 is prime, and thus the only minimal prime in this family.
* Case (A,B):
** AB is prime, and thus the only minimal prime in this family.
* Case (B,1):
** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes)
**** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1
***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1
****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime)
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1
****** The smallest prime of the form B{A}01 is BAA01
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
**** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1
***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1
****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1
******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1
******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes.
****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1
******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime)
***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes)
****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B04{A}1 is B04A1
****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime)
**** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1
***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1
****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1
******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime.
******* The smallest prime of the form B{B}AA1 is BBBAA1
****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1
******* The smallest prime of the form B{B}AA1 is BBBAA1
**** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes)
***** The smallest prime of the form B{B}B1 is BBBB1
***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form B{B}BAA1 is BBBAA1
* Case (B,5):
** B5 is prime, and thus the only minimal prime in this family.
* Case (B,7):
** B7 is prime, and thus the only minimal prime in this family.
* Case (B,B):
** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes)
*** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B
**** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B
***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime.
***** For the B{0,B}9B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes)
******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B
******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 &gt; 1, 10^n+5 ≥ 10^1+5 = 15 &gt; 1, this factorization is nontrivial), thus cannot be prime.
***** For the B{0,B}99B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes)
******* The smallest prime of the form B{0}99B is B00099B
******* The smallest prime of the form B{B}99B is BBBBBB99B
***** For the B{0,B}999B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes)
******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes)
******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime)

== Examples of families which can be ruled out as contain no primes &gt; ''b'' ==

It is not known if this problem is solvable:

Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''?

It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime &gt; ''b'' or not. We use two different heuristic strategies to show that such families contain no primes &gt; ''b''.

In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;.

Examples of first strategy: (we can show that the corresponding numbers are &gt; all elements in ''S'', if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial)

* In base 10, all numbers of the form 4{6}9 are divisible by 7
* In base 6, all numbers of the form 4{0}1 are divisible by 5
* In base 15, all numbers of the form 9{6}8 are divisible by 11
* In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5}
* In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3}
* In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5}
* In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13}
* In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17}
* In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13}
* In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13}

Examples of second strategy: (we can show that both factors are &gt; 1, if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial)

* In base 9, all numbers of the form {1} factored as difference of squares
* In base 8, all numbers of the form 1{0}1 factored as sum of cubes
* In base 9, all numbers of the form 3{8} factored as difference of squares
* In base 16, all numbers of the form 8{F} factored as difference of squares
* In base 16, all numbers of the form {F}7 factored as difference of squares
* In base 9, all numbers of the form 3{1} factored as difference of squares
* In base 16, all numbers of the form {4}1 factored as difference of squares
* In base 16, all numbers of the form 1{5} factored as difference of squares
* In base 16, all numbers of the from {C}D factored as ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;
* In base 16, all numbers of the form B{4}1 factored as difference of squares

Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are &gt; all elements in S, and for the part of the second strategy, both factors are &gt; 1, if n makes corresponding numbers &gt; b, thus these factorizations are nontrivial)

* In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd
* In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even

== Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes &gt; ''b'' ==

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-powers factorization

=== 1{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 1{0}3 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== 1{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 1{0}5 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 5: Finite covering set {5}

=== 1{0}6 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== 1{0}7 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 7: Finite covering set {7}

=== 1{0}z ===

(none)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

(none)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {1} ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with ''r''&gt;1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2))

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' == 0 mod 2: Finite covering set {2}

=== 1{2} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 1{3} ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361)
* ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865)

=== 1{4} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization

=== 1{z} ===

(none)

=== 2{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 2{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {2}1 ===

* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 2{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 3{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {3}1 ===

* ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121)

=== 3{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 4{0}1 ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 4{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {4}1 ===

* ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225)

=== 4{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 5{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 5{z} ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}
* ''b'' = 6''m''&lt;sup&gt;2&lt;/sup&gt; with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014)

=== 6{0}1 ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== 6{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 7{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 7{z} ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993)
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Difference-of-cubes factorization

=== 8{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022)
* ''b'' = 467: Finite covering set {3, 5, 7, 19, 37}
* ''b'' = 722: Finite covering set {3, 5, 13, 73, 109}
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Sum-of-cubes factorization
* ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2

=== 8{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 9{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 9{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{0}1 ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{z} ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 307: Finite covering set {5, 11, 29}
* ''b'' = 901: Finite covering set {7, 11, 13, 19}

=== C{0}1 ===

* ''b'' == 1 mod 13: Finite covering set {13}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 296, 901: Finite covering set {7, 11, 13, 19}
* ''b'' = 562, 828, 900: Finite covering set {7, 13, 19}
* ''b'' = 563: Finite covering set {5, 7, 13, 19, 29}
* ''b'' = 597: Finite covering set {5, 13, 29}

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

(none)

=== {y}z ===

(none)

=== z{0}1 ===

(none)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3))
* ''b'' = 4''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt; (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}t ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 7: Finite covering set {7}

=== {z}u ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== {z}v ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 5: Finite covering set {5}

=== {z}w ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== {z}x ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== {z}y ===

* ''b'' == 0 mod 2: Finite covering set {2}

== Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) ==

Format: base (length)

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

(none)

=== 1{0}2 ===

(none)

=== 1{0}3 ===

(none)

=== 1{0}4 ===

53 (13403)

113 (10647)

=== 1{0}z ===

113 (20089)

123 (64371)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

(none)

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

208 (26682)

607 (11032)

828 (19659)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

201 (31276)

222 (52727)

227 (36323)

327 (135983)

425 (11231)

710 (24112)

717 (37508)

719 (13420)

=== {1} ===

152 (270217)

184 (16703)

200 (17807)

311 (36497)

326 (26713)

331 (25033)

371 (15527)

485 (99523)

629 (32233)

649 (43987)

670 (18617)

684 (22573)

691 (62903)

693 (41189)

731 (15427)

752 (32833)

872 (10093)

932 (20431)

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

(none)

=== 1{z} ===

107 (21911)

170 (166429)

278 (43909)

303 (40175)

383 (20957)

515 (58467)

522 (62289)

578 (129469)

590 (15527)

647 (21577)

662 (16591)

698 (127559)

704 (62035)

845 (39407)

938 (40423)

969 (24097)

989 (26869)

=== 2{0}1 ===

101 (192276)

206 (46206)

218 (333926)

236 (161230)

257 (12184)

305 (16808)

467 (126776)

578 (44166)

626 (174204)

695 (94626)

752 (26164)

788 (72918)

869 (49150)

887 (27772)

899 (15732)

932 (13644)

=== 2{z} ===

432 (16003)

=== 3{0}1 ===

912 (132174)

=== 3{z} ===

72 (1119850)

212 (34414)

218 (23050)

270 (89662)

303 (198358)

312 (51566)

422 (21738)

480 (93610)

513 (38032)

527 (46074)

566 (23874)

650 (498102)

686 (16584)

758 (15574)

783 (12508)

800 (33838)

921 (98668)

947 (10056)

=== 4{0}1 ===

107 (32587)

227 (13347)

257 (160423)

355 (10990)

410 (144079)

440 (56087)

452 (14155)

482 (30691)

542 (15983)

579 (67776)

608 (20707)

635 (11723)

650 (96223)

679 (69450)

737 (269303)

740 (58043)

789 (149140)

797 (468703)

920 (103687)

934 (101404)

962 (84235)

=== 4{z} ===

14 (19699)

68 (13575)

254 (15451)

800 (20509)

=== 5{0}1 ===

326 (400786)

350 (20392)

554 (10630)

662 (13390)

926 (40036)

=== 5{z} ===

258 (212135)

272 (148427)

299 (64898)

307 (26263)

354 (25566)

433 (283919)

635 (36163)

678 (40859)

692 (45447)

719 (20552)

768 (70214)

857 (23083)

867 (61411)

972 (36703)

=== 6{0}1 ===

108 (16318)

129 (16797)

409 (369833)

522 (52604)

587 (24120)

643 (164916)

762 (11152)

789 (27297)

986 (21634)

=== 6{z} ===

68 (25396)

332 (15222)

338 (42868)

362 (146342)

488 (33164)

566 (164828)

980 (50878)

986 (12506)

1016 (23336)

=== 7{0}1 ===

398 (17473)

1004 (54849)

=== 7{z} ===

97 (192336)

170 (15423)

194 (38361)

202 (155772)

282 (21413)

283 (164769)

332 (13205)

412 (29792)

560 (19905)

639 (10668)

655 (53009)

811 (31784)

814 (17366)

866 (108591)

908 (61797)

962 (31841)

992 (10605)

997 (15815)

=== 8{0}1 ===

23 (119216)

53 (227184)

158 (123476)

254 (67716)

320 (52004)

410 (279992)

425 (94662)

513 (19076)

518 (11768)

596 (148446)

641 (87702)

684 (23387)

695 (39626)

785 (900326)

788 (11408)

893 (86772)

908 (243440)

920 (107822)

962 (47222)

998 (81240)

1013 (43872)

=== 8{z} ===

138 (35686)

412 (12154)

788 (11326)

990 (23032)

=== 9{0}1 ===

248 (39511)

592 (96870)

=== 9{z} ===

431 (43574)

446 (152028)

458 (126262)

599 (11776)

846 (12781)

=== A{0}1 ===

173 (264235)

198 (47665)

311 (314807)

341 (106009)

449 (18507)

492 (42843)

605 (12395)

708 (17563)

710 (31039)

743 (285479)

744 (137056)

786 (68169)

800 (15105)

802 (149320)

879 (25004)

929 (13065)

977 (125873)

986 (48279)

1004 (10645)

=== A{z} ===

368 (10867)

488 (10231)

534 (80328)

662 (13307)

978 (14066)

=== B{0}1 ===

710 (15272)

740 (33520)

878 (227482)

=== B{z} ===

153 (21660)

186 (112718)

439 (18752)

593 (16064)

602 (36518)

707 (10573)

717 (67707)

=== C{0}1 ===

68 (656922)

219 (29231)

230 (94751)

312 (21163)

334 (83334)

353 (20262)

359 (61295)

457 (10024)

481 (45941)

501 (20140)

593 (42779)

600 (11242)

604 (17371)

641 (26422)

700 (91953)

887 (13961)

919 (45359)

923 (64365)

992 (10300)

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

(none)

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

38 (136212)

83 (21496)

113 (286644)

188 (13508)

401 (103670)

417 (21003)

458 (46900)

494 (21580)

518 (129372)

527 (65822)

602 (17644)

608 (36228)

638 (74528)

663 (47557)

723 (24536)

758 (50564)

833 (12220)

904 (13430)

938 (50008)

950 (16248)

=== z{0}1 ===

202 (46774)

251 (102979)

272 (16681)

297 (14314)

298 (60671)

326 (64757)

347 (69661)

363 (142877)

452 (71941)

543 (10042)

564 (38065)

634 (84823)

788 (13541)

869 (12289)

890 (37377)

953 (60995)

1004 (29685)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

53 (21942)

124 (16426)

175 (31626)

188 (22036)

316 (48538)

365 (25578)

373 (24006)

434 (10090)

530 (11086)

545 (12346)

560 (15072)

596 (12762)

701 (12576)

706 (10656)

821 (13536)

833 (17116)

966 (14820)

983 (11272)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}y ===

317 (13896)

== Bases 2≤''b''≤1024 which have these families as unsolved families ==

Unsolved families are families which are neither primes (&gt;''b'') found nor can be ruled out as contain no primes &gt; ''b''

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608)

1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000)

1{0}3: 646, 718, 998 (length limit: 2000)

1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000)

1{0}5

1{0}6

1{0}7

1{0}8

1{0}9

1{0}A

1{0}B

1{0}C

1{0}D

1{0}E

1{0}F

1{0}G

1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000)

1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000)

10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000)

11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000)

{1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000)

{1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000)

11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z})

{1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000)

{1}z

1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000)

1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995

1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000)

1{z}: 581, 992, 1019 (length limit: ≥100000)

2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000)

2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000)

2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000)

{2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000)

2{z}: 588, 972 (length limit: ≥100000)

3{0}1: 718 (length limit: ≥100000)

3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000)

3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000)

{3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000)

3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000)

4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000)

4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000)

{4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000)

4{z}: 338, 998 (length limit: ≥100000)

5{0}1: 308, 512, 824 (length limit: ≥100000)

5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000)

6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000)

6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000)

7{0}1: (none)

7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000)

8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000)

8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000)

9{0}1: 724, 884 (length limit: ≥100000)

9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000)

A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000)

A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000)

B{0}1: 560, 770, 968 (length limit: ≥100000)

B{z}: 263, 615, 912, 978 (length limit: ≥100000)

C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000)

C{z}

D{0}1

D{z}

E{0}1

E{z}

F{0}1

F{z}

G{0}1

{&amp;#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000)

{&amp;#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143)

&amp;#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000)

${&amp;#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2)

x{z}

y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000)

z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000)

{y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000)

{z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326)

zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z})

{z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000)

{z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1)

{z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000)

{z}k

{z}l

{z}m

{z}n

{z}o

{z}p

{z}q

{z}r

{z}s

{z}t

{z}u

{z}v

{z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000)

{z}x: (none)

{z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000)

== List of lengths for quasi-minimal primes in some simple families ==

[https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024]

NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases &lt;=7, family {z}x in bases &lt;=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime &gt; base) in this base)

RC: this family can be proven to only contain composite numbers (only count numbers &gt; base)

unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers &gt; base)

Background color: red for title (bases or families), green for length &gt; 10000, orange for 2500 &lt; length ≤ 10000, white for length ≤ 2500, cyan for &quot;RC&quot;, pink for &quot;NB&quot;, yellow for &quot;unknown&quot;.

Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families.

== References ==

* [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem]

* [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are &gt; ''b'' for bases ''b''≤16]

* [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary]

* [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia]

* [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime&gt;base is not required) in The Prime Puzzles &amp; Problems Connection]

* [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles &amp; Problems Connection]

* [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16]

* [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime&gt;base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29]

* [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime&gt;base is not required, search limits of lengths: 10000 for all b]

* [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime&gt;base is not required]

* [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime&gt;base is not required]

* [http://www.prothsearch.com/sierp.html the Sierpinski problem]

* [http://www.prothsearch.com/rieselprob.html the Riesel problem]

* [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem]

* [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030]

* [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100]

* [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem]

* [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024]

* [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project]

* [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project]

* [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count]

* [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty]

* [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers]

* [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares]

* [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software]

* [https://sourceforge.net/projects/openpfgw/ pfgw software]

* [http://jpenne.free.fr/index2.html llr software]

* [http://www.ellipsa.eu/public/primo/primo.html PRIMO software]

* [https://primes.utm.edu/prove/index.html website for primality proving]

* [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!]

* [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime&gt;base is not required]

* [https://primes.utm.edu/primes/lists/all.txt top proven primes]

* [http://www.primenumbers.net/prptop/prptop.php top PRPs]

* [http://factordb.com online factor database, including many primes which are minimal primes in a small base]</text>
      <sha1>2ljkqt358vq0ijg4k7uk2gazjsardvp</sha1>
    </revision>
    <revision>
      <id>2412022</id>
      <parentid>2412016</parentid>
      <timestamp>2022-08-05T03:31:43Z</timestamp>
      <contributor>
        <ip>2402:7500:901:2B90:BD69:8142:3D96:F08F</ip>
      </contributor>
      <comment>/* Unsolved families */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1959682" xml:space="preserve">A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime &gt; ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime &gt; 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 &gt; 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 &gt; 10 by changing the order.

(using A−Z to represent digit values 10 to 35)

For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')).

I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E&lt;sub&gt;800873&lt;/sub&gt;, and base 36 has quasi-minimal prime P&lt;sub&gt;81993&lt;/sub&gt;SZ).

Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to:

* Prove that all elements in ''S'', when read as base ''b'' representation, are primes &gt; ''b''.

* Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are &gt; ''b'', are composite.

* Prove that all primes &gt; ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are &gt; ''b'', are composite).

e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to:

* Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes &gt; 10.

* Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are &gt; 10 are composite.

* Prove that all primes &gt; 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers &gt; 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite).

==Condensed table==

{|class=wikitable
|''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form of largest known quasi-minimal prime base ''b''
|-
|2||1||11||2||3
|-
|3||3||111||3||13
|-
|4||5||221||3||41
|-
|5||22||10&lt;sub&gt;93&lt;/sub&gt;13||96||5&lt;sup&gt;95&lt;/sup&gt;+8
|-
|6||11||40041||5||5209
|-
|7||71||3&lt;sub&gt;16&lt;/sub&gt;1||17||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2
|-
|8||75||4&lt;sub&gt;220&lt;/sub&gt;7||221||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||1161||3×9&lt;sup&gt;1160&lt;/sup&gt;+10
|-
|10||77||50&lt;sub&gt;28&lt;/sub&gt;27||31||5×10&lt;sup&gt;30&lt;/sup&gt;+27
|-
|11&lt;sup&gt;*&lt;/sup&gt;||1068||57&lt;sub&gt;62668&lt;/sub&gt;||62669||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10
|-
|12||106||40&lt;sub&gt;39&lt;/sub&gt;77||42||4×12&lt;sup&gt;41&lt;/sup&gt;+91
|-
|13&lt;sup&gt;*&lt;/sup&gt;||3195~3197||80&lt;sub&gt;32017&lt;/sub&gt;111||32021||8×13&lt;sup&gt;32020&lt;/sup&gt;+183
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||19699||5×14&lt;sup&gt;19698&lt;/sup&gt;−1
|-
|15||1284||7&lt;sub&gt;155&lt;/sub&gt;97||157||(15&lt;sup&gt;157&lt;/sup&gt;+59)/2
|-
|16&lt;sup&gt;*&lt;/sup&gt;||2346~2347||4&lt;sub&gt;72785&lt;/sub&gt;DD||72787||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15
|-
|17&lt;sup&gt;*&lt;/sup&gt;||10408~10428||F70&lt;sub&gt;186767&lt;/sub&gt;1||186770||262×17&lt;sup&gt;186768&lt;/sup&gt;+1
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||6271||12×18&lt;sup&gt;6270&lt;/sup&gt;+221
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||6271||16×20&lt;sup&gt;6270&lt;/sup&gt;+13
|-
|21&lt;sup&gt;*&lt;/sup&gt;||13375~13396||CF&lt;sub&gt;479147&lt;/sub&gt;0K||479150||(51×21&lt;sup&gt;479149&lt;/sup&gt;−1243)/4
|-
|22&lt;sup&gt;*&lt;/sup&gt;||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||22003||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||8134||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49
|-
|30&lt;sup&gt;*&lt;/sup&gt;||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||34206||25×30&lt;sup&gt;34205&lt;/sup&gt;−1
|-
|36&lt;sup&gt;*&lt;/sup&gt;||35257~35263||P&lt;sub&gt;81993&lt;/sub&gt;SZ||81995||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7
|}

&lt;sup&gt;*&lt;/sup&gt; Data assumes the primality of the [[w:probable prime|probable prime]]s.

Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes &gt; 10&lt;sup&gt;25000&lt;/sup&gt;), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below.

Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists))

Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists))

==Data for quasi-minimal primes==

===Base 2===

11

===Base 3===

12, 21, 111

===Base 4===

11, 13, 23, 31, 221

===Base 5===

12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013

===Base 6===

11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041

===Base 7===

14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331

===Base 8===

13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447

===Base 9===

12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011

===Base 10===

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027

===Base 11===

12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, 2888882883, 2888888883, 3555555509, 3577777077, 3700000001, 4000000005, 40000005AA, 5377777707, 5555505553, 555555580A, 600000A999, 7000100047, 7000704777, 7007777107, 7057777777, 7070007774, 7077707774, 7077777074, 7100000704, 7470000041, 7701000004, 7707077774, 7770707774, 7777707074, 8888822883, 9555555503, 9900000979, 9999770007, A000144444, A900000066, A999999971, 10000000477, 33333333337, 44444444447, 44444444777, 55A55555552, 60000008883, 68888888306, 68888888883, 70000003999, 70000007717, 70004777777, 70477777777, 77007770004, 77700000477, 77707000704, 77707770074, 77707777774, 77777077774, 77777770004, 83000000006, 97000000999, A0000000001, A0014444444, A4777777771, 100000000057, 305007777777, 305777777777, 370000000007, 377777770007, 377777777107, 700000007474, 707077000074, 707077777774, 707777777717, 770000010004, 771007000007, 777070700004, 777700000704, A95555555555, A99999777777, 1000000003007, 40000000A0041, 58A5555555555, 7004400000004, 7700000000104, 7707000007047, 7707707000004, 7777007000004, 7777700000004, 7777770077704, 7777777710077, 9977777777717, A000000014444, A044444444441, A144444444411, 40000000000401, 45557777777777, 4555AAAAAAAAAA, 59077777777777, 70007777777771, 70077070000074, 70700000004777, 77000007700704, 77700000700047, 77777777770704, 88888888830006, 90900000000799, A0000044444441, 300000000005777, 302000000000002, 55555555A555552, 700000000000174, 770000000000474, 771700000000007, 777070000000047, 777777777771777, 777777777777177, 990000000000799, A00000000444441, 1000000000000073, 1000000000000404, 4700000000000404, 5777777770777777, 6000000000000083, 7077777777777771, 7707000700000047, 7770000000000084, 7770000000007047, 8888888888888306, 8888888888888322, 9707777777777777, 11111111111111111, 14444444444441111, 44444444444444111, 70000000000000004, 70000000000040044, 70000000007477777, 77777777777770044, 77777777777771007, 77777777777777717, A1444444444444444, A5555555555555509, A9999999999999996, 320000000000000002, 597777777777777707, 707700700000000074, 770000000000077704, 805055555555555555, 888888888888888883, 997700000000000007, 1444444444444444444, 5077777777777777077, 7777777777777777771, 7777777777777777793, 8550555555555555555, 8555505555555555555, 9777777777777777773, 4000000000000000A041, 555555555555555550503, 5555555555555555A5552, 55AAAAAAAAAAAAAAAAA58, 855555555055555555555, 45AAAAAAAAAAAAAAAAAAAA, 5307777777777777777777, 7707777777777777777704, 7900000000000000000005, 9777777777777777777707, A999999999999999999999, 10000000000000000000747, 970000000000000000000777, 999900000000000000000007, 3577777777777777777777777, 5555555555555555555555A52, 7000000000000000000777771, 7000000000000000077777771, AAAAAAAAAAAAAAAAAAA000058, 10000000000000000000000044, 77700000000000000000000008, 500777777777777777777777777, 777777777777777777777770774, 855555555555555555555055555, A44444444444444444444444441, 1500000000000000000000000007, 40000000000000000000000000041, 440000000000000000000000000001, 70000000000000000000000000007771, 999999999999999999999999999999991, 95555555555555555555555555555555553, 1900000000000000000000000000000000001, 7777777777777777777777777777777777474, 7777777777777777777777777777777777704, 10000000000000000000000000000000000000307, 50777777777777777777777777777777777777707, 475777777777777777777777777777777777777777, 555555555555555555555555555555555555555A2A, 5900000000000000000000000000000000000000003, A477777777777777777777777777777777777777777, 90000000000000000000000000000000000000009799, 444444444444444444444444444444444444444444441, 577777777777777777777777777777777777777777707777, 9700000000000000000000000000000000000000000000000007, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA0058, 8055555555555555555555555555555555555555555555555555555555555, A9997777777777777777777777777777777777777777777777777777777777, 44777777777777777777777777777777777777777777777777777777777777777, 99777777777777777777777777777777777777777777777777777777777777777, 577077777777777777777777777777777777777777777777777777777777777777, 835000000000000000000000000000000000000000000000000000000000000000007, 74700000000000000000000000000000000000000000000000000000000000000000000000001, 100000000000000000000000000000000000000000000000000000000000000000000000000035, 555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555558A, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000037, 57777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777077, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA058, 55555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555053, 3266666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000051, 500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000057, 555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555552A, 5077777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, 8555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555505, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA58, 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===Base 12===

11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077

===Base 13===

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===Base 14===

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===Base 15===

12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797

===Base 16===

11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD

===Base 17===

12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 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====Additional known quasi-minimal primes (not necessarily the next)====

570&lt;sub&gt;51310&lt;/sub&gt;1, 49&lt;sub&gt;111333&lt;/sub&gt;, 970&lt;sub&gt;166047&lt;/sub&gt;1, F70&lt;sub&gt;186767&lt;/sub&gt;1

===Base 18===

11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, 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===Base 20===

13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, GGJ0J, GGJJJ, GJ00J, GJGJJ, GJJJ7, H00EH, H024B, H04B1, H0E4B, H0F41, H0H11, H0HEH, H4E0B, H6I11, HAAA3, HAAG3, HAGA3, HB44B, HBBBB, HBHHH, HGGA3, HGGG3, HH001, HH061, HH3AH, HH6F1, HHG63, HHIA3, HI011, HIII3, I0029, I02I9, I085H, I0A41, I0F29, I2229, I2929, I2I29, I2I99, I2II9, I33A3, I3EHH, I422B, I424B, I4EEB, I4IEB, I80G1, I9IA9, IA099, IA309, IA841, IA939, IA9A9, IAF09, IAF99, IAI99, IAIA9, IB0A1, IB4IB, IB54B, IBB4B, IBBIB, IBE0H, IBH0H, IC929, ICC29, IF841, IFAG1, IGA81, IHB4B, II0A9, II42B, II44B, IIB5B, III4B, IIIA3, IIIIB, J00G1, J00JB, J0601, J06D1, J06I1, J0861, J08B1, J0B01, J0CGJ, J0E0B, J0GA1, J0GG1, J0I2B, J0J2B, J0JIB, J55CJ, J60A1, J60G1, J6D81, JA777, JA7A7, JAJG1, JAJJ1, JAJJ7, JB00B, JB08B, JC00J, JC0GJ, JEECJ, JGGGJ, JGGJJ, JJ0B1, JJ0JB, JJ55J, JJAG1, JJAJ1, JJC0J, JJE0B, JJEEB, JJGG1, JJGGJ, JJGJJ, JJJ01, JJJEB, JJJJB, 17555J, 175EEJ, 1E0007, 1JE55J, 1JJJ5J, 20005H, 2000HH, 222I29, 22E889, 22EE89, 24222B, 2999I9, 29III9, 2A000H, 2B0I0H, 2I2999, 2I9I29, 2III29, 333AF3, 336IA3, 36A3F3, 36I3A3, 388E8H, 3AF333, 3EHHHH, 3HEHHH, 3HHEHH, 40008B, 40054B, 40405B, 40448B, 415E5J, 44440B, 44452B, 44524B, 44E08B, 45444B, 4AAAE7, 4AEEE7, 4E4447, 4GEEE7, 4I544B, 4IEEEB, 500EFJ, 500FCJ, 5055FJ, 50AFFJ, 50EEFJ, 50EF5J, 50F5FJ, 50FFCJ, 52224B, 54EEEB, 5558AH, 555CAJ, 5585AH, 55F55J, 55FFFJ, 5888AH, 588A5H, 5E555J, 5F055J, 5F05CJ, 5F5F5J, 5FFFCJ, 5JEEEB, 60006D, 6000F1, 6000G3, 6001G1, 600A63, 600G1D, 600G6D, 603A33, 606G6D, 608001, 608F89, 608IF9, 60A333, 60A363, 60AAA3, 60CCCD, 60D04D, 60D0DD, 60F8I9, 60FF89, 60GCG3, 636AF3, 63A333, 63A3F3, 63AFF3, 63IIA3, 64AAAD, 660089, 660A33, 660DDD, 660F89, 662289, 666A33, 666D4D, 66A6A3, 66AF33, 66AF63, 66D0DD, 66FF89, 6A33F3, 6AFF33, 6AG6A3, 6AGA63, 6D000D, 6D004D, 6D0D4D, 6F6689, 7000G7, 7060DD, 70AAAD, 70D0CD, 733C33, 7366C3, 73AAA3, 7600DD, 760CDD, 766333, 76664D, 766FC3, 76CCDD, 76DD4D, 770G07, 771JEJ, 7771EJ, 777481, 7777I1, 777F81, 77C7JJ, 77EC77, 77EEC7, 77EJJJ, 77GCG7, 77JECJ, 7A7E77, 7AAAG3, 7AAF63, 7AEEE7, 7C7GG7, 7CCCAD, 7CCCG7, 7CGJJJ, 7D000D, 7J000D, 7JCGGJ, 80005B, 8000G1, 8000I1, 800GA1, 800I01, 80GGA1, 84405B, 84454B, 888EAH, 88E8AH, 8A4441, 8B8EAH, 8E8IF9, 8I0AG1, 902229, 909A09, 909C29, 909FA9, 90IC29, 9222I9, 92I999, 944EE7, 94AEE7, 977GG7, 97AEE7, 97GGG7, 990I29, 999I29, 99I299, 99IIA9, 9A9447, 9A9AF9, 9A9FF9, 9CII29, 9EEEE7, 9FA9A9, 9I9C29, 9I9I29, 9ICI29, 9IIIA9, A1555J, A1E55J, A333C9, A336A3, A33889, A33F63, A36663, A444E7, A4AAE7, A555FJ, A666A3, A6AAA3, A7A7E7, A7AE77, A944E7, A9A9F9, A9AAA9, AA3389, AA9AF9, AAA099, AAA3C9, AAA7E7, AAA939, AAACG9, AAAFF9, AAAI99, AAAIC9, AAE4E7, AAG9A9, AAGAA3, ADE55J, AF6663, AF9FF9, AGA963, AGG6A3, AH4441, AI000H, AI99C9, AI9AA9, AII099, B0004B, B0054B, B0080B, B00H0H, B04IIB, B05B8B, B0A6I1, B0BB8B, B0BIBB, B0E0IH, B0HA0H, B0I0HH, B0I4IB, B0IIBB, B888AH, BB058B, BB05BB, BB080B, BB0I4B, BBB04B, BBB08B, BBIB8B, BE88AH, BEHA0H, BEHHAH, BHEA0H, BHEHAH, BHHA0H, BIE00H, C007A7, C00GG7, C00JA7, C00JAJ, C00JC7, C0C7A7, C0JAC7, C0JJC7, C3G999, CAJJ77, CC0629, CC0929, CCC007, CCCCC7, CCCCG3, CCCGG7, CCCI29, CCI029, CD4GGJ, CE0009, CEFF69, CEFFF9, CEI609, CII029, CIII29, CJACC7, CJAJC7, CJJJA7, D000D7, D000J7, D004CD, D0D007, DCCDC7, DDD04D, DDDAI1, DEJJJJ, DG000D, DG5E5J, DGGG5J, DJ00CJ, DJEEE7, DJJJ5J, E00097, E00A0H, E00AA9, E02289, E028I9, E029I9, E055AJ, E0774J, E0777J, E07JCJ, E0AF5J, E0BI8B, E0E4CJ, E0EA5J, E0ECJJ, E0FFF9, E0H0HH, E0HA0H, E0IF29, E0J5AJ, E0JJAJ, E29299, E444E7, E4EEE7, E4IIIB, E66289, E66629, E66909, E66F69, E69609, E69999, E7774J, E77C77, E7C7C7, E8B8AH, E90029, E90229, E90909, E90FF9, E92299, E94EE7, E97EE7, E990F9, E99I09, E9F669, E9I029, EA8AF9, EAAAI9, EAIAA9, EAJJJJ, EBIBIB, EC7CC7, EE08F9, EE408B, EE8IF9, EEAF5J, EEE84B, EEEC07, EEEC77, EEEE0B, EEF55J, EEFFFJ, EEJ5AJ, EFE009, EFEF09, EFF669, EFFFF9, EGGGG9, EIF669, F004AJ, F00A41, F00JA1, F04481, F055CJ, F0A15J, F0EI09, F0G01J, F0J0GJ, F0J0JJ, F0JJ5J, F14441, F68F89, F68IF9, F800G1, F8E8I9, F99299, FA0009, FA1JJJ, FAAIA9, FAII09, FAII99, FCC929, FCE229, FCE609, FCEF29, FE68F9, FE8FA9, FF0E09, FF68F9, FF9FA9, FFC4AJ, FFC929, FFCE09, FFE009, FFEE89, FFEF09, FFFA99, FFFFFJ, FFFGJJ, FG1JJJ, FI99A9, FIA999, FIAAA9, FIAII9, FIIA09, G00093, G000EJ, G000G9, G00781, G007G3, G00C09, G07GC3, G09033, G0C903, G0CGG3, G0G903, G0G933, G0GCC3, G0GGC9, G30003, G36003, G5000J, G5E0EJ, G60303, G6AGG3, G6GAA3, G7AAA3, G7G363, G90AA9, G90C63, G9AAA9, GAAAA3, GAAGG3, GC0003, GCCC93, GCCCC3, GCG903, GDEEE7, GE0007, GG0CG3, GGA8G1, GGC6G3, GGCCG3, GGCGC9, GGG0I9, GGG363, GGG6G3, GGG75J, GGG8G1, GGG963, GGGAA3, GGGCG3, GGGCG9, H044EB, H0BIBB, H0E00B, H0F011, H0IIEB, H2444B, H400EB, H44IEB, H4EEEB, HBB4IB, HE000B, HEE40B, HEEE4B, HEHHAH, HF0011, HF4441, HH3HEH, HHGAA3, HHGAG3, HHHAG3, HHHEAH, HI0001, HIBIBB, HIEBBB, I0000H, I000A9, I002HH, I00A09, I00A99, I00AG1, I00AI9, I00H3H, I02999, I03HEH, I09AF9, I0A009, I0E0HH, I0I299, I0IIA9, I4445B, I4I45B, I90I29, I99029, I99A09, I9I029, IA3999, IB000H, IB00EH, IB0HEH, IB5B8B, IF2999, II0929, II92I9, II99A9, II9C29, IIAFF9, IIF299, III5BB, III8BB, J0000B, J00081, J000IB, J0010D, J006A1, J00BI1, J00I8B, J00IA1, J05EEJ, J06G81, J0C0JJ, J0E0CJ, J0EEEB, J0J0EB, J5EAEJ, J5EEAJ, J608I1, JE505J, JEEEEB, JJ0001, JJ0JCJ, JJ0JGJ, JJAJC7, JJJAE7, JJJBI1, JJJI8B, 14GGGG1, 1J5555J, 1JGGGG1, 205555H, 20I000H, 222222B, 2929999, 2BI000H, 3333A33, 3333G09, 333A3A3, 333AAA3, 336AAA3, 336AF33, 33II3A3, 363AF33, 3GIIII9, 400445B, 404454B, 404800B, 415555J, 440045B, 440080B, 442222B, 444004B, 444422B, 444444B, 4444G47, 444GGG7, 444I4IB, 44AAA47, 44I44IB, 44I4IIB, 4AAAAA7, 4EEE4E7, 4I4454B, 5000AFJ, 525555H, 555552H, 555555J, 555585H, 58BBBBB, 5BB8BBB, 5E0E55J, 5EBBBBB, 60003A3, 60008I9, 6000DDD, 6006DDD, 60080A1, 60080I1, 600A3A3, 600D8I1, 600F841, 60CCDDD, 60CGGG3, 60DDD0D, 60DDDDD, 60GGG63, 60GGGC3, 66600A3, 66603A3, 66608I9, 6666089, 66666A3, 6666C29, 6666F89, 666AFF3, 66DDDDD, 6800001, 6AGGGG3, 6CCCC29, 6G6AAA3, 6GAGGG3, 6GCCC63, 6GGG633, 700000D, 70000CD, 7000EC7, 70070C7, 700EEC7, 7070EC7, 70EEEC7, 76666CD, 77333A3, 7733A33, 775EE5J, 777A777, 777CGJJ, 777GE5J, 77A7777, 77JCJJJ, 77JJJCJ, 77JJJJJ, 7A77777, 7A777A7, 7AAAA63, 7AAAAAD, 7C00007, 7E77JCJ, 7EE7JCJ, 7EEC007, 7EEEEC7, 7G5E55J, 7GE555J, 7GJJJJJ, 7J6666D, 9000029, 9000299, 9009I29, 900I2I9, 90FAAA9, 90I0299, 90I0I29, 90I9929, 90II029, 90II299, 90II2I9, 94444E7, 97777E7, 9902I99, 99902I9, 9992929, 9992999, 9999929, 99I0C29, 99I9029, 99I92I9, 99III29, 9AAAF09, 9AFFFF9, 9I0AAA9, 9IAAA09, 9II0I29, 9II9929, 9III029, 9III2I9, 9IIIC29, A5F555J, A777E77, A994447, A9AAF99, AA00089, AA6A6A3, AAA0009, AAA3999, AAA9A99, AAA9AA9, AAAAA47, AAAAA77, AAAAA99, AAAAG99, AAAAIA9, AAAD447, AAAGAC9, AAGAAAD, AAGAAC9, AF00009, AGAAA99, AGAGG63, AGGGA63, AI0C999, AJ777A7, B000B8B, B000HEH, B000I0H, B00I00H, B0B805B, B0HHHEH, B0IHE0H, B0III8B, B8B000B, BB8BBBB, BBB8B0B, BEH0HHH, BHHHEHH, C00000J, C000C07, C066629, C0AJJJJ, C0C0A47, C0G000J, C0JAJJJ, C0JJAJJ, C0JJJJJ, C6CCG33, CC004A7, CC00A47, CC6CG33, CC92229, CCCC7G7, CCCGGG3, CE66609, CEF6669, D00007D, D00007J, D00071J, D000C0J, D000I01, D00700D, D00C007, D00C0C7, D00G00D, D0C00C7, D0CDCC7, D444447, D7JJJJJ, DC0000J, DDD7DDD, DDDCCC7, DJ0000J, DJ000EJ, DJJJJ0J, DJJJJJJ, E000009, E00084B, E000889, E000909, E000HAH, E008AF9, E04000B, E0AA8F9, E0CJJJJ, E0HHHHH, E40400B, E6660F9, E666609, E669969, E699669, E7CCC07, E996669, E99I669, E9FAAA9, EB8BBBB, EBBII8B, EBH0HHH, EE092I9, EE44IIB, EE4E4E7, EEBB8BB, EEC0007, EEE0IBB, EEE0IIB, EEEBB8B, EEEE44B, EEJCJJJ, EF00F09, EF0555J, EFAAAA9, EFF0009, EFF00F9, EFF60F9, EI0AAA9, EI660F9, EIIBB8B, EJ0CJJJ, EJ5055J, EJJ0CJJ, F0E00F9, F0E88I9, F0JJJJ1, F299929, F4G555J, F5555CJ, F5555FJ, F6668I9, F8GGGG1, F929999, F999C29, FA84441, FAAA9A9, FCE00F9, FCEF009, FCFEE29, FE00F09, FEE2889, FEF0009, FFAA9A9, FFCEE29, FFFF4AJ, FIII9A9, FIIIAI9, FJJJJ81, G000009, G0000J7, G000303, G000AA9, G000CC3, G000GG3, G003003, G0090C3, G00CG03, G00D007, G00G0G3, G00GG03, G00GG93, G080001, G0G00I9, G0G09C3, G0G0C93, G0G0GG9, G0GG093, G0GGC63, G0GGGG9, G44GG47, G6AAGA3, G999999, GAA9AA9, GCCCGG9, GCGGCC9, GCGGG03, GEE4447, GEEEE47, GG0C007, GG0GC09, GG0GGG3, GG6AAA3, GG6AGA3, GG6GAG3, GGEEE47, GGG00C9, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 21===

12, 18, 1A, 1G, 1K, 21, 25, 2B, 2H, 2J, 34, 38, 3A, 3G, 3K, 45, 4D, 4H, 4J, 52, 54, 58, 61, 65, 6B, 6D, 72, 74, 7A, 7G, 7K, 85, 8B, 8D, 92, 94, 98, 9A, A1, AD, AH, AJ, B2, B8, BA, BK, C5, CB, CH, CJ, D4, D8, DA, DK, ED, EH, EJ, F2, FG, G1, GB, GD, GH, H2, HA, HG, I1, I5, IB, IJ, J2, JA, JK, K1, KB, KD, KJ, 111, 115, 11H, 135, 13H, 13J, 151, 15B, 15H, 164, 175, 17B, 17D, 17J, 191, 19B, 19D, 19H, 1B1, 1B5, 1BB, 1BJ, 1D5, 1DD, 1DJ, 1E4, 1F1, 1F5, 1FD, 1FH, 1HB, 1HD, 1I4, 1JD, 1JH, 1JJ, 22D, 232, 29G, 29K, 2DG, 2F4, 2FK, 31H, 322, 33D, 351, 355, 35B, 35J, 362, 371, 37B, 37D, 37H, 37J, 39B, 39J, 3B5, 3BD, 3BH, 3D1, 3D5, 3DB, 3DD, 3DH, 3E2, 3FJ, 3HD, 3HH, 3HJ, 3J1, 3JB, 3JJ, 414, 481, 49K, 4B4, 4BG, 4C1, 4EB, 4F4, 4F8, 4FA, 4FK, 50G, 51B, 51D, 51H, 531, 535, 53D, 53J, 551, 56A, 56G, 56K, 575, 57J, 595, 59H, 5B1, 5B5, 5BB, 5CA, 5CG, 5CK, 5F1, 5FB, 5FJ, 5GA, 5GG, 5HH, 5IA, 5J5, 5JD, 5JH, 60H, 632, 66H, 66J, 678, 68J, 69G, 6F8, 6FA, 6GJ, 6H8, 6HK, 6IH, 6J4, 6JG, 6KH, 711, 71B, 71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, GGJ, GJ4, GK5, H0K, H15, H1B, H1J, H31, H3D, H3H, H48, H51, H55, H5J, H6K, H75, H84, H88, H91, H95, H9D, H9H, HBD, HC4, HC8, HDJ, HF5, HFB, HFH, HHD, HHJ, HI4, HI8, HJ5, HJB, HKK, I0D, I2D, I6H, I78, I8H, I9K, IFA, IFK, IID, J08, J1J, J31, J35, J3J, J44, J5H, J68, J6G, J71, J7B, J7D, J7H, J8G, J95, J9D, JBD, JBH, JBJ, JCG, JDB, JDH, JE4, JE8, JEG, JF5, JFD, JFJ, JG4, JGG, JH1, JH5, JHB, JHH, JI4, JJ1, JJ5, K0H, K5G, K6H, K9K, KG5, KH4, KKH, 101B, 103D, 1044, 1055, 10DH, 10FB, 10H1, 10H5, 10J1, 110J, 11CD, 11EB, 11ID, 130D, 13C1, 13EB, 141B, 1444, 1471, 14FB, 1505, 15CD, 167H, 16FJ, 16HH, 176H, 17C1, 17IH, 190J, 1BF4, 1BID, 1C04, 1C3D, 1CD1, 1D6H, 1DE1, 1DEB, 1E1B, 1E71, 1E95, 1EFB, 1F0B, 1F4B, 1FB4, 1H01, 1H05, 1H41, 1H6H, 1H6J, 1HC1, 1HE1, 1HF4, 1IDH, 1IHH, 1JE5, 1JF4, 20CD, 20D2, 20ID, 2278, 22F8, 239D, 23FD, 2478, 2708, 27FD, 293D, 29DD, 2A78, 2C0D, 2CF8, 2F0A, 2F9D, 2FAA, 2FDD, 2GFA, 2IF8, 3015, 301D, 301J, 3031, 3035, 305D, 30F1, 30FB, 30FD, 30HB, 30I2, 3105, 31C1, 329D, 330H, 336J, 33E1, 33EB, 33IH, 35CD, 363H, 390H, 39E5, 39ID, 3B0J, 3B6J, 3BC1, 3BEB, 3CC2, 3CF1, 3CFD, 3D32, 3E3B, 3E91, 3EF1, 3EFB, 3F05, 3FC1, 3FE1, 3FEB, 3H01, 3H0B, 3HC1, 3I3H, 3I9D, 404B, 4078, 40E1, 411B, 4131, 4171, 41FB, 4331, 4391, 43BB, 43FB, 43I2, 4432, 444B, 44E1, 477B, 4931, 4971, 499B, 49CG, 4A32, 4A4B, 4A78, 4A9G, 4B1B, 4B31, 4B3B, 4BFB, 4C78, 4E32, 4F1B, 4FB1, 5011, 509D, 50BD, 50DB, 50DD, 50EK, 50FD, 50H1, 50HJ, 50IK, 50JJ, 5101, 5105, 51E1, 530B, 53EB, 53IH, 550D, 550H, 55AB, 55BG, 55DG, 55E5, 55KH, 563H, 56JJ, 5701, 570H, 57EB, 57ID, 57IH, 590D, 59C1, 59CD, 5A3B, 5A7B, 5AGK, 5AIG, 5AKG, 5B0D, 5B9G, 5BCD, 5BGJ, 5BIH, 5C91, 5CD1, 5D0D, 5D0J, 5D6H, 5D6J, 5DC1, 5DG5, 5DIH, 5DJG, 5E0A, 5E11, 5E3B, 5E71, 5EAK, 5EEA, 5EKK, 5F0H, 5F9K, 5G0K, 5G9J, 5GF5, 5GIK, 5H0B, 5H5K, 5H9K, 5HE5, 5HID, 5I3H, 5I7D, 5IKG, 5J6J, 5JE1, 5KAG, 5KEK, 5KFH, 5KH5, 60FK, 63FH, 67JH, 69JH, 6E9K, 6F0K, 6F64, 6FJH, 6H04, 6HE4, 6HFJ, 6IF4, 6J9H, 6J9J, 7015, 7031, 703B, 7055, 705J, 7088, 709B, 709D, 709H, 70DJ, 70I8, 710J, 716J, 730B, 73C1, 73E5, 7501, 769H, 76FJ, 76JJ, 770D, 77E1, 77E5, 77EB, 7888, 78FH, 78H1, 78JH, 790H, 79CD, 79IH, 7B0J, 7B6H, 7C31, 7C91, 7C9D, 7E35, 7E75, 7E91, 7EF1, 7F01, 7F78, 7FCD, 7FH8, 7FIH, 7FJ8, 7H81, 7HC1, 7HF8, 7J6J, 7J78, 7JC1, 7JF8, 7JH8, 7JJ8, 809K, 80H8, 81H1, 81HH, 81J1, 839H, 83FH, 869K, 871H, 8731, 8771, 8788, 879J, 87F1, 87FJ, 87JH, 888H, 8911, 8931, 896K, 8A78, 8A9K, 8CC1, 8E9K, 8F3H, 8F9H, 8FF1, 8FHH, 8FJH, 8G9K, 8GJ8, 8H3J, 8H7H, 8H8K, 8HEK, 8HF1, 8HFJ, 8HH1, 8HJJ, 8HK8, 8J7J, 8J91, 8JC8, 8JFH, 8JJH, 8K78, 8K8H, 8KH8, 901J, 9035, 903B, 9071, 90CG, 90GG, 90JD, 9101, 91E5, 9301, 93E1, 93EB, 95CD, 95DG, 95FK, 967J, 969H, 96HH, 96HJ, 96JJ, 9705, 97C1, 9905, 995K, 999G, 99CD, 99EB, 99K5, 9B5G, 9BE5, 9BID, 9C0G, 9C11, 9C71, 9C9D, 9CEG, 9CGG, 9D05, 9D5G, 9DDG, 9DE5, 9E35, 9E3B, 9E71, 9EB5, 9EEG, 9EF1, 9EFB, 9F0J, 9F6H, 9F9K, 9FC1, 9FIH, 9FKH, 9G35, 9G95, 9GIK, 9GJJ, 9GKG, 9H0H, 9H5K, 9H6H, 9HKH, 9I6K, 9I9H, 9IDD, 9J6J, 9JEB, 9JGJ, 9JID, 9JJG, 9K35, 9K95, 9KEG, A33B, A3F5, A432, A478, A53B, A5IK, A788, A795, A878, A975, A99B, A9CG, A9GG, AAAB, AB44, AB55, ABFB, AC32, AC78, ACFA, AE32, AE5A, AE9K, AEEB, AEFA, AEFK, AF75, AFCA, AFCK, AFEA, AFFB, AG9G, AGFA, AI32, B064, B091, B09H, B0B1, B0D5, B0DH, B0EG, B0FB, B0FH, B0H5, B0HD, B114, B141, B14B, B1B4, B1E1, B411, B431, B444, B4B1, B59G, B5G5, B5JG, B67H, B67J, B69H, B6FH, B6FJ, B6HH, B6JH, B96H, B9DG, B9G5, BB14, BBB4, BC1D, BC31, BCDD, BD05, BD0H, BD6H, BDC1, BDE5, BDIH, BE0G, BE31, BEBB, BEE4, BEIG, BF0B, BF0H, BF6J, BFEB, BFIH, BG0G, BG55, BG95, BGG4, BGGG, BGIG, BGJ5, BH0B, BHEB, BIEG, BIGG, BIHH, BJ01, BJ0J, BJC1, BJF4, BJID, C041, C081, C09G, C0E1, C0FK, C104, C131, C171, C1C4, C1D1, C2F8, C2FA, C39D, C3F1, C478, C6FK, C7FD, C8C1, C90G, C911, C971, C99D, CAFK, CC78, CC81, CCE1, CCF8, CD02, CD31, CDFD, CE01, CE32, CF0A, CF48, CF64, CF68, CF6A, CF91, CFAK, CFDD, CFEA, CFEK, CIF4, CK32, CK9G, CKFA, D01D, D01H, D07B, D0DB, D0F1, D0FD, D0HD, D0HH, D10D, D10H, D16H, D1EB, D1IH, D23D, D3E1, D56H, D63J, D6CG, D6IG, D70B, D70D, D70J, D99G, D9GJ, D9ID, D9JG, DBC1, DBCD, DBE1, DBJG, DC11, DC9D, DCF1, DCFD, DCGG, DCIG, DD0H, DD6J, DDGJ, DDID, DDJG, DE31, DE75, DFC1, DFID, DFIH, DG22, DG6G, DG75, DG9J, DGCG, DGIG, DH6H, DHE5, DI6G, DICG, DIDH, DIFD, DIFH, DIGG, DJCD, DJDG, DJG5, E032, E05A, E0EB, E11B, E2F8, E31B, E331, E375, E395, E3C2, E3F1, E3F5, E3FB, E3I2, E53B, E555, E591, E59B, E5EG, E6F4, E755, E775, E89K, E915, E935, E93B, E96K, E9FB, EA32, EAEB, EAFK, EB44, EB95, ECFA, ECFK, EE32, EEEB, EEFK, EF08, EF35, EF48, EF55, EF64, EF68, EF75, EF95, EFB1, EFCA, EFCK, EFEA, EFIA, EFKK, EGF4, EIF4, EKF4, EKF8, EKFA, F008, F051, F075, F095, F09H, F0CA, F0CK, F0DD, F0FB, F0H1, F0IA, F0JJ, F105, F10B, F10D, F16J, F1B4, F1EB, F1ID, F1J4, F36H, F468, F47B, F491, F4C8, F501, F50B, F648, F66A, F69H, F705, F778, F864, F8F1, F90D, F99K, F9C1, F9HK, FA6A, FA75, FA7B, FA95, FAEA, FAFB, FB0D, FB14, FBCD, FBE1, FBJ4, FC1D, FC68, FC9D, FCAA, FCCK, FCD1, FCEA, FD6H, FDIH, FE0K, FE1B, FE64, FE95, 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====Additional known quasi-minimal primes (not necessarily the next)====

40&lt;sub&gt;47333&lt;/sub&gt;9G, CF&lt;sub&gt;479147&lt;/sub&gt;0K

===Base 22===

11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, IHH, IIJ, IJ7, IL7, J07, J55, J69, J8J, J99, J9J, JA5, JAH, JE7, JEH, JHH, JI9, JJ5, JJ9, JJH, JK7, K2J, K51, K5D, K75, K81, KA1, KB1, KB7, KBJ, KD1, KEJ, KG5, KIF, KJ5, KKD, KKJ, L0D, L47, L7F, L97, LAF, LD3, LD9, LDD, LEF, LGD, LI7, LJ7, LJJ, LLD, 104H, 10D5, 1205, 12B5, 140H, 1433, 144H, 14AH, 14B3, 16ED, 1AIH, 1B43, 1DD5, 1DDD, 1E6D, 1EGD, 1G05, 1GDD, 1GED, 1GGD, 1HB3, 1HHH, 1IAH, 200L, 2015, 2051, 20A1, 20DJ, 20GD, 20IL, 21B5, 21DD, 220D, 226D, 228D, 22B5, 22G5, 22K5, 22KD, 2555, 2557, 2581, 25C1, 26A1, 26B1, 2725, 2755, 2801, 2861, 288D, 28B1, 28KD, 2AA1, 2B25, 2B51, 2BB1, 2C81, 2D6D, 2DA1, 2DDJ, 2DGD, 2G0J, 2GB5, 2GDD, 2GGJ, 2I0L, 2I6L, 2ICL, 2J05, 2JK5, 2K07, 2K2D, 2K55, 2K6D, 2KB5, 2KI7, 2L2D, 2L8D, 2LK7, 302L, 30G3, 320L, 32IL, 332L, 33G3, 33G9, 36AF, 36EF, 382J, 388J, 39EL, 3AAF, 3BBJ, 3CG9, 3E2J, 3E6F, 3E6L, 3EEF, 3FAF, 3G69, 3GI3, 3GI9, 3IG9, 3LEL, 3LG3, 4025, 404H, 4063, 4075, 40AH, 40B5, 40B7, 40GH, 4225, 4363, 4447, 444H, 447H, 4487, 44B7, 44H7, 4525, 4555, 45B5, 4663, 4777, 47GH, 4807, 4B0H, 4BB5, 4BG3, 4EBH, 4G4H, 4GB3, 4HH3, 4I03, 4I63, 4IGH, 5069, 5077, 50KD, 5255, 52IJ, 5455, 5477, 5507, 5527, 556F, 557F, 5587, 56EF, 56FF, 56GD, 56KD, 5771, 57C1, 57EF, 5807, 580J, 589J, 58E7, 58IJ, 58K7, 5905, 5945, 5957, 5969, 5989, 598J, 5999, 59EJ, 59K7, 5AEF, 5B25, 5B55, 5BA5, 5BI7, 5BIJ, 5BK5, 5BK7, 5C21, 5EAF, 5EE7, 5F09, 5F6F, 5F95, 5FKD, 5G6D, 5G89, 5I09, 5I0F, 5I27, 5I6F, 5I89, 5IB7, 5IBJ, 5IFF, 5J77, 5K07, 5K25, 5K27, 5K6F, 5K87, 5KK7, 6013, 60A1, 60AF, 60EL, 6281, 62B1, 62KD, 63G9, 6403, 6643, 66EL, 66G9, 68KD, 69B3, 6A01, 6A0F, 6AAF, 6AFF, 6B21, 6B8D, 6BBD, 6BK1, 6D03, 6D43, 6D69, 6D6F, 6D93, 6D9F, 6E21, 6E81, 6E8D, 6EBD, 6ECL, 6ELL, 6FD9, 6GBD, 6GG9, 6IEF, 6K21, 6K8D, 6KBD, 6KE1, 6L43, 6LB3, 6LDF, 6LEL, 6LG9, 700F, 7027, 7057, 70EH, 70LF, 710H, 71G5, 7207, 7225, 7255, 727L, 7505, 7507, 755F, 75EF, 75F5, 75I7, 766F, 7681, 76CF, 7771, 7781, 77IL, 7861, 7A0H, 7A4H, 7BB5, 7C6F, 7EEH, 7EHL, 7FG5, 7G45, 7HEL, 7I0L, 7I27, 7IE7, 7IGH, 7K07, 7K61, 7K77, 7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, H0D1, H0HJ, H0LL, H1HH, H20D, H22D, H26D, H2I7, H2K7, H30L, H3KJ, H40H, H447, H4G3, H4H3, H4HH, H66L, H6ED, 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5IIIIIIIIIIIIIIIIIIIIIIIIIIF, D555555555555555555555550555, EEAAAAAAAAAAAAAAAAAAAAAAAAAF, HHHHHHHHHHHHHHHHHHHHHHHHEEBH, K66666666666666666666666666F, LLLLLLLLLLLLLLLLLLLLLLLLEEB7, D5555555555555555555555555A55, GGGGGGGGGGGGGGGGGGGGGGGGGGGG3, GIG0000000000000000000000000H, HH00000000000000000000000001H, K0000000000000000000000005KEF, 5BBBBBBBBBBBBBBBBBBBBBBBBBBBBD, HB0000000000000000000000000001, K000000000000000000000000505EF, L7777777777777777777777777772L, 2000000000000000000000000000CB1, C8CCCCCCCCCCCCCCCCCCCCCCCCCCCC9, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKFF, JE0000000000000000000000000000J, K000000000000000000000000000261, A0000000000000000000000000004I4H, HD000000000000000000000000000001, K000000000000000000000000000EC01, K0FFFFFFFFFFFFFFFFFFFFFFFFFFFFCF, D0002222222222222222222222222222D, FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL2L, I700000000000000000000000000000GH, K00000000000000000000000000000E61, 20000000000000000000000000000000JJ, DD5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL0KE7, 44HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, F0000000000000000000000000000000000000000000000000000000L89, J000000000000000000000000000000000000000000000000000000JJIJ, K222222222222222222222222222222222222222222222222222222222DD, KCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCF, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKCF, D55KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, K000000000000000000000000000000000000000000000000000000000000077, 22222222222222222222222222222222222222222222222222222222222222222D, K000000000000000000000000000000000000000000000000000000000000008IJ, DFA00000000000000000000000000000000000000000000000000000000000000005, CC4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJE0J, 777777777777777777777777777777777777777777777777777777777777777777777777EK7, 6000000000000000000000000000000000000000000000000000000000000000000000000000000043, KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKB5, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEEH, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 50000000000000000000000000000000000000000000000000000000000000000000000000000000002C1, K0000000000000000000000000000000000000000000000000000000000000000000000000000000000055EF, H700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000K1, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0I7, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, A400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, E0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000071, 7LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIL, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLK77, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEI7, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIB3, I7G00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 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77EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEK7, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJKJ, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777772L, BKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK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KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK5

===Base 24===

15, 17, 1D, 1H, 1J, 1N, 25, 2B, 2D, 2J, 2N, 31, 37, 3B, 3H, 41, 45, 47, 4B, 4D, 4H, 57, 5B, 5H, 5J, 65, 67, 6D, 6J, 6N, 75, 7B, 7D, 7N, 81, 85, 87, 8J, 97, 9B, 9D, 9H, 9N, A1, AB, AH, AN, B5, B7, BD, BH, BJ, C5, CJ, CN, D1, D5, DJ, E1, EB, ED, EH, EN, F7, FD, FJ, FN, G5, GD, GH, H1, HB, HD, HN, I1, I7, IB, IH, J1, J5, J7, JB, JN, K7, KB, KJ, KN, L5, LH, LJ, MD, MJ, N5, NB, NH, NJ, 101, 10B, 111, 1F1, 1FB, 1GB, 1LB, 201, 221, 22H, 261, 271, 277, 28H, 2A7, 2C7, 2G7, 2H7, 2L1, 2L7, 2MH, 305, 30D, 30J, 33N, 34N, 35D, 35N, 38D, 395, 3A5, 3AJ, 3CD, 3DD, 3DN, 3E5, 3EJ, 3GJ, 3IJ, 3JJ, 3K5, 3KD, 3ND, 43N, 44N, 49J, 4EJ, 4GJ, 4GN, 4NN, 50N, 535, 54N, 551, 55N, 5C1, 5CD, 5E5, 5K1, 5KD, 5LN, 5M5, 5N1, 601, 60B, 61B, 66H, 68B, 691, 6CH, 6FH, 6GB, 6HH, 6MH, 70H, 70J, 711, 761, 771, 77H, 77J, 78H, 7C7, 7CH, 7FH, 7G7, 7H7, 7HH, 7IJ, 7JJ, 7K1, 7M1, 7M7, 80D, 82H, 83N, 88D, 88H, 8AD, 8CD, 8DB, 8DD, 8DH, 8DN, 8GB, 8KD, 8MB, 8MH, 905, 911, 921, 935, 955, 99J, 9AJ, 9G1, 9JJ, 9K5, 9L1, 9M5, A0J, A3J, A95, AA7, AD7, AE5, AG7, AGJ, AI5, AIJ, AJD, AL7, ALD, B01, B0N, B11, B61, B6B, B8N, B91, BIN, BL1, BLN, BNN, C1B, C21, C27, C2H, C3D, C61, C8H, C91, CA7, CB1, CBB, CC7, CCB, CCD, CDD, CFB, CG1, CGB, CK1, CL1, CMB, CMH, D0B, D3D, D3N, D4N, D6B, D6H, D7H, D8B, D8N, DAD, DCD, DCH, DDH, DDN, DG7, DGB, DID, DMN, DND, E05, E4J, EA7, EEJ, EF5, EGJ, EI5, EJJ, EM5, EM7, F01, F21, F51, F8H, F95, FC1, FF1, FFB, FKH, FM5, G0B, G0N, G11, G3N, G6B, G77, G7J, G8B, G8N, G91, GA7, GBB, GC7, GFB, GG1, GGJ, GGN, GK1, GL1, GLN, GMN, GN1, GNN, H0J, H2H, H3J, H4J, H77, HA5, HA7, HE5, HFH, HIJ, HJJ, HKH, HL7, HMH, I0N, I3D, I3J, I3N, I4N, I5D, I95, IA5, IAJ, IE5, IEJ, IF5, IGJ, IJD, IK5, IKD, J0D, J4J, J8D, JAD, JDH, JEJ, JFH, JHH, JKD, JMH, K35, K6H, KCD, KFH, KH5, KLD, KM1, L01, L0B, L0D, L0N, L61, L6B, L8D, LA7, LC7, LDD, LF1, LG7, LGB, LGN, LID, LK1, LKD, LL1, LLB, LLD, LMN, LNN, M0H, M11, M21, M4N, M71, M91, M95, MA5, MA7, MBN, MC7, MF1, MF5, MFB, MFH, MG7, MI5, MIN, ML1, ML7, MLB, MMH, N01, N21, N4N, N71, N8N, NC1, ND7, NE7, NG1, NID, NK1, NL7, NMN, NN7, 11CB, 11MB, 1291, 12G1, 16C1, 16CB, 16K1, 186B, 18CB, 19K1, 1BK1, 1C8B, 1K91, 1KC1, 1KL1, 1L21, 1LC1, 1LM1, 1M61, 1M8B, 1MG1, 206H, 20CH, 20M7, 21C1, 21M1, 2207, 260H, 26KH, 2991, 2C6H, 2CC1, 2CM1, 2F11, 2FHH, 2MC1, 2MK1, 2MM1, 308N, 30GN, 30IN, 30LN, 30MN, 333J, 33JD, 33LD, 343J, 344J, 35I5, 380N, 393J, 394J, 3A3D, 3FI5, 3IMN, 3J3D, 3JID, 3L3D, 3L8N, 3M0N, 3M55, 3NGN, 404J, 408N, 40LN, 434J, 44AJ, 4ILN, 4JAJ, 4L8N, 5091, 5095, 50F1, 50I5, 51L1, 5211, 5291, 52G1, 53ID, 53MN, 5595, 55AD, 56F1, 588N, 58MN, 58ND, 5961, 5991, 5A5D, 5AAD, 5F91, 5GF1, 5GIN, 5I05, 5I55, 5I8D, 5IDD, 5IDN, 5IIN, 5IMN, 5KI5, 5M61, 5M8N, 5N3N, 602H, 6211, 62F1, 62G1, 66C1, 66FB, 66M1, 66MB, 6B21, 6BM1, 6BMB, 6C6B, 6CF1, 6CLB, 6FG1, 6K21, 6K2H, 6KG1, 6KKH, 6L21, 6LCB, 6LM1, 6MB1, 6MBB, 6MG1, 6MK1, 7001, 7027, 7207, 726H, 739J, 793J, 79C1, 7A4J, 7A9J, 7AE7, 7C01, 7CC1, 7FL1, 7G21, 7G9J, 7GAJ, 7GC1, 7HGJ, 7J2H, 7J6H, 7MKH, 800B, 800H, 804N, 806H, 808N, 80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, NK3D, NK8D, 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5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, 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EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEG7, 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===Base 30===

11, 17, 1B, 1D, 1H, 1N, 1T, 21, 27, 2B, 2D, 2J, 2N, 2T, 37, 3B, 3D, 3H, 3J, 3N, 47, 4B, 4H, 4J, 4T, 51, 57, 5D, 5H, 5N, 5T, 61, 6B, 6D, 6H, 6J, 71, 7D, 7H, 7J, 7N, 7T, 81, 8B, 8H, 8N, 8T, 91, 97, 9B, 9D, 9N, A7, AB, AD, AH, B1, B7, BH, BJ, BN, BT, C7, CD, CJ, CN, CT, D7, DB, DJ, DT, E1, EB, ED, EJ, EN, ET, F7, FB, FD, FH, FT, G7, GB, GJ, GN, GT, HB, HD, I1, I7, IH, IN, IT, J1, J7, JH, JN, JT, K1, K7, KD, KH, KJ, L1, LB, LD, LH, LN, LT, M1, MD, MH, MN, N1, NB, NJ, NT, O7, OD, OJ, ON, P1, P7, PB, PJ, PN, Q7, QH, QT, R1, RB, RD, RH, RJ, RT, SD, SH, SJ, SN, T7, TB, TD, TH, 10J, 15J, 1IJ, 1JJ, 1LJ, 1MJ, 1QJ, 22H, 29H, 2EH, 2GH, 30T, 331, 33T, 36T, 39T, 3A1, 3C1, 3G1, 3KT, 3MT, 3OT, 3S1, 3T1, 40D, 40N, 431, 44D, 46N, 48D, 4AN, 4DD, 4DN, 4F1, 4FN, 4GD, 4ID, 4PD, 4S1, 50J, 58J, 59J, 5CB, 5FJ, 5IB, 5IJ, 5JB, 5MB, 5MJ, 5OB, 5SB, 607, 63T, 687, 6E7, 6KT, 6L7, 6M7, 6MT, 6NN, 6QN, 6R7, 6S7, 6ST, 6TT, 70B, 77B, 787, 7KB, 7M7, 7MB, 7SB, 807, 80D, 80J, 84D, 85J, 877, 88J, 89J, 8DD, 8FJ, 8ID, 8IJ, 8JJ, 8M7, 8MJ, 8PD, 8QD, 8R7, 90H, 93T, 95J, 99H, 99J, 9AJ, 9AT, 9EH, 9GH, 9HH, 9HJ, 9JJ, 9MJ, 9OH, 9OT, 9PH, 9ST, 9TT, A01, A0T, A1J, A31, A6N, A6T, AAJ, AAN, AFN, AKN, AKT, ALJ, AMJ, AMT, AO1, AOT, AQ1, AQN, ARN, AT1, B5B, BBD, BCB, BDD, BID, BOB, BPD, BQB, C31, C9H, CC1, CCB, CCH, CF1, CH1, CIB, CKB, CMB, D01, D0H, D2H, D41, D4D, D6N, D8D, D9H, DDH, DDN, DGH, DH1, DHH, DID, DKN, DNN, DO1, DOH, DQN, DS1, EEH, EHH, EM7, EOH, EPH, ER7, F0N, F31, F5J, F8J, F9J, FLJ, FO1, FQ1, FQN, FS1, G01, G4D, G8D, GF1, GGH, GO1, GOH, GQD, GS1, H07, H0J, H0T, H1J, H2H, H31, H4N, H8J, HA1, HAJ, HAT, HC1, HE7, HEH, HFN, HGH, HH7, HIJ, HJJ, HKN, HL7, HNH, HPH, HQN, HS1, HTJ, HTN, I4D, I5B, I5J, I8D, IAJ, IDD, IGD, IIJ, IKB, IMB, IMJ, IOB, IPD, IQD, J8J, J9J, JAJ, JBD, JCB, JFJ, JIJ, JJB, JJD, JLJ, JPD, JQB, K3T, K4N, KAT, KBB, KCB, KMB, KNN, KOB, KOT, KQN, KST, KTT, L0J, L5J, L67, LAJ, LJJ, LQJ, LR7, M0J, M3T, M5B, M7B, M87, M9J, MAT, MFJ, MIJ, MJJ, MKB, MMJ, MOT, MQJ, N07, N0H, N67, N6N, N87, NAN, NDH, NGD, NHH, NKN, NN7, NNH, NPH, NQD, NQN, NR7, O01, O0B, O0H, OAT, OC1, OCH, OEH, OF1, OH1, OKT, OMB, OOT, OPH, OQ1, OQB, OS1, OST, P3T, P6T, P9H, PCH, PDH, PEH, PGD, PHH, PHT, PID, PMT, PPT, PQD, PST, PTT, Q5B, Q6N, Q9J, QAJ, QBB, QBD, QC1, QDN, QFJ, QFN, QGD, QJB, QKN, QLJ, QMB, QND, QNN, QO1, QQ1, QQN, QSB, R4N, R77, R87, RAN, RKN, RM7, RNN, RR7, RS7, S0T, S41, S6T, S87, SBB, SC1, SM7, SMT, SOB, SQ1, SR7, SS1, STT, T0J, T6T, T9T, TAN, TFN, TKN, TKT, TNN, TO1, TOT, TPT, TQ1, TQN, TTN, 18AJ, 19FJ, 1AFJ, 1FAJ, 20CH, 20PH, 2C0H, 2CHH, 2COH, 2H0H, 2HCH, 2HHH, 2POH, 2PPH, 3001, 34Q1, 3F41, 3FF1, 3QF1, 3SPT, 3SST, 3TTT, 40A1, 4441, 44KN, 44QN, 4AC1, 4C01, 4CA1, 4DG1, 4GA1, 4GC1, 4NND, 4NRN, 4OA1, 4Q4N, 4QA1, 4QRN, 4RQN, 550B, 555B, 55KB, 5A5J, 5AJJ, 5BKB, 5JQJ, 5KKB, 5QQB, 5QQJ, 604N, 606T, 60FN, 60KN, 60OT, 60PT, 660N, 660T, 664N, 66AT, 66FN, 66KN, 66TN, 6A0N, 6A9T, 6AAT, 6FKN, 6FRN, 6K0N, 6KFN, 6KKN, 6O6T, 6PAT, 6RFN, 6RRN, 6T6N, 6TRN, 7067, 70R7, 75BB, 77R7, 7C5B, 7CQB, 7E67, 7IQB, 7O5B, 7OIB, 7QOB, 7R07, 7RE7, 8667, 88L7, 88S7, 8E87, 8EE7, 8EL7, 8J8D, 8LL7, 8LS7, 906T, 908J, 90FJ, 90QJ, 90TJ, 92CH, 966T, 99MT, 99PT, 9I0J, 9I8J, 9LFJ, 9LIJ, 9M6T, 9MKT, 9MMT, 9PKT, 9QQJ, 9TLJ, A04N, A0QJ, A3AT, A3ST, A4A1, A4G1, A5QJ, A8QJ, A90J, A9PT, AA3T, AA41, AAAT, AAF1, AAG1, AAPT, AC41, ACS1, AF0J, AFC1, AFIJ, AG41, AGA1, AGG1, AI8J, AIJJ, AIQJ, AJ5J, AJJJ, AQ8J, AQJJ, AQQJ, AS3T, AT4N, AT5J, ATST, B04D, B0QD, BBIB, BBKB, BI0B, BK0B, BKIB, BKSB, BMIB, BMSB, BQ4D, BQQD, BSMB, C00B, C05B, C0A1, C0BB, C0EH, C0GH, C0Q1, C0QB, C2OH, C2PH, C441, C4Q1, C50B, CB0B, CBSB, CEGH, CG2H, CGHH, CHHH, CHOH, COGH, COO1, COOB, COOH, CQ01, CQ0B, CQQB, CQS1, CS0B, D00N, D0PD, D44N, DA0N, DCEH, DCG1, DCQ1, DDC1, DDF1, DDPD, DDQ1, DECH, DF4N, DFA1, DFRN, DG0D, DGD1, DGQ1, DNEH, DPDD, DQA1, DQPD, DRFN, DRRN, E20H, E667, E767, E8L7, E8S7, ECGH, EG2H, EH67, EH77, ES67, F001, F01J, F0MJ, F0QJ, F44N, F4G1, F4RN, F6RN, FA0J, FAC1, FAF1, FAIJ, FC01, FCG1, FFAJ, FFC1, FFG1, FFIJ, FGA1, FJJJ, FJQJ, FKAN, FR6N, FRFN, G09H, G0EH, G0HH, G20H, G341, G3Q1, G4G1, G92H, GA41, GAC1, GC0H, GCEH, GCG1, GCPH, GD31, GDA1, GDG1, GDPD, GE2H, GE9H, GGG1, GGGD, GHCH, GHHH, GP0D, GP2H, GPPD, GQ31, GQA1, GQG1, H00N, H0CH, H3ST, H4O1, H667, H66N, H677, H69T, H99T, H9CH, H9KT, H9LJ, H9PT, H9QJ, HH6N, HHF1, HHFJ, HHH1, HHHT, HHLJ, HHOH, HHQ1, HHST, HK6T, HL9J, HLMJ, HMKT, HMMT, HNM7, HNRN, HOPT, HQ01, HQ5J, HR0N, HR6N, HS77, HSKT, HT01, I0BB, I0JD, I0JJ, I98J, I9LJ, I9QJ, ICBB, IF0J, IFFJ, IFQJ, IIBD, IJBB, IJID, IL8J, ILFJ, IQ8J, IQJJ, IQQJ, IS0B, ISQB, J00D, J05J, J0BB, J0MJ, J0OB, J4QD, J50B, J5QJ, JBKB, JBMB, JDGD, JDQD, JGDD, JI0D, JIIB, JISB, JKIB, JKKB, JM0B, JOIB, JQ4D, JQDD, JQID, JQJJ, JQMJ, JSIB, JSMB, K06N, K0KB, K0KT, K0MT, K0PT, K0SB, K5QB, K60T, K66T, K6AN, K6FN, K6PT, K6RN, K96T, K99T, KA0N, KF6N, KFKN, KI0B, KK0T, KK6N, KK6T, KKFN, KKKB, KKKT, KKQB, KKTN, KMMT, KQ0B, KQQB, KR0N, KS5B, L0M7, L8E7, L98J, LFFJ, LI8J, LIFJ, LL87, LL9J, LLIJ, LLM7, LM77, LML7, LMM7, M09T, M0E7, M0IB, M0L7, M0M7, M0TT, M55J, M5LJ, M60T, M69T, M707, M767, M777, M7E7, M7S7, M8LJ, M96T, M9KT, MC0B, MCOB, ME07, MIBB, MJ0B, MJBB, MJSB, MLLJ, MLS7, MM07, MM0T, MML7, MMOB, MMR7, MOBB, MOCB, MP0T, MP9T, MPKT, MQ0B, MQIB, MQOB, MS07, MSPT, MSSB, MT8J, MTTT, N00D, N04D, N04N, N0FN, N0ND, N0RN, N2OH, N92H, NCGH, ND0D, ND4N, NDPD, NEGH, NEL7, NF4N, NFRN, NGCH, NGEH, NHM7, NME7, NML7, NMM7, NN0D, NN4D, NN4N, NNDN, NNID, NNND, NNRN, NP4D, NSL7, NSS7, O00T, O03T, O0MT, O0PT, O2HH, O2OH, O341, O4A1, O4G1, O4O1, O5KB, O90T, OA41, OBSB, OC5B, OCOB, OG2H, OG31, OH9H, OH9T, OHHH, OHTT, OICB, OISB, OM0T, OM6T, OMPT, OMTT, OO2H, OO9H, OOA1, OOCB, OOGH, OOKB, OOO1, OSKB, OTMT, P00T, P04D, P08D, P09T, P0KT, P90T, PAAT, PKKT, PO0T, PP0D, PPPD, Q00B, Q0AN, Q0D1, Q0F1, Q0IB, Q0JJ, Q0MJ, Q0OB, Q55J, Q5QJ, Q88D, QA0N, QA41, QA4N, QDF1, QDPD, QF41, QFA1, QG31, QIQJ, QJ0J, QJ8D, QJDD, QJJJ, QJMJ, QJQD, QKKB, QKQB, QOKB, QOOB, QP0D, QPDD, QQ8D, QQID, QQJD, QQPD, QQQB, QS01, QSA1, R00N, R067, R06N, R0FN, R0L7, R60N, RE67, RFRN, RRFN, RRRN, S00B, S3AT, S3ST, S50B, S5QB, S99T, SAAT, SC0B, SE67, SGG1, SICB, SIQB, SK5B, SKIB, SKKT, SKPT, SKQB, SM0B, SMMB, SMSB, SOA1, SOG1, SPOT, SQCB, SQQB, SS0B, SS67, ST31, STG1, T00N, T03T, T041, T04N, T0AT, T0C1, T0MT, T0ST, T1AJ, T3F1, T4A1, T4RN, T5JJ, T66N, T8AJ, T9FJ, T9QJ, TA3T, TAA1, TAG1, TAIJ, TCS1, TFA1, TFAJ, TFF1, TFFJ, TFG1, TGA1, TI9J, TIJJ, TIQJ, TL8J, TL9J, TM0T, TM5J, TM8J, TMAJ, TMLJ, TMST, TSF1, TSG1, TT01, TT5J, TT8J, TTA1, TTC1, TTFJ, TTG1, TTLJ, TTMJ, 30441, 30O41, 30OO1, 34O41, 3AATT, 3ASAT, 3O441, 3Q041, 40G41, 40GQ1, 40O41, 40OG1, 40OO1, 44001, 440C1, 440G1, 4444N, 44CG1, 44GQ1, 4AAA1, 4COG1, 4DAA1, 4GQ41, 4KRRN, 4OGG1, 4Q041, 4QGG1, 5005B, 500BB, 500KB, 500QB, 50K0B, 50Q0B, 50QKB, 555AJ, 555LJ, 555QJ, 55J5J, 5B00B, 5B0BB, 5J55J, 5J5JJ, 5JJ5J, 5JJJJ, 5K00B, 5QJ5J, 5QK0B, 6006N, 6009T, 600AT, 6066N, 60R0N, 666N7, 6696T, 66OPT, 67767, 6900T, 696PT, 69P0T, 6F6AN, 6O9PT, 6OP0T, 6OP9T, 6P99T, 755QB, 77767, 7CBBB, 7IBIB, 7IIBB, 7IICB, 7QIIB, 8888D, 88E67, 8L887, 8SSE7, 9099T, 90KKT, 90KMT, 90MPT, 9690T, 96P9T, 990KT, 9990T, 9999T, 99K6T, 9FFFJ, 9HKMT, 9HMPT, 9ILLJ, 9KKMT, 9KKPT, 9M00T, 9T8QJ, 9TFIJ, A008J, A00NN, A055J, A0FJJ, A0I0J, A0I9J, A0J0J, A44C1, A555J, A9FFJ, A9QIJ, A9T8J, AFFA1, AFFFJ, AFFJJ, AFFQJ, AJ00J, AP99T, ASA9T, ASFF1, ASP9T, ASSAT, AT3TT, ATFQJ, ATQIJ, ATT9J, ATTQJ, B088D, B08GD, B0GGD, B0SKB, B8GGD, BG00D, BIISB, BISIB, BS0IB, BS0SB, BSSKB, BSSSB, C0041, C0G41, C0H0H, C0PPH, C0S01, C4001, C4AG1, C4OG1, C5BBB, CG00H, CGE0H, CGGA1, CGQ41, COSSB, CP20H, CPGPH, CPP2H, CPPOH, CPPPH, CS001, CS55B, CSSQB, D00GD, D0A4N, D0FAN, D0GDD, D0N0D, DAAA1, DAAC1, DAFG1, DDAA1, DDGGD, DFFAN, DFGC1, DGDGD, DGG31, DGGA1, DH0AN, DPPPH, DQDD1, DQGG1, E0L87, E88E7, EC00H, EC02H, EE867, EE887, EEL87, ELE87, F00IJ, F0441, F0AG1, F0F0J, F0F41, F0FF1, F0GG1, F4041, F44A1, F44C1, F4A41, F64KN, F6K6N, FAAA1, FAFQJ, FCAA1, FF041, FF0F1, FF64N, FF6KN, FFA41, FFA4N, FFF4N, FFFF1, FFFFN, FFKRN, FFQMJ, FIJ0J, FJ00J, FKKKN, FNNFN, FQQ0J, FQQMJ, G00DD, G00DH, G00PD, G02PH, G0DDD, G0PDD, GAAA1, GC2HH, GDDDD, GDGDD, GDPPH, GG0PD, GGCA1, GGCQ1, GGDD1, GH4Q1, GHHG1, GII0D, GQ441, H009H, H00G1, H00H1, H04Q1, H0ANN, H0H01, H0H9H, H0HG1, H0HO1, H0O41, H0OHH, H0QG1, H40G1, H4G41, H4GG1, H60AN, HAN0N, HF0G1, HFF41, HFFMJ, HFFQJ, HGQ41, HH3TT, HH401, HH441, HH4G1, HH55J, HH66T, HH6OT, HH96T, HHA0N, HHANN, HHC0H, HHCHH, HHH0N, HHHCH, HHHHN, HHHNN, HHKMT, HHO41, HHP9T, HHPKT, HHTMT, HHTT1, HK9MT, HKKMT, HLLFJ, HM66T, HM7R7, HMM67, HMM77, HMMM7, HMTST, HO9HT, HOO41, HOOOH, HOT3T, HOTT1, HPO9T, HSO3T, HTGG1, I000J, I009J, I00BD, I00QJ, I00SB, I08QJ, I0I0D, I0IQB, I0JIB, I0Q0B, I0SCB, I0SIB, I0SSB, IB00D, IBBSB, IBISB, ICSSB, II0CB, II0ID, II0SB, IIIBB, IIQ0B, IIQIB, IISIB, IJ0SB, IJJQJ, ILLLJ, IQ00J, IQC0B, IQIQB, IQQIB, ISSCB, J00MB, J08GD, J0IID, J0QQD, J55BB, J55JJ, J5J5J, J5JJJ, J88GD, JB0SB, JBB0B, JBBSB, JDDDD, JG0ID, JGG0D, JIIID, JJ0QJ, JJ5JJ, JQ08D, JQ0QJ, JQQ0D, JQQ5J, JS55B, JSK0B, JSS5B, K000T, K006T, K00TN, K0FAN, K505B, K6T0N, K9KMT, K9KPT, K9P0T, KIQIB, KK00N, KK05B, KK0AN, KK0KN, KKM9T, KS0IB, KS0QB, KSQIB, KTR6N, KTRRN, L0087, L08S7, LE087, LEE87, LL8LJ, LMEE7, LMSE7, LMSS7, M00CB, M00PT, M066T, M0CQB, M0K6T, M0MQB, M0MSB, M0QQB, M0SMB, M0SST, M666T, M900T, MBBSB, MEE77, MEL77, MELE7, MES77, MESE7, MESS7, MI00B, MIICB, MIISB, MIQCB, MK9PT, ML7L7, MLEL7, MLLE7, MLME7, MM677, MMBIB, MMBSB, MMCQB, MME77, MMEE7, MMICB, MMISB, MMK6T, MMKKT, MMM9T, MMMMT, MMMTT, MMQQB, MMSCB, MMSKT, MMSMB, MMSS7, MMTST, MOIIB, MOOIB, MOSIB, MQCQB, MR007, MR667, MRL07, MS7L7, MSEL7, MSK9T, MSL77, MSSL7, MT00T, MTMMT, N0DDD, N4NNN, N7LE7, N7S77, NE2CH, NE9CH, NEC2H, NEE77, NFFNN, NFNFN, NII0D, NL777, NL7L7, NLES7, NLLL7, NLS77, NOG9H, NRR0N, O6P9T, O9H6T, O9HPT, O9P9T, OCBBB, OG441, OGAG1, OGGA1, OHH6T, OHOOH, OKIIB, OKK5B, OKKIB, OO5BB, OOBIB, OOIIB, OSCSB, OT3TT, OT441, OTG41, OTGG1, P00PH, P0D0D, P0G2H, P0OGH, P0PGH, PA99T, PGP0H, POOOH, PP20H, PP88D, PPP0H, Q0001, Q000N, Q001J, Q00ID, Q00PD, Q044N, Q04G1, Q0I0D, Q0PPD, Q40G1, Q444N, Q44RN, Q4AA1, Q8QQJ, QAAS1, QAFF1, QAFG1, QASG1, QDGG1, QFGG1, QI00D, QIICB, QIIQB, QIQCB, QOIIB, QPP4D, QQ08J, QQ0CB, QQC0B, QQI0J, QQJ5J, QSFG1, R00E7, R0E07, R0NE7, R6F6N, R6FFN, REE07, RELE7, RFF6N, RL0E7, RLE07, RLEE7, RLLE7, RQR0N, RR0QN, RRQ0N, S03O1, S0AF1, S0AG1, S0O31, S0QIB, S0SIB, S30F1, S30O1, S7QIB, SA3PT, SAAA1, SAFG1, SASST, SCSQB, SF0G1, SFFF1, SI0SB, SISSB, SKSSB, SSCQB, SSCSB, SSMIB, SSPAT, SSSKT, SSSMB, SSSSB, STAF1, STF01, T0001, T0031, T0AF1, T0AS1, T0G31, T0R6N, T0T31, T0TF1, T3AAT, T40G1, T4CG1, T4G41, T5LLJ, T8LLJ, TA441, TA98J, TAFJJ, TAQ5J, TASST, TATAT, TC401, TCGG1, TFQIJ, TFQJJ, TFQMJ, TG441, TGC41, TI8LJ, TLLMJ, TMMMT, TMTMT, TQ8QJ, TSS3T, TT3AT, TT9IJ, TTAAT, TTQIJ, TTS31, TTS3T, TTT9J, TTTST, 20000H, 200OOH, 3440O1, 3TAAST, 404CQ1, 4KKKKN, 4KKKRN, 4QQQQD, 505BBB, 50BB0B, 6000AN, 6444RN, 66666N, 666O9T, 66999T, 669P9T, 66N777, 6A444N, 6FF66N, 6FFF6N, 6R666N, 766767, 77S677, 7IBBBB, 8888E7, 8LLLLJ, 8SSSL7, 9000MT, 90K90T, 90KP0T, 90M90T, 99000T, 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===Base 36===

11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 2UJ, 2WN, 2XN, 30Z, 337, 33B, 33H, 33P, 34H, 34P, 37D, 38Z, 39H, 3AB, 3AP, 3AZ, 3D7, 3DH, 3FD, 3FZ, 3HD, 3HH, 3KZ, 3L7, 3LZ, 3MB, 3O7, 3OZ, 3PD, 3PP, 3Q7, 3RH, 3S7, 3SZ, 3UP, 3UZ, 3XB, 40P, 43B, 43H, 46V, 48B, 48V, 49B, 4AP, 4BB, 4E1, 4FP, 4HB, 4HH, 4HP, 4IH, 4MB, 4OP, 4OV, 4PH, 4RH, 4VB, 4VH, 4WH, 4WP, 4WV, 4X1, 4XH, 55T, 565, 58Z, 5C5, 5E7, 5ET, 5EZ, 5IN, 5KT, 5L7, 5MP, 5NN, 5O5, 5O7, 5OP, 5R7, 5RT, 5RZ, 5ST, 5SZ, 5TN, 5TZ, 5W7, 5XN, 5YN, 61T, 625, 62J, 64V, 661, 66J, 691, 6AV, 6EV, 6IJ, 6J1, 6K5, 6O1, 6OT, 6QJ, 6QT, 6RV, 6SJ, 6T1, 6TT, 6U5, 6UV, 6VV, 6W5, 6Y1, 6YT, 72D, 737, 77D, 791, 7A1, 7AZ, 7D7, 7GD, 7I1, 7IN, 7LN, 7M7, 7MN, 7N7, 7NN, 7RN, 7RZ, 7WN, 7X7, 7Z1, 801, 80V, 82D, 83B, 841, 84H, 877, 881, 887, 88B, 88V, 88Z, 89H, 89V, 8A1, 8AB, 8B7, 8BH, 8D1, 8DH, 8EB, 8EV, 8GD, 8GZ, 8HD, 8IV, 8L7, 8LZ, 8M1, 8MB, 8MD, 8O7, 8OB, 8Q7, 8QD, 8RD, 8S7, 8SH, 8SZ, 8UZ, 8W7, 8WV, 8XV, 8Y1, 90H, 91J, 93B, 93H, 94V, 981, 98H, 99J, 9AH, 9BB, 9C5, 9IH, 9IV, 9JV, 9KH, 9M1, 9OB, 9QJ, 9R1, 9RH, 9SH, 9UJ, 9VB, 9VJ, 9W5, 9X1, 9Y1, 9YB, 9YJ, 9YV, A25, A3P, A3Z, A4H, A61, A81, A8B, AAB, AAH, ABB, AC5, AEZ, AHP, AK1, AKB, AKH, ALZ, AMB, AO5, AOZ, APH, AQP, AR1, ARV, ARZ, AUV, AV5, AVB, AVV, AWV, AX1, AXB, AZ1, B2J, B4B, B4V, B6V, B87, B8H, B9B, BAB, BAH, BBH, BBV, BE7, BEB, BGJ, BHB, BIJ, BJB, BJT, BLJ, BOB, BOT, BQ7, BRV, BS7, BTT, BTV, BVB, BVJ, BVT, BWV, BX7, BXH, BYH, BYV, C25, C2J, C2N, C55, C65, C6J, C95, CCN, CDD, CED, CF5, CFJ, CGD, CIN, CJD, CO5, CRN, CUJ, CXJ, CXN, D37, D3P, D41, D55, D5P, D6T, D77, D81, D9H, DA1, DCD, DD1, DD5, DDH, DDP, DE7, DEP, DF5, DFD, DG7, DHH, DI1, DK1, DK5, DKT, DO1, DOP, DP1, DPD, DQ7, DR7, DRH, DS7, DTT, DU1, DWD, DX7, DYH, E0P, E2D, E3Z, E41, E4P, E71, E81, E87, E8B, EAZ, EDP, EED, EEN, EEV, EFZ, EGT, EI1, EKZ, ELB, EMB, EMN, EN7, EO1, EOT, EPP, EPZ, EQ7, ERN, EST, ETN, ETV, EUP, EUZ, EVT, EWD, EWN, EX1, EYB, EYN, EZD, EZP, F2J, F3B, F3Z, F4P, F65, F8B, F8Z, FCJ, FD5, FEZ, FFB, FFD, FG5, FGD, FGZ, FIJ, FJ5, FJJ, FLZ, FPD, FQD, FSZ, FXB, G07, G0D, G0Z, G3D, G3Z, G55, G6T, G7D, G7Z, G87, G8Z, GA5, GCJ, GD7, GE7, GET, GG5, GG7, GGT, GGZ, GI7, GJD, GKZ, GL7, GLT, GQT, GRT, GS7, GST, GU5, GUZ, GW5, GXJ, GZ7, GZZ, H3D, H3H, H4H, H9B, HAH, HED, HGD, HHP, HIH, HKH, HKP, HLN, HNH, HOB, HOP, HQP, HRD, HRH, HRN, HSH, HWD, HWH, HWP, HYN, I41, I6J, I71, I7N, I87, I8H, I9H, I9J, IA1, IAV, IE1, IFJ, IG7, IHH, II1, IIH, IIV, IJV, IK1, IL7, ILJ, ILN, IM1, INN, IOV, IRH, IUV, IX1, IXH, IXV, IYJ, J2D, J4V, J55, J61, J6J, J8B, J8V, J95, J9J, J9V, JA5, JBB, JD5, JDT, JF5, JFJ, JGJ, JGT, JIV, JJ1, JJD, JJV, JK5, JKT, JLB, JLT, JMN, JN5, JNJ, JQJ, JQN, JQT, JRD, JTB, JVJ, JVN, JVV, JWN, JY1, JYJ, K0B, K25, K3P, K65, K81, K95, K9H, KAH, KC5, KEP, KEZ, KG5, KHP, KK1, KKT, KLB, KLP, KLZ, KM1, KMB, KMP, KOT, KP1, KQP, KR1, KRT, KRZ, KTT, KW5, KX1, KZB, L0N, L0P, L8Z, L9B, LAZ, LBJ, LEN, LET, LFB, LFZ, LG7, LGZ, LIJ, LJJ, LKB, LKP, LLB, LLP, LLT, LMB, LMN, LN7, LO7, LOT, LPZ, LQT, LRN, LTN, LTT, LWP, LX7, LYN, M01, M0P, M2D, M2N, M37, M3B, M41, M61, M77, M87, M8D, M91, ME1, ME7, MEB, MFB, MFP, MGD, MI7, MKB, MM7, MMN, MN7, MNN, MO7, MOB, MOP, MPP, MQP, MS7, MW7, MXN, MYN, N5N, N65, N87, N8D, N8H, N95, NCD, NCJ, NDH, NE7, NFJ, NG5, NG7, NGJ, NJ5, NN7, NND, NO5, NQD, NQJ, NRN, NU5, NWH, NWN, NXH, NXN, NYJ, O0Z, O25, O37, O3B, O3P, O3Z, O41, O4B, O5Z, O61, O71, O7Z, O81, OA5, OAP, OBB, OBV, OC5, OEZ, OF5, OG7, OKP, OKZ, OMB, OO5, OOZ, OP5, OPP, OQT, OR1, OR7, OS7, OST, OTB, OTZ, OU5, OVV, OW5, OX7, OXB, OXV, OYV, OZ5, OZ7, P01, P0D, P0N, P3P, P4H, P4P, P5N, P65, P6V, P8V, PD1, PED, PEZ, PHH, PHP, PI1, PIN, PLN, PLP, PLZ, POP, PP1, PPH, PPV, PQD, PQN, PRV, PV5, PVH, PVV, PWH, PWP, PX1, PXV, PYH, PYN, Q07, Q0P, Q2N, Q37, Q3P, Q6J, Q6T, Q7D, Q8D, Q9J, QCD, QCJ, QD7, QED, QFP, QGT, QI7, QIN, QJN, QLJ, QMD, QMN, QMP, QND, QNJ, QOT, QQJ, QTN, QWN, QYJ, QYT, R0V, R0Z, R37, R3H, R4H, R77, R7D, R7N, R7Z, R81, R8V, R91, RA1, RCD, RCN, RD1, RED, REV, REZ, RGT, RGZ, RHD, RIV, RKH, RKZ, RLN, RMD, RNH, RO7, RQN, RRD, RRT, RRZ, RS7, RSH, RT1, RTV, RU1, RUZ, RVN, RVT, RW7, RWH, RX7, RY1, S0J, S6J, S87, S9H, SAZ, SCJ, SET, SFJ, SG7, SGZ, SI7, SM7, SO7, SOT, SQJ, SQT, SRH, SSH, STT, SW7, SX7, SXH, SYH, T0N, T0Z, T1D, T1T, T6V, T8Z, TBB, TBV, TCN, TD1, TEV, TGT, TKT, TLB, TM1, TO1, TOB, TQN, TR1, TRD, TT1, TTB, TTN, TUZ, TVB, TVN, TVT, TWD, TWV, TXB, TXV, TYV, U2J, U3Z, U61, U8V, U95, UA1, UC5, UEP, UEZ, UFJ, UG5, UJ5, ULZ, UO5, UOP, UOZ, UQP, URV, URZ, UUJ, UW5, UWV, UXJ, UXV, UYJ, V0H, V25, V4V, V55, V6V, V9J, V9V, VBB, VBJ, VCJ, VET, VGJ, VHP, VIH, VIN, VJJ, VJN, VLT, VMP, VNJ, VOB, VP5, VQJ, VQT, VRT, VRV, VSH, VSJ, VST, VTB, VTN, VUP, VW5, VWN, VXH, VXN, VYB, W07, W25, W3D, W3H, W4P, W4V, W57, W6V, W7D, W8H, W95, W9H, WAH, WAV, WCN, WD7, WDD, WDH, WE7, WEN, WF5, WGD, WHH, WK5, WKH, WN7, WNN, WPP, WPV, WQP, WR7, WRD, WRH, WRN, WS7, WU5, WUP, WVN, WWH, WWP, WX7, WXH, WXN, WYN, WYV, X0J, X2J, X2N, X4B, X4H, X4V, X6J, X87, X8B, X91, X9B, XAV, XCN, XEB, XFB, XHB, XHH, XHN, XKB, XLJ, XNH, XO1, XQ7, XS7, XSH, XUJ, XWN, XWV, XX7, XXH, XXV, Y0N, Y1J, Y1T, Y2N, Y3H, Y61, Y8V, Y91, YBV, YCN, YEB, YFJ, YHB, YHN, YIH, YJN, YJT, YLJ, YLN, YMB, YMN, YNH, YOB, YOV, YRH, YTB, YTN, YTT, YTV, YVB, YVH, YWH, YWV, YXB, YY1, YYJ, YYT, YYV, Z01, Z3D, Z3Z, Z71, Z8B, ZB7, ZBB, ZD5, ZDD, ZDP, ZG7, ZGD, ZGZ, ZKB, ZLP, ZM1, ZO5, ZP1, ZPD, ZQD, ZU1, ZX1, ZXB, ZZD, 102J, 106T, 10ET, 10GJ, 10LJ, 10TD, 12FJ, 12LJ, 13ED, 13GD, 13MD, 192J, 1D0T, 1DET, 1E3D, 1EFD, 1F0D, 1FWD, 1G0J, 1GGJ, 1GIJ, 1GUJ, 1I0J, 1ISJ, 1JOT, 1JTT, 1M0D, 1M3D, 1MFD, 1MWD, 1OKT, 1OOT, 1Q3D, 1QDT, 1QKT, 1QTD, 1QWD, 1SLJ, 1TDT, 1TFD, 1TOT, 1TTD, 1TTT, 1UGJ, 1ULJ, 1WFD, 1WWD, 20CJ, 20FJ, 20G5, 20IN, 20JD, 20MD, 20RN, 20WD, 20YN, 2295, 25RN, 260J, 26LJ, 288D, 28DD, 28FD, 296J, 29G5, 29K5, 29LJ, 2C05, 2C0J, 2C0N, 2CJJ, 2CK5, 2CLJ, 2CSJ, 2D25, 2DC5, 2DDD, 2DW5, 2F0J, 2FDD, 2FMD, 2FSJ, 2G65, 2G95, 2GK5, 2I0N, 2JC5, 2JCJ, 2JWD, 2K05, 2L6J, 2L9J, 2N0N, 2NIN, 2NN5, 2NSJ, 2NW5, 2NYN, 2Q0N, 2QNN, 2R0N, 2RFD, 2RWD, 2SJJ, 2SXJ, 2SYJ, 2W05, 2WC5, 2WD5, 2WMD, 2XCJ, 2YCJ, 2YJJ, 2YRN, 2YSJ, 300D, 302D, 304B, 3077, 3087, 308D, 30BH, 30DD, 30FP, 30G7, 30GD, 30MD, 30QP, 30YH, 320D, 333D, 333Z, 33WD, 344B, 34FB, 373Z, 37E7, 37EZ, 37G7, 3807, 38BB, 38E7, 38G7, 38HB, 38I7, 38X7, 390B, 394B, 399B, 39EB, 39FB, 39KB, 3ASH, 3AWH, 3B8B, 3BG7, 3BLB, 3BR7, 3BW7, 3C8D, 3D0P, 3D2D, 3DKP, 3E3D, 3E77, 3EBB, 3EFB, 3EKB, 3EQP, 3ERD, 3ERZ, 3EZZ, 3F4B, 3FEP, 3G2D, 3GCD, 3GED, 3GM7, 3GR7, 3GWD, 3GZD, 3H0B, 3HEP, 3HKB, 3IAH, 3ISH, 3K4B, 3KBH, 3KEB, 3KKB, 3L4B, 3L8B, 3M3D, 3MEP, 3MG7, 3MWP, 3OFP, 3OKB, 3OMP, 3OOB, 3OYB, 3P8H, 3PXH, 3QDP, 3QQD, 3QRD, 3QWD, 3R3D, 3R3Z, 3R8D, 3RG7, 3RM7, 3RQD, 3RR7, 3RZ7, 3SIH, 3W0H, 3WED, 3WM7, 3WQD, 3WW7, 3WYH, 3X8H, 3XE7, 3XG7, 3XKH, 3Y0B, 3Y4B, 3Y8H, 3YAH, 3YSH, 3YYB, 3Z4B, 3ZFP, 404V, 4061, 40EB, 40I1, 40IV, 40KH, 40M1, 40R1, 40XB, 40XV, 43QP, 4441, 444H, 444P, 4481, 448H, 4491, 449H, 44EB, 44EV, 44K1, 44KB, 44KP, 44M1, 44R1, 44RV, 44SH, 44U1, 44VP, 44YB, 46I1, 46K1, 46M1, 480H, 488H, 48AH, 48I1, 48K1, 48U1, 4991, 49EV, 49I1, 49O1, 49UV, 4A0H, 4A41, 4A4B, 4A8H, 4A91, 4ALB, 4ASH, 4BXV, 4E4B, 4EEP, 4EFB, 4ELP, 4EUV, 4EXB, 4FOB, 4I01, 4I0V, 4I4V, 4I61, 4I81, 4I91, 4IEV, 4IY1, 4K0H, 4K91, 4KI1, 4KKH, 4KKP, 4KUP, 4L3P, 4L4P, 4LAB, 4LMP, 4MA1, 4MK1, 4MMP, 4MP1, 4MU1, 4O0B, 4OFB, 4OKB, 4OM1, 4OU1, 4OYB, 4P41, 4P81, 4PO1, 4PYV, 4QLP, 4QQP, 4RAV, 4RK1, 4RRV, 4S8H, 4U3P, 4U91, 4U9V, 4UIV, 4UUP, 4VAV, 4VLP, 4VVV, 4X0B, 4X0V, 4XAB, 4XBV, 4XVV, 4XYV, 4Y0V, 4Y41, 4Y4H, 4Y4V, 4Y81, 4YAV, 4YEV, 4YIV, 4YK1, 4YKB, 4YM1, 4YR1, 4YUV, 4YYB, 5025, 503Z, 5095, 50G5, 50PN, 50QN, 50WP, 50Z7, 5205, 52RN, 53FP, 53M7, 53PZ, 53X7, 53ZZ, 54PP, 550N, 553P, 554P, 5557, 555N, 557Z, 5595, 55EN, 55G5, 55GZ, 55LZ, 55OZ, 55QN, 55Z7, 572N, 577Z, 5787, 57CN, 57FZ, 57QN, 57S7, 57ZZ, 58I7, 58X7, 5955, 5A95, 5AEP, 5CMN, 5CWN, 5E2N, 5EAP, 5ECN, 5EKP, 5EQN, 5F95, 5FLP, 5FQP, 5G05, 5GK5, 5GQ7, 5I37, 5II7, 5IM7, 5L3P, 5LCN, 5LFP, 5LGT, 5LKZ, 5LUP, 5LZZ, 5M07, 5N55, 5NQ7, 5NX7, 5OAZ, 5OFZ, 5OLT, 5OOT, 5P25, 5P95, 5PG5, 5PGZ, 5PPN, 5PQP, 5PW5, 5Q0N, 5Q87, 5QCN, 5QKP, 5QN7, 5QPN, 5QPP, 5QQT, 5QTT, 5QWP, 5QX7, 5S37, 5SS7, 5T6T, 5TQT, 5U4P, 5UGZ, 5W0P, 5W2N, 5W5P, 5WLN, 5WP5, 5WQN, 5XM7, 5Y0T, 5YOT, 5Z25, 5Z4P, 5Z57, 5ZG5, 5ZLZ, 5ZQ7, 5ZWP, 6005, 600V, 601J, 6055, 606V, 60A1, 60C5, 60GT, 60JJ, 60LJ, 60M1, 60R1, 60VT, 60YV, 610J, 61FJ, 6401, 64K1, 6505, 665T, 6665, 66A5, 66F5, 66G5, 66GT, 66KT, 66V5, 66XV, 68OV, 68YV, 690J, 69G5, 69O5, 69XJ, 69XV, 6A55, 6AA1, 6AA5, 6AF5, 6C9J, 6CA5, 6CGJ, 6CYJ, 6EK1, 6ER1, 6FLJ, 6FXJ, 6G0J, 6G65, 6GGJ, 6GJ5, 6GLJ, 6I81, 6I9V, 6J65, 6J6V, 6JET, 6JST, 6JWV, 6K01, 6KET, 6KST, 6L0J, 6L6T, 6LFJ, 6LRT, 6LXJ, 6MA1, 6MM1, 6MX1, 6O8V, 6OV5, 6R0T, 6RET, 6RKT, 6S6T, 6T8V, 6U01, 6U9J, 6UE1, 6UGJ, 6UJJ, 6ULJ, 6UU1, 6V0T, 6VA5, 6VF5, 6VG5, 6VKT, 6VXJ, 6WWV, 6WXV, 6X41, 6X6V, 6XIV, 6XJV, 6XOV, 6Y0V, 6YCJ, 6YXV, 7001, 702N, 7041, 707N, 7081, 70GZ, 70I7, 70KZ, 70ND, 70O7, 70OZ, 70R7, 70U1, 70UZ, 70WD, 70YN, 71QD, 71WD, 720N, 72CN, 72YN, 733Z, 73ED, 73ZD, 7401, 7441, 74M1, 74O1, 74R1, 7641, 76K1, 76M1, 76R1, 76U1, 76X1, 773Z, 7741, 7771, 7781, 77FZ, 77G7, 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8KRH, 8KU1, 8KYH, 8KZ1, 8L4B, 8LBB, 8MG7, 8MR7, 8O4V, 8OI1, 8OK1, 8OOV, 8OX1, 8RAV, 8RHH, 8RUV, 8RX1, 8U4V, 8UK1, 8V4B, 8V9B, 8VHH, 8VRH, 8VVV, 8VYV, 8W0H, 8WFD, 8X07, 8XAH, 8XR1, 8XU1, 8XYB, 8Y6V, 8Y8H, 8YBB, 8YFB, 8YHH, 8YKB, 8YLB, 8YUV, 8YVV, 8YYH, 8Z1D, 8Z4B, 8Z8D, 8Z91, 8ZCD, 8ZEZ, 8ZFD, 8ZFZ, 8ZK1, 8ZO1, 8ZX7, 90A5, 90BJ, 90BV, 90EB, 90EV, 90GJ, 90IJ, 90J5, 90JB, 90LB, 90O1, 90OV, 90XB, 90XV, 9225, 92K5, 9441, 94BH, 94EB, 94KB, 94XB, 9505, 9601, 960V, 96A1, 96J5, 96K1, 96V5, 96WV, 96XV, 984B, 98AV, 98BV, 98FB, 994H, 998B, 998V, 999B, 99A1, 99A5, 99E1, 99G5, 99O1, 99O5, 99WH, 9A05, 9A55, 9A95, 9AA5, 9ABV, 9AE1, 9AEV, 9AI1, 9AO1, 9AOV, 9B0J, 9B0V, 9B8V, 9B9H, 9BCJ, 9BFJ, 9BSJ, 9BVV, 9CCJ, 9CLJ, 9CSJ, 9E01, 9E4B, 9EA1, 9EAB, 9EAV, 9EK1, 9F05, 9F25, 9FA5, 9FBJ, 9FFJ, 9FJB, 9FSJ, 9FXJ, 9G25, 9G2J, 9GIJ, 9GSJ, 9HFB, 9HLB, 9IIJ, 9IJ1, 9J01, 9J0B, 9J0J, 9J9B, 9JCJ, 9JE1, 9K05, 9K41, 9K55, 9K61, 9K8B, 9KA5, 9KI1, 9KK5, 9KU5, 9L2J, 9LCJ, 9O05, 9O0V, 9O95, 9OI1, 9OO1, 9OOV, 9OWV, 9RRV, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ

====Additional known quasi-minimal primes (not necessarily the next)====

P&lt;sub&gt;81993&lt;/sub&gt;SZ

==Unsolved families==

Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')) by covering congruence, algebraic factorization, or combine of them.

{|class=wikitable
|base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family
|-
|13||9{5}||(113×13&lt;sup&gt;''n''&lt;/sup&gt;−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|13||A{3}A||(41×13&lt;sup&gt;''n''+1&lt;/sup&gt;+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|16||{3}AF||(16&lt;sup&gt;''n''+2&lt;/sup&gt;+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1{7}||(23×17&lt;sup&gt;''n''&lt;/sup&gt;−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1F{0}7||32×17&lt;sup&gt;''n''+1&lt;/sup&gt;+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||4{7}A||(71×17&lt;sup&gt;''n''+1&lt;/sup&gt;+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||51{0}D||86×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||70F{0}D||2038×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||8{B}9||(139×17&lt;sup&gt;''n''+1&lt;/sup&gt;−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||9{5}9||(149×17&lt;sup&gt;''n''+1&lt;/sup&gt;+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||95{F}||(2543×17&lt;sup&gt;''n''&lt;/sup&gt;−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||A{D}F||(173×17&lt;sup&gt;''n''+1&lt;/sup&gt;+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}B3||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+190||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}DB||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+232||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2BE||(11×17&lt;sup&gt;''n''+3&lt;/sup&gt;−41579)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2E||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;−2411)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}E9||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+773)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}EE||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+853)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||D0G{D}||(60381×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2860381*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||E9{B}||(3963×17&lt;sup&gt;''n''&lt;/sup&gt;−11)/16||29000||[http://factordb.com/index.php?query=%283963*17%5En-11%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||F1{9}||(4105×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16||1000000||[http://factordb.com/index.php?query=%284105*17%5En-9%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||7{K}Z||(53×36&lt;sup&gt;''n''+1&lt;/sup&gt;+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||B{0}EUV||11×36&lt;sup&gt;''n''+3&lt;/sup&gt;+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||HM{0}N||634×36&lt;sup&gt;''n''+1&lt;/sup&gt;+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||N{0}YYN||23×36&lt;sup&gt;''n''+3&lt;/sup&gt;+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||O{L}Z||(123×36&lt;sup&gt;''n''+1&lt;/sup&gt;+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||S{0}8H||28×36&lt;sup&gt;''n''+2&lt;/sup&gt;+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|}

For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E

For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D}

=== Base 17 ===

All searched to length 29000

* 1{7}
* 1F{0}7
* 4{7}A
* 51{0}D
* 70F{0}D
* 8{B}9
* 9{5}9
* 95{F}
* A{D}F
* B{0}B3
* B{0}DB
* {B}2BE
* {B}2E
* {B}E9
* {B}EE
* D0G{D}
* E9{B}
* F1{9}
* FD0{D}
* G{7}F

=== Base 21 ===

All searched to length 20000

* 2{7}9D
* 2F{C}A
* 4{3}B
* 5{0}DJ
* {5}FEK
* {7}ID
* 99{0}99H
* {9}0D
* {9}D
* B0{H}6H
* B3{0}EB
* B9{0}E5
* B{D}B
* B{H}6H
* DH{D}
* F{9}D
* {F}35
* G{0}FK
* H{0}7771
* H{D}
* {J}6J

=== Base 36 ===

All searched to length 20000

* 7{K}Z
* B{0}EUV
* HM{0}N
* N{0}YYN
* O{L}Z
* S{0}8H

==Primality certificates for the proven primes &gt; 10&lt;sup&gt;299&lt;/sup&gt;==

See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]]

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime
|-
|9||149||76&lt;sub&gt;329&lt;/sub&gt;2||(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642]
|-
|9||150||27&lt;sub&gt;686&lt;/sub&gt;07||(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486]
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||3×9&lt;sup&gt;1160&lt;/sup&gt;+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423]
|-
|11||1065||A&lt;sub&gt;713&lt;/sub&gt;58||11&lt;sup&gt;715&lt;/sup&gt;−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487]
|-
|11||1066||7&lt;sub&gt;759&lt;/sub&gt;44||(7×11&lt;sup&gt;761&lt;/sup&gt;−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840]
|-
|11||1067||557&lt;sub&gt;1011&lt;/sub&gt;||(607×11&lt;sup&gt;1011&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522]
|-
|13||3165||50&lt;sub&gt;270&lt;/sub&gt;44||5×13&lt;sup&gt;272&lt;/sup&gt;+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005]
|-
|13||3166||9&lt;sub&gt;271&lt;/sub&gt;095||(3×13&lt;sup&gt;274&lt;/sup&gt;−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654]
|-
|13||3167||10&lt;sub&gt;286&lt;/sub&gt;7771||13&lt;sup&gt;290&lt;/sup&gt;+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633]
|-
|13||3168||9&lt;sub&gt;308&lt;/sub&gt;1||(3×13&lt;sup&gt;309&lt;/sup&gt;−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=308&amp;c0=-&amp;EN= 13&lt;sup&gt;308&lt;/sup&gt;−1]
|-
|13||3169||B&lt;sub&gt;341&lt;/sub&gt;C4||(11×13&lt;sup&gt;343&lt;/sup&gt;+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618]
|-
|13||3170||8B&lt;sub&gt;343&lt;/sub&gt;||(107×13&lt;sup&gt;343&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736]
|-
|13||3171||710&lt;sub&gt;371&lt;/sub&gt;111||92×13&lt;sup&gt;374&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609]
|-
|13||3172||75&lt;sub&gt;375&lt;/sub&gt;7||(89×13&lt;sup&gt;376&lt;/sup&gt;+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596]
|-
|13||3173||9B0&lt;sub&gt;391&lt;/sub&gt;9||128×13&lt;sup&gt;392&lt;/sup&gt;+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790]
|-
|13||3174||7B0B&lt;sub&gt;397&lt;/sub&gt;||(15923×13&lt;sup&gt;397&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574]
|-
|13||3175||10&lt;sub&gt;414&lt;/sub&gt;93||13&lt;sup&gt;416&lt;/sup&gt;+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240]
|-
|13||3176||81010&lt;sub&gt;415&lt;/sub&gt;1||17746×13&lt;sup&gt;416&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3177||8110&lt;sub&gt;435&lt;/sub&gt;1||1366×13&lt;sup&gt;436&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3178||B7&lt;sub&gt;486&lt;/sub&gt;||(139×13&lt;sup&gt;486&lt;/sup&gt;−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892]
|-
|13||3179||B&lt;sub&gt;563&lt;/sub&gt;C||(11×13&lt;sup&gt;564&lt;/sup&gt;+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=564&amp;c0=-&amp;EN= 13&lt;sup&gt;564&lt;/sup&gt;−1]
|-
|13||3180||1B&lt;sub&gt;576&lt;/sub&gt;||(23×13&lt;sup&gt;576&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=576&amp;c0=-&amp;EN= 13&lt;sup&gt;576&lt;/sup&gt;−1]
|-
|13||3181||80&lt;sub&gt;693&lt;/sub&gt;87||8×13&lt;sup&gt;695&lt;/sup&gt;+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532]
|-
|13||3182||CC5&lt;sub&gt;713&lt;/sub&gt;||(2021×13&lt;sup&gt;713&lt;/sup&gt;−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353]
|-
|13||3183||B&lt;sub&gt;834&lt;/sub&gt;74||(11×13&lt;sup&gt;836&lt;/sup&gt;−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871]
|-
|13||3184||9&lt;sub&gt;968&lt;/sub&gt;B||(3×13&lt;sup&gt;969&lt;/sup&gt;+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244]
|-
|13||3185||10&lt;sub&gt;1295&lt;/sub&gt;181||13&lt;sup&gt;1298&lt;/sup&gt;+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013]
|-
|13||3186||9&lt;sub&gt;1362&lt;/sub&gt;5||(3×13&lt;sup&gt;1363&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776]
|-
|13||3187||7&lt;sub&gt;1504&lt;/sub&gt;1||(7×13&lt;sup&gt;1505&lt;/sup&gt;−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755]
|-
|13||3188||930&lt;sub&gt;1551&lt;/sub&gt;1||120×13&lt;sup&gt;1552&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3189||720&lt;sub&gt;2297&lt;/sub&gt;2||93×13&lt;sup&gt;2298&lt;/sup&gt;+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910]
|-
|13||3190||1770&lt;sub&gt;2703&lt;/sub&gt;17||267×13&lt;sup&gt;2705&lt;/sup&gt;+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825]
|-
|13||3191||390&lt;sub&gt;6266&lt;/sub&gt;1||48×13&lt;sup&gt;6267&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3192||B0&lt;sub&gt;6540&lt;/sub&gt;BBA||11×13&lt;sup&gt;6543&lt;/sup&gt;+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906]
|-
|13||3193||C&lt;sub&gt;10631&lt;/sub&gt;92||13&lt;sup&gt;10633&lt;/sup&gt;−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750]
|-
|14||649||34D&lt;sub&gt;708&lt;/sub&gt;||47×14&lt;sup&gt;708&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||5×14&lt;sup&gt;19698&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2328||880&lt;sub&gt;246&lt;/sub&gt;7||136×16&lt;sup&gt;247&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is &lt; 10&lt;sup&gt;299&lt;/sup&gt;
|-
|16||2329||D4&lt;sub&gt;263&lt;/sub&gt;D||(199×16&lt;sup&gt;264&lt;/sup&gt;+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238]
|-
|16||2330||E0&lt;sub&gt;261&lt;/sub&gt;4DD||14×16&lt;sup&gt;264&lt;/sup&gt;+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352]
|-
|16||2331||8C0&lt;sub&gt;290&lt;/sub&gt;ED||140×16&lt;sup&gt;292&lt;/sup&gt;+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307]
|-
|16||2332||DA&lt;sub&gt;305&lt;/sub&gt;5||(41×16&lt;sup&gt;306&lt;/sup&gt;−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284]
|-
|16||2333||CE80&lt;sub&gt;422&lt;/sub&gt;D||3304×16&lt;sup&gt;423&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257]
|-
|16||2334||5F&lt;sub&gt;544&lt;/sub&gt;6F||6×16&lt;sup&gt;546&lt;/sup&gt;−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967]
|-
|16||2335||88F&lt;sub&gt;545&lt;/sub&gt;||137×16&lt;sup&gt;545&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2336||BE0&lt;sub&gt;792&lt;/sub&gt;BB||190×16&lt;sup&gt;794&lt;/sup&gt;+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938]
|-
|16||2337||D9&lt;sub&gt;1052&lt;/sub&gt;||(68×16&lt;sup&gt;1052&lt;/sup&gt;−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020]
|-
|16||2338||FAF&lt;sub&gt;1062&lt;/sub&gt;45||251×16&lt;sup&gt;1064&lt;/sup&gt;−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610]
|-
|16||2339||F8&lt;sub&gt;1517&lt;/sub&gt;F||(233×16&lt;sup&gt;1518&lt;/sup&gt;+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824]
|-
|16||2340||20&lt;sub&gt;1713&lt;/sub&gt;321||2×16&lt;sup&gt;1716&lt;/sup&gt;+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735]
|-
|16||2341||300F&lt;sub&gt;1960&lt;/sub&gt;AF||769×16&lt;sup&gt;1962&lt;/sup&gt;−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750]
|-
|16||2342||90&lt;sub&gt;3542&lt;/sub&gt;91||9×16&lt;sup&gt;3544&lt;/sup&gt;+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191]
|-
|16||2343||5BC&lt;sub&gt;3700&lt;/sub&gt;D||(459×16&lt;sup&gt;3701&lt;/sup&gt;+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322]
|-
|16||2344||D0B&lt;sub&gt;17804&lt;/sub&gt;||(3131×16&lt;sup&gt;17804&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511]
|-
|18||547||80&lt;sub&gt;298&lt;/sub&gt;B||8×18&lt;sup&gt;299&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is an algebraic factor of ''N''+1, factordb entry of 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is [http://factordb.com/index.php?id=1100000000900149167]
|-
|18||548||H&lt;sub&gt;766&lt;/sub&gt;FH||18&lt;sup&gt;768&lt;/sup&gt;−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490]
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||12×18&lt;sup&gt;6270&lt;/sup&gt;+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437]
|-
|20||3301||H&lt;sub&gt;247&lt;/sub&gt;A0H||(17×20&lt;sup&gt;250&lt;/sup&gt;−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619]
|-
|20||3302||7&lt;sub&gt;249&lt;/sub&gt;A7||(7×20&lt;sup&gt;251&lt;/sup&gt;+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602]
|-
|20||3303||J7&lt;sub&gt;270&lt;/sub&gt;||(368×20&lt;sup&gt;270&lt;/sup&gt;−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462]
|-
|20||3304||J&lt;sub&gt;330&lt;/sub&gt;CCC7||20&lt;sup&gt;334&lt;/sup&gt;−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572]
|-
|20||3305||40&lt;sub&gt;387&lt;/sub&gt;404B||4×20&lt;sup&gt;391&lt;/sup&gt;+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563]
|-
|20||3306||EC0&lt;sub&gt;429&lt;/sub&gt;7||292×20&lt;sup&gt;430&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702]
|-
|20||3307||G&lt;sub&gt;447&lt;/sub&gt;99||(16×20&lt;sup&gt;449&lt;/sup&gt;−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753]
|-
|20||3308||3A&lt;sub&gt;527&lt;/sub&gt;3||(67×20&lt;sup&gt;528&lt;/sup&gt;−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531]
|-
|20||3309||E&lt;sub&gt;566&lt;/sub&gt;C7||(14×20&lt;sup&gt;568&lt;/sup&gt;−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516]
|-
|20||3310||JCJ&lt;sub&gt;629&lt;/sub&gt;||393×20&lt;sup&gt;629&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|20||3311||J&lt;sub&gt;655&lt;/sub&gt;05J||20&lt;sup&gt;658&lt;/sup&gt;−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052]
|-
|20||3312||50&lt;sub&gt;1163&lt;/sub&gt;AJ||5×20&lt;sup&gt;1165&lt;/sup&gt;+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412]
|-
|20||3313||CD&lt;sub&gt;2449&lt;/sub&gt;||(241×20&lt;sup&gt;2449&lt;/sup&gt;−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915]
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||16×20&lt;sup&gt;6270&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457]
|-
|22||7984||I7G0&lt;sub&gt;254&lt;/sub&gt;H||8882×22&lt;sup&gt;255&lt;/sup&gt;+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788]
|-
|22||7985||D0&lt;sub&gt;255&lt;/sub&gt;5EEF||13×22&lt;sup&gt;259&lt;/sup&gt;+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932]
|-
|22||7986||IK&lt;sub&gt;322&lt;/sub&gt;F||(398×22&lt;sup&gt;323&lt;/sup&gt;−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145]
|-
|22||7987||C0&lt;sub&gt;340&lt;/sub&gt;G9||12×22&lt;sup&gt;342&lt;/sup&gt;+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159]
|-
|22||7988||77E&lt;sub&gt;348&lt;/sub&gt;K7||(485×22&lt;sup&gt;350&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779]
|-
|22||7989||J&lt;sub&gt;379&lt;/sub&gt;KJ||(19×22&lt;sup&gt;381&lt;/sup&gt;+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027]
|-
|22||7990||J&lt;sub&gt;388&lt;/sub&gt;EJ||(19×22&lt;sup&gt;390&lt;/sup&gt;−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729]
|-
|22||7991||DJ&lt;sub&gt;400&lt;/sub&gt;||(292×22&lt;sup&gt;400&lt;/sup&gt;−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110]
|-
|22||7992||E&lt;sub&gt;404&lt;/sub&gt;K7||(2×22&lt;sup&gt;406&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298]
|-
|22||7993||66F&lt;sub&gt;453&lt;/sub&gt;B3||(971×22&lt;sup&gt;455&lt;/sup&gt;−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809]
|-
|22||7994||L0&lt;sub&gt;454&lt;/sub&gt;B63||21×22&lt;sup&gt;457&lt;/sup&gt;+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331]
|-
|22||7995||L&lt;sub&gt;483&lt;/sub&gt;G3||22&lt;sup&gt;485&lt;/sup&gt;−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730]
|-
|22||7996||E60&lt;sub&gt;496&lt;/sub&gt;L||314×22&lt;sup&gt;497&lt;/sup&gt;+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239]
|-
|22||7997||I&lt;sub&gt;626&lt;/sub&gt;AF||(6×22&lt;sup&gt;628&lt;/sup&gt;−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334]
|-
|22||7998||K0&lt;sub&gt;760&lt;/sub&gt;EC1||20×22&lt;sup&gt;763&lt;/sup&gt;+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415]
|-
|22||7999||J0&lt;sub&gt;767&lt;/sub&gt;IGGJ||19×22&lt;sup&gt;771&lt;/sup&gt;+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567]
|-
|22||8000||7&lt;sub&gt;959&lt;/sub&gt;K7||(22&lt;sup&gt;961&lt;/sup&gt;+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817]
|-
|22||8001||L&lt;sub&gt;2385&lt;/sub&gt;KE7||22&lt;sup&gt;2388&lt;/sup&gt;−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774]
|-
|22||8002||7&lt;sub&gt;3815&lt;/sub&gt;2L||(22&lt;sup&gt;3817&lt;/sup&gt;−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839]
|-
|24||3400||I0&lt;sub&gt;241&lt;/sub&gt;I5||18×24&lt;sup&gt;243&lt;/sup&gt;+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037]
|-
|24||3401||D0&lt;sub&gt;259&lt;/sub&gt;KKD||13×24&lt;sup&gt;262&lt;/sup&gt;+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725]
|-
|24||3402||C7&lt;sub&gt;298&lt;/sub&gt;||(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235]
|-
|24||3403||20&lt;sub&gt;313&lt;/sub&gt;7||2×24&lt;sup&gt;314&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241]
|-
|24||3404||BC0&lt;sub&gt;331&lt;/sub&gt;B||276×24&lt;sup&gt;332&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842]
|-
|24||3405||N&lt;sub&gt;2644&lt;/sub&gt;LLN||24&lt;sup&gt;2647&lt;/sup&gt;−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089]
|-
|24||3406||D&lt;sub&gt;2698&lt;/sub&gt;LD||(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876]
|-
|24||3407||A0&lt;sub&gt;2951&lt;/sub&gt;8ID||10×24&lt;sup&gt;2954&lt;/sup&gt;+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654]
|-
|24||3408||88N&lt;sub&gt;5951&lt;/sub&gt;||201×24&lt;sup&gt;5951&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606]
|-
|30||2613||AN&lt;sub&gt;206&lt;/sub&gt;||(313×30&lt;sup&gt;206&lt;/sup&gt;−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073]
|-
|30||2614||M&lt;sub&gt;241&lt;/sub&gt;QB||(22×30&lt;sup&gt;243&lt;/sup&gt;+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295]
|-
|30||2615||M0&lt;sub&gt;547&lt;/sub&gt;SS7||22×30&lt;sup&gt;550&lt;/sup&gt;+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988]
|-
|30||2616||C0&lt;sub&gt;1022&lt;/sub&gt;1||12×30&lt;sup&gt;1023&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|30||2617||5&lt;sub&gt;4882&lt;/sub&gt;J||(5×30&lt;sup&gt;4883&lt;/sup&gt;+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423]
|-
|30||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||25×30&lt;sup&gt;34205&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|}

==Unproven PRPs==

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this PRP
|-
|11||1068||57&lt;sub&gt;62668&lt;/sub&gt;||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000003573679860]
|-
|13||3194||C5&lt;sub&gt;23755&lt;/sub&gt;C||(149×13&lt;sup&gt;23756&lt;/sup&gt;+79)/12||[http://factordb.com/index.php?id=1100000003590647776]
|-
|13||3195||80&lt;sub&gt;32017&lt;/sub&gt;111||8×13&lt;sup&gt;32020&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000000490878060]
|-
|16||2345||DB&lt;sub&gt;32234&lt;/sub&gt;||(206×16&lt;sup&gt;32234&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000002383583629]
|-
|16||2346||4&lt;sub&gt;72785&lt;/sub&gt;DD||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15||[http://factordb.com/index.php?id=1100000003615909841]
|-
|22||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21||[http://factordb.com/index.php?id=1100000003594696838]
|-
|30||2618||I0&lt;sub&gt;24608&lt;/sub&gt;D||18×30&lt;sup&gt;24609&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003593967511]
|}

All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10&lt;sup&gt;16&lt;/sup&gt;. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]])

==Proof==

===Base 2===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.

===Base 3===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (2,1), (2,2)

* Case (1,1):
** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes)
*** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime.

===Base 4===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (2,1), (2,3), (3,1), (3,3)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.

===Base 5===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4)

* Case (1,1):
** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes)
*** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes)
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes)
***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb])
***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes)
**** The smallest prime of the form 1{4}4 is 14444.
**** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes)
*** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes)
**** For the 3{0,3}1 family, we can separate this family to four families:
***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01.
****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime.
***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031.
***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001.
***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31.
****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime.
**** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.
* Case (3,4):
** 34 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes)
**** The smallest prime of the form 4{4}1 is 44441.
**** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime.
* Case (4,2):
** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime.

===Base 6===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes)
*** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes)
**** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 4{0}41 is 40041
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes)
*** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.

===Base 7===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

* Case (1,1):
** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes)
*** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {}
**** If there are &gt;=3 number of 1 and 5 in the {}:
***** If there is 111 in the {}, then we have the prime 11111
***** If there is 115 in the {}, then the prime 115 is a subsequence
***** If there is 151 in the {}, then the prime 115 is a subsequence
***** If there is 155 in the {}, then the prime 155 is a subsequence
***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime
***** If there is 515 in the {}, then the prime 515 is a subsequence
***** If there is 551 in the {}, then the prime 551 is a subsequence
***** If there is 555 in the {}, then the prime 551 is a subsequence
**** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1
***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1
****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1
******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21
******** The smallest prime of the form 11{0}21 is 1100021
******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime
******** The smallest prime of the form 1001{0}21 is 100121
********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families
***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1
****** Since 1051 is prime, we only need to consider the family 15{0}1
******* The smallest prime of the form 15{0}1 is 150001
* Case (1,2):
** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes)
*** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {}
**** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime.
**** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime.
**** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2
***** Since 1022 and 1202 are primes, we only need to consider the number 122
****** 122 is not prime.
**** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime.
* Case (1,3):
** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** 14 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime.
* Case (1,6):
** 16 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime.
* Case (2,2):
** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,6):
** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes)
*** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes)
**** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's)
***** For the 3{0,1}1 family:
****** If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
****** If there are no 1's between (3,1), then the form will be 3{0}1
******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime.
****** If there are no 0's between (3,1), then the form will be 3{1}1
******* The smallest prime of the form 3{1}1 is 31111
****** If there are exactly 1 0's between (3,1), then there must be &lt;3 1's between (3,1), or 31111 will be a subsequence.
******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime.
******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011
******** Neither 3101 nor 3011 is prime.
******* If there are no 1's between (3,1), then the number must be 301
******** 301 is not prime.
***** For the 3{3}3{0,1}1 family:
****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence.
****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1
******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence.
******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime.
******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime.
******* If there are no 1's between (33,1), then the form will be 33{0}1
******** The smallest prime of the form 33{0}1 is 33001
****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1
******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence.
******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301
******** The smallest prime of the form 3{3}301 is 33333301
******* If there are no 0's between (3{3}3,1), then the form is 3{3}31
******** The smallest prime of the form 3{3}31 is 33333333333333331
**** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1
***** Numbers containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1,3}3{0,1,3}1
******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes)
******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes)
********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes)
********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence.
********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime.
********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1
*********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime.
********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime.
***** Numbers not containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1}1
******* If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1
******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime.
******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1
******** If there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111
********* None of 31151, 31511, 35111 are primes.
******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 351
********* 351 is not prime.
******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111
********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes.
********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime.
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1
********* Neither 3051 nor 3501 is prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** If there are &gt;=2 5's in {}, then 553 will be a subsequence.
*** If there are no 5's in {}, then the family will be 3{0,3,6}3
**** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
*** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3
**** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes)
***** The smallest prime of the form 3{0}53 is 300053
* Case (3,4):
** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes)
*** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes)
**** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes)
**** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes)
* Case (3,5):
** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes)
*** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence.
*** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence.
*** If there are at least two 1's in {}, then 115 will be a subsequence.
*** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5
**** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime.
*** If there is no 1's in {}, then the family will be 3{0,4,5}5
**** If there are at least to 4's in {}, then 344 and 445 will be subsequences.
**** If there is no 4's in {}, then the family will be 3{0,5}5
***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5
****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime.
****** The smallest prime of the form 3{5}5 is 35555
**** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5
***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes)
****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime.
* Case (3,6):
** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,2):
** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes)
*** If there is no 5's in {}, then the family will be 4{0,2,4,6}4
**** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4
**** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes)
***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence.
***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4
****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes)
***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4
****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4
******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes)
* Case (4,5):
** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** If there are at least two 5's in {}, then 4555 will be a subsequence.
*** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there is no 5's in {}, then the family will be 4{0}5
**** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime.
* Case (4,6):
** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (5,1):
** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 3's in {}, then 533 will be a subsequence.
*** If there is no 3's in {}, then the family will be 5{0,1}1
**** Since 5011 is prime, we only need to consider the family 5{1}{0}1
***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes)
****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime.
****** The smallest prime of the form 51{0}1 is 5100000001
****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime.
****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime.
*** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1
**** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence.
***** Thus we only need to consider the family 5{0}3{0,1}1
****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime.
****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime.
****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence.
****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1
******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0)
******** The smallest prime of the form 5311{0}1 is 531101
* Case (5,2):
** 52 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 1's in {}, then 113 will be a subsequence.
*** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
*** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime.
* Case (5,4):
** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes)
*** If there are no 5's in {}, then the family will be 5{0,3}4
**** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime.
*** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence.
*** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there are at least two 5's in {}, then 5554 will be a subsequence.
* Case (5,5):
** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes)
*** If there are no 4's in {}, then the family will be 5{0,5}5
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
*** If there are no 5's in {}, then the family will be 5{0,4}5
**** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime.
*** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence.
* Case (5,6):
** 56 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,2):
** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (6,3):
** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,4):
** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are at least three 3's in {}, then 3334 will be a subsequence.
*** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4
**** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence.
**** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence.
**** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence.
**** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence.
**** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence.
**** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence.
**** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence.
**** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence.
**** Thus the number can only be 634
***** 634 is not prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,6):
** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}6
**** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime.
*** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6
**** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime.
*** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence.
*** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence.

===Base 8===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7)

* Case (1,1):
** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes)
**** 171 is not prime.
**** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 &gt; 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 &gt; 1, this factorization is nontrivial), thus cannot be prime.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes)
*** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime)
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes)
*** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes)
*** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes)
**** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime.
**** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes)
***** The smallest prime of the form 33{4}1 is 3344441
***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime.
**** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes)
***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime.
***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes)
****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes.
* Case (3,3):
** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes)
*** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes)
**** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime)
**** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461
***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes)
****** The smallest prime of the form 4{4}41 is 444444441
****** The smallest prime of the form 4{4}641 is 444641
***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes)
****** The smallest prime of the form 4{4}411 is 444444441
****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes)
***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461
****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime)
**** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes)
***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime)
***** The smallest prime of the form 4{4}641 is 444641
* Case (4,3):
** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes)
*** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes)
**** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime.
**** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime.
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes)
*** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes)
**** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb])
**** The smallest prime of the form 4{4}77 is 4444477
**** The smallest prime of the form 4{7}7 is 47777
**** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime)
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes)
*** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes)
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
**** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes)
***** 500525 is not prime.
***** The smallest prime of the form 5{5}25 is 555555555555525
***** The smallest prime of the form 5{5}025 is 55555025
***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes)
***** The smallest prime of the form 5{5}0525 is 5550525
***** The smallest prime of the form 5{5}00525 is 5500525
***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes)
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence)
**** The form is 6{0,1,7}4{0,1,7}1
***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1
****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411
******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41
******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41
********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.
********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime.
********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime.
******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411
******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime)
*** Numbers not containing 4:
**** The form is 6{0,1,7}1
***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1
****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime.
****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1
******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1
******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.
******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime)
* Case (6,3):
** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,7):
** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them
**** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17
***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime.
**** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77
***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime.
**** If there is neither 1 nor 7 between them, then the form is 6{0,4}7
***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes)
****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime.
****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime.
* Case (7,1):
** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes)
*** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes)
**** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes)
***** The smallest prime of the form 7{7}1 is 7777777777771
***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime)
***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime.
***** The smallest prime of the form 7{7}4444441 is 77774444441
****** Since this prime has just 4 7's, we only need to consider the families with &lt;=3 7's
******* The smallest prime of the form 7{4}1 is 744444441
******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime.
******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes)
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.

===Base 10===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,9):
** 19 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 2{0}21 is 20021
**** The smallest prime of the form 22{0}1 is 22000001
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes)
*** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes)
**** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime.
**** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime.
* Case (2,9):
** 29 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,9):
** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** 47 is prime, and thus the only minimal prime in this family.
* Case (4,9):
** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime.
* Case (5,1):
** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes)
**** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes)
***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 5{5}1 is 555555555551
**** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581
***** 581 is not prime.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,7):
** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027
**** The smallest prime of the form 52{0}7 is 5200007
* Case (5,9):
** 59 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,3):
** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,9):
** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49
***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49
****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes)
******* The smallest prime of the form 6{6}49 is 666649
******** Since this prime has just 4 6's, we only need to consider the families with &lt;=3 6's
********* The smallest prime of the form 6{0}49 is 60000049
********* The smallest prime of the form 66{0}49 is 66000049
********* The smallest prime of the form 666{0}49 is 66600049
* Case (7,1):
** 71 is prime, and thus the only minimal prime in this family.
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,9):
** 79 is prime, and thus the only minimal prime in this family.
* Case (8,1):
** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes)
**** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes)
***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime)
***** The smallest prime of the form 80{5}1 is 80555551
* Case (8,3):
** 83 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime.
* Case (8,9):
** 89 is prime, and thus the only minimal prime in this family.
* Case (9,1):
** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes)
**** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021
***** None of 921, 9201, 9021 are primes.
**** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801
***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
* Case (9,3):
** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (9,7):
** 97 is prime, and thus the only minimal prime in this family.
* Case (9,9):
** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9
***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes)
****** The smallest prime of the form 94{6}9 is 946669

===Base 12===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,B):
** 1B is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes)
*** The smallest prime of the form 2{0}1 is 2001
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (2,B):
** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes)
**** None of 20B, 22B, 29B, 209B, 220B are primes.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,B):
** 3B is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes)
**** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 44{A}1 is 44AAA1
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes)
**** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077
* Case (4,B):
** 4B is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes)
*** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime.
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (5,B):
** 5B is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,5):
** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes)
**** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 6{0}A5 is 600A5
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,B):
** 6B is prime, and thus the only minimal prime in this family.
* Case (7,1):
** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes)
*** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1
**** None of 741, 7B1, 74B1 are primes.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,B):
** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes)
**** The smallest prime of the form 7{9}B is 7999B
**** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes)
**** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime)
* Case (8,1):
** 81 is prime, and thus the only minimal prime in this family.
* Case (8,5):
** 85 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** 87 is prime, and thus the only minimal prime in this family.
* Case (8,B):
** 8B is prime, and thus the only minimal prime in this family.
* Case (9,1):
** 91 is prime, and thus the only minimal prime in this family.
* Case (9,5):
** 95 is prime, and thus the only minimal prime in this family.
* Case (9,7):
** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes)
**** None of 947, 977, 997, 9477, 9977 are primes.
* Case (9,B):
** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes)
**** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes)
***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime)
***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime)
**** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes)
***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes)
****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime.
****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime)
****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime.
**** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B
***** None of 99B, 999B, 979B, 9799B, 9979B are primes.
* Case (A,1):
** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes)
**** For the A{A}{0}1 family:
***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form AA{0}1 is AA000001
***** The smallest prime of the form AAA{0}1 is AAA0001
***** The smallest prime of the form AAAA{0}1 is AAAA1
****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc.
**** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime.
* Case (A,5):
** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes)
*** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes)
**** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form A{0}65 is A00065
* Case (A,7):
** A7 is prime, and thus the only minimal prime in this family.
* Case (A,B):
** AB is prime, and thus the only minimal prime in this family.
* Case (B,1):
** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes)
**** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1
***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1
****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime)
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1
****** The smallest prime of the form B{A}01 is BAA01
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
**** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1
***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1
****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1
******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1
******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes.
****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1
******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime)
***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes)
****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B04{A}1 is B04A1
****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime)
**** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1
***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1
****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1
******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime.
******* The smallest prime of the form B{B}AA1 is BBBAA1
****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1
******* The smallest prime of the form B{B}AA1 is BBBAA1
**** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes)
***** The smallest prime of the form B{B}B1 is BBBB1
***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form B{B}BAA1 is BBBAA1
* Case (B,5):
** B5 is prime, and thus the only minimal prime in this family.
* Case (B,7):
** B7 is prime, and thus the only minimal prime in this family.
* Case (B,B):
** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes)
*** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B
**** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B
***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime.
***** For the B{0,B}9B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes)
******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B
******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 &gt; 1, 10^n+5 ≥ 10^1+5 = 15 &gt; 1, this factorization is nontrivial), thus cannot be prime.
***** For the B{0,B}99B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes)
******* The smallest prime of the form B{0}99B is B00099B
******* The smallest prime of the form B{B}99B is BBBBBB99B
***** For the B{0,B}999B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes)
******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes)
******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime)

== Examples of families which can be ruled out as contain no primes &gt; ''b'' ==

It is not known if this problem is solvable:

Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''?

It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime &gt; ''b'' or not. We use two different heuristic strategies to show that such families contain no primes &gt; ''b''.

In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;.

Examples of first strategy: (we can show that the corresponding numbers are &gt; all elements in ''S'', if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial)

* In base 10, all numbers of the form 4{6}9 are divisible by 7
* In base 6, all numbers of the form 4{0}1 are divisible by 5
* In base 15, all numbers of the form 9{6}8 are divisible by 11
* In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5}
* In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3}
* In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5}
* In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13}
* In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17}
* In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13}
* In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13}

Examples of second strategy: (we can show that both factors are &gt; 1, if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial)

* In base 9, all numbers of the form {1} factored as difference of squares
* In base 8, all numbers of the form 1{0}1 factored as sum of cubes
* In base 9, all numbers of the form 3{8} factored as difference of squares
* In base 16, all numbers of the form 8{F} factored as difference of squares
* In base 16, all numbers of the form {F}7 factored as difference of squares
* In base 9, all numbers of the form 3{1} factored as difference of squares
* In base 16, all numbers of the form {4}1 factored as difference of squares
* In base 16, all numbers of the form 1{5} factored as difference of squares
* In base 16, all numbers of the from {C}D factored as ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;
* In base 16, all numbers of the form B{4}1 factored as difference of squares

Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are &gt; all elements in S, and for the part of the second strategy, both factors are &gt; 1, if n makes corresponding numbers &gt; b, thus these factorizations are nontrivial)

* In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd
* In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even

== Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes &gt; ''b'' ==

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-powers factorization

=== 1{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 1{0}3 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== 1{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 1{0}5 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 5: Finite covering set {5}

=== 1{0}6 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== 1{0}7 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 7: Finite covering set {7}

=== 1{0}z ===

(none)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

(none)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {1} ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with ''r''&gt;1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2))

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' == 0 mod 2: Finite covering set {2}

=== 1{2} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 1{3} ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361)
* ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865)

=== 1{4} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization

=== 1{z} ===

(none)

=== 2{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 2{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {2}1 ===

* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 2{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 3{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {3}1 ===

* ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121)

=== 3{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 4{0}1 ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 4{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {4}1 ===

* ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225)

=== 4{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 5{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 5{z} ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}
* ''b'' = 6''m''&lt;sup&gt;2&lt;/sup&gt; with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014)

=== 6{0}1 ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== 6{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 7{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 7{z} ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993)
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Difference-of-cubes factorization

=== 8{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022)
* ''b'' = 467: Finite covering set {3, 5, 7, 19, 37}
* ''b'' = 722: Finite covering set {3, 5, 13, 73, 109}
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Sum-of-cubes factorization
* ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2

=== 8{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 9{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 9{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{0}1 ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{z} ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 307: Finite covering set {5, 11, 29}
* ''b'' = 901: Finite covering set {7, 11, 13, 19}

=== C{0}1 ===

* ''b'' == 1 mod 13: Finite covering set {13}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 296, 901: Finite covering set {7, 11, 13, 19}
* ''b'' = 562, 828, 900: Finite covering set {7, 13, 19}
* ''b'' = 563: Finite covering set {5, 7, 13, 19, 29}
* ''b'' = 597: Finite covering set {5, 13, 29}

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

(none)

=== {y}z ===

(none)

=== z{0}1 ===

(none)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3))
* ''b'' = 4''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt; (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}t ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 7: Finite covering set {7}

=== {z}u ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== {z}v ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 5: Finite covering set {5}

=== {z}w ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== {z}x ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== {z}y ===

* ''b'' == 0 mod 2: Finite covering set {2}

== Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) ==

Format: base (length)

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

(none)

=== 1{0}2 ===

(none)

=== 1{0}3 ===

(none)

=== 1{0}4 ===

53 (13403)

113 (10647)

=== 1{0}z ===

113 (20089)

123 (64371)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

(none)

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

208 (26682)

607 (11032)

828 (19659)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

201 (31276)

222 (52727)

227 (36323)

327 (135983)

425 (11231)

710 (24112)

717 (37508)

719 (13420)

=== {1} ===

152 (270217)

184 (16703)

200 (17807)

311 (36497)

326 (26713)

331 (25033)

371 (15527)

485 (99523)

629 (32233)

649 (43987)

670 (18617)

684 (22573)

691 (62903)

693 (41189)

731 (15427)

752 (32833)

872 (10093)

932 (20431)

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

(none)

=== 1{z} ===

107 (21911)

170 (166429)

278 (43909)

303 (40175)

383 (20957)

515 (58467)

522 (62289)

578 (129469)

590 (15527)

647 (21577)

662 (16591)

698 (127559)

704 (62035)

845 (39407)

938 (40423)

969 (24097)

989 (26869)

=== 2{0}1 ===

101 (192276)

206 (46206)

218 (333926)

236 (161230)

257 (12184)

305 (16808)

467 (126776)

578 (44166)

626 (174204)

695 (94626)

752 (26164)

788 (72918)

869 (49150)

887 (27772)

899 (15732)

932 (13644)

=== 2{z} ===

432 (16003)

=== 3{0}1 ===

912 (132174)

=== 3{z} ===

72 (1119850)

212 (34414)

218 (23050)

270 (89662)

303 (198358)

312 (51566)

422 (21738)

480 (93610)

513 (38032)

527 (46074)

566 (23874)

650 (498102)

686 (16584)

758 (15574)

783 (12508)

800 (33838)

921 (98668)

947 (10056)

=== 4{0}1 ===

107 (32587)

227 (13347)

257 (160423)

355 (10990)

410 (144079)

440 (56087)

452 (14155)

482 (30691)

542 (15983)

579 (67776)

608 (20707)

635 (11723)

650 (96223)

679 (69450)

737 (269303)

740 (58043)

789 (149140)

797 (468703)

920 (103687)

934 (101404)

962 (84235)

=== 4{z} ===

14 (19699)

68 (13575)

254 (15451)

800 (20509)

=== 5{0}1 ===

326 (400786)

350 (20392)

554 (10630)

662 (13390)

926 (40036)

=== 5{z} ===

258 (212135)

272 (148427)

299 (64898)

307 (26263)

354 (25566)

433 (283919)

635 (36163)

678 (40859)

692 (45447)

719 (20552)

768 (70214)

857 (23083)

867 (61411)

972 (36703)

=== 6{0}1 ===

108 (16318)

129 (16797)

409 (369833)

522 (52604)

587 (24120)

643 (164916)

762 (11152)

789 (27297)

986 (21634)

=== 6{z} ===

68 (25396)

332 (15222)

338 (42868)

362 (146342)

488 (33164)

566 (164828)

980 (50878)

986 (12506)

1016 (23336)

=== 7{0}1 ===

398 (17473)

1004 (54849)

=== 7{z} ===

97 (192336)

170 (15423)

194 (38361)

202 (155772)

282 (21413)

283 (164769)

332 (13205)

412 (29792)

560 (19905)

639 (10668)

655 (53009)

811 (31784)

814 (17366)

866 (108591)

908 (61797)

962 (31841)

992 (10605)

997 (15815)

=== 8{0}1 ===

23 (119216)

53 (227184)

158 (123476)

254 (67716)

320 (52004)

410 (279992)

425 (94662)

513 (19076)

518 (11768)

596 (148446)

641 (87702)

684 (23387)

695 (39626)

785 (900326)

788 (11408)

893 (86772)

908 (243440)

920 (107822)

962 (47222)

998 (81240)

1013 (43872)

=== 8{z} ===

138 (35686)

412 (12154)

788 (11326)

990 (23032)

=== 9{0}1 ===

248 (39511)

592 (96870)

=== 9{z} ===

431 (43574)

446 (152028)

458 (126262)

599 (11776)

846 (12781)

=== A{0}1 ===

173 (264235)

198 (47665)

311 (314807)

341 (106009)

449 (18507)

492 (42843)

605 (12395)

708 (17563)

710 (31039)

743 (285479)

744 (137056)

786 (68169)

800 (15105)

802 (149320)

879 (25004)

929 (13065)

977 (125873)

986 (48279)

1004 (10645)

=== A{z} ===

368 (10867)

488 (10231)

534 (80328)

662 (13307)

978 (14066)

=== B{0}1 ===

710 (15272)

740 (33520)

878 (227482)

=== B{z} ===

153 (21660)

186 (112718)

439 (18752)

593 (16064)

602 (36518)

707 (10573)

717 (67707)

=== C{0}1 ===

68 (656922)

219 (29231)

230 (94751)

312 (21163)

334 (83334)

353 (20262)

359 (61295)

457 (10024)

481 (45941)

501 (20140)

593 (42779)

600 (11242)

604 (17371)

641 (26422)

700 (91953)

887 (13961)

919 (45359)

923 (64365)

992 (10300)

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

(none)

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

38 (136212)

83 (21496)

113 (286644)

188 (13508)

401 (103670)

417 (21003)

458 (46900)

494 (21580)

518 (129372)

527 (65822)

602 (17644)

608 (36228)

638 (74528)

663 (47557)

723 (24536)

758 (50564)

833 (12220)

904 (13430)

938 (50008)

950 (16248)

=== z{0}1 ===

202 (46774)

251 (102979)

272 (16681)

297 (14314)

298 (60671)

326 (64757)

347 (69661)

363 (142877)

452 (71941)

543 (10042)

564 (38065)

634 (84823)

788 (13541)

869 (12289)

890 (37377)

953 (60995)

1004 (29685)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

53 (21942)

124 (16426)

175 (31626)

188 (22036)

316 (48538)

365 (25578)

373 (24006)

434 (10090)

530 (11086)

545 (12346)

560 (15072)

596 (12762)

701 (12576)

706 (10656)

821 (13536)

833 (17116)

966 (14820)

983 (11272)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}y ===

317 (13896)

== Bases 2≤''b''≤1024 which have these families as unsolved families ==

Unsolved families are families which are neither primes (&gt;''b'') found nor can be ruled out as contain no primes &gt; ''b''

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608)

1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000)

1{0}3: 646, 718, 998 (length limit: 2000)

1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000)

1{0}5

1{0}6

1{0}7

1{0}8

1{0}9

1{0}A

1{0}B

1{0}C

1{0}D

1{0}E

1{0}F

1{0}G

1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000)

1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000)

10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000)

11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000)

{1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000)

{1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000)

11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z})

{1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000)

{1}z

1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000)

1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995

1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000)

1{z}: 581, 992, 1019 (length limit: ≥100000)

2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000)

2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000)

2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000)

{2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000)

2{z}: 588, 972 (length limit: ≥100000)

3{0}1: 718 (length limit: ≥100000)

3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000)

3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000)

{3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000)

3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000)

4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000)

4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000)

{4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000)

4{z}: 338, 998 (length limit: ≥100000)

5{0}1: 308, 512, 824 (length limit: ≥100000)

5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000)

6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000)

6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000)

7{0}1: (none)

7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000)

8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000)

8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000)

9{0}1: 724, 884 (length limit: ≥100000)

9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000)

A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000)

A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000)

B{0}1: 560, 770, 968 (length limit: ≥100000)

B{z}: 263, 615, 912, 978 (length limit: ≥100000)

C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000)

C{z}

D{0}1

D{z}

E{0}1

E{z}

F{0}1

F{z}

G{0}1

{&amp;#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000)

{&amp;#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143)

&amp;#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000)

${&amp;#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2)

x{z}

y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000)

z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000)

{y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000)

{z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326)

zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z})

{z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000)

{z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1)

{z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000)

{z}k

{z}l

{z}m

{z}n

{z}o

{z}p

{z}q

{z}r

{z}s

{z}t

{z}u

{z}v

{z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000)

{z}x: (none)

{z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000)

== List of lengths for quasi-minimal primes in some simple families ==

[https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024]

NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases &lt;=7, family {z}x in bases &lt;=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime &gt; base) in this base)

RC: this family can be proven to only contain composite numbers (only count numbers &gt; base)

unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers &gt; base)

Background color: red for title (bases or families), green for length &gt; 10000, orange for 2500 &lt; length ≤ 10000, white for length ≤ 2500, cyan for &quot;RC&quot;, pink for &quot;NB&quot;, yellow for &quot;unknown&quot;.

Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families.

== References ==

* [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem]

* [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are &gt; ''b'' for bases ''b''≤16]

* [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary]

* [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia]

* [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime&gt;base is not required) in The Prime Puzzles &amp; Problems Connection]

* [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles &amp; Problems Connection]

* [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16]

* [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime&gt;base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29]

* [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime&gt;base is not required, search limits of lengths: 10000 for all b]

* [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime&gt;base is not required]

* [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime&gt;base is not required]

* [http://www.prothsearch.com/sierp.html the Sierpinski problem]

* [http://www.prothsearch.com/rieselprob.html the Riesel problem]

* [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem]

* [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030]

* [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100]

* [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem]

* [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024]

* [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project]

* [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project]

* [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count]

* [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty]

* [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers]

* [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares]

* [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software]

* [https://sourceforge.net/projects/openpfgw/ pfgw software]

* [http://jpenne.free.fr/index2.html llr software]

* [http://www.ellipsa.eu/public/primo/primo.html PRIMO software]

* [https://primes.utm.edu/prove/index.html website for primality proving]

* [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!]

* [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime&gt;base is not required]

* [https://primes.utm.edu/primes/lists/all.txt top proven primes]

* [http://www.primenumbers.net/prptop/prptop.php top PRPs]

* [http://factordb.com online factor database, including many primes which are minimal primes in a small base]</text>
      <sha1>1wviwikslymvr0nx8zq88dqe9ynzo0v</sha1>
    </revision>
    <revision>
      <id>2412023</id>
      <parentid>2412022</parentid>
      <timestamp>2022-08-05T03:37:01Z</timestamp>
      <contributor>
        <ip>2402:7500:901:2B90:BD69:8142:3D96:F08F</ip>
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      <comment>/* Unsolved families */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1960139" xml:space="preserve">A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime &gt; ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime &gt; 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 &gt; 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 &gt; 10 by changing the order.

(using A−Z to represent digit values 10 to 35)

For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')).

I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E&lt;sub&gt;800873&lt;/sub&gt;, and base 36 has quasi-minimal prime P&lt;sub&gt;81993&lt;/sub&gt;SZ).

Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to:

* Prove that all elements in ''S'', when read as base ''b'' representation, are primes &gt; ''b''.

* Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are &gt; ''b'', are composite.

* Prove that all primes &gt; ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are &gt; ''b'', are composite).

e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to:

* Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes &gt; 10.

* Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are &gt; 10 are composite.

* Prove that all primes &gt; 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers &gt; 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite).

==Condensed table==

{|class=wikitable
|''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form of largest known quasi-minimal prime base ''b''
|-
|2||1||11||2||3
|-
|3||3||111||3||13
|-
|4||5||221||3||41
|-
|5||22||10&lt;sub&gt;93&lt;/sub&gt;13||96||5&lt;sup&gt;95&lt;/sup&gt;+8
|-
|6||11||40041||5||5209
|-
|7||71||3&lt;sub&gt;16&lt;/sub&gt;1||17||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2
|-
|8||75||4&lt;sub&gt;220&lt;/sub&gt;7||221||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||1161||3×9&lt;sup&gt;1160&lt;/sup&gt;+10
|-
|10||77||50&lt;sub&gt;28&lt;/sub&gt;27||31||5×10&lt;sup&gt;30&lt;/sup&gt;+27
|-
|11&lt;sup&gt;*&lt;/sup&gt;||1068||57&lt;sub&gt;62668&lt;/sub&gt;||62669||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10
|-
|12||106||40&lt;sub&gt;39&lt;/sub&gt;77||42||4×12&lt;sup&gt;41&lt;/sup&gt;+91
|-
|13&lt;sup&gt;*&lt;/sup&gt;||3195~3197||80&lt;sub&gt;32017&lt;/sub&gt;111||32021||8×13&lt;sup&gt;32020&lt;/sup&gt;+183
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||19699||5×14&lt;sup&gt;19698&lt;/sup&gt;−1
|-
|15||1284||7&lt;sub&gt;155&lt;/sub&gt;97||157||(15&lt;sup&gt;157&lt;/sup&gt;+59)/2
|-
|16&lt;sup&gt;*&lt;/sup&gt;||2346~2347||4&lt;sub&gt;72785&lt;/sub&gt;DD||72787||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15
|-
|17&lt;sup&gt;*&lt;/sup&gt;||10408~10428||F70&lt;sub&gt;186767&lt;/sub&gt;1||186770||262×17&lt;sup&gt;186768&lt;/sup&gt;+1
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||6271||12×18&lt;sup&gt;6270&lt;/sup&gt;+221
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||6271||16×20&lt;sup&gt;6270&lt;/sup&gt;+13
|-
|21&lt;sup&gt;*&lt;/sup&gt;||13375~13396||CF&lt;sub&gt;479147&lt;/sub&gt;0K||479150||(51×21&lt;sup&gt;479149&lt;/sup&gt;−1243)/4
|-
|22&lt;sup&gt;*&lt;/sup&gt;||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||22003||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||8134||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49
|-
|30&lt;sup&gt;*&lt;/sup&gt;||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||34206||25×30&lt;sup&gt;34205&lt;/sup&gt;−1
|-
|36&lt;sup&gt;*&lt;/sup&gt;||35257~35263||P&lt;sub&gt;81993&lt;/sub&gt;SZ||81995||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7
|}

&lt;sup&gt;*&lt;/sup&gt; Data assumes the primality of the [[w:probable prime|probable prime]]s.

Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes &gt; 10&lt;sup&gt;25000&lt;/sup&gt;), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below.

Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists))

Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists))

==Data for quasi-minimal primes==

===Base 2===

11

===Base 3===

12, 21, 111

===Base 4===

11, 13, 23, 31, 221

===Base 5===

12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013

===Base 6===

11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041

===Base 7===

14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331

===Base 8===

13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447

===Base 9===

12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011

===Base 10===

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027

===Base 11===

12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, 2888882883, 2888888883, 3555555509, 3577777077, 3700000001, 4000000005, 40000005AA, 5377777707, 5555505553, 555555580A, 600000A999, 7000100047, 7000704777, 7007777107, 7057777777, 7070007774, 7077707774, 7077777074, 7100000704, 7470000041, 7701000004, 7707077774, 7770707774, 7777707074, 8888822883, 9555555503, 9900000979, 9999770007, A000144444, A900000066, A999999971, 10000000477, 33333333337, 44444444447, 44444444777, 55A55555552, 60000008883, 68888888306, 68888888883, 70000003999, 70000007717, 70004777777, 70477777777, 77007770004, 77700000477, 77707000704, 77707770074, 77707777774, 77777077774, 77777770004, 83000000006, 97000000999, A0000000001, A0014444444, A4777777771, 100000000057, 305007777777, 305777777777, 370000000007, 377777770007, 377777777107, 700000007474, 707077000074, 707077777774, 707777777717, 770000010004, 771007000007, 777070700004, 777700000704, A95555555555, A99999777777, 1000000003007, 40000000A0041, 58A5555555555, 7004400000004, 7700000000104, 7707000007047, 7707707000004, 7777007000004, 7777700000004, 7777770077704, 7777777710077, 9977777777717, A000000014444, A044444444441, A144444444411, 40000000000401, 45557777777777, 4555AAAAAAAAAA, 59077777777777, 70007777777771, 70077070000074, 70700000004777, 77000007700704, 77700000700047, 77777777770704, 88888888830006, 90900000000799, A0000044444441, 300000000005777, 302000000000002, 55555555A555552, 700000000000174, 770000000000474, 771700000000007, 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7707777777777777777704, 7900000000000000000005, 9777777777777777777707, A999999999999999999999, 10000000000000000000747, 970000000000000000000777, 999900000000000000000007, 3577777777777777777777777, 5555555555555555555555A52, 7000000000000000000777771, 7000000000000000077777771, AAAAAAAAAAAAAAAAAAA000058, 10000000000000000000000044, 77700000000000000000000008, 500777777777777777777777777, 777777777777777777777770774, 855555555555555555555055555, A44444444444444444444444441, 1500000000000000000000000007, 40000000000000000000000000041, 440000000000000000000000000001, 70000000000000000000000000007771, 999999999999999999999999999999991, 95555555555555555555555555555555553, 1900000000000000000000000000000000001, 7777777777777777777777777777777777474, 7777777777777777777777777777777777704, 10000000000000000000000000000000000000307, 50777777777777777777777777777777777777707, 475777777777777777777777777777777777777777, 555555555555555555555555555555555555555A2A, 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===Base 12===

11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077

===Base 13===

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===Base 14===

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===Base 15===

12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797

===Base 16===

11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD

===Base 17===

12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 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====Additional known quasi-minimal primes (not necessarily the next)====

570&lt;sub&gt;51310&lt;/sub&gt;1, 49&lt;sub&gt;111333&lt;/sub&gt;, 970&lt;sub&gt;166047&lt;/sub&gt;1, F70&lt;sub&gt;186767&lt;/sub&gt;1

===Base 18===

11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, 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===Base 20===

13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, GGJ0J, GGJJJ, GJ00J, GJGJJ, GJJJ7, H00EH, H024B, H04B1, H0E4B, H0F41, H0H11, H0HEH, H4E0B, H6I11, HAAA3, HAAG3, HAGA3, HB44B, HBBBB, HBHHH, HGGA3, HGGG3, HH001, HH061, HH3AH, HH6F1, HHG63, HHIA3, HI011, HIII3, I0029, I02I9, I085H, I0A41, I0F29, I2229, I2929, I2I29, I2I99, I2II9, I33A3, I3EHH, I422B, I424B, I4EEB, I4IEB, I80G1, I9IA9, IA099, IA309, IA841, IA939, IA9A9, IAF09, IAF99, IAI99, IAIA9, IB0A1, IB4IB, IB54B, IBB4B, IBBIB, IBE0H, IBH0H, IC929, ICC29, IF841, IFAG1, IGA81, IHB4B, II0A9, II42B, II44B, IIB5B, III4B, IIIA3, IIIIB, J00G1, J00JB, J0601, J06D1, J06I1, J0861, J08B1, J0B01, J0CGJ, J0E0B, J0GA1, J0GG1, J0I2B, J0J2B, J0JIB, J55CJ, J60A1, J60G1, J6D81, JA777, JA7A7, JAJG1, JAJJ1, JAJJ7, JB00B, JB08B, JC00J, JC0GJ, JEECJ, JGGGJ, JGGJJ, JJ0B1, JJ0JB, JJ55J, JJAG1, JJAJ1, JJC0J, JJE0B, JJEEB, JJGG1, JJGGJ, JJGJJ, JJJ01, JJJEB, JJJJB, 17555J, 175EEJ, 1E0007, 1JE55J, 1JJJ5J, 20005H, 2000HH, 222I29, 22E889, 22EE89, 24222B, 2999I9, 29III9, 2A000H, 2B0I0H, 2I2999, 2I9I29, 2III29, 333AF3, 336IA3, 36A3F3, 36I3A3, 388E8H, 3AF333, 3EHHHH, 3HEHHH, 3HHEHH, 40008B, 40054B, 40405B, 40448B, 415E5J, 44440B, 44452B, 44524B, 44E08B, 45444B, 4AAAE7, 4AEEE7, 4E4447, 4GEEE7, 4I544B, 4IEEEB, 500EFJ, 500FCJ, 5055FJ, 50AFFJ, 50EEFJ, 50EF5J, 50F5FJ, 50FFCJ, 52224B, 54EEEB, 5558AH, 555CAJ, 5585AH, 55F55J, 55FFFJ, 5888AH, 588A5H, 5E555J, 5F055J, 5F05CJ, 5F5F5J, 5FFFCJ, 5JEEEB, 60006D, 6000F1, 6000G3, 6001G1, 600A63, 600G1D, 600G6D, 603A33, 606G6D, 608001, 608F89, 608IF9, 60A333, 60A363, 60AAA3, 60CCCD, 60D04D, 60D0DD, 60F8I9, 60FF89, 60GCG3, 636AF3, 63A333, 63A3F3, 63AFF3, 63IIA3, 64AAAD, 660089, 660A33, 660DDD, 660F89, 662289, 666A33, 666D4D, 66A6A3, 66AF33, 66AF63, 66D0DD, 66FF89, 6A33F3, 6AFF33, 6AG6A3, 6AGA63, 6D000D, 6D004D, 6D0D4D, 6F6689, 7000G7, 7060DD, 70AAAD, 70D0CD, 733C33, 7366C3, 73AAA3, 7600DD, 760CDD, 766333, 76664D, 766FC3, 76CCDD, 76DD4D, 770G07, 771JEJ, 7771EJ, 777481, 7777I1, 777F81, 77C7JJ, 77EC77, 77EEC7, 77EJJJ, 77GCG7, 77JECJ, 7A7E77, 7AAAG3, 7AAF63, 7AEEE7, 7C7GG7, 7CCCAD, 7CCCG7, 7CGJJJ, 7D000D, 7J000D, 7JCGGJ, 80005B, 8000G1, 8000I1, 800GA1, 800I01, 80GGA1, 84405B, 84454B, 888EAH, 88E8AH, 8A4441, 8B8EAH, 8E8IF9, 8I0AG1, 902229, 909A09, 909C29, 909FA9, 90IC29, 9222I9, 92I999, 944EE7, 94AEE7, 977GG7, 97AEE7, 97GGG7, 990I29, 999I29, 99I299, 99IIA9, 9A9447, 9A9AF9, 9A9FF9, 9CII29, 9EEEE7, 9FA9A9, 9I9C29, 9I9I29, 9ICI29, 9IIIA9, A1555J, A1E55J, A333C9, A336A3, A33889, A33F63, A36663, A444E7, A4AAE7, A555FJ, A666A3, A6AAA3, A7A7E7, A7AE77, A944E7, A9A9F9, A9AAA9, AA3389, AA9AF9, AAA099, AAA3C9, AAA7E7, AAA939, AAACG9, AAAFF9, AAAI99, AAAIC9, AAE4E7, AAG9A9, AAGAA3, ADE55J, AF6663, AF9FF9, AGA963, AGG6A3, AH4441, AI000H, AI99C9, AI9AA9, AII099, B0004B, B0054B, B0080B, B00H0H, B04IIB, B05B8B, B0A6I1, B0BB8B, B0BIBB, B0E0IH, B0HA0H, B0I0HH, B0I4IB, B0IIBB, B888AH, BB058B, BB05BB, BB080B, BB0I4B, BBB04B, BBB08B, BBIB8B, BE88AH, BEHA0H, BEHHAH, BHEA0H, BHEHAH, BHHA0H, BIE00H, C007A7, C00GG7, C00JA7, C00JAJ, C00JC7, C0C7A7, C0JAC7, C0JJC7, C3G999, CAJJ77, CC0629, CC0929, CCC007, CCCCC7, CCCCG3, CCCGG7, CCCI29, CCI029, CD4GGJ, CE0009, CEFF69, CEFFF9, CEI609, CII029, CIII29, CJACC7, CJAJC7, CJJJA7, D000D7, D000J7, D004CD, D0D007, DCCDC7, DDD04D, DDDAI1, DEJJJJ, DG000D, DG5E5J, DGGG5J, DJ00CJ, DJEEE7, DJJJ5J, E00097, E00A0H, E00AA9, E02289, E028I9, E029I9, E055AJ, E0774J, E0777J, E07JCJ, E0AF5J, E0BI8B, E0E4CJ, E0EA5J, E0ECJJ, E0FFF9, E0H0HH, E0HA0H, E0IF29, E0J5AJ, E0JJAJ, E29299, E444E7, E4EEE7, E4IIIB, E66289, E66629, E66909, E66F69, E69609, E69999, E7774J, E77C77, E7C7C7, E8B8AH, E90029, E90229, E90909, E90FF9, E92299, E94EE7, E97EE7, E990F9, E99I09, E9F669, E9I029, EA8AF9, EAAAI9, EAIAA9, EAJJJJ, EBIBIB, EC7CC7, EE08F9, EE408B, EE8IF9, EEAF5J, EEE84B, EEEC07, EEEC77, EEEE0B, EEF55J, EEFFFJ, EEJ5AJ, EFE009, EFEF09, EFF669, EFFFF9, EGGGG9, EIF669, F004AJ, F00A41, F00JA1, F04481, F055CJ, F0A15J, F0EI09, F0G01J, F0J0GJ, F0J0JJ, F0JJ5J, F14441, F68F89, F68IF9, F800G1, F8E8I9, F99299, FA0009, FA1JJJ, FAAIA9, FAII09, FAII99, FCC929, FCE229, FCE609, FCEF29, FE68F9, FE8FA9, FF0E09, FF68F9, FF9FA9, FFC4AJ, FFC929, FFCE09, FFE009, FFEE89, FFEF09, FFFA99, FFFFFJ, FFFGJJ, FG1JJJ, FI99A9, FIA999, FIAAA9, FIAII9, FIIA09, G00093, G000EJ, G000G9, G00781, G007G3, G00C09, G07GC3, G09033, G0C903, G0CGG3, G0G903, G0G933, G0GCC3, G0GGC9, G30003, G36003, G5000J, G5E0EJ, G60303, G6AGG3, G6GAA3, G7AAA3, G7G363, G90AA9, G90C63, G9AAA9, GAAAA3, GAAGG3, GC0003, GCCC93, GCCCC3, GCG903, GDEEE7, GE0007, GG0CG3, GGA8G1, GGC6G3, GGCCG3, GGCGC9, GGG0I9, GGG363, GGG6G3, GGG75J, GGG8G1, GGG963, GGGAA3, GGGCG3, GGGCG9, H044EB, H0BIBB, H0E00B, H0F011, H0IIEB, H2444B, H400EB, H44IEB, H4EEEB, HBB4IB, HE000B, HEE40B, HEEE4B, HEHHAH, HF0011, HF4441, HH3HEH, HHGAA3, HHGAG3, HHHAG3, HHHEAH, HI0001, HIBIBB, HIEBBB, I0000H, I000A9, I002HH, I00A09, I00A99, I00AG1, I00AI9, I00H3H, I02999, I03HEH, I09AF9, I0A009, I0E0HH, I0I299, I0IIA9, I4445B, I4I45B, I90I29, I99029, I99A09, I9I029, IA3999, IB000H, IB00EH, IB0HEH, IB5B8B, IF2999, II0929, II92I9, II99A9, II9C29, IIAFF9, IIF299, III5BB, III8BB, J0000B, J00081, J000IB, J0010D, J006A1, J00BI1, J00I8B, J00IA1, J05EEJ, J06G81, J0C0JJ, J0E0CJ, J0EEEB, J0J0EB, J5EAEJ, J5EEAJ, J608I1, JE505J, JEEEEB, JJ0001, JJ0JCJ, JJ0JGJ, JJAJC7, JJJAE7, JJJBI1, JJJI8B, 14GGGG1, 1J5555J, 1JGGGG1, 205555H, 20I000H, 222222B, 2929999, 2BI000H, 3333A33, 3333G09, 333A3A3, 333AAA3, 336AAA3, 336AF33, 33II3A3, 363AF33, 3GIIII9, 400445B, 404454B, 404800B, 415555J, 440045B, 440080B, 442222B, 444004B, 444422B, 444444B, 4444G47, 444GGG7, 444I4IB, 44AAA47, 44I44IB, 44I4IIB, 4AAAAA7, 4EEE4E7, 4I4454B, 5000AFJ, 525555H, 555552H, 555555J, 555585H, 58BBBBB, 5BB8BBB, 5E0E55J, 5EBBBBB, 60003A3, 60008I9, 6000DDD, 6006DDD, 60080A1, 60080I1, 600A3A3, 600D8I1, 600F841, 60CCDDD, 60CGGG3, 60DDD0D, 60DDDDD, 60GGG63, 60GGGC3, 66600A3, 66603A3, 66608I9, 6666089, 66666A3, 6666C29, 6666F89, 666AFF3, 66DDDDD, 6800001, 6AGGGG3, 6CCCC29, 6G6AAA3, 6GAGGG3, 6GCCC63, 6GGG633, 700000D, 70000CD, 7000EC7, 70070C7, 700EEC7, 7070EC7, 70EEEC7, 76666CD, 77333A3, 7733A33, 775EE5J, 777A777, 777CGJJ, 777GE5J, 77A7777, 77JCJJJ, 77JJJCJ, 77JJJJJ, 7A77777, 7A777A7, 7AAAA63, 7AAAAAD, 7C00007, 7E77JCJ, 7EE7JCJ, 7EEC007, 7EEEEC7, 7G5E55J, 7GE555J, 7GJJJJJ, 7J6666D, 9000029, 9000299, 9009I29, 900I2I9, 90FAAA9, 90I0299, 90I0I29, 90I9929, 90II029, 90II299, 90II2I9, 94444E7, 97777E7, 9902I99, 99902I9, 9992929, 9992999, 9999929, 99I0C29, 99I9029, 99I92I9, 99III29, 9AAAF09, 9AFFFF9, 9I0AAA9, 9IAAA09, 9II0I29, 9II9929, 9III029, 9III2I9, 9IIIC29, A5F555J, A777E77, A994447, A9AAF99, AA00089, AA6A6A3, AAA0009, AAA3999, AAA9A99, AAA9AA9, AAAAA47, AAAAA77, AAAAA99, AAAAG99, AAAAIA9, AAAD447, AAAGAC9, AAGAAAD, AAGAAC9, AF00009, AGAAA99, AGAGG63, AGGGA63, AI0C999, AJ777A7, B000B8B, B000HEH, B000I0H, B00I00H, B0B805B, B0HHHEH, B0IHE0H, B0III8B, B8B000B, BB8BBBB, BBB8B0B, BEH0HHH, BHHHEHH, C00000J, C000C07, C066629, C0AJJJJ, C0C0A47, C0G000J, C0JAJJJ, C0JJAJJ, C0JJJJJ, C6CCG33, CC004A7, CC00A47, CC6CG33, CC92229, CCCC7G7, CCCGGG3, CE66609, CEF6669, D00007D, D00007J, D00071J, D000C0J, D000I01, D00700D, D00C007, D00C0C7, D00G00D, D0C00C7, D0CDCC7, D444447, D7JJJJJ, DC0000J, DDD7DDD, DDDCCC7, DJ0000J, DJ000EJ, DJJJJ0J, DJJJJJJ, E000009, E00084B, E000889, E000909, E000HAH, E008AF9, E04000B, E0AA8F9, E0CJJJJ, E0HHHHH, E40400B, E6660F9, E666609, E669969, E699669, E7CCC07, E996669, E99I669, E9FAAA9, EB8BBBB, EBBII8B, EBH0HHH, EE092I9, EE44IIB, EE4E4E7, EEBB8BB, EEC0007, EEE0IBB, EEE0IIB, EEEBB8B, EEEE44B, EEJCJJJ, EF00F09, EF0555J, EFAAAA9, EFF0009, EFF00F9, EFF60F9, EI0AAA9, EI660F9, EIIBB8B, EJ0CJJJ, EJ5055J, EJJ0CJJ, F0E00F9, F0E88I9, F0JJJJ1, F299929, F4G555J, F5555CJ, F5555FJ, F6668I9, F8GGGG1, F929999, F999C29, FA84441, FAAA9A9, FCE00F9, FCEF009, FCFEE29, FE00F09, FEE2889, FEF0009, FFAA9A9, FFCEE29, FFFF4AJ, FIII9A9, FIIIAI9, FJJJJ81, G000009, G0000J7, G000303, G000AA9, G000CC3, G000GG3, G003003, G0090C3, G00CG03, G00D007, G00G0G3, G00GG03, G00GG93, G080001, G0G00I9, G0G09C3, G0G0C93, G0G0GG9, G0GG093, G0GGC63, G0GGGG9, G44GG47, G6AAGA3, G999999, GAA9AA9, GCCCGG9, GCGGCC9, GCGGG03, GEE4447, GEEEE47, GG0C007, GG0GC09, GG0GGG3, GG6AAA3, GG6AGA3, GG6GAG3, GGEEE47, GGG00C9, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 21===

12, 18, 1A, 1G, 1K, 21, 25, 2B, 2H, 2J, 34, 38, 3A, 3G, 3K, 45, 4D, 4H, 4J, 52, 54, 58, 61, 65, 6B, 6D, 72, 74, 7A, 7G, 7K, 85, 8B, 8D, 92, 94, 98, 9A, A1, AD, AH, AJ, B2, B8, BA, BK, C5, CB, CH, CJ, D4, D8, DA, DK, ED, EH, EJ, F2, FG, G1, GB, GD, GH, H2, HA, HG, I1, I5, IB, IJ, J2, JA, JK, K1, KB, KD, KJ, 111, 115, 11H, 135, 13H, 13J, 151, 15B, 15H, 164, 175, 17B, 17D, 17J, 191, 19B, 19D, 19H, 1B1, 1B5, 1BB, 1BJ, 1D5, 1DD, 1DJ, 1E4, 1F1, 1F5, 1FD, 1FH, 1HB, 1HD, 1I4, 1JD, 1JH, 1JJ, 22D, 232, 29G, 29K, 2DG, 2F4, 2FK, 31H, 322, 33D, 351, 355, 35B, 35J, 362, 371, 37B, 37D, 37H, 37J, 39B, 39J, 3B5, 3BD, 3BH, 3D1, 3D5, 3DB, 3DD, 3DH, 3E2, 3FJ, 3HD, 3HH, 3HJ, 3J1, 3JB, 3JJ, 414, 481, 49K, 4B4, 4BG, 4C1, 4EB, 4F4, 4F8, 4FA, 4FK, 50G, 51B, 51D, 51H, 531, 535, 53D, 53J, 551, 56A, 56G, 56K, 575, 57J, 595, 59H, 5B1, 5B5, 5BB, 5CA, 5CG, 5CK, 5F1, 5FB, 5FJ, 5GA, 5GG, 5HH, 5IA, 5J5, 5JD, 5JH, 60H, 632, 66H, 66J, 678, 68J, 69G, 6F8, 6FA, 6GJ, 6H8, 6HK, 6IH, 6J4, 6JG, 6KH, 711, 71B, 71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, GGJ, GJ4, GK5, H0K, H15, H1B, H1J, H31, H3D, H3H, H48, H51, H55, H5J, H6K, H75, H84, H88, H91, H95, H9D, H9H, HBD, HC4, HC8, HDJ, HF5, HFB, HFH, HHD, HHJ, HI4, HI8, HJ5, HJB, HKK, I0D, I2D, I6H, I78, I8H, I9K, IFA, IFK, IID, J08, J1J, J31, J35, J3J, J44, J5H, J68, J6G, J71, J7B, J7D, J7H, J8G, J95, J9D, JBD, JBH, JBJ, JCG, JDB, JDH, JE4, JE8, JEG, JF5, JFD, JFJ, JG4, JGG, JH1, JH5, JHB, JHH, JI4, JJ1, JJ5, K0H, K5G, K6H, K9K, KG5, KH4, KKH, 101B, 103D, 1044, 1055, 10DH, 10FB, 10H1, 10H5, 10J1, 110J, 11CD, 11EB, 11ID, 130D, 13C1, 13EB, 141B, 1444, 1471, 14FB, 1505, 15CD, 167H, 16FJ, 16HH, 176H, 17C1, 17IH, 190J, 1BF4, 1BID, 1C04, 1C3D, 1CD1, 1D6H, 1DE1, 1DEB, 1E1B, 1E71, 1E95, 1EFB, 1F0B, 1F4B, 1FB4, 1H01, 1H05, 1H41, 1H6H, 1H6J, 1HC1, 1HE1, 1HF4, 1IDH, 1IHH, 1JE5, 1JF4, 20CD, 20D2, 20ID, 2278, 22F8, 239D, 23FD, 2478, 2708, 27FD, 293D, 29DD, 2A78, 2C0D, 2CF8, 2F0A, 2F9D, 2FAA, 2FDD, 2GFA, 2IF8, 3015, 301D, 301J, 3031, 3035, 305D, 30F1, 30FB, 30FD, 30HB, 30I2, 3105, 31C1, 329D, 330H, 336J, 33E1, 33EB, 33IH, 35CD, 363H, 390H, 39E5, 39ID, 3B0J, 3B6J, 3BC1, 3BEB, 3CC2, 3CF1, 3CFD, 3D32, 3E3B, 3E91, 3EF1, 3EFB, 3F05, 3FC1, 3FE1, 3FEB, 3H01, 3H0B, 3HC1, 3I3H, 3I9D, 404B, 4078, 40E1, 411B, 4131, 4171, 41FB, 4331, 4391, 43BB, 43FB, 43I2, 4432, 444B, 44E1, 477B, 4931, 4971, 499B, 49CG, 4A32, 4A4B, 4A78, 4A9G, 4B1B, 4B31, 4B3B, 4BFB, 4C78, 4E32, 4F1B, 4FB1, 5011, 509D, 50BD, 50DB, 50DD, 50EK, 50FD, 50H1, 50HJ, 50IK, 50JJ, 5101, 5105, 51E1, 530B, 53EB, 53IH, 550D, 550H, 55AB, 55BG, 55DG, 55E5, 55KH, 563H, 56JJ, 5701, 570H, 57EB, 57ID, 57IH, 590D, 59C1, 59CD, 5A3B, 5A7B, 5AGK, 5AIG, 5AKG, 5B0D, 5B9G, 5BCD, 5BGJ, 5BIH, 5C91, 5CD1, 5D0D, 5D0J, 5D6H, 5D6J, 5DC1, 5DG5, 5DIH, 5DJG, 5E0A, 5E11, 5E3B, 5E71, 5EAK, 5EEA, 5EKK, 5F0H, 5F9K, 5G0K, 5G9J, 5GF5, 5GIK, 5H0B, 5H5K, 5H9K, 5HE5, 5HID, 5I3H, 5I7D, 5IKG, 5J6J, 5JE1, 5KAG, 5KEK, 5KFH, 5KH5, 60FK, 63FH, 67JH, 69JH, 6E9K, 6F0K, 6F64, 6FJH, 6H04, 6HE4, 6HFJ, 6IF4, 6J9H, 6J9J, 7015, 7031, 703B, 7055, 705J, 7088, 709B, 709D, 709H, 70DJ, 70I8, 710J, 716J, 730B, 73C1, 73E5, 7501, 769H, 76FJ, 76JJ, 770D, 77E1, 77E5, 77EB, 7888, 78FH, 78H1, 78JH, 790H, 79CD, 79IH, 7B0J, 7B6H, 7C31, 7C91, 7C9D, 7E35, 7E75, 7E91, 7EF1, 7F01, 7F78, 7FCD, 7FH8, 7FIH, 7FJ8, 7H81, 7HC1, 7HF8, 7J6J, 7J78, 7JC1, 7JF8, 7JH8, 7JJ8, 809K, 80H8, 81H1, 81HH, 81J1, 839H, 83FH, 869K, 871H, 8731, 8771, 8788, 879J, 87F1, 87FJ, 87JH, 888H, 8911, 8931, 896K, 8A78, 8A9K, 8CC1, 8E9K, 8F3H, 8F9H, 8FF1, 8FHH, 8FJH, 8G9K, 8GJ8, 8H3J, 8H7H, 8H8K, 8HEK, 8HF1, 8HFJ, 8HH1, 8HJJ, 8HK8, 8J7J, 8J91, 8JC8, 8JFH, 8JJH, 8K78, 8K8H, 8KH8, 901J, 9035, 903B, 9071, 90CG, 90GG, 90JD, 9101, 91E5, 9301, 93E1, 93EB, 95CD, 95DG, 95FK, 967J, 969H, 96HH, 96HJ, 96JJ, 9705, 97C1, 9905, 995K, 999G, 99CD, 99EB, 99K5, 9B5G, 9BE5, 9BID, 9C0G, 9C11, 9C71, 9C9D, 9CEG, 9CGG, 9D05, 9D5G, 9DDG, 9DE5, 9E35, 9E3B, 9E71, 9EB5, 9EEG, 9EF1, 9EFB, 9F0J, 9F6H, 9F9K, 9FC1, 9FIH, 9FKH, 9G35, 9G95, 9GIK, 9GJJ, 9GKG, 9H0H, 9H5K, 9H6H, 9HKH, 9I6K, 9I9H, 9IDD, 9J6J, 9JEB, 9JGJ, 9JID, 9JJG, 9K35, 9K95, 9KEG, A33B, A3F5, A432, A478, A53B, A5IK, A788, A795, A878, A975, A99B, A9CG, A9GG, AAAB, AB44, AB55, ABFB, AC32, AC78, ACFA, AE32, AE5A, AE9K, AEEB, AEFA, AEFK, AF75, AFCA, AFCK, AFEA, AFFB, AG9G, AGFA, AI32, B064, B091, B09H, B0B1, B0D5, B0DH, B0EG, B0FB, B0FH, B0H5, B0HD, B114, B141, B14B, B1B4, B1E1, B411, B431, B444, B4B1, B59G, B5G5, B5JG, B67H, B67J, B69H, B6FH, B6FJ, B6HH, B6JH, B96H, B9DG, B9G5, BB14, BBB4, BC1D, BC31, BCDD, BD05, BD0H, BD6H, BDC1, BDE5, BDIH, BE0G, BE31, BEBB, BEE4, BEIG, BF0B, BF0H, BF6J, BFEB, BFIH, BG0G, BG55, BG95, BGG4, BGGG, BGIG, BGJ5, BH0B, BHEB, BIEG, BIGG, BIHH, BJ01, BJ0J, BJC1, BJF4, BJID, C041, C081, C09G, C0E1, C0FK, C104, C131, C171, C1C4, C1D1, C2F8, C2FA, C39D, C3F1, C478, C6FK, C7FD, C8C1, C90G, C911, C971, C99D, CAFK, CC78, CC81, CCE1, CCF8, CD02, CD31, CDFD, CE01, CE32, CF0A, CF48, CF64, CF68, CF6A, CF91, CFAK, CFDD, CFEA, CFEK, CIF4, CK32, CK9G, CKFA, D01D, D01H, D07B, D0DB, D0F1, D0FD, D0HD, D0HH, D10D, D10H, D16H, D1EB, D1IH, D23D, D3E1, D56H, D63J, D6CG, D6IG, D70B, D70D, D70J, D99G, D9GJ, D9ID, D9JG, DBC1, DBCD, DBE1, DBJG, DC11, DC9D, DCF1, DCFD, DCGG, DCIG, DD0H, DD6J, DDGJ, DDID, DDJG, DE31, DE75, DFC1, DFID, DFIH, DG22, DG6G, DG75, DG9J, DGCG, DGIG, DH6H, DHE5, DI6G, DICG, DIDH, DIFD, DIFH, DIGG, DJCD, DJDG, DJG5, E032, E05A, E0EB, E11B, E2F8, E31B, E331, E375, E395, E3C2, E3F1, E3F5, E3FB, E3I2, E53B, E555, E591, E59B, E5EG, E6F4, E755, E775, E89K, E915, E935, E93B, E96K, E9FB, EA32, EAEB, EAFK, EB44, EB95, ECFA, ECFK, EE32, EEEB, EEFK, EF08, EF35, EF48, EF55, EF64, EF68, EF75, EF95, EFB1, EFCA, EFCK, EFEA, EFIA, EFKK, EGF4, EIF4, EKF4, EKF8, EKFA, F008, F051, F075, F095, F09H, F0CA, F0CK, F0DD, F0FB, F0H1, F0IA, F0JJ, F105, F10B, F10D, F16J, F1B4, F1EB, F1ID, F1J4, F36H, F468, F47B, F491, F4C8, F501, F50B, F648, F66A, F69H, F705, F778, F864, F8F1, F90D, F99K, F9C1, F9HK, FA6A, FA75, FA7B, FA95, FAEA, FAFB, FB0D, FB14, FBCD, FBE1, FBJ4, FC1D, FC68, FC9D, FCAA, FCCK, FCD1, FCEA, FD6H, FDIH, FE0K, FE1B, FE64, FE95, 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====Additional known quasi-minimal primes (not necessarily the next)====

40&lt;sub&gt;47333&lt;/sub&gt;9G, CF&lt;sub&gt;479147&lt;/sub&gt;0K

===Base 22===

11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, IHH, IIJ, IJ7, IL7, J07, J55, J69, J8J, J99, J9J, JA5, JAH, JE7, JEH, JHH, JI9, JJ5, JJ9, JJH, JK7, K2J, K51, K5D, K75, K81, KA1, KB1, KB7, KBJ, KD1, KEJ, KG5, KIF, KJ5, KKD, KKJ, L0D, L47, L7F, L97, LAF, LD3, LD9, LDD, LEF, LGD, LI7, LJ7, LJJ, LLD, 104H, 10D5, 1205, 12B5, 140H, 1433, 144H, 14AH, 14B3, 16ED, 1AIH, 1B43, 1DD5, 1DDD, 1E6D, 1EGD, 1G05, 1GDD, 1GED, 1GGD, 1HB3, 1HHH, 1IAH, 200L, 2015, 2051, 20A1, 20DJ, 20GD, 20IL, 21B5, 21DD, 220D, 226D, 228D, 22B5, 22G5, 22K5, 22KD, 2555, 2557, 2581, 25C1, 26A1, 26B1, 2725, 2755, 2801, 2861, 288D, 28B1, 28KD, 2AA1, 2B25, 2B51, 2BB1, 2C81, 2D6D, 2DA1, 2DDJ, 2DGD, 2G0J, 2GB5, 2GDD, 2GGJ, 2I0L, 2I6L, 2ICL, 2J05, 2JK5, 2K07, 2K2D, 2K55, 2K6D, 2KB5, 2KI7, 2L2D, 2L8D, 2LK7, 302L, 30G3, 320L, 32IL, 332L, 33G3, 33G9, 36AF, 36EF, 382J, 388J, 39EL, 3AAF, 3BBJ, 3CG9, 3E2J, 3E6F, 3E6L, 3EEF, 3FAF, 3G69, 3GI3, 3GI9, 3IG9, 3LEL, 3LG3, 4025, 404H, 4063, 4075, 40AH, 40B5, 40B7, 40GH, 4225, 4363, 4447, 444H, 447H, 4487, 44B7, 44H7, 4525, 4555, 45B5, 4663, 4777, 47GH, 4807, 4B0H, 4BB5, 4BG3, 4EBH, 4G4H, 4GB3, 4HH3, 4I03, 4I63, 4IGH, 5069, 5077, 50KD, 5255, 52IJ, 5455, 5477, 5507, 5527, 556F, 557F, 5587, 56EF, 56FF, 56GD, 56KD, 5771, 57C1, 57EF, 5807, 580J, 589J, 58E7, 58IJ, 58K7, 5905, 5945, 5957, 5969, 5989, 598J, 5999, 59EJ, 59K7, 5AEF, 5B25, 5B55, 5BA5, 5BI7, 5BIJ, 5BK5, 5BK7, 5C21, 5EAF, 5EE7, 5F09, 5F6F, 5F95, 5FKD, 5G6D, 5G89, 5I09, 5I0F, 5I27, 5I6F, 5I89, 5IB7, 5IBJ, 5IFF, 5J77, 5K07, 5K25, 5K27, 5K6F, 5K87, 5KK7, 6013, 60A1, 60AF, 60EL, 6281, 62B1, 62KD, 63G9, 6403, 6643, 66EL, 66G9, 68KD, 69B3, 6A01, 6A0F, 6AAF, 6AFF, 6B21, 6B8D, 6BBD, 6BK1, 6D03, 6D43, 6D69, 6D6F, 6D93, 6D9F, 6E21, 6E81, 6E8D, 6EBD, 6ECL, 6ELL, 6FD9, 6GBD, 6GG9, 6IEF, 6K21, 6K8D, 6KBD, 6KE1, 6L43, 6LB3, 6LDF, 6LEL, 6LG9, 700F, 7027, 7057, 70EH, 70LF, 710H, 71G5, 7207, 7225, 7255, 727L, 7505, 7507, 755F, 75EF, 75F5, 75I7, 766F, 7681, 76CF, 7771, 7781, 77IL, 7861, 7A0H, 7A4H, 7BB5, 7C6F, 7EEH, 7EHL, 7FG5, 7G45, 7HEL, 7I0L, 7I27, 7IE7, 7IGH, 7K07, 7K61, 7K77, 7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, H0D1, H0HJ, H0LL, H1HH, H20D, H22D, H26D, H2I7, H2K7, H30L, H3KJ, H40H, H447, H4G3, H4H3, H4HH, H66L, H6ED, 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5IIIIIIIIIIIIIIIIIIIIIIIIIIF, D555555555555555555555550555, EEAAAAAAAAAAAAAAAAAAAAAAAAAF, HHHHHHHHHHHHHHHHHHHHHHHHEEBH, K66666666666666666666666666F, LLLLLLLLLLLLLLLLLLLLLLLLEEB7, D5555555555555555555555555A55, GGGGGGGGGGGGGGGGGGGGGGGGGGGG3, GIG0000000000000000000000000H, HH00000000000000000000000001H, K0000000000000000000000005KEF, 5BBBBBBBBBBBBBBBBBBBBBBBBBBBBD, HB0000000000000000000000000001, K000000000000000000000000505EF, L7777777777777777777777777772L, 2000000000000000000000000000CB1, C8CCCCCCCCCCCCCCCCCCCCCCCCCCCC9, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKFF, JE0000000000000000000000000000J, K000000000000000000000000000261, A0000000000000000000000000004I4H, HD000000000000000000000000000001, K000000000000000000000000000EC01, K0FFFFFFFFFFFFFFFFFFFFFFFFFFFFCF, D0002222222222222222222222222222D, FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL2L, I700000000000000000000000000000GH, K00000000000000000000000000000E61, 20000000000000000000000000000000JJ, DD5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL0KE7, 44HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, F0000000000000000000000000000000000000000000000000000000L89, J000000000000000000000000000000000000000000000000000000JJIJ, K222222222222222222222222222222222222222222222222222222222DD, KCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCF, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKCF, D55KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, K000000000000000000000000000000000000000000000000000000000000077, 22222222222222222222222222222222222222222222222222222222222222222D, K000000000000000000000000000000000000000000000000000000000000008IJ, DFA00000000000000000000000000000000000000000000000000000000000000005, CC4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJE0J, 777777777777777777777777777777777777777777777777777777777777777777777777EK7, 6000000000000000000000000000000000000000000000000000000000000000000000000000000043, KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKB5, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEEH, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 50000000000000000000000000000000000000000000000000000000000000000000000000000000002C1, K0000000000000000000000000000000000000000000000000000000000000000000000000000000000055EF, H700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000K1, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0I7, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, A400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, E0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000071, 7LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIL, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLK77, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEI7, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIB3, I7G00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 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77EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEK7, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJKJ, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777772L, BKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK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KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK5

===Base 24===

15, 17, 1D, 1H, 1J, 1N, 25, 2B, 2D, 2J, 2N, 31, 37, 3B, 3H, 41, 45, 47, 4B, 4D, 4H, 57, 5B, 5H, 5J, 65, 67, 6D, 6J, 6N, 75, 7B, 7D, 7N, 81, 85, 87, 8J, 97, 9B, 9D, 9H, 9N, A1, AB, AH, AN, B5, B7, BD, BH, BJ, C5, CJ, CN, D1, D5, DJ, E1, EB, ED, EH, EN, F7, FD, FJ, FN, G5, GD, GH, H1, HB, HD, HN, I1, I7, IB, IH, J1, J5, J7, JB, JN, K7, KB, KJ, KN, L5, LH, LJ, MD, MJ, N5, NB, NH, NJ, 101, 10B, 111, 1F1, 1FB, 1GB, 1LB, 201, 221, 22H, 261, 271, 277, 28H, 2A7, 2C7, 2G7, 2H7, 2L1, 2L7, 2MH, 305, 30D, 30J, 33N, 34N, 35D, 35N, 38D, 395, 3A5, 3AJ, 3CD, 3DD, 3DN, 3E5, 3EJ, 3GJ, 3IJ, 3JJ, 3K5, 3KD, 3ND, 43N, 44N, 49J, 4EJ, 4GJ, 4GN, 4NN, 50N, 535, 54N, 551, 55N, 5C1, 5CD, 5E5, 5K1, 5KD, 5LN, 5M5, 5N1, 601, 60B, 61B, 66H, 68B, 691, 6CH, 6FH, 6GB, 6HH, 6MH, 70H, 70J, 711, 761, 771, 77H, 77J, 78H, 7C7, 7CH, 7FH, 7G7, 7H7, 7HH, 7IJ, 7JJ, 7K1, 7M1, 7M7, 80D, 82H, 83N, 88D, 88H, 8AD, 8CD, 8DB, 8DD, 8DH, 8DN, 8GB, 8KD, 8MB, 8MH, 905, 911, 921, 935, 955, 99J, 9AJ, 9G1, 9JJ, 9K5, 9L1, 9M5, A0J, A3J, A95, AA7, AD7, AE5, AG7, AGJ, AI5, AIJ, AJD, AL7, ALD, B01, B0N, B11, B61, B6B, B8N, B91, BIN, BL1, BLN, BNN, C1B, C21, C27, C2H, C3D, C61, C8H, C91, CA7, CB1, CBB, CC7, CCB, CCD, CDD, CFB, CG1, CGB, CK1, CL1, CMB, CMH, D0B, D3D, D3N, D4N, D6B, D6H, D7H, D8B, D8N, DAD, DCD, DCH, DDH, DDN, DG7, DGB, DID, DMN, DND, E05, E4J, EA7, EEJ, EF5, EGJ, EI5, EJJ, EM5, EM7, F01, F21, F51, F8H, F95, FC1, FF1, FFB, FKH, FM5, G0B, G0N, G11, G3N, G6B, G77, G7J, G8B, G8N, G91, GA7, GBB, GC7, GFB, GG1, GGJ, GGN, GK1, GL1, GLN, GMN, GN1, GNN, H0J, H2H, H3J, H4J, H77, HA5, HA7, HE5, HFH, HIJ, HJJ, HKH, HL7, HMH, I0N, I3D, I3J, I3N, I4N, I5D, I95, IA5, IAJ, IE5, IEJ, IF5, IGJ, IJD, IK5, IKD, J0D, J4J, J8D, JAD, JDH, JEJ, JFH, JHH, JKD, JMH, K35, K6H, KCD, KFH, KH5, KLD, KM1, L01, L0B, L0D, L0N, L61, L6B, L8D, LA7, LC7, LDD, LF1, LG7, LGB, LGN, LID, LK1, LKD, LL1, LLB, LLD, LMN, LNN, M0H, M11, M21, M4N, M71, M91, M95, MA5, MA7, MBN, MC7, MF1, MF5, MFB, MFH, MG7, MI5, MIN, ML1, ML7, MLB, MMH, N01, N21, N4N, N71, N8N, NC1, ND7, NE7, NG1, NID, NK1, NL7, NMN, NN7, 11CB, 11MB, 1291, 12G1, 16C1, 16CB, 16K1, 186B, 18CB, 19K1, 1BK1, 1C8B, 1K91, 1KC1, 1KL1, 1L21, 1LC1, 1LM1, 1M61, 1M8B, 1MG1, 206H, 20CH, 20M7, 21C1, 21M1, 2207, 260H, 26KH, 2991, 2C6H, 2CC1, 2CM1, 2F11, 2FHH, 2MC1, 2MK1, 2MM1, 308N, 30GN, 30IN, 30LN, 30MN, 333J, 33JD, 33LD, 343J, 344J, 35I5, 380N, 393J, 394J, 3A3D, 3FI5, 3IMN, 3J3D, 3JID, 3L3D, 3L8N, 3M0N, 3M55, 3NGN, 404J, 408N, 40LN, 434J, 44AJ, 4ILN, 4JAJ, 4L8N, 5091, 5095, 50F1, 50I5, 51L1, 5211, 5291, 52G1, 53ID, 53MN, 5595, 55AD, 56F1, 588N, 58MN, 58ND, 5961, 5991, 5A5D, 5AAD, 5F91, 5GF1, 5GIN, 5I05, 5I55, 5I8D, 5IDD, 5IDN, 5IIN, 5IMN, 5KI5, 5M61, 5M8N, 5N3N, 602H, 6211, 62F1, 62G1, 66C1, 66FB, 66M1, 66MB, 6B21, 6BM1, 6BMB, 6C6B, 6CF1, 6CLB, 6FG1, 6K21, 6K2H, 6KG1, 6KKH, 6L21, 6LCB, 6LM1, 6MB1, 6MBB, 6MG1, 6MK1, 7001, 7027, 7207, 726H, 739J, 793J, 79C1, 7A4J, 7A9J, 7AE7, 7C01, 7CC1, 7FL1, 7G21, 7G9J, 7GAJ, 7GC1, 7HGJ, 7J2H, 7J6H, 7MKH, 800B, 800H, 804N, 806H, 808N, 80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, NK3D, NK8D, 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5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, 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EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEG7, 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===Base 30===

11, 17, 1B, 1D, 1H, 1N, 1T, 21, 27, 2B, 2D, 2J, 2N, 2T, 37, 3B, 3D, 3H, 3J, 3N, 47, 4B, 4H, 4J, 4T, 51, 57, 5D, 5H, 5N, 5T, 61, 6B, 6D, 6H, 6J, 71, 7D, 7H, 7J, 7N, 7T, 81, 8B, 8H, 8N, 8T, 91, 97, 9B, 9D, 9N, A7, AB, AD, AH, B1, B7, BH, BJ, BN, BT, C7, CD, CJ, CN, CT, D7, DB, DJ, DT, E1, EB, ED, EJ, EN, ET, F7, FB, FD, FH, FT, G7, GB, GJ, GN, GT, HB, HD, I1, I7, IH, IN, IT, J1, J7, JH, JN, JT, K1, K7, KD, KH, KJ, L1, LB, LD, LH, LN, LT, M1, MD, MH, MN, N1, NB, NJ, NT, O7, OD, OJ, ON, P1, P7, PB, PJ, PN, Q7, QH, QT, R1, RB, RD, RH, RJ, RT, SD, SH, SJ, SN, T7, TB, TD, TH, 10J, 15J, 1IJ, 1JJ, 1LJ, 1MJ, 1QJ, 22H, 29H, 2EH, 2GH, 30T, 331, 33T, 36T, 39T, 3A1, 3C1, 3G1, 3KT, 3MT, 3OT, 3S1, 3T1, 40D, 40N, 431, 44D, 46N, 48D, 4AN, 4DD, 4DN, 4F1, 4FN, 4GD, 4ID, 4PD, 4S1, 50J, 58J, 59J, 5CB, 5FJ, 5IB, 5IJ, 5JB, 5MB, 5MJ, 5OB, 5SB, 607, 63T, 687, 6E7, 6KT, 6L7, 6M7, 6MT, 6NN, 6QN, 6R7, 6S7, 6ST, 6TT, 70B, 77B, 787, 7KB, 7M7, 7MB, 7SB, 807, 80D, 80J, 84D, 85J, 877, 88J, 89J, 8DD, 8FJ, 8ID, 8IJ, 8JJ, 8M7, 8MJ, 8PD, 8QD, 8R7, 90H, 93T, 95J, 99H, 99J, 9AJ, 9AT, 9EH, 9GH, 9HH, 9HJ, 9JJ, 9MJ, 9OH, 9OT, 9PH, 9ST, 9TT, A01, A0T, A1J, A31, A6N, A6T, AAJ, AAN, AFN, AKN, AKT, ALJ, AMJ, AMT, AO1, AOT, AQ1, AQN, ARN, AT1, B5B, BBD, BCB, BDD, BID, BOB, BPD, BQB, C31, C9H, CC1, CCB, CCH, CF1, CH1, CIB, CKB, CMB, D01, D0H, D2H, D41, D4D, D6N, D8D, D9H, DDH, DDN, DGH, DH1, DHH, DID, DKN, DNN, DO1, DOH, DQN, DS1, EEH, EHH, EM7, EOH, EPH, ER7, F0N, F31, F5J, F8J, F9J, FLJ, FO1, FQ1, FQN, FS1, G01, G4D, G8D, GF1, GGH, GO1, GOH, GQD, GS1, H07, H0J, H0T, H1J, H2H, H31, H4N, H8J, HA1, HAJ, HAT, HC1, HE7, HEH, HFN, HGH, HH7, HIJ, HJJ, HKN, HL7, HNH, HPH, HQN, HS1, HTJ, HTN, I4D, I5B, I5J, I8D, IAJ, IDD, IGD, IIJ, IKB, IMB, IMJ, IOB, IPD, IQD, J8J, J9J, JAJ, JBD, JCB, JFJ, JIJ, JJB, JJD, JLJ, JPD, JQB, K3T, K4N, KAT, KBB, KCB, KMB, KNN, KOB, KOT, KQN, KST, KTT, L0J, L5J, L67, LAJ, LJJ, LQJ, LR7, M0J, M3T, M5B, M7B, M87, M9J, MAT, MFJ, MIJ, MJJ, MKB, MMJ, MOT, MQJ, N07, N0H, N67, N6N, N87, NAN, NDH, NGD, NHH, NKN, NN7, NNH, NPH, NQD, NQN, NR7, O01, O0B, O0H, OAT, OC1, OCH, OEH, OF1, OH1, OKT, OMB, OOT, OPH, OQ1, OQB, OS1, OST, P3T, P6T, P9H, PCH, PDH, PEH, PGD, PHH, PHT, PID, PMT, PPT, PQD, PST, PTT, Q5B, Q6N, Q9J, QAJ, QBB, QBD, QC1, QDN, QFJ, QFN, QGD, QJB, QKN, QLJ, QMB, QND, QNN, QO1, QQ1, QQN, QSB, R4N, R77, R87, RAN, RKN, RM7, RNN, RR7, RS7, S0T, S41, S6T, S87, SBB, SC1, SM7, SMT, SOB, SQ1, SR7, SS1, STT, T0J, T6T, T9T, TAN, TFN, TKN, TKT, TNN, TO1, TOT, TPT, TQ1, TQN, TTN, 18AJ, 19FJ, 1AFJ, 1FAJ, 20CH, 20PH, 2C0H, 2CHH, 2COH, 2H0H, 2HCH, 2HHH, 2POH, 2PPH, 3001, 34Q1, 3F41, 3FF1, 3QF1, 3SPT, 3SST, 3TTT, 40A1, 4441, 44KN, 44QN, 4AC1, 4C01, 4CA1, 4DG1, 4GA1, 4GC1, 4NND, 4NRN, 4OA1, 4Q4N, 4QA1, 4QRN, 4RQN, 550B, 555B, 55KB, 5A5J, 5AJJ, 5BKB, 5JQJ, 5KKB, 5QQB, 5QQJ, 604N, 606T, 60FN, 60KN, 60OT, 60PT, 660N, 660T, 664N, 66AT, 66FN, 66KN, 66TN, 6A0N, 6A9T, 6AAT, 6FKN, 6FRN, 6K0N, 6KFN, 6KKN, 6O6T, 6PAT, 6RFN, 6RRN, 6T6N, 6TRN, 7067, 70R7, 75BB, 77R7, 7C5B, 7CQB, 7E67, 7IQB, 7O5B, 7OIB, 7QOB, 7R07, 7RE7, 8667, 88L7, 88S7, 8E87, 8EE7, 8EL7, 8J8D, 8LL7, 8LS7, 906T, 908J, 90FJ, 90QJ, 90TJ, 92CH, 966T, 99MT, 99PT, 9I0J, 9I8J, 9LFJ, 9LIJ, 9M6T, 9MKT, 9MMT, 9PKT, 9QQJ, 9TLJ, A04N, A0QJ, A3AT, A3ST, A4A1, A4G1, A5QJ, A8QJ, A90J, A9PT, AA3T, AA41, AAAT, AAF1, AAG1, AAPT, AC41, ACS1, AF0J, AFC1, AFIJ, AG41, AGA1, AGG1, AI8J, AIJJ, AIQJ, AJ5J, AJJJ, AQ8J, AQJJ, AQQJ, AS3T, AT4N, AT5J, ATST, B04D, B0QD, BBIB, BBKB, BI0B, BK0B, BKIB, BKSB, BMIB, BMSB, BQ4D, BQQD, BSMB, C00B, C05B, C0A1, C0BB, C0EH, C0GH, C0Q1, C0QB, C2OH, C2PH, C441, C4Q1, C50B, CB0B, CBSB, CEGH, CG2H, CGHH, CHHH, CHOH, COGH, COO1, COOB, COOH, CQ01, CQ0B, CQQB, CQS1, CS0B, D00N, D0PD, D44N, DA0N, DCEH, DCG1, DCQ1, DDC1, DDF1, DDPD, DDQ1, DECH, DF4N, DFA1, DFRN, DG0D, DGD1, DGQ1, DNEH, DPDD, DQA1, DQPD, DRFN, DRRN, E20H, E667, E767, E8L7, E8S7, ECGH, EG2H, EH67, EH77, ES67, F001, F01J, F0MJ, F0QJ, F44N, F4G1, F4RN, F6RN, FA0J, FAC1, FAF1, FAIJ, FC01, FCG1, FFAJ, FFC1, FFG1, FFIJ, FGA1, FJJJ, FJQJ, FKAN, FR6N, FRFN, G09H, G0EH, G0HH, G20H, G341, G3Q1, G4G1, G92H, GA41, GAC1, GC0H, GCEH, GCG1, GCPH, GD31, GDA1, GDG1, GDPD, GE2H, GE9H, GGG1, GGGD, GHCH, GHHH, GP0D, GP2H, GPPD, GQ31, GQA1, GQG1, H00N, H0CH, H3ST, H4O1, H667, H66N, H677, H69T, H99T, H9CH, H9KT, H9LJ, H9PT, H9QJ, HH6N, HHF1, HHFJ, HHH1, HHHT, HHLJ, HHOH, HHQ1, HHST, HK6T, HL9J, HLMJ, HMKT, HMMT, HNM7, HNRN, HOPT, HQ01, HQ5J, HR0N, HR6N, HS77, HSKT, HT01, I0BB, I0JD, I0JJ, I98J, I9LJ, I9QJ, ICBB, IF0J, IFFJ, IFQJ, IIBD, IJBB, IJID, IL8J, ILFJ, IQ8J, IQJJ, IQQJ, IS0B, ISQB, J00D, J05J, J0BB, J0MJ, J0OB, J4QD, J50B, J5QJ, JBKB, JBMB, JDGD, JDQD, JGDD, JI0D, JIIB, JISB, JKIB, JKKB, JM0B, JOIB, JQ4D, JQDD, JQID, JQJJ, JQMJ, JSIB, JSMB, K06N, K0KB, K0KT, K0MT, K0PT, K0SB, K5QB, K60T, K66T, K6AN, K6FN, K6PT, K6RN, K96T, K99T, KA0N, KF6N, KFKN, KI0B, KK0T, KK6N, KK6T, KKFN, KKKB, KKKT, KKQB, KKTN, KMMT, KQ0B, KQQB, KR0N, KS5B, L0M7, L8E7, L98J, LFFJ, LI8J, LIFJ, LL87, LL9J, LLIJ, LLM7, LM77, LML7, LMM7, M09T, M0E7, M0IB, M0L7, M0M7, M0TT, M55J, M5LJ, M60T, M69T, M707, M767, M777, M7E7, M7S7, M8LJ, M96T, M9KT, MC0B, MCOB, ME07, MIBB, MJ0B, MJBB, MJSB, MLLJ, MLS7, MM07, MM0T, MML7, MMOB, MMR7, MOBB, MOCB, MP0T, MP9T, MPKT, MQ0B, MQIB, MQOB, MS07, MSPT, MSSB, MT8J, MTTT, N00D, N04D, N04N, N0FN, N0ND, N0RN, N2OH, N92H, NCGH, ND0D, ND4N, NDPD, NEGH, NEL7, NF4N, NFRN, NGCH, NGEH, NHM7, NME7, NML7, NMM7, NN0D, NN4D, NN4N, NNDN, NNID, NNND, NNRN, NP4D, NSL7, NSS7, O00T, O03T, O0MT, O0PT, O2HH, O2OH, O341, O4A1, O4G1, O4O1, O5KB, O90T, OA41, OBSB, OC5B, OCOB, OG2H, OG31, OH9H, OH9T, OHHH, OHTT, OICB, OISB, OM0T, OM6T, OMPT, OMTT, OO2H, OO9H, OOA1, OOCB, OOGH, OOKB, OOO1, OSKB, OTMT, P00T, P04D, P08D, P09T, P0KT, P90T, PAAT, PKKT, PO0T, PP0D, PPPD, Q00B, Q0AN, Q0D1, Q0F1, Q0IB, Q0JJ, Q0MJ, Q0OB, Q55J, Q5QJ, Q88D, QA0N, QA41, QA4N, QDF1, QDPD, QF41, QFA1, QG31, QIQJ, QJ0J, QJ8D, QJDD, QJJJ, QJMJ, QJQD, QKKB, QKQB, QOKB, QOOB, QP0D, QPDD, QQ8D, QQID, QQJD, QQPD, QQQB, QS01, QSA1, R00N, R067, R06N, R0FN, R0L7, R60N, RE67, RFRN, RRFN, RRRN, S00B, S3AT, S3ST, S50B, S5QB, S99T, SAAT, SC0B, SE67, SGG1, SICB, SIQB, SK5B, SKIB, SKKT, SKPT, SKQB, SM0B, SMMB, SMSB, SOA1, SOG1, SPOT, SQCB, SQQB, SS0B, SS67, ST31, STG1, T00N, T03T, T041, T04N, T0AT, T0C1, T0MT, T0ST, T1AJ, T3F1, T4A1, T4RN, T5JJ, T66N, T8AJ, T9FJ, T9QJ, TA3T, TAA1, TAG1, TAIJ, TCS1, TFA1, TFAJ, TFF1, TFFJ, TFG1, TGA1, TI9J, TIJJ, TIQJ, TL8J, TL9J, TM0T, TM5J, TM8J, TMAJ, TMLJ, TMST, TSF1, TSG1, TT01, TT5J, TT8J, TTA1, TTC1, TTFJ, TTG1, TTLJ, TTMJ, 30441, 30O41, 30OO1, 34O41, 3AATT, 3ASAT, 3O441, 3Q041, 40G41, 40GQ1, 40O41, 40OG1, 40OO1, 44001, 440C1, 440G1, 4444N, 44CG1, 44GQ1, 4AAA1, 4COG1, 4DAA1, 4GQ41, 4KRRN, 4OGG1, 4Q041, 4QGG1, 5005B, 500BB, 500KB, 500QB, 50K0B, 50Q0B, 50QKB, 555AJ, 555LJ, 555QJ, 55J5J, 5B00B, 5B0BB, 5J55J, 5J5JJ, 5JJ5J, 5JJJJ, 5K00B, 5QJ5J, 5QK0B, 6006N, 6009T, 600AT, 6066N, 60R0N, 666N7, 6696T, 66OPT, 67767, 6900T, 696PT, 69P0T, 6F6AN, 6O9PT, 6OP0T, 6OP9T, 6P99T, 755QB, 77767, 7CBBB, 7IBIB, 7IIBB, 7IICB, 7QIIB, 8888D, 88E67, 8L887, 8SSE7, 9099T, 90KKT, 90KMT, 90MPT, 9690T, 96P9T, 990KT, 9990T, 9999T, 99K6T, 9FFFJ, 9HKMT, 9HMPT, 9ILLJ, 9KKMT, 9KKPT, 9M00T, 9T8QJ, 9TFIJ, A008J, A00NN, A055J, A0FJJ, A0I0J, A0I9J, A0J0J, A44C1, A555J, A9FFJ, A9QIJ, A9T8J, AFFA1, AFFFJ, AFFJJ, AFFQJ, AJ00J, AP99T, ASA9T, ASFF1, ASP9T, ASSAT, AT3TT, ATFQJ, ATQIJ, ATT9J, ATTQJ, B088D, B08GD, B0GGD, B0SKB, B8GGD, BG00D, BIISB, BISIB, BS0IB, BS0SB, BSSKB, BSSSB, C0041, C0G41, C0H0H, C0PPH, C0S01, C4001, C4AG1, C4OG1, C5BBB, CG00H, CGE0H, CGGA1, CGQ41, COSSB, CP20H, CPGPH, CPP2H, CPPOH, CPPPH, CS001, CS55B, CSSQB, D00GD, D0A4N, D0FAN, D0GDD, D0N0D, DAAA1, DAAC1, DAFG1, DDAA1, DDGGD, DFFAN, DFGC1, DGDGD, DGG31, DGGA1, DH0AN, DPPPH, DQDD1, DQGG1, E0L87, E88E7, EC00H, EC02H, EE867, EE887, EEL87, ELE87, F00IJ, F0441, F0AG1, F0F0J, F0F41, F0FF1, F0GG1, F4041, F44A1, F44C1, F4A41, F64KN, F6K6N, FAAA1, FAFQJ, FCAA1, FF041, FF0F1, FF64N, FF6KN, FFA41, FFA4N, FFF4N, FFFF1, FFFFN, FFKRN, FFQMJ, FIJ0J, FJ00J, FKKKN, FNNFN, FQQ0J, FQQMJ, G00DD, G00DH, G00PD, G02PH, G0DDD, G0PDD, GAAA1, GC2HH, GDDDD, GDGDD, GDPPH, GG0PD, GGCA1, GGCQ1, GGDD1, GH4Q1, GHHG1, GII0D, GQ441, H009H, H00G1, H00H1, H04Q1, H0ANN, H0H01, H0H9H, H0HG1, H0HO1, H0O41, H0OHH, H0QG1, H40G1, H4G41, H4GG1, H60AN, HAN0N, HF0G1, HFF41, HFFMJ, HFFQJ, HGQ41, HH3TT, HH401, HH441, HH4G1, HH55J, HH66T, HH6OT, HH96T, HHA0N, HHANN, HHC0H, HHCHH, HHH0N, HHHCH, HHHHN, HHHNN, HHKMT, HHO41, HHP9T, HHPKT, HHTMT, HHTT1, HK9MT, HKKMT, HLLFJ, HM66T, HM7R7, HMM67, HMM77, HMMM7, HMTST, HO9HT, HOO41, HOOOH, HOT3T, HOTT1, HPO9T, HSO3T, HTGG1, I000J, I009J, I00BD, I00QJ, I00SB, I08QJ, I0I0D, I0IQB, I0JIB, I0Q0B, I0SCB, I0SIB, I0SSB, IB00D, IBBSB, IBISB, ICSSB, II0CB, II0ID, II0SB, IIIBB, IIQ0B, IIQIB, IISIB, IJ0SB, IJJQJ, ILLLJ, IQ00J, IQC0B, IQIQB, IQQIB, ISSCB, J00MB, J08GD, J0IID, J0QQD, J55BB, J55JJ, J5J5J, J5JJJ, J88GD, JB0SB, JBB0B, JBBSB, JDDDD, JG0ID, JGG0D, JIIID, JJ0QJ, JJ5JJ, JQ08D, JQ0QJ, JQQ0D, JQQ5J, JS55B, JSK0B, JSS5B, K000T, K006T, K00TN, K0FAN, K505B, K6T0N, K9KMT, K9KPT, K9P0T, KIQIB, KK00N, KK05B, KK0AN, KK0KN, KKM9T, KS0IB, KS0QB, KSQIB, KTR6N, KTRRN, L0087, L08S7, LE087, LEE87, LL8LJ, LMEE7, LMSE7, LMSS7, M00CB, M00PT, M066T, M0CQB, M0K6T, M0MQB, M0MSB, M0QQB, M0SMB, M0SST, M666T, M900T, MBBSB, MEE77, MEL77, MELE7, MES77, MESE7, MESS7, MI00B, MIICB, MIISB, MIQCB, MK9PT, ML7L7, MLEL7, MLLE7, MLME7, MM677, MMBIB, MMBSB, MMCQB, MME77, MMEE7, MMICB, MMISB, MMK6T, MMKKT, MMM9T, MMMMT, MMMTT, MMQQB, MMSCB, MMSKT, MMSMB, MMSS7, MMTST, MOIIB, MOOIB, MOSIB, MQCQB, MR007, MR667, MRL07, MS7L7, MSEL7, MSK9T, MSL77, MSSL7, MT00T, MTMMT, N0DDD, N4NNN, N7LE7, N7S77, NE2CH, NE9CH, NEC2H, NEE77, NFFNN, NFNFN, NII0D, NL777, NL7L7, NLES7, NLLL7, NLS77, NOG9H, NRR0N, O6P9T, O9H6T, O9HPT, O9P9T, OCBBB, OG441, OGAG1, OGGA1, OHH6T, OHOOH, OKIIB, OKK5B, OKKIB, OO5BB, OOBIB, OOIIB, OSCSB, OT3TT, OT441, OTG41, OTGG1, P00PH, P0D0D, P0G2H, P0OGH, P0PGH, PA99T, PGP0H, POOOH, PP20H, PP88D, PPP0H, Q0001, Q000N, Q001J, Q00ID, Q00PD, Q044N, Q04G1, Q0I0D, Q0PPD, Q40G1, Q444N, Q44RN, Q4AA1, Q8QQJ, QAAS1, QAFF1, QAFG1, QASG1, QDGG1, QFGG1, QI00D, QIICB, QIIQB, QIQCB, QOIIB, QPP4D, QQ08J, QQ0CB, QQC0B, QQI0J, QQJ5J, QSFG1, R00E7, R0E07, R0NE7, R6F6N, R6FFN, REE07, RELE7, RFF6N, RL0E7, RLE07, RLEE7, RLLE7, RQR0N, RR0QN, RRQ0N, S03O1, S0AF1, S0AG1, S0O31, S0QIB, S0SIB, S30F1, S30O1, S7QIB, SA3PT, SAAA1, SAFG1, SASST, SCSQB, SF0G1, SFFF1, SI0SB, SISSB, SKSSB, SSCQB, SSCSB, SSMIB, SSPAT, SSSKT, SSSMB, SSSSB, STAF1, STF01, T0001, T0031, T0AF1, T0AS1, T0G31, T0R6N, T0T31, T0TF1, T3AAT, T40G1, T4CG1, T4G41, T5LLJ, T8LLJ, TA441, TA98J, TAFJJ, TAQ5J, TASST, TATAT, TC401, TCGG1, TFQIJ, TFQJJ, TFQMJ, TG441, TGC41, TI8LJ, TLLMJ, TMMMT, TMTMT, TQ8QJ, TSS3T, TT3AT, TT9IJ, TTAAT, TTQIJ, TTS31, TTS3T, TTT9J, TTTST, 20000H, 200OOH, 3440O1, 3TAAST, 404CQ1, 4KKKKN, 4KKKRN, 4QQQQD, 505BBB, 50BB0B, 6000AN, 6444RN, 66666N, 666O9T, 66999T, 669P9T, 66N777, 6A444N, 6FF66N, 6FFF6N, 6R666N, 766767, 77S677, 7IBBBB, 8888E7, 8LLLLJ, 8SSSL7, 9000MT, 90K90T, 90KP0T, 90M90T, 99000T, 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===Base 36===

11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 2UJ, 2WN, 2XN, 30Z, 337, 33B, 33H, 33P, 34H, 34P, 37D, 38Z, 39H, 3AB, 3AP, 3AZ, 3D7, 3DH, 3FD, 3FZ, 3HD, 3HH, 3KZ, 3L7, 3LZ, 3MB, 3O7, 3OZ, 3PD, 3PP, 3Q7, 3RH, 3S7, 3SZ, 3UP, 3UZ, 3XB, 40P, 43B, 43H, 46V, 48B, 48V, 49B, 4AP, 4BB, 4E1, 4FP, 4HB, 4HH, 4HP, 4IH, 4MB, 4OP, 4OV, 4PH, 4RH, 4VB, 4VH, 4WH, 4WP, 4WV, 4X1, 4XH, 55T, 565, 58Z, 5C5, 5E7, 5ET, 5EZ, 5IN, 5KT, 5L7, 5MP, 5NN, 5O5, 5O7, 5OP, 5R7, 5RT, 5RZ, 5ST, 5SZ, 5TN, 5TZ, 5W7, 5XN, 5YN, 61T, 625, 62J, 64V, 661, 66J, 691, 6AV, 6EV, 6IJ, 6J1, 6K5, 6O1, 6OT, 6QJ, 6QT, 6RV, 6SJ, 6T1, 6TT, 6U5, 6UV, 6VV, 6W5, 6Y1, 6YT, 72D, 737, 77D, 791, 7A1, 7AZ, 7D7, 7GD, 7I1, 7IN, 7LN, 7M7, 7MN, 7N7, 7NN, 7RN, 7RZ, 7WN, 7X7, 7Z1, 801, 80V, 82D, 83B, 841, 84H, 877, 881, 887, 88B, 88V, 88Z, 89H, 89V, 8A1, 8AB, 8B7, 8BH, 8D1, 8DH, 8EB, 8EV, 8GD, 8GZ, 8HD, 8IV, 8L7, 8LZ, 8M1, 8MB, 8MD, 8O7, 8OB, 8Q7, 8QD, 8RD, 8S7, 8SH, 8SZ, 8UZ, 8W7, 8WV, 8XV, 8Y1, 90H, 91J, 93B, 93H, 94V, 981, 98H, 99J, 9AH, 9BB, 9C5, 9IH, 9IV, 9JV, 9KH, 9M1, 9OB, 9QJ, 9R1, 9RH, 9SH, 9UJ, 9VB, 9VJ, 9W5, 9X1, 9Y1, 9YB, 9YJ, 9YV, A25, A3P, A3Z, A4H, A61, A81, A8B, AAB, AAH, ABB, AC5, AEZ, AHP, AK1, AKB, AKH, ALZ, AMB, AO5, AOZ, APH, AQP, AR1, ARV, ARZ, AUV, AV5, AVB, AVV, AWV, AX1, AXB, AZ1, B2J, B4B, B4V, B6V, B87, B8H, B9B, BAB, BAH, BBH, BBV, BE7, BEB, BGJ, BHB, BIJ, BJB, BJT, BLJ, BOB, BOT, BQ7, BRV, BS7, BTT, BTV, BVB, BVJ, BVT, BWV, BX7, BXH, BYH, BYV, C25, C2J, C2N, C55, C65, C6J, C95, CCN, CDD, CED, CF5, CFJ, CGD, CIN, CJD, CO5, CRN, CUJ, CXJ, CXN, D37, D3P, D41, D55, D5P, D6T, D77, D81, D9H, DA1, DCD, DD1, DD5, DDH, DDP, DE7, DEP, DF5, DFD, DG7, DHH, DI1, DK1, DK5, DKT, DO1, DOP, DP1, DPD, DQ7, DR7, DRH, DS7, DTT, DU1, DWD, DX7, DYH, E0P, E2D, E3Z, E41, E4P, E71, E81, E87, E8B, EAZ, EDP, EED, EEN, EEV, EFZ, EGT, EI1, EKZ, ELB, EMB, EMN, EN7, EO1, EOT, EPP, EPZ, EQ7, ERN, EST, ETN, ETV, EUP, EUZ, EVT, EWD, EWN, EX1, EYB, EYN, EZD, EZP, F2J, F3B, F3Z, F4P, F65, F8B, F8Z, FCJ, FD5, FEZ, FFB, FFD, FG5, FGD, FGZ, FIJ, FJ5, FJJ, FLZ, FPD, FQD, FSZ, FXB, G07, G0D, G0Z, G3D, G3Z, G55, G6T, G7D, G7Z, G87, G8Z, GA5, GCJ, GD7, GE7, GET, GG5, GG7, GGT, GGZ, GI7, GJD, GKZ, GL7, GLT, GQT, GRT, GS7, GST, GU5, GUZ, GW5, GXJ, GZ7, GZZ, H3D, H3H, H4H, H9B, HAH, HED, HGD, HHP, HIH, HKH, HKP, HLN, HNH, HOB, HOP, HQP, HRD, HRH, HRN, HSH, HWD, HWH, HWP, HYN, I41, I6J, I71, I7N, I87, I8H, I9H, I9J, IA1, IAV, IE1, IFJ, IG7, IHH, II1, IIH, IIV, IJV, IK1, IL7, ILJ, ILN, IM1, INN, IOV, IRH, IUV, IX1, IXH, IXV, IYJ, J2D, J4V, J55, J61, J6J, J8B, J8V, J95, J9J, J9V, JA5, JBB, JD5, JDT, JF5, JFJ, JGJ, JGT, JIV, JJ1, JJD, JJV, JK5, JKT, JLB, JLT, JMN, JN5, JNJ, JQJ, JQN, JQT, JRD, JTB, JVJ, JVN, JVV, JWN, JY1, JYJ, K0B, K25, K3P, K65, K81, K95, K9H, KAH, KC5, KEP, KEZ, KG5, KHP, KK1, KKT, KLB, KLP, KLZ, KM1, KMB, KMP, KOT, KP1, KQP, KR1, KRT, KRZ, KTT, KW5, KX1, KZB, L0N, L0P, L8Z, L9B, LAZ, LBJ, LEN, LET, LFB, LFZ, LG7, LGZ, LIJ, LJJ, LKB, LKP, LLB, LLP, LLT, LMB, LMN, LN7, LO7, LOT, LPZ, LQT, LRN, LTN, LTT, LWP, LX7, LYN, M01, M0P, M2D, M2N, M37, M3B, M41, M61, M77, M87, M8D, M91, ME1, ME7, MEB, MFB, MFP, MGD, MI7, MKB, MM7, MMN, MN7, MNN, MO7, MOB, MOP, MPP, MQP, MS7, MW7, MXN, MYN, N5N, N65, N87, N8D, N8H, N95, NCD, NCJ, NDH, NE7, NFJ, NG5, NG7, NGJ, NJ5, NN7, NND, NO5, NQD, NQJ, NRN, NU5, NWH, NWN, NXH, NXN, NYJ, O0Z, O25, O37, O3B, O3P, O3Z, O41, O4B, O5Z, O61, O71, O7Z, O81, OA5, OAP, OBB, OBV, OC5, OEZ, OF5, OG7, OKP, OKZ, OMB, OO5, OOZ, OP5, OPP, OQT, OR1, OR7, OS7, OST, OTB, OTZ, OU5, OVV, OW5, OX7, OXB, OXV, OYV, OZ5, OZ7, P01, P0D, P0N, P3P, P4H, P4P, P5N, P65, P6V, P8V, PD1, PED, PEZ, PHH, PHP, PI1, PIN, PLN, PLP, PLZ, POP, PP1, PPH, PPV, PQD, PQN, PRV, PV5, PVH, PVV, PWH, PWP, PX1, PXV, PYH, PYN, Q07, Q0P, Q2N, Q37, Q3P, Q6J, Q6T, Q7D, Q8D, Q9J, QCD, QCJ, QD7, QED, QFP, QGT, QI7, QIN, QJN, QLJ, QMD, QMN, QMP, QND, QNJ, QOT, QQJ, QTN, QWN, QYJ, QYT, R0V, R0Z, R37, R3H, R4H, R77, R7D, R7N, R7Z, R81, R8V, R91, RA1, RCD, RCN, RD1, RED, REV, REZ, RGT, RGZ, RHD, RIV, RKH, RKZ, RLN, RMD, RNH, RO7, RQN, RRD, RRT, RRZ, RS7, RSH, RT1, RTV, RU1, RUZ, RVN, RVT, RW7, RWH, RX7, RY1, S0J, S6J, S87, S9H, SAZ, SCJ, SET, SFJ, SG7, SGZ, SI7, SM7, SO7, SOT, SQJ, SQT, SRH, SSH, STT, SW7, SX7, SXH, SYH, T0N, T0Z, T1D, T1T, T6V, T8Z, TBB, TBV, TCN, TD1, TEV, TGT, TKT, TLB, TM1, TO1, TOB, TQN, TR1, TRD, TT1, TTB, TTN, TUZ, TVB, TVN, TVT, TWD, TWV, TXB, TXV, TYV, U2J, U3Z, U61, U8V, U95, UA1, UC5, UEP, UEZ, UFJ, UG5, UJ5, ULZ, UO5, UOP, UOZ, UQP, URV, URZ, UUJ, UW5, UWV, UXJ, UXV, UYJ, V0H, V25, V4V, V55, V6V, V9J, V9V, VBB, VBJ, VCJ, VET, VGJ, VHP, VIH, VIN, VJJ, VJN, VLT, VMP, VNJ, VOB, VP5, VQJ, VQT, VRT, VRV, VSH, VSJ, VST, VTB, VTN, VUP, VW5, VWN, VXH, VXN, VYB, W07, W25, W3D, W3H, W4P, W4V, W57, W6V, W7D, W8H, W95, W9H, WAH, WAV, WCN, WD7, WDD, WDH, WE7, WEN, WF5, WGD, WHH, WK5, WKH, WN7, WNN, WPP, WPV, WQP, WR7, WRD, WRH, WRN, WS7, WU5, WUP, WVN, WWH, WWP, WX7, WXH, WXN, WYN, WYV, X0J, X2J, X2N, X4B, X4H, X4V, X6J, X87, X8B, X91, X9B, XAV, XCN, XEB, XFB, XHB, XHH, XHN, XKB, XLJ, XNH, XO1, XQ7, XS7, XSH, XUJ, XWN, XWV, XX7, XXH, XXV, Y0N, Y1J, Y1T, Y2N, Y3H, Y61, Y8V, Y91, YBV, YCN, YEB, YFJ, YHB, YHN, YIH, YJN, YJT, YLJ, YLN, YMB, YMN, YNH, YOB, YOV, YRH, YTB, YTN, YTT, YTV, YVB, YVH, YWH, YWV, YXB, YY1, YYJ, YYT, YYV, Z01, Z3D, Z3Z, Z71, Z8B, ZB7, ZBB, ZD5, ZDD, ZDP, ZG7, ZGD, ZGZ, ZKB, ZLP, ZM1, ZO5, ZP1, ZPD, ZQD, ZU1, ZX1, ZXB, ZZD, 102J, 106T, 10ET, 10GJ, 10LJ, 10TD, 12FJ, 12LJ, 13ED, 13GD, 13MD, 192J, 1D0T, 1DET, 1E3D, 1EFD, 1F0D, 1FWD, 1G0J, 1GGJ, 1GIJ, 1GUJ, 1I0J, 1ISJ, 1JOT, 1JTT, 1M0D, 1M3D, 1MFD, 1MWD, 1OKT, 1OOT, 1Q3D, 1QDT, 1QKT, 1QTD, 1QWD, 1SLJ, 1TDT, 1TFD, 1TOT, 1TTD, 1TTT, 1UGJ, 1ULJ, 1WFD, 1WWD, 20CJ, 20FJ, 20G5, 20IN, 20JD, 20MD, 20RN, 20WD, 20YN, 2295, 25RN, 260J, 26LJ, 288D, 28DD, 28FD, 296J, 29G5, 29K5, 29LJ, 2C05, 2C0J, 2C0N, 2CJJ, 2CK5, 2CLJ, 2CSJ, 2D25, 2DC5, 2DDD, 2DW5, 2F0J, 2FDD, 2FMD, 2FSJ, 2G65, 2G95, 2GK5, 2I0N, 2JC5, 2JCJ, 2JWD, 2K05, 2L6J, 2L9J, 2N0N, 2NIN, 2NN5, 2NSJ, 2NW5, 2NYN, 2Q0N, 2QNN, 2R0N, 2RFD, 2RWD, 2SJJ, 2SXJ, 2SYJ, 2W05, 2WC5, 2WD5, 2WMD, 2XCJ, 2YCJ, 2YJJ, 2YRN, 2YSJ, 300D, 302D, 304B, 3077, 3087, 308D, 30BH, 30DD, 30FP, 30G7, 30GD, 30MD, 30QP, 30YH, 320D, 333D, 333Z, 33WD, 344B, 34FB, 373Z, 37E7, 37EZ, 37G7, 3807, 38BB, 38E7, 38G7, 38HB, 38I7, 38X7, 390B, 394B, 399B, 39EB, 39FB, 39KB, 3ASH, 3AWH, 3B8B, 3BG7, 3BLB, 3BR7, 3BW7, 3C8D, 3D0P, 3D2D, 3DKP, 3E3D, 3E77, 3EBB, 3EFB, 3EKB, 3EQP, 3ERD, 3ERZ, 3EZZ, 3F4B, 3FEP, 3G2D, 3GCD, 3GED, 3GM7, 3GR7, 3GWD, 3GZD, 3H0B, 3HEP, 3HKB, 3IAH, 3ISH, 3K4B, 3KBH, 3KEB, 3KKB, 3L4B, 3L8B, 3M3D, 3MEP, 3MG7, 3MWP, 3OFP, 3OKB, 3OMP, 3OOB, 3OYB, 3P8H, 3PXH, 3QDP, 3QQD, 3QRD, 3QWD, 3R3D, 3R3Z, 3R8D, 3RG7, 3RM7, 3RQD, 3RR7, 3RZ7, 3SIH, 3W0H, 3WED, 3WM7, 3WQD, 3WW7, 3WYH, 3X8H, 3XE7, 3XG7, 3XKH, 3Y0B, 3Y4B, 3Y8H, 3YAH, 3YSH, 3YYB, 3Z4B, 3ZFP, 404V, 4061, 40EB, 40I1, 40IV, 40KH, 40M1, 40R1, 40XB, 40XV, 43QP, 4441, 444H, 444P, 4481, 448H, 4491, 449H, 44EB, 44EV, 44K1, 44KB, 44KP, 44M1, 44R1, 44RV, 44SH, 44U1, 44VP, 44YB, 46I1, 46K1, 46M1, 480H, 488H, 48AH, 48I1, 48K1, 48U1, 4991, 49EV, 49I1, 49O1, 49UV, 4A0H, 4A41, 4A4B, 4A8H, 4A91, 4ALB, 4ASH, 4BXV, 4E4B, 4EEP, 4EFB, 4ELP, 4EUV, 4EXB, 4FOB, 4I01, 4I0V, 4I4V, 4I61, 4I81, 4I91, 4IEV, 4IY1, 4K0H, 4K91, 4KI1, 4KKH, 4KKP, 4KUP, 4L3P, 4L4P, 4LAB, 4LMP, 4MA1, 4MK1, 4MMP, 4MP1, 4MU1, 4O0B, 4OFB, 4OKB, 4OM1, 4OU1, 4OYB, 4P41, 4P81, 4PO1, 4PYV, 4QLP, 4QQP, 4RAV, 4RK1, 4RRV, 4S8H, 4U3P, 4U91, 4U9V, 4UIV, 4UUP, 4VAV, 4VLP, 4VVV, 4X0B, 4X0V, 4XAB, 4XBV, 4XVV, 4XYV, 4Y0V, 4Y41, 4Y4H, 4Y4V, 4Y81, 4YAV, 4YEV, 4YIV, 4YK1, 4YKB, 4YM1, 4YR1, 4YUV, 4YYB, 5025, 503Z, 5095, 50G5, 50PN, 50QN, 50WP, 50Z7, 5205, 52RN, 53FP, 53M7, 53PZ, 53X7, 53ZZ, 54PP, 550N, 553P, 554P, 5557, 555N, 557Z, 5595, 55EN, 55G5, 55GZ, 55LZ, 55OZ, 55QN, 55Z7, 572N, 577Z, 5787, 57CN, 57FZ, 57QN, 57S7, 57ZZ, 58I7, 58X7, 5955, 5A95, 5AEP, 5CMN, 5CWN, 5E2N, 5EAP, 5ECN, 5EKP, 5EQN, 5F95, 5FLP, 5FQP, 5G05, 5GK5, 5GQ7, 5I37, 5II7, 5IM7, 5L3P, 5LCN, 5LFP, 5LGT, 5LKZ, 5LUP, 5LZZ, 5M07, 5N55, 5NQ7, 5NX7, 5OAZ, 5OFZ, 5OLT, 5OOT, 5P25, 5P95, 5PG5, 5PGZ, 5PPN, 5PQP, 5PW5, 5Q0N, 5Q87, 5QCN, 5QKP, 5QN7, 5QPN, 5QPP, 5QQT, 5QTT, 5QWP, 5QX7, 5S37, 5SS7, 5T6T, 5TQT, 5U4P, 5UGZ, 5W0P, 5W2N, 5W5P, 5WLN, 5WP5, 5WQN, 5XM7, 5Y0T, 5YOT, 5Z25, 5Z4P, 5Z57, 5ZG5, 5ZLZ, 5ZQ7, 5ZWP, 6005, 600V, 601J, 6055, 606V, 60A1, 60C5, 60GT, 60JJ, 60LJ, 60M1, 60R1, 60VT, 60YV, 610J, 61FJ, 6401, 64K1, 6505, 665T, 6665, 66A5, 66F5, 66G5, 66GT, 66KT, 66V5, 66XV, 68OV, 68YV, 690J, 69G5, 69O5, 69XJ, 69XV, 6A55, 6AA1, 6AA5, 6AF5, 6C9J, 6CA5, 6CGJ, 6CYJ, 6EK1, 6ER1, 6FLJ, 6FXJ, 6G0J, 6G65, 6GGJ, 6GJ5, 6GLJ, 6I81, 6I9V, 6J65, 6J6V, 6JET, 6JST, 6JWV, 6K01, 6KET, 6KST, 6L0J, 6L6T, 6LFJ, 6LRT, 6LXJ, 6MA1, 6MM1, 6MX1, 6O8V, 6OV5, 6R0T, 6RET, 6RKT, 6S6T, 6T8V, 6U01, 6U9J, 6UE1, 6UGJ, 6UJJ, 6ULJ, 6UU1, 6V0T, 6VA5, 6VF5, 6VG5, 6VKT, 6VXJ, 6WWV, 6WXV, 6X41, 6X6V, 6XIV, 6XJV, 6XOV, 6Y0V, 6YCJ, 6YXV, 7001, 702N, 7041, 707N, 7081, 70GZ, 70I7, 70KZ, 70ND, 70O7, 70OZ, 70R7, 70U1, 70UZ, 70WD, 70YN, 71QD, 71WD, 720N, 72CN, 72YN, 733Z, 73ED, 73ZD, 7401, 7441, 74M1, 74O1, 74R1, 7641, 76K1, 76M1, 76R1, 76U1, 76X1, 773Z, 7741, 7771, 7781, 77FZ, 77G7, 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8KRH, 8KU1, 8KYH, 8KZ1, 8L4B, 8LBB, 8MG7, 8MR7, 8O4V, 8OI1, 8OK1, 8OOV, 8OX1, 8RAV, 8RHH, 8RUV, 8RX1, 8U4V, 8UK1, 8V4B, 8V9B, 8VHH, 8VRH, 8VVV, 8VYV, 8W0H, 8WFD, 8X07, 8XAH, 8XR1, 8XU1, 8XYB, 8Y6V, 8Y8H, 8YBB, 8YFB, 8YHH, 8YKB, 8YLB, 8YUV, 8YVV, 8YYH, 8Z1D, 8Z4B, 8Z8D, 8Z91, 8ZCD, 8ZEZ, 8ZFD, 8ZFZ, 8ZK1, 8ZO1, 8ZX7, 90A5, 90BJ, 90BV, 90EB, 90EV, 90GJ, 90IJ, 90J5, 90JB, 90LB, 90O1, 90OV, 90XB, 90XV, 9225, 92K5, 9441, 94BH, 94EB, 94KB, 94XB, 9505, 9601, 960V, 96A1, 96J5, 96K1, 96V5, 96WV, 96XV, 984B, 98AV, 98BV, 98FB, 994H, 998B, 998V, 999B, 99A1, 99A5, 99E1, 99G5, 99O1, 99O5, 99WH, 9A05, 9A55, 9A95, 9AA5, 9ABV, 9AE1, 9AEV, 9AI1, 9AO1, 9AOV, 9B0J, 9B0V, 9B8V, 9B9H, 9BCJ, 9BFJ, 9BSJ, 9BVV, 9CCJ, 9CLJ, 9CSJ, 9E01, 9E4B, 9EA1, 9EAB, 9EAV, 9EK1, 9F05, 9F25, 9FA5, 9FBJ, 9FFJ, 9FJB, 9FSJ, 9FXJ, 9G25, 9G2J, 9GIJ, 9GSJ, 9HFB, 9HLB, 9IIJ, 9IJ1, 9J01, 9J0B, 9J0J, 9J9B, 9JCJ, 9JE1, 9K05, 9K41, 9K55, 9K61, 9K8B, 9KA5, 9KI1, 9KK5, 9KU5, 9L2J, 9LCJ, 9O05, 9O0V, 9O95, 9OI1, 9OO1, 9OOV, 9OWV, 9RRV, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ

====Additional known quasi-minimal primes (not necessarily the next)====

P&lt;sub&gt;81993&lt;/sub&gt;SZ

==Unsolved families==

Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')) by covering congruence, algebraic factorization, or combine of them.

{|class=wikitable
|base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family
|-
|13||9{5}||(113×13&lt;sup&gt;''n''&lt;/sup&gt;−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|13||A{3}A||(41×13&lt;sup&gt;''n''+1&lt;/sup&gt;+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|16||{3}AF||(16&lt;sup&gt;''n''+2&lt;/sup&gt;+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1{7}||(23×17&lt;sup&gt;''n''&lt;/sup&gt;−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1F{0}7||32×17&lt;sup&gt;''n''+1&lt;/sup&gt;+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||4{7}A||(71×17&lt;sup&gt;''n''+1&lt;/sup&gt;+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||51{0}D||86×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||70F{0}D||2038×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||8{B}9||(139×17&lt;sup&gt;''n''+1&lt;/sup&gt;−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||9{5}9||(149×17&lt;sup&gt;''n''+1&lt;/sup&gt;+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||95{F}||(2543×17&lt;sup&gt;''n''&lt;/sup&gt;−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||A{D}F||(173×17&lt;sup&gt;''n''+1&lt;/sup&gt;+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}B3||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+190||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}DB||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+232||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2BE||(11×17&lt;sup&gt;''n''+3&lt;/sup&gt;−41579)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2E||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;−2411)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}E9||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+773)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}EE||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+853)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||D0G{D}||(60381×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2860381*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||E9{B}||(3963×17&lt;sup&gt;''n''&lt;/sup&gt;−11)/16||29000||[http://factordb.com/index.php?query=%283963*17%5En-11%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||F1{9}||(4105×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16||1000000||[http://factordb.com/index.php?query=%284105*17%5En-9%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||FD0{D}||(72909×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2872909*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||G{7}F||(263×17&lt;sup&gt;''n''&lt;/sup&gt;+121)/16||29000||[http://factordb.com/index.php?query=%28263*17%5E%28n%2B1%29%2B121%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||7{K}Z||(53×36&lt;sup&gt;''n''+1&lt;/sup&gt;+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||B{0}EUV||11×36&lt;sup&gt;''n''+3&lt;/sup&gt;+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||HM{0}N||634×36&lt;sup&gt;''n''+1&lt;/sup&gt;+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||N{0}YYN||23×36&lt;sup&gt;''n''+3&lt;/sup&gt;+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||O{L}Z||(123×36&lt;sup&gt;''n''+1&lt;/sup&gt;+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||S{0}8H||28×36&lt;sup&gt;''n''+2&lt;/sup&gt;+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|}

For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E

For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D}

=== Base 17 ===

All searched to length 29000

* 1{7}
* 1F{0}7
* 4{7}A
* 51{0}D
* 70F{0}D
* 8{B}9
* 9{5}9
* 95{F}
* A{D}F
* B{0}B3
* B{0}DB
* {B}2BE
* {B}2E
* {B}E9
* {B}EE
* D0G{D}
* E9{B}
* F1{9}
* FD0{D}
* G{7}F

=== Base 21 ===

All searched to length 20000

* 2{7}9D
* 2F{C}A
* 4{3}B
* 5{0}DJ
* {5}FEK
* {7}ID
* 99{0}99H
* {9}0D
* {9}D
* B0{H}6H
* B3{0}EB
* B9{0}E5
* B{D}B
* B{H}6H
* DH{D}
* F{9}D
* {F}35
* G{0}FK
* H{0}7771
* H{D}
* {J}6J

=== Base 36 ===

All searched to length 20000

* 7{K}Z
* B{0}EUV
* HM{0}N
* N{0}YYN
* O{L}Z
* S{0}8H

==Primality certificates for the proven primes &gt; 10&lt;sup&gt;299&lt;/sup&gt;==

See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]]

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime
|-
|9||149||76&lt;sub&gt;329&lt;/sub&gt;2||(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642]
|-
|9||150||27&lt;sub&gt;686&lt;/sub&gt;07||(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486]
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||3×9&lt;sup&gt;1160&lt;/sup&gt;+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423]
|-
|11||1065||A&lt;sub&gt;713&lt;/sub&gt;58||11&lt;sup&gt;715&lt;/sup&gt;−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487]
|-
|11||1066||7&lt;sub&gt;759&lt;/sub&gt;44||(7×11&lt;sup&gt;761&lt;/sup&gt;−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840]
|-
|11||1067||557&lt;sub&gt;1011&lt;/sub&gt;||(607×11&lt;sup&gt;1011&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522]
|-
|13||3165||50&lt;sub&gt;270&lt;/sub&gt;44||5×13&lt;sup&gt;272&lt;/sup&gt;+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005]
|-
|13||3166||9&lt;sub&gt;271&lt;/sub&gt;095||(3×13&lt;sup&gt;274&lt;/sup&gt;−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654]
|-
|13||3167||10&lt;sub&gt;286&lt;/sub&gt;7771||13&lt;sup&gt;290&lt;/sup&gt;+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633]
|-
|13||3168||9&lt;sub&gt;308&lt;/sub&gt;1||(3×13&lt;sup&gt;309&lt;/sup&gt;−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=308&amp;c0=-&amp;EN= 13&lt;sup&gt;308&lt;/sup&gt;−1]
|-
|13||3169||B&lt;sub&gt;341&lt;/sub&gt;C4||(11×13&lt;sup&gt;343&lt;/sup&gt;+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618]
|-
|13||3170||8B&lt;sub&gt;343&lt;/sub&gt;||(107×13&lt;sup&gt;343&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736]
|-
|13||3171||710&lt;sub&gt;371&lt;/sub&gt;111||92×13&lt;sup&gt;374&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609]
|-
|13||3172||75&lt;sub&gt;375&lt;/sub&gt;7||(89×13&lt;sup&gt;376&lt;/sup&gt;+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596]
|-
|13||3173||9B0&lt;sub&gt;391&lt;/sub&gt;9||128×13&lt;sup&gt;392&lt;/sup&gt;+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790]
|-
|13||3174||7B0B&lt;sub&gt;397&lt;/sub&gt;||(15923×13&lt;sup&gt;397&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574]
|-
|13||3175||10&lt;sub&gt;414&lt;/sub&gt;93||13&lt;sup&gt;416&lt;/sup&gt;+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240]
|-
|13||3176||81010&lt;sub&gt;415&lt;/sub&gt;1||17746×13&lt;sup&gt;416&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3177||8110&lt;sub&gt;435&lt;/sub&gt;1||1366×13&lt;sup&gt;436&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3178||B7&lt;sub&gt;486&lt;/sub&gt;||(139×13&lt;sup&gt;486&lt;/sup&gt;−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892]
|-
|13||3179||B&lt;sub&gt;563&lt;/sub&gt;C||(11×13&lt;sup&gt;564&lt;/sup&gt;+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=564&amp;c0=-&amp;EN= 13&lt;sup&gt;564&lt;/sup&gt;−1]
|-
|13||3180||1B&lt;sub&gt;576&lt;/sub&gt;||(23×13&lt;sup&gt;576&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=576&amp;c0=-&amp;EN= 13&lt;sup&gt;576&lt;/sup&gt;−1]
|-
|13||3181||80&lt;sub&gt;693&lt;/sub&gt;87||8×13&lt;sup&gt;695&lt;/sup&gt;+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532]
|-
|13||3182||CC5&lt;sub&gt;713&lt;/sub&gt;||(2021×13&lt;sup&gt;713&lt;/sup&gt;−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353]
|-
|13||3183||B&lt;sub&gt;834&lt;/sub&gt;74||(11×13&lt;sup&gt;836&lt;/sup&gt;−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871]
|-
|13||3184||9&lt;sub&gt;968&lt;/sub&gt;B||(3×13&lt;sup&gt;969&lt;/sup&gt;+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244]
|-
|13||3185||10&lt;sub&gt;1295&lt;/sub&gt;181||13&lt;sup&gt;1298&lt;/sup&gt;+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013]
|-
|13||3186||9&lt;sub&gt;1362&lt;/sub&gt;5||(3×13&lt;sup&gt;1363&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776]
|-
|13||3187||7&lt;sub&gt;1504&lt;/sub&gt;1||(7×13&lt;sup&gt;1505&lt;/sup&gt;−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755]
|-
|13||3188||930&lt;sub&gt;1551&lt;/sub&gt;1||120×13&lt;sup&gt;1552&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3189||720&lt;sub&gt;2297&lt;/sub&gt;2||93×13&lt;sup&gt;2298&lt;/sup&gt;+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910]
|-
|13||3190||1770&lt;sub&gt;2703&lt;/sub&gt;17||267×13&lt;sup&gt;2705&lt;/sup&gt;+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825]
|-
|13||3191||390&lt;sub&gt;6266&lt;/sub&gt;1||48×13&lt;sup&gt;6267&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3192||B0&lt;sub&gt;6540&lt;/sub&gt;BBA||11×13&lt;sup&gt;6543&lt;/sup&gt;+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906]
|-
|13||3193||C&lt;sub&gt;10631&lt;/sub&gt;92||13&lt;sup&gt;10633&lt;/sup&gt;−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750]
|-
|14||649||34D&lt;sub&gt;708&lt;/sub&gt;||47×14&lt;sup&gt;708&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||5×14&lt;sup&gt;19698&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2328||880&lt;sub&gt;246&lt;/sub&gt;7||136×16&lt;sup&gt;247&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is &lt; 10&lt;sup&gt;299&lt;/sup&gt;
|-
|16||2329||D4&lt;sub&gt;263&lt;/sub&gt;D||(199×16&lt;sup&gt;264&lt;/sup&gt;+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238]
|-
|16||2330||E0&lt;sub&gt;261&lt;/sub&gt;4DD||14×16&lt;sup&gt;264&lt;/sup&gt;+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352]
|-
|16||2331||8C0&lt;sub&gt;290&lt;/sub&gt;ED||140×16&lt;sup&gt;292&lt;/sup&gt;+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307]
|-
|16||2332||DA&lt;sub&gt;305&lt;/sub&gt;5||(41×16&lt;sup&gt;306&lt;/sup&gt;−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284]
|-
|16||2333||CE80&lt;sub&gt;422&lt;/sub&gt;D||3304×16&lt;sup&gt;423&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257]
|-
|16||2334||5F&lt;sub&gt;544&lt;/sub&gt;6F||6×16&lt;sup&gt;546&lt;/sup&gt;−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967]
|-
|16||2335||88F&lt;sub&gt;545&lt;/sub&gt;||137×16&lt;sup&gt;545&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2336||BE0&lt;sub&gt;792&lt;/sub&gt;BB||190×16&lt;sup&gt;794&lt;/sup&gt;+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938]
|-
|16||2337||D9&lt;sub&gt;1052&lt;/sub&gt;||(68×16&lt;sup&gt;1052&lt;/sup&gt;−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020]
|-
|16||2338||FAF&lt;sub&gt;1062&lt;/sub&gt;45||251×16&lt;sup&gt;1064&lt;/sup&gt;−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610]
|-
|16||2339||F8&lt;sub&gt;1517&lt;/sub&gt;F||(233×16&lt;sup&gt;1518&lt;/sup&gt;+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824]
|-
|16||2340||20&lt;sub&gt;1713&lt;/sub&gt;321||2×16&lt;sup&gt;1716&lt;/sup&gt;+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735]
|-
|16||2341||300F&lt;sub&gt;1960&lt;/sub&gt;AF||769×16&lt;sup&gt;1962&lt;/sup&gt;−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750]
|-
|16||2342||90&lt;sub&gt;3542&lt;/sub&gt;91||9×16&lt;sup&gt;3544&lt;/sup&gt;+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191]
|-
|16||2343||5BC&lt;sub&gt;3700&lt;/sub&gt;D||(459×16&lt;sup&gt;3701&lt;/sup&gt;+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322]
|-
|16||2344||D0B&lt;sub&gt;17804&lt;/sub&gt;||(3131×16&lt;sup&gt;17804&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511]
|-
|18||547||80&lt;sub&gt;298&lt;/sub&gt;B||8×18&lt;sup&gt;299&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is an algebraic factor of ''N''+1, factordb entry of 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is [http://factordb.com/index.php?id=1100000000900149167]
|-
|18||548||H&lt;sub&gt;766&lt;/sub&gt;FH||18&lt;sup&gt;768&lt;/sup&gt;−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490]
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||12×18&lt;sup&gt;6270&lt;/sup&gt;+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437]
|-
|20||3301||H&lt;sub&gt;247&lt;/sub&gt;A0H||(17×20&lt;sup&gt;250&lt;/sup&gt;−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619]
|-
|20||3302||7&lt;sub&gt;249&lt;/sub&gt;A7||(7×20&lt;sup&gt;251&lt;/sup&gt;+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602]
|-
|20||3303||J7&lt;sub&gt;270&lt;/sub&gt;||(368×20&lt;sup&gt;270&lt;/sup&gt;−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462]
|-
|20||3304||J&lt;sub&gt;330&lt;/sub&gt;CCC7||20&lt;sup&gt;334&lt;/sup&gt;−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572]
|-
|20||3305||40&lt;sub&gt;387&lt;/sub&gt;404B||4×20&lt;sup&gt;391&lt;/sup&gt;+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563]
|-
|20||3306||EC0&lt;sub&gt;429&lt;/sub&gt;7||292×20&lt;sup&gt;430&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702]
|-
|20||3307||G&lt;sub&gt;447&lt;/sub&gt;99||(16×20&lt;sup&gt;449&lt;/sup&gt;−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753]
|-
|20||3308||3A&lt;sub&gt;527&lt;/sub&gt;3||(67×20&lt;sup&gt;528&lt;/sup&gt;−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531]
|-
|20||3309||E&lt;sub&gt;566&lt;/sub&gt;C7||(14×20&lt;sup&gt;568&lt;/sup&gt;−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516]
|-
|20||3310||JCJ&lt;sub&gt;629&lt;/sub&gt;||393×20&lt;sup&gt;629&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|20||3311||J&lt;sub&gt;655&lt;/sub&gt;05J||20&lt;sup&gt;658&lt;/sup&gt;−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052]
|-
|20||3312||50&lt;sub&gt;1163&lt;/sub&gt;AJ||5×20&lt;sup&gt;1165&lt;/sup&gt;+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412]
|-
|20||3313||CD&lt;sub&gt;2449&lt;/sub&gt;||(241×20&lt;sup&gt;2449&lt;/sup&gt;−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915]
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||16×20&lt;sup&gt;6270&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457]
|-
|22||7984||I7G0&lt;sub&gt;254&lt;/sub&gt;H||8882×22&lt;sup&gt;255&lt;/sup&gt;+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788]
|-
|22||7985||D0&lt;sub&gt;255&lt;/sub&gt;5EEF||13×22&lt;sup&gt;259&lt;/sup&gt;+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932]
|-
|22||7986||IK&lt;sub&gt;322&lt;/sub&gt;F||(398×22&lt;sup&gt;323&lt;/sup&gt;−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145]
|-
|22||7987||C0&lt;sub&gt;340&lt;/sub&gt;G9||12×22&lt;sup&gt;342&lt;/sup&gt;+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159]
|-
|22||7988||77E&lt;sub&gt;348&lt;/sub&gt;K7||(485×22&lt;sup&gt;350&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779]
|-
|22||7989||J&lt;sub&gt;379&lt;/sub&gt;KJ||(19×22&lt;sup&gt;381&lt;/sup&gt;+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027]
|-
|22||7990||J&lt;sub&gt;388&lt;/sub&gt;EJ||(19×22&lt;sup&gt;390&lt;/sup&gt;−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729]
|-
|22||7991||DJ&lt;sub&gt;400&lt;/sub&gt;||(292×22&lt;sup&gt;400&lt;/sup&gt;−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110]
|-
|22||7992||E&lt;sub&gt;404&lt;/sub&gt;K7||(2×22&lt;sup&gt;406&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298]
|-
|22||7993||66F&lt;sub&gt;453&lt;/sub&gt;B3||(971×22&lt;sup&gt;455&lt;/sup&gt;−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809]
|-
|22||7994||L0&lt;sub&gt;454&lt;/sub&gt;B63||21×22&lt;sup&gt;457&lt;/sup&gt;+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331]
|-
|22||7995||L&lt;sub&gt;483&lt;/sub&gt;G3||22&lt;sup&gt;485&lt;/sup&gt;−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730]
|-
|22||7996||E60&lt;sub&gt;496&lt;/sub&gt;L||314×22&lt;sup&gt;497&lt;/sup&gt;+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239]
|-
|22||7997||I&lt;sub&gt;626&lt;/sub&gt;AF||(6×22&lt;sup&gt;628&lt;/sup&gt;−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334]
|-
|22||7998||K0&lt;sub&gt;760&lt;/sub&gt;EC1||20×22&lt;sup&gt;763&lt;/sup&gt;+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415]
|-
|22||7999||J0&lt;sub&gt;767&lt;/sub&gt;IGGJ||19×22&lt;sup&gt;771&lt;/sup&gt;+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567]
|-
|22||8000||7&lt;sub&gt;959&lt;/sub&gt;K7||(22&lt;sup&gt;961&lt;/sup&gt;+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817]
|-
|22||8001||L&lt;sub&gt;2385&lt;/sub&gt;KE7||22&lt;sup&gt;2388&lt;/sup&gt;−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774]
|-
|22||8002||7&lt;sub&gt;3815&lt;/sub&gt;2L||(22&lt;sup&gt;3817&lt;/sup&gt;−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839]
|-
|24||3400||I0&lt;sub&gt;241&lt;/sub&gt;I5||18×24&lt;sup&gt;243&lt;/sup&gt;+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037]
|-
|24||3401||D0&lt;sub&gt;259&lt;/sub&gt;KKD||13×24&lt;sup&gt;262&lt;/sup&gt;+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725]
|-
|24||3402||C7&lt;sub&gt;298&lt;/sub&gt;||(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235]
|-
|24||3403||20&lt;sub&gt;313&lt;/sub&gt;7||2×24&lt;sup&gt;314&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241]
|-
|24||3404||BC0&lt;sub&gt;331&lt;/sub&gt;B||276×24&lt;sup&gt;332&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842]
|-
|24||3405||N&lt;sub&gt;2644&lt;/sub&gt;LLN||24&lt;sup&gt;2647&lt;/sup&gt;−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089]
|-
|24||3406||D&lt;sub&gt;2698&lt;/sub&gt;LD||(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876]
|-
|24||3407||A0&lt;sub&gt;2951&lt;/sub&gt;8ID||10×24&lt;sup&gt;2954&lt;/sup&gt;+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654]
|-
|24||3408||88N&lt;sub&gt;5951&lt;/sub&gt;||201×24&lt;sup&gt;5951&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606]
|-
|30||2613||AN&lt;sub&gt;206&lt;/sub&gt;||(313×30&lt;sup&gt;206&lt;/sup&gt;−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073]
|-
|30||2614||M&lt;sub&gt;241&lt;/sub&gt;QB||(22×30&lt;sup&gt;243&lt;/sup&gt;+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295]
|-
|30||2615||M0&lt;sub&gt;547&lt;/sub&gt;SS7||22×30&lt;sup&gt;550&lt;/sup&gt;+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988]
|-
|30||2616||C0&lt;sub&gt;1022&lt;/sub&gt;1||12×30&lt;sup&gt;1023&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|30||2617||5&lt;sub&gt;4882&lt;/sub&gt;J||(5×30&lt;sup&gt;4883&lt;/sup&gt;+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423]
|-
|30||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||25×30&lt;sup&gt;34205&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|}

==Unproven PRPs==

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this PRP
|-
|11||1068||57&lt;sub&gt;62668&lt;/sub&gt;||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000003573679860]
|-
|13||3194||C5&lt;sub&gt;23755&lt;/sub&gt;C||(149×13&lt;sup&gt;23756&lt;/sup&gt;+79)/12||[http://factordb.com/index.php?id=1100000003590647776]
|-
|13||3195||80&lt;sub&gt;32017&lt;/sub&gt;111||8×13&lt;sup&gt;32020&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000000490878060]
|-
|16||2345||DB&lt;sub&gt;32234&lt;/sub&gt;||(206×16&lt;sup&gt;32234&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000002383583629]
|-
|16||2346||4&lt;sub&gt;72785&lt;/sub&gt;DD||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15||[http://factordb.com/index.php?id=1100000003615909841]
|-
|22||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21||[http://factordb.com/index.php?id=1100000003594696838]
|-
|30||2618||I0&lt;sub&gt;24608&lt;/sub&gt;D||18×30&lt;sup&gt;24609&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003593967511]
|}

All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10&lt;sup&gt;16&lt;/sup&gt;. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]])

==Proof==

===Base 2===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.

===Base 3===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (2,1), (2,2)

* Case (1,1):
** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes)
*** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime.

===Base 4===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (2,1), (2,3), (3,1), (3,3)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.

===Base 5===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4)

* Case (1,1):
** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes)
*** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes)
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes)
***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb])
***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes)
**** The smallest prime of the form 1{4}4 is 14444.
**** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes)
*** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes)
**** For the 3{0,3}1 family, we can separate this family to four families:
***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01.
****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime.
***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031.
***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001.
***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31.
****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime.
**** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.
* Case (3,4):
** 34 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes)
**** The smallest prime of the form 4{4}1 is 44441.
**** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime.
* Case (4,2):
** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime.

===Base 6===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes)
*** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes)
**** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 4{0}41 is 40041
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes)
*** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.

===Base 7===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

* Case (1,1):
** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes)
*** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {}
**** If there are &gt;=3 number of 1 and 5 in the {}:
***** If there is 111 in the {}, then we have the prime 11111
***** If there is 115 in the {}, then the prime 115 is a subsequence
***** If there is 151 in the {}, then the prime 115 is a subsequence
***** If there is 155 in the {}, then the prime 155 is a subsequence
***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime
***** If there is 515 in the {}, then the prime 515 is a subsequence
***** If there is 551 in the {}, then the prime 551 is a subsequence
***** If there is 555 in the {}, then the prime 551 is a subsequence
**** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1
***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1
****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1
******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21
******** The smallest prime of the form 11{0}21 is 1100021
******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime
******** The smallest prime of the form 1001{0}21 is 100121
********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families
***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1
****** Since 1051 is prime, we only need to consider the family 15{0}1
******* The smallest prime of the form 15{0}1 is 150001
* Case (1,2):
** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes)
*** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {}
**** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime.
**** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime.
**** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2
***** Since 1022 and 1202 are primes, we only need to consider the number 122
****** 122 is not prime.
**** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime.
* Case (1,3):
** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** 14 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime.
* Case (1,6):
** 16 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime.
* Case (2,2):
** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,6):
** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes)
*** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes)
**** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's)
***** For the 3{0,1}1 family:
****** If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
****** If there are no 1's between (3,1), then the form will be 3{0}1
******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime.
****** If there are no 0's between (3,1), then the form will be 3{1}1
******* The smallest prime of the form 3{1}1 is 31111
****** If there are exactly 1 0's between (3,1), then there must be &lt;3 1's between (3,1), or 31111 will be a subsequence.
******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime.
******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011
******** Neither 3101 nor 3011 is prime.
******* If there are no 1's between (3,1), then the number must be 301
******** 301 is not prime.
***** For the 3{3}3{0,1}1 family:
****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence.
****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1
******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence.
******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime.
******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime.
******* If there are no 1's between (33,1), then the form will be 33{0}1
******** The smallest prime of the form 33{0}1 is 33001
****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1
******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence.
******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301
******** The smallest prime of the form 3{3}301 is 33333301
******* If there are no 0's between (3{3}3,1), then the form is 3{3}31
******** The smallest prime of the form 3{3}31 is 33333333333333331
**** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1
***** Numbers containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1,3}3{0,1,3}1
******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes)
******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes)
********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes)
********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence.
********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime.
********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1
*********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime.
********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime.
***** Numbers not containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1}1
******* If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1
******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime.
******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1
******** If there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111
********* None of 31151, 31511, 35111 are primes.
******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 351
********* 351 is not prime.
******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111
********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes.
********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime.
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1
********* Neither 3051 nor 3501 is prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** If there are &gt;=2 5's in {}, then 553 will be a subsequence.
*** If there are no 5's in {}, then the family will be 3{0,3,6}3
**** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
*** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3
**** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes)
***** The smallest prime of the form 3{0}53 is 300053
* Case (3,4):
** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes)
*** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes)
**** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes)
**** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes)
* Case (3,5):
** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes)
*** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence.
*** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence.
*** If there are at least two 1's in {}, then 115 will be a subsequence.
*** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5
**** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime.
*** If there is no 1's in {}, then the family will be 3{0,4,5}5
**** If there are at least to 4's in {}, then 344 and 445 will be subsequences.
**** If there is no 4's in {}, then the family will be 3{0,5}5
***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5
****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime.
****** The smallest prime of the form 3{5}5 is 35555
**** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5
***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes)
****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime.
* Case (3,6):
** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,2):
** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes)
*** If there is no 5's in {}, then the family will be 4{0,2,4,6}4
**** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4
**** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes)
***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence.
***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4
****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes)
***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4
****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4
******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes)
* Case (4,5):
** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** If there are at least two 5's in {}, then 4555 will be a subsequence.
*** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there is no 5's in {}, then the family will be 4{0}5
**** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime.
* Case (4,6):
** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (5,1):
** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 3's in {}, then 533 will be a subsequence.
*** If there is no 3's in {}, then the family will be 5{0,1}1
**** Since 5011 is prime, we only need to consider the family 5{1}{0}1
***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes)
****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime.
****** The smallest prime of the form 51{0}1 is 5100000001
****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime.
****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime.
*** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1
**** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence.
***** Thus we only need to consider the family 5{0}3{0,1}1
****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime.
****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime.
****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence.
****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1
******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0)
******** The smallest prime of the form 5311{0}1 is 531101
* Case (5,2):
** 52 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 1's in {}, then 113 will be a subsequence.
*** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
*** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime.
* Case (5,4):
** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes)
*** If there are no 5's in {}, then the family will be 5{0,3}4
**** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime.
*** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence.
*** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there are at least two 5's in {}, then 5554 will be a subsequence.
* Case (5,5):
** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes)
*** If there are no 4's in {}, then the family will be 5{0,5}5
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
*** If there are no 5's in {}, then the family will be 5{0,4}5
**** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime.
*** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence.
* Case (5,6):
** 56 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,2):
** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (6,3):
** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,4):
** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are at least three 3's in {}, then 3334 will be a subsequence.
*** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4
**** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence.
**** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence.
**** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence.
**** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence.
**** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence.
**** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence.
**** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence.
**** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence.
**** Thus the number can only be 634
***** 634 is not prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,6):
** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}6
**** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime.
*** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6
**** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime.
*** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence.
*** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence.

===Base 8===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7)

* Case (1,1):
** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes)
**** 171 is not prime.
**** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 &gt; 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 &gt; 1, this factorization is nontrivial), thus cannot be prime.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes)
*** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime)
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes)
*** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes)
*** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes)
**** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime.
**** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes)
***** The smallest prime of the form 33{4}1 is 3344441
***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime.
**** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes)
***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime.
***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes)
****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes.
* Case (3,3):
** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes)
*** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes)
**** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime)
**** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461
***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes)
****** The smallest prime of the form 4{4}41 is 444444441
****** The smallest prime of the form 4{4}641 is 444641
***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes)
****** The smallest prime of the form 4{4}411 is 444444441
****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes)
***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461
****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime)
**** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes)
***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime)
***** The smallest prime of the form 4{4}641 is 444641
* Case (4,3):
** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes)
*** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes)
**** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime.
**** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime.
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes)
*** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes)
**** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb])
**** The smallest prime of the form 4{4}77 is 4444477
**** The smallest prime of the form 4{7}7 is 47777
**** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime)
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes)
*** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes)
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
**** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes)
***** 500525 is not prime.
***** The smallest prime of the form 5{5}25 is 555555555555525
***** The smallest prime of the form 5{5}025 is 55555025
***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes)
***** The smallest prime of the form 5{5}0525 is 5550525
***** The smallest prime of the form 5{5}00525 is 5500525
***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes)
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence)
**** The form is 6{0,1,7}4{0,1,7}1
***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1
****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411
******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41
******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41
********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.
********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime.
********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime.
******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411
******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime)
*** Numbers not containing 4:
**** The form is 6{0,1,7}1
***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1
****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime.
****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1
******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1
******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.
******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime)
* Case (6,3):
** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,7):
** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them
**** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17
***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime.
**** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77
***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime.
**** If there is neither 1 nor 7 between them, then the form is 6{0,4}7
***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes)
****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime.
****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime.
* Case (7,1):
** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes)
*** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes)
**** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes)
***** The smallest prime of the form 7{7}1 is 7777777777771
***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime)
***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime.
***** The smallest prime of the form 7{7}4444441 is 77774444441
****** Since this prime has just 4 7's, we only need to consider the families with &lt;=3 7's
******* The smallest prime of the form 7{4}1 is 744444441
******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime.
******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes)
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.

===Base 10===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,9):
** 19 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 2{0}21 is 20021
**** The smallest prime of the form 22{0}1 is 22000001
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes)
*** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes)
**** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime.
**** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime.
* Case (2,9):
** 29 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,9):
** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** 47 is prime, and thus the only minimal prime in this family.
* Case (4,9):
** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime.
* Case (5,1):
** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes)
**** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes)
***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 5{5}1 is 555555555551
**** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581
***** 581 is not prime.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,7):
** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027
**** The smallest prime of the form 52{0}7 is 5200007
* Case (5,9):
** 59 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,3):
** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,9):
** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49
***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49
****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes)
******* The smallest prime of the form 6{6}49 is 666649
******** Since this prime has just 4 6's, we only need to consider the families with &lt;=3 6's
********* The smallest prime of the form 6{0}49 is 60000049
********* The smallest prime of the form 66{0}49 is 66000049
********* The smallest prime of the form 666{0}49 is 66600049
* Case (7,1):
** 71 is prime, and thus the only minimal prime in this family.
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,9):
** 79 is prime, and thus the only minimal prime in this family.
* Case (8,1):
** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes)
**** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes)
***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime)
***** The smallest prime of the form 80{5}1 is 80555551
* Case (8,3):
** 83 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime.
* Case (8,9):
** 89 is prime, and thus the only minimal prime in this family.
* Case (9,1):
** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes)
**** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021
***** None of 921, 9201, 9021 are primes.
**** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801
***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
* Case (9,3):
** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (9,7):
** 97 is prime, and thus the only minimal prime in this family.
* Case (9,9):
** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9
***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes)
****** The smallest prime of the form 94{6}9 is 946669

===Base 12===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,B):
** 1B is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes)
*** The smallest prime of the form 2{0}1 is 2001
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (2,B):
** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes)
**** None of 20B, 22B, 29B, 209B, 220B are primes.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,B):
** 3B is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes)
**** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 44{A}1 is 44AAA1
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes)
**** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077
* Case (4,B):
** 4B is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes)
*** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime.
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (5,B):
** 5B is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,5):
** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes)
**** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 6{0}A5 is 600A5
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,B):
** 6B is prime, and thus the only minimal prime in this family.
* Case (7,1):
** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes)
*** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1
**** None of 741, 7B1, 74B1 are primes.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,B):
** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes)
**** The smallest prime of the form 7{9}B is 7999B
**** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes)
**** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime)
* Case (8,1):
** 81 is prime, and thus the only minimal prime in this family.
* Case (8,5):
** 85 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** 87 is prime, and thus the only minimal prime in this family.
* Case (8,B):
** 8B is prime, and thus the only minimal prime in this family.
* Case (9,1):
** 91 is prime, and thus the only minimal prime in this family.
* Case (9,5):
** 95 is prime, and thus the only minimal prime in this family.
* Case (9,7):
** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes)
**** None of 947, 977, 997, 9477, 9977 are primes.
* Case (9,B):
** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes)
**** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes)
***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime)
***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime)
**** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes)
***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes)
****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime.
****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime)
****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime.
**** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B
***** None of 99B, 999B, 979B, 9799B, 9979B are primes.
* Case (A,1):
** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes)
**** For the A{A}{0}1 family:
***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form AA{0}1 is AA000001
***** The smallest prime of the form AAA{0}1 is AAA0001
***** The smallest prime of the form AAAA{0}1 is AAAA1
****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc.
**** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime.
* Case (A,5):
** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes)
*** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes)
**** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form A{0}65 is A00065
* Case (A,7):
** A7 is prime, and thus the only minimal prime in this family.
* Case (A,B):
** AB is prime, and thus the only minimal prime in this family.
* Case (B,1):
** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes)
**** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1
***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1
****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime)
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1
****** The smallest prime of the form B{A}01 is BAA01
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
**** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1
***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1
****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1
******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1
******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes.
****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1
******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime)
***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes)
****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B04{A}1 is B04A1
****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime)
**** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1
***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1
****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1
******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime.
******* The smallest prime of the form B{B}AA1 is BBBAA1
****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1
******* The smallest prime of the form B{B}AA1 is BBBAA1
**** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes)
***** The smallest prime of the form B{B}B1 is BBBB1
***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form B{B}BAA1 is BBBAA1
* Case (B,5):
** B5 is prime, and thus the only minimal prime in this family.
* Case (B,7):
** B7 is prime, and thus the only minimal prime in this family.
* Case (B,B):
** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes)
*** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B
**** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B
***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime.
***** For the B{0,B}9B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes)
******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B
******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 &gt; 1, 10^n+5 ≥ 10^1+5 = 15 &gt; 1, this factorization is nontrivial), thus cannot be prime.
***** For the B{0,B}99B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes)
******* The smallest prime of the form B{0}99B is B00099B
******* The smallest prime of the form B{B}99B is BBBBBB99B
***** For the B{0,B}999B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes)
******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes)
******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime)

== Examples of families which can be ruled out as contain no primes &gt; ''b'' ==

It is not known if this problem is solvable:

Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''?

It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime &gt; ''b'' or not. We use two different heuristic strategies to show that such families contain no primes &gt; ''b''.

In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;.

Examples of first strategy: (we can show that the corresponding numbers are &gt; all elements in ''S'', if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial)

* In base 10, all numbers of the form 4{6}9 are divisible by 7
* In base 6, all numbers of the form 4{0}1 are divisible by 5
* In base 15, all numbers of the form 9{6}8 are divisible by 11
* In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5}
* In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3}
* In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5}
* In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13}
* In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17}
* In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13}
* In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13}

Examples of second strategy: (we can show that both factors are &gt; 1, if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial)

* In base 9, all numbers of the form {1} factored as difference of squares
* In base 8, all numbers of the form 1{0}1 factored as sum of cubes
* In base 9, all numbers of the form 3{8} factored as difference of squares
* In base 16, all numbers of the form 8{F} factored as difference of squares
* In base 16, all numbers of the form {F}7 factored as difference of squares
* In base 9, all numbers of the form 3{1} factored as difference of squares
* In base 16, all numbers of the form {4}1 factored as difference of squares
* In base 16, all numbers of the form 1{5} factored as difference of squares
* In base 16, all numbers of the from {C}D factored as ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;
* In base 16, all numbers of the form B{4}1 factored as difference of squares

Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are &gt; all elements in S, and for the part of the second strategy, both factors are &gt; 1, if n makes corresponding numbers &gt; b, thus these factorizations are nontrivial)

* In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd
* In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even

== Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes &gt; ''b'' ==

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-powers factorization

=== 1{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 1{0}3 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== 1{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 1{0}5 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 5: Finite covering set {5}

=== 1{0}6 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== 1{0}7 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 7: Finite covering set {7}

=== 1{0}z ===

(none)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

(none)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {1} ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with ''r''&gt;1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2))

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' == 0 mod 2: Finite covering set {2}

=== 1{2} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 1{3} ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361)
* ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865)

=== 1{4} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization

=== 1{z} ===

(none)

=== 2{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 2{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {2}1 ===

* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 2{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 3{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {3}1 ===

* ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121)

=== 3{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 4{0}1 ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 4{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {4}1 ===

* ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225)

=== 4{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 5{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 5{z} ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}
* ''b'' = 6''m''&lt;sup&gt;2&lt;/sup&gt; with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014)

=== 6{0}1 ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== 6{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 7{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 7{z} ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993)
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Difference-of-cubes factorization

=== 8{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022)
* ''b'' = 467: Finite covering set {3, 5, 7, 19, 37}
* ''b'' = 722: Finite covering set {3, 5, 13, 73, 109}
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Sum-of-cubes factorization
* ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2

=== 8{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 9{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 9{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{0}1 ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{z} ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 307: Finite covering set {5, 11, 29}
* ''b'' = 901: Finite covering set {7, 11, 13, 19}

=== C{0}1 ===

* ''b'' == 1 mod 13: Finite covering set {13}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 296, 901: Finite covering set {7, 11, 13, 19}
* ''b'' = 562, 828, 900: Finite covering set {7, 13, 19}
* ''b'' = 563: Finite covering set {5, 7, 13, 19, 29}
* ''b'' = 597: Finite covering set {5, 13, 29}

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

(none)

=== {y}z ===

(none)

=== z{0}1 ===

(none)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3))
* ''b'' = 4''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt; (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}t ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 7: Finite covering set {7}

=== {z}u ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== {z}v ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 5: Finite covering set {5}

=== {z}w ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== {z}x ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== {z}y ===

* ''b'' == 0 mod 2: Finite covering set {2}

== Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) ==

Format: base (length)

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

(none)

=== 1{0}2 ===

(none)

=== 1{0}3 ===

(none)

=== 1{0}4 ===

53 (13403)

113 (10647)

=== 1{0}z ===

113 (20089)

123 (64371)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

(none)

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

208 (26682)

607 (11032)

828 (19659)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

201 (31276)

222 (52727)

227 (36323)

327 (135983)

425 (11231)

710 (24112)

717 (37508)

719 (13420)

=== {1} ===

152 (270217)

184 (16703)

200 (17807)

311 (36497)

326 (26713)

331 (25033)

371 (15527)

485 (99523)

629 (32233)

649 (43987)

670 (18617)

684 (22573)

691 (62903)

693 (41189)

731 (15427)

752 (32833)

872 (10093)

932 (20431)

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

(none)

=== 1{z} ===

107 (21911)

170 (166429)

278 (43909)

303 (40175)

383 (20957)

515 (58467)

522 (62289)

578 (129469)

590 (15527)

647 (21577)

662 (16591)

698 (127559)

704 (62035)

845 (39407)

938 (40423)

969 (24097)

989 (26869)

=== 2{0}1 ===

101 (192276)

206 (46206)

218 (333926)

236 (161230)

257 (12184)

305 (16808)

467 (126776)

578 (44166)

626 (174204)

695 (94626)

752 (26164)

788 (72918)

869 (49150)

887 (27772)

899 (15732)

932 (13644)

=== 2{z} ===

432 (16003)

=== 3{0}1 ===

912 (132174)

=== 3{z} ===

72 (1119850)

212 (34414)

218 (23050)

270 (89662)

303 (198358)

312 (51566)

422 (21738)

480 (93610)

513 (38032)

527 (46074)

566 (23874)

650 (498102)

686 (16584)

758 (15574)

783 (12508)

800 (33838)

921 (98668)

947 (10056)

=== 4{0}1 ===

107 (32587)

227 (13347)

257 (160423)

355 (10990)

410 (144079)

440 (56087)

452 (14155)

482 (30691)

542 (15983)

579 (67776)

608 (20707)

635 (11723)

650 (96223)

679 (69450)

737 (269303)

740 (58043)

789 (149140)

797 (468703)

920 (103687)

934 (101404)

962 (84235)

=== 4{z} ===

14 (19699)

68 (13575)

254 (15451)

800 (20509)

=== 5{0}1 ===

326 (400786)

350 (20392)

554 (10630)

662 (13390)

926 (40036)

=== 5{z} ===

258 (212135)

272 (148427)

299 (64898)

307 (26263)

354 (25566)

433 (283919)

635 (36163)

678 (40859)

692 (45447)

719 (20552)

768 (70214)

857 (23083)

867 (61411)

972 (36703)

=== 6{0}1 ===

108 (16318)

129 (16797)

409 (369833)

522 (52604)

587 (24120)

643 (164916)

762 (11152)

789 (27297)

986 (21634)

=== 6{z} ===

68 (25396)

332 (15222)

338 (42868)

362 (146342)

488 (33164)

566 (164828)

980 (50878)

986 (12506)

1016 (23336)

=== 7{0}1 ===

398 (17473)

1004 (54849)

=== 7{z} ===

97 (192336)

170 (15423)

194 (38361)

202 (155772)

282 (21413)

283 (164769)

332 (13205)

412 (29792)

560 (19905)

639 (10668)

655 (53009)

811 (31784)

814 (17366)

866 (108591)

908 (61797)

962 (31841)

992 (10605)

997 (15815)

=== 8{0}1 ===

23 (119216)

53 (227184)

158 (123476)

254 (67716)

320 (52004)

410 (279992)

425 (94662)

513 (19076)

518 (11768)

596 (148446)

641 (87702)

684 (23387)

695 (39626)

785 (900326)

788 (11408)

893 (86772)

908 (243440)

920 (107822)

962 (47222)

998 (81240)

1013 (43872)

=== 8{z} ===

138 (35686)

412 (12154)

788 (11326)

990 (23032)

=== 9{0}1 ===

248 (39511)

592 (96870)

=== 9{z} ===

431 (43574)

446 (152028)

458 (126262)

599 (11776)

846 (12781)

=== A{0}1 ===

173 (264235)

198 (47665)

311 (314807)

341 (106009)

449 (18507)

492 (42843)

605 (12395)

708 (17563)

710 (31039)

743 (285479)

744 (137056)

786 (68169)

800 (15105)

802 (149320)

879 (25004)

929 (13065)

977 (125873)

986 (48279)

1004 (10645)

=== A{z} ===

368 (10867)

488 (10231)

534 (80328)

662 (13307)

978 (14066)

=== B{0}1 ===

710 (15272)

740 (33520)

878 (227482)

=== B{z} ===

153 (21660)

186 (112718)

439 (18752)

593 (16064)

602 (36518)

707 (10573)

717 (67707)

=== C{0}1 ===

68 (656922)

219 (29231)

230 (94751)

312 (21163)

334 (83334)

353 (20262)

359 (61295)

457 (10024)

481 (45941)

501 (20140)

593 (42779)

600 (11242)

604 (17371)

641 (26422)

700 (91953)

887 (13961)

919 (45359)

923 (64365)

992 (10300)

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

(none)

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

38 (136212)

83 (21496)

113 (286644)

188 (13508)

401 (103670)

417 (21003)

458 (46900)

494 (21580)

518 (129372)

527 (65822)

602 (17644)

608 (36228)

638 (74528)

663 (47557)

723 (24536)

758 (50564)

833 (12220)

904 (13430)

938 (50008)

950 (16248)

=== z{0}1 ===

202 (46774)

251 (102979)

272 (16681)

297 (14314)

298 (60671)

326 (64757)

347 (69661)

363 (142877)

452 (71941)

543 (10042)

564 (38065)

634 (84823)

788 (13541)

869 (12289)

890 (37377)

953 (60995)

1004 (29685)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

53 (21942)

124 (16426)

175 (31626)

188 (22036)

316 (48538)

365 (25578)

373 (24006)

434 (10090)

530 (11086)

545 (12346)

560 (15072)

596 (12762)

701 (12576)

706 (10656)

821 (13536)

833 (17116)

966 (14820)

983 (11272)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}y ===

317 (13896)

== Bases 2≤''b''≤1024 which have these families as unsolved families ==

Unsolved families are families which are neither primes (&gt;''b'') found nor can be ruled out as contain no primes &gt; ''b''

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608)

1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000)

1{0}3: 646, 718, 998 (length limit: 2000)

1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000)

1{0}5

1{0}6

1{0}7

1{0}8

1{0}9

1{0}A

1{0}B

1{0}C

1{0}D

1{0}E

1{0}F

1{0}G

1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000)

1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000)

10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000)

11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000)

{1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000)

{1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000)

11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z})

{1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000)

{1}z

1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000)

1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995

1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000)

1{z}: 581, 992, 1019 (length limit: ≥100000)

2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000)

2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000)

2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000)

{2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000)

2{z}: 588, 972 (length limit: ≥100000)

3{0}1: 718 (length limit: ≥100000)

3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000)

3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000)

{3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000)

3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000)

4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000)

4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000)

{4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000)

4{z}: 338, 998 (length limit: ≥100000)

5{0}1: 308, 512, 824 (length limit: ≥100000)

5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000)

6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000)

6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000)

7{0}1: (none)

7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000)

8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000)

8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000)

9{0}1: 724, 884 (length limit: ≥100000)

9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000)

A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000)

A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000)

B{0}1: 560, 770, 968 (length limit: ≥100000)

B{z}: 263, 615, 912, 978 (length limit: ≥100000)

C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000)

C{z}

D{0}1

D{z}

E{0}1

E{z}

F{0}1

F{z}

G{0}1

{&amp;#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000)

{&amp;#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143)

&amp;#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000)

${&amp;#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2)

x{z}

y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000)

z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000)

{y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000)

{z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326)

zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z})

{z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000)

{z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1)

{z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000)

{z}k

{z}l

{z}m

{z}n

{z}o

{z}p

{z}q

{z}r

{z}s

{z}t

{z}u

{z}v

{z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000)

{z}x: (none)

{z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000)

== List of lengths for quasi-minimal primes in some simple families ==

[https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024]

NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases &lt;=7, family {z}x in bases &lt;=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime &gt; base) in this base)

RC: this family can be proven to only contain composite numbers (only count numbers &gt; base)

unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers &gt; base)

Background color: red for title (bases or families), green for length &gt; 10000, orange for 2500 &lt; length ≤ 10000, white for length ≤ 2500, cyan for &quot;RC&quot;, pink for &quot;NB&quot;, yellow for &quot;unknown&quot;.

Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families.

== References ==

* [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem]

* [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are &gt; ''b'' for bases ''b''≤16]

* [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary]

* [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia]

* [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime&gt;base is not required) in The Prime Puzzles &amp; Problems Connection]

* [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles &amp; Problems Connection]

* [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16]

* [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime&gt;base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29]

* [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime&gt;base is not required, search limits of lengths: 10000 for all b]

* [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime&gt;base is not required]

* [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime&gt;base is not required]

* [http://www.prothsearch.com/sierp.html the Sierpinski problem]

* [http://www.prothsearch.com/rieselprob.html the Riesel problem]

* [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem]

* [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030]

* [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100]

* [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem]

* [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024]

* [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project]

* [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project]

* [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count]

* [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty]

* [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers]

* [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares]

* [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software]

* [https://sourceforge.net/projects/openpfgw/ pfgw software]

* [http://jpenne.free.fr/index2.html llr software]

* [http://www.ellipsa.eu/public/primo/primo.html PRIMO software]

* [https://primes.utm.edu/prove/index.html website for primality proving]

* [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!]

* [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime&gt;base is not required]

* [https://primes.utm.edu/primes/lists/all.txt top proven primes]

* [http://www.primenumbers.net/prptop/prptop.php top PRPs]

* [http://factordb.com online factor database, including many primes which are minimal primes in a small base]</text>
      <sha1>hn22432vf8qf4psxk6rchdfbsrsxdkn</sha1>
    </revision>
    <revision>
      <id>2412024</id>
      <parentid>2412023</parentid>
      <timestamp>2022-08-05T03:40:47Z</timestamp>
      <contributor>
        <ip>2402:7500:901:2B90:BD69:8142:3D96:F08F</ip>
      </contributor>
      <comment>/* Unsolved families */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1960599" xml:space="preserve">A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime &gt; ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime &gt; 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 &gt; 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 &gt; 10 by changing the order.

(using A−Z to represent digit values 10 to 35)

For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')).

I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E&lt;sub&gt;800873&lt;/sub&gt;, and base 36 has quasi-minimal prime P&lt;sub&gt;81993&lt;/sub&gt;SZ).

Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to:

* Prove that all elements in ''S'', when read as base ''b'' representation, are primes &gt; ''b''.

* Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are &gt; ''b'', are composite.

* Prove that all primes &gt; ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are &gt; ''b'', are composite).

e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to:

* Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes &gt; 10.

* Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are &gt; 10 are composite.

* Prove that all primes &gt; 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers &gt; 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite).

==Condensed table==

{|class=wikitable
|''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form of largest known quasi-minimal prime base ''b''
|-
|2||1||11||2||3
|-
|3||3||111||3||13
|-
|4||5||221||3||41
|-
|5||22||10&lt;sub&gt;93&lt;/sub&gt;13||96||5&lt;sup&gt;95&lt;/sup&gt;+8
|-
|6||11||40041||5||5209
|-
|7||71||3&lt;sub&gt;16&lt;/sub&gt;1||17||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2
|-
|8||75||4&lt;sub&gt;220&lt;/sub&gt;7||221||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||1161||3×9&lt;sup&gt;1160&lt;/sup&gt;+10
|-
|10||77||50&lt;sub&gt;28&lt;/sub&gt;27||31||5×10&lt;sup&gt;30&lt;/sup&gt;+27
|-
|11&lt;sup&gt;*&lt;/sup&gt;||1068||57&lt;sub&gt;62668&lt;/sub&gt;||62669||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10
|-
|12||106||40&lt;sub&gt;39&lt;/sub&gt;77||42||4×12&lt;sup&gt;41&lt;/sup&gt;+91
|-
|13&lt;sup&gt;*&lt;/sup&gt;||3195~3197||80&lt;sub&gt;32017&lt;/sub&gt;111||32021||8×13&lt;sup&gt;32020&lt;/sup&gt;+183
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||19699||5×14&lt;sup&gt;19698&lt;/sup&gt;−1
|-
|15||1284||7&lt;sub&gt;155&lt;/sub&gt;97||157||(15&lt;sup&gt;157&lt;/sup&gt;+59)/2
|-
|16&lt;sup&gt;*&lt;/sup&gt;||2346~2347||4&lt;sub&gt;72785&lt;/sub&gt;DD||72787||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15
|-
|17&lt;sup&gt;*&lt;/sup&gt;||10408~10428||F70&lt;sub&gt;186767&lt;/sub&gt;1||186770||262×17&lt;sup&gt;186768&lt;/sup&gt;+1
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||6271||12×18&lt;sup&gt;6270&lt;/sup&gt;+221
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||6271||16×20&lt;sup&gt;6270&lt;/sup&gt;+13
|-
|21&lt;sup&gt;*&lt;/sup&gt;||13375~13396||CF&lt;sub&gt;479147&lt;/sub&gt;0K||479150||(51×21&lt;sup&gt;479149&lt;/sup&gt;−1243)/4
|-
|22&lt;sup&gt;*&lt;/sup&gt;||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||22003||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||8134||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49
|-
|30&lt;sup&gt;*&lt;/sup&gt;||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||34206||25×30&lt;sup&gt;34205&lt;/sup&gt;−1
|-
|36&lt;sup&gt;*&lt;/sup&gt;||35257~35263||P&lt;sub&gt;81993&lt;/sub&gt;SZ||81995||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7
|}

&lt;sup&gt;*&lt;/sup&gt; Data assumes the primality of the [[w:probable prime|probable prime]]s.

Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes &gt; 10&lt;sup&gt;25000&lt;/sup&gt;), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below.

Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists))

Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists))

==Data for quasi-minimal primes==

===Base 2===

11

===Base 3===

12, 21, 111

===Base 4===

11, 13, 23, 31, 221

===Base 5===

12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013

===Base 6===

11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041

===Base 7===

14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331

===Base 8===

13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447

===Base 9===

12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011

===Base 10===

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027

===Base 11===

12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, 2888882883, 2888888883, 3555555509, 3577777077, 3700000001, 4000000005, 40000005AA, 5377777707, 5555505553, 555555580A, 600000A999, 7000100047, 7000704777, 7007777107, 7057777777, 7070007774, 7077707774, 7077777074, 7100000704, 7470000041, 7701000004, 7707077774, 7770707774, 7777707074, 8888822883, 9555555503, 9900000979, 9999770007, A000144444, A900000066, A999999971, 10000000477, 33333333337, 44444444447, 44444444777, 55A55555552, 60000008883, 68888888306, 68888888883, 70000003999, 70000007717, 70004777777, 70477777777, 77007770004, 77700000477, 77707000704, 77707770074, 77707777774, 77777077774, 77777770004, 83000000006, 97000000999, A0000000001, A0014444444, A4777777771, 100000000057, 305007777777, 305777777777, 370000000007, 377777770007, 377777777107, 700000007474, 707077000074, 707077777774, 707777777717, 770000010004, 771007000007, 777070700004, 777700000704, A95555555555, A99999777777, 1000000003007, 40000000A0041, 58A5555555555, 7004400000004, 7700000000104, 7707000007047, 7707707000004, 7777007000004, 7777700000004, 7777770077704, 7777777710077, 9977777777717, A000000014444, A044444444441, A144444444411, 40000000000401, 45557777777777, 4555AAAAAAAAAA, 59077777777777, 70007777777771, 70077070000074, 70700000004777, 77000007700704, 77700000700047, 77777777770704, 88888888830006, 90900000000799, A0000044444441, 300000000005777, 302000000000002, 55555555A555552, 700000000000174, 770000000000474, 771700000000007, 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7707777777777777777704, 7900000000000000000005, 9777777777777777777707, A999999999999999999999, 10000000000000000000747, 970000000000000000000777, 999900000000000000000007, 3577777777777777777777777, 5555555555555555555555A52, 7000000000000000000777771, 7000000000000000077777771, AAAAAAAAAAAAAAAAAAA000058, 10000000000000000000000044, 77700000000000000000000008, 500777777777777777777777777, 777777777777777777777770774, 855555555555555555555055555, A44444444444444444444444441, 1500000000000000000000000007, 40000000000000000000000000041, 440000000000000000000000000001, 70000000000000000000000000007771, 999999999999999999999999999999991, 95555555555555555555555555555555553, 1900000000000000000000000000000000001, 7777777777777777777777777777777777474, 7777777777777777777777777777777777704, 10000000000000000000000000000000000000307, 50777777777777777777777777777777777777707, 475777777777777777777777777777777777777777, 555555555555555555555555555555555555555A2A, 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===Base 12===

11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077

===Base 13===

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===Base 14===

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===Base 15===

12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797

===Base 16===

11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD

===Base 17===

12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 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====Additional known quasi-minimal primes (not necessarily the next)====

570&lt;sub&gt;51310&lt;/sub&gt;1, 49&lt;sub&gt;111333&lt;/sub&gt;, 970&lt;sub&gt;166047&lt;/sub&gt;1, F70&lt;sub&gt;186767&lt;/sub&gt;1

===Base 18===

11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, 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===Base 20===

13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, GGJ0J, GGJJJ, GJ00J, GJGJJ, GJJJ7, H00EH, H024B, H04B1, H0E4B, H0F41, H0H11, H0HEH, H4E0B, H6I11, HAAA3, HAAG3, HAGA3, HB44B, HBBBB, HBHHH, HGGA3, HGGG3, HH001, HH061, HH3AH, HH6F1, HHG63, HHIA3, HI011, HIII3, I0029, I02I9, I085H, I0A41, I0F29, I2229, I2929, I2I29, I2I99, I2II9, I33A3, I3EHH, I422B, I424B, I4EEB, I4IEB, I80G1, I9IA9, IA099, IA309, IA841, IA939, IA9A9, IAF09, IAF99, IAI99, IAIA9, IB0A1, IB4IB, IB54B, IBB4B, IBBIB, IBE0H, IBH0H, IC929, ICC29, IF841, IFAG1, IGA81, IHB4B, II0A9, II42B, II44B, IIB5B, III4B, IIIA3, IIIIB, J00G1, J00JB, J0601, J06D1, J06I1, J0861, J08B1, J0B01, J0CGJ, J0E0B, J0GA1, J0GG1, J0I2B, J0J2B, J0JIB, J55CJ, J60A1, J60G1, J6D81, JA777, JA7A7, JAJG1, JAJJ1, JAJJ7, JB00B, JB08B, JC00J, JC0GJ, JEECJ, JGGGJ, JGGJJ, JJ0B1, JJ0JB, JJ55J, JJAG1, JJAJ1, JJC0J, JJE0B, JJEEB, JJGG1, JJGGJ, JJGJJ, JJJ01, JJJEB, JJJJB, 17555J, 175EEJ, 1E0007, 1JE55J, 1JJJ5J, 20005H, 2000HH, 222I29, 22E889, 22EE89, 24222B, 2999I9, 29III9, 2A000H, 2B0I0H, 2I2999, 2I9I29, 2III29, 333AF3, 336IA3, 36A3F3, 36I3A3, 388E8H, 3AF333, 3EHHHH, 3HEHHH, 3HHEHH, 40008B, 40054B, 40405B, 40448B, 415E5J, 44440B, 44452B, 44524B, 44E08B, 45444B, 4AAAE7, 4AEEE7, 4E4447, 4GEEE7, 4I544B, 4IEEEB, 500EFJ, 500FCJ, 5055FJ, 50AFFJ, 50EEFJ, 50EF5J, 50F5FJ, 50FFCJ, 52224B, 54EEEB, 5558AH, 555CAJ, 5585AH, 55F55J, 55FFFJ, 5888AH, 588A5H, 5E555J, 5F055J, 5F05CJ, 5F5F5J, 5FFFCJ, 5JEEEB, 60006D, 6000F1, 6000G3, 6001G1, 600A63, 600G1D, 600G6D, 603A33, 606G6D, 608001, 608F89, 608IF9, 60A333, 60A363, 60AAA3, 60CCCD, 60D04D, 60D0DD, 60F8I9, 60FF89, 60GCG3, 636AF3, 63A333, 63A3F3, 63AFF3, 63IIA3, 64AAAD, 660089, 660A33, 660DDD, 660F89, 662289, 666A33, 666D4D, 66A6A3, 66AF33, 66AF63, 66D0DD, 66FF89, 6A33F3, 6AFF33, 6AG6A3, 6AGA63, 6D000D, 6D004D, 6D0D4D, 6F6689, 7000G7, 7060DD, 70AAAD, 70D0CD, 733C33, 7366C3, 73AAA3, 7600DD, 760CDD, 766333, 76664D, 766FC3, 76CCDD, 76DD4D, 770G07, 771JEJ, 7771EJ, 777481, 7777I1, 777F81, 77C7JJ, 77EC77, 77EEC7, 77EJJJ, 77GCG7, 77JECJ, 7A7E77, 7AAAG3, 7AAF63, 7AEEE7, 7C7GG7, 7CCCAD, 7CCCG7, 7CGJJJ, 7D000D, 7J000D, 7JCGGJ, 80005B, 8000G1, 8000I1, 800GA1, 800I01, 80GGA1, 84405B, 84454B, 888EAH, 88E8AH, 8A4441, 8B8EAH, 8E8IF9, 8I0AG1, 902229, 909A09, 909C29, 909FA9, 90IC29, 9222I9, 92I999, 944EE7, 94AEE7, 977GG7, 97AEE7, 97GGG7, 990I29, 999I29, 99I299, 99IIA9, 9A9447, 9A9AF9, 9A9FF9, 9CII29, 9EEEE7, 9FA9A9, 9I9C29, 9I9I29, 9ICI29, 9IIIA9, A1555J, A1E55J, A333C9, A336A3, A33889, A33F63, A36663, A444E7, A4AAE7, A555FJ, A666A3, A6AAA3, A7A7E7, A7AE77, A944E7, A9A9F9, A9AAA9, AA3389, AA9AF9, AAA099, AAA3C9, AAA7E7, AAA939, AAACG9, AAAFF9, AAAI99, AAAIC9, AAE4E7, AAG9A9, AAGAA3, ADE55J, AF6663, AF9FF9, AGA963, AGG6A3, AH4441, AI000H, AI99C9, AI9AA9, AII099, B0004B, B0054B, B0080B, B00H0H, B04IIB, B05B8B, B0A6I1, B0BB8B, B0BIBB, B0E0IH, B0HA0H, B0I0HH, B0I4IB, B0IIBB, B888AH, BB058B, BB05BB, BB080B, BB0I4B, BBB04B, BBB08B, BBIB8B, BE88AH, BEHA0H, BEHHAH, BHEA0H, BHEHAH, BHHA0H, BIE00H, C007A7, C00GG7, C00JA7, C00JAJ, C00JC7, C0C7A7, C0JAC7, C0JJC7, C3G999, CAJJ77, CC0629, CC0929, CCC007, CCCCC7, CCCCG3, CCCGG7, CCCI29, CCI029, CD4GGJ, CE0009, CEFF69, CEFFF9, CEI609, CII029, CIII29, CJACC7, CJAJC7, CJJJA7, D000D7, D000J7, D004CD, D0D007, DCCDC7, DDD04D, DDDAI1, DEJJJJ, DG000D, DG5E5J, DGGG5J, DJ00CJ, DJEEE7, DJJJ5J, E00097, E00A0H, E00AA9, E02289, E028I9, E029I9, E055AJ, E0774J, E0777J, E07JCJ, E0AF5J, E0BI8B, E0E4CJ, E0EA5J, E0ECJJ, E0FFF9, E0H0HH, E0HA0H, E0IF29, E0J5AJ, E0JJAJ, E29299, E444E7, E4EEE7, E4IIIB, E66289, E66629, E66909, E66F69, E69609, E69999, E7774J, E77C77, E7C7C7, E8B8AH, E90029, E90229, E90909, E90FF9, E92299, E94EE7, E97EE7, E990F9, E99I09, E9F669, E9I029, EA8AF9, EAAAI9, EAIAA9, EAJJJJ, EBIBIB, EC7CC7, EE08F9, EE408B, EE8IF9, EEAF5J, EEE84B, EEEC07, EEEC77, EEEE0B, EEF55J, EEFFFJ, EEJ5AJ, EFE009, EFEF09, EFF669, EFFFF9, EGGGG9, EIF669, F004AJ, F00A41, F00JA1, F04481, F055CJ, F0A15J, F0EI09, F0G01J, F0J0GJ, F0J0JJ, F0JJ5J, F14441, F68F89, F68IF9, F800G1, F8E8I9, F99299, FA0009, FA1JJJ, FAAIA9, FAII09, FAII99, FCC929, FCE229, FCE609, FCEF29, FE68F9, FE8FA9, FF0E09, FF68F9, FF9FA9, FFC4AJ, FFC929, FFCE09, FFE009, FFEE89, FFEF09, FFFA99, FFFFFJ, FFFGJJ, FG1JJJ, FI99A9, FIA999, FIAAA9, FIAII9, FIIA09, G00093, G000EJ, G000G9, G00781, G007G3, G00C09, G07GC3, G09033, G0C903, G0CGG3, G0G903, G0G933, G0GCC3, G0GGC9, G30003, G36003, G5000J, G5E0EJ, G60303, G6AGG3, G6GAA3, G7AAA3, G7G363, G90AA9, G90C63, G9AAA9, GAAAA3, GAAGG3, GC0003, GCCC93, GCCCC3, GCG903, GDEEE7, GE0007, GG0CG3, GGA8G1, GGC6G3, GGCCG3, GGCGC9, GGG0I9, GGG363, GGG6G3, GGG75J, GGG8G1, GGG963, GGGAA3, GGGCG3, GGGCG9, H044EB, H0BIBB, H0E00B, H0F011, H0IIEB, H2444B, H400EB, H44IEB, H4EEEB, HBB4IB, HE000B, HEE40B, HEEE4B, HEHHAH, HF0011, HF4441, HH3HEH, HHGAA3, HHGAG3, HHHAG3, HHHEAH, HI0001, HIBIBB, HIEBBB, I0000H, I000A9, I002HH, I00A09, I00A99, I00AG1, I00AI9, I00H3H, I02999, I03HEH, I09AF9, I0A009, I0E0HH, I0I299, I0IIA9, I4445B, I4I45B, I90I29, I99029, I99A09, I9I029, IA3999, IB000H, IB00EH, IB0HEH, IB5B8B, IF2999, II0929, II92I9, II99A9, II9C29, IIAFF9, IIF299, III5BB, III8BB, J0000B, J00081, J000IB, J0010D, J006A1, J00BI1, J00I8B, J00IA1, J05EEJ, J06G81, J0C0JJ, J0E0CJ, J0EEEB, J0J0EB, J5EAEJ, J5EEAJ, J608I1, JE505J, JEEEEB, JJ0001, JJ0JCJ, JJ0JGJ, JJAJC7, JJJAE7, JJJBI1, JJJI8B, 14GGGG1, 1J5555J, 1JGGGG1, 205555H, 20I000H, 222222B, 2929999, 2BI000H, 3333A33, 3333G09, 333A3A3, 333AAA3, 336AAA3, 336AF33, 33II3A3, 363AF33, 3GIIII9, 400445B, 404454B, 404800B, 415555J, 440045B, 440080B, 442222B, 444004B, 444422B, 444444B, 4444G47, 444GGG7, 444I4IB, 44AAA47, 44I44IB, 44I4IIB, 4AAAAA7, 4EEE4E7, 4I4454B, 5000AFJ, 525555H, 555552H, 555555J, 555585H, 58BBBBB, 5BB8BBB, 5E0E55J, 5EBBBBB, 60003A3, 60008I9, 6000DDD, 6006DDD, 60080A1, 60080I1, 600A3A3, 600D8I1, 600F841, 60CCDDD, 60CGGG3, 60DDD0D, 60DDDDD, 60GGG63, 60GGGC3, 66600A3, 66603A3, 66608I9, 6666089, 66666A3, 6666C29, 6666F89, 666AFF3, 66DDDDD, 6800001, 6AGGGG3, 6CCCC29, 6G6AAA3, 6GAGGG3, 6GCCC63, 6GGG633, 700000D, 70000CD, 7000EC7, 70070C7, 700EEC7, 7070EC7, 70EEEC7, 76666CD, 77333A3, 7733A33, 775EE5J, 777A777, 777CGJJ, 777GE5J, 77A7777, 77JCJJJ, 77JJJCJ, 77JJJJJ, 7A77777, 7A777A7, 7AAAA63, 7AAAAAD, 7C00007, 7E77JCJ, 7EE7JCJ, 7EEC007, 7EEEEC7, 7G5E55J, 7GE555J, 7GJJJJJ, 7J6666D, 9000029, 9000299, 9009I29, 900I2I9, 90FAAA9, 90I0299, 90I0I29, 90I9929, 90II029, 90II299, 90II2I9, 94444E7, 97777E7, 9902I99, 99902I9, 9992929, 9992999, 9999929, 99I0C29, 99I9029, 99I92I9, 99III29, 9AAAF09, 9AFFFF9, 9I0AAA9, 9IAAA09, 9II0I29, 9II9929, 9III029, 9III2I9, 9IIIC29, A5F555J, A777E77, A994447, A9AAF99, AA00089, AA6A6A3, AAA0009, AAA3999, AAA9A99, AAA9AA9, AAAAA47, AAAAA77, AAAAA99, AAAAG99, AAAAIA9, AAAD447, AAAGAC9, AAGAAAD, AAGAAC9, AF00009, AGAAA99, AGAGG63, AGGGA63, AI0C999, AJ777A7, B000B8B, B000HEH, B000I0H, B00I00H, B0B805B, B0HHHEH, B0IHE0H, B0III8B, B8B000B, BB8BBBB, BBB8B0B, BEH0HHH, BHHHEHH, C00000J, C000C07, C066629, C0AJJJJ, C0C0A47, C0G000J, C0JAJJJ, C0JJAJJ, C0JJJJJ, C6CCG33, CC004A7, CC00A47, CC6CG33, CC92229, CCCC7G7, CCCGGG3, CE66609, CEF6669, D00007D, D00007J, D00071J, D000C0J, D000I01, D00700D, D00C007, D00C0C7, D00G00D, D0C00C7, D0CDCC7, D444447, D7JJJJJ, DC0000J, DDD7DDD, DDDCCC7, DJ0000J, DJ000EJ, DJJJJ0J, DJJJJJJ, E000009, E00084B, E000889, E000909, E000HAH, E008AF9, E04000B, E0AA8F9, E0CJJJJ, E0HHHHH, E40400B, E6660F9, E666609, E669969, E699669, E7CCC07, E996669, E99I669, E9FAAA9, EB8BBBB, EBBII8B, EBH0HHH, EE092I9, EE44IIB, EE4E4E7, EEBB8BB, EEC0007, EEE0IBB, EEE0IIB, EEEBB8B, EEEE44B, EEJCJJJ, EF00F09, EF0555J, EFAAAA9, EFF0009, EFF00F9, EFF60F9, EI0AAA9, EI660F9, EIIBB8B, EJ0CJJJ, EJ5055J, EJJ0CJJ, F0E00F9, F0E88I9, F0JJJJ1, F299929, F4G555J, F5555CJ, F5555FJ, F6668I9, F8GGGG1, F929999, F999C29, FA84441, FAAA9A9, FCE00F9, FCEF009, FCFEE29, FE00F09, FEE2889, FEF0009, FFAA9A9, FFCEE29, FFFF4AJ, FIII9A9, FIIIAI9, FJJJJ81, G000009, G0000J7, G000303, G000AA9, G000CC3, G000GG3, G003003, G0090C3, G00CG03, G00D007, G00G0G3, G00GG03, G00GG93, G080001, G0G00I9, G0G09C3, G0G0C93, G0G0GG9, G0GG093, G0GGC63, G0GGGG9, G44GG47, G6AAGA3, G999999, GAA9AA9, GCCCGG9, GCGGCC9, GCGGG03, GEE4447, GEEEE47, GG0C007, GG0GC09, GG0GGG3, GG6AAA3, GG6AGA3, GG6GAG3, GGEEE47, GGG00C9, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 21===

12, 18, 1A, 1G, 1K, 21, 25, 2B, 2H, 2J, 34, 38, 3A, 3G, 3K, 45, 4D, 4H, 4J, 52, 54, 58, 61, 65, 6B, 6D, 72, 74, 7A, 7G, 7K, 85, 8B, 8D, 92, 94, 98, 9A, A1, AD, AH, AJ, B2, B8, BA, BK, C5, CB, CH, CJ, D4, D8, DA, DK, ED, EH, EJ, F2, FG, G1, GB, GD, GH, H2, HA, HG, I1, I5, IB, IJ, J2, JA, JK, K1, KB, KD, KJ, 111, 115, 11H, 135, 13H, 13J, 151, 15B, 15H, 164, 175, 17B, 17D, 17J, 191, 19B, 19D, 19H, 1B1, 1B5, 1BB, 1BJ, 1D5, 1DD, 1DJ, 1E4, 1F1, 1F5, 1FD, 1FH, 1HB, 1HD, 1I4, 1JD, 1JH, 1JJ, 22D, 232, 29G, 29K, 2DG, 2F4, 2FK, 31H, 322, 33D, 351, 355, 35B, 35J, 362, 371, 37B, 37D, 37H, 37J, 39B, 39J, 3B5, 3BD, 3BH, 3D1, 3D5, 3DB, 3DD, 3DH, 3E2, 3FJ, 3HD, 3HH, 3HJ, 3J1, 3JB, 3JJ, 414, 481, 49K, 4B4, 4BG, 4C1, 4EB, 4F4, 4F8, 4FA, 4FK, 50G, 51B, 51D, 51H, 531, 535, 53D, 53J, 551, 56A, 56G, 56K, 575, 57J, 595, 59H, 5B1, 5B5, 5BB, 5CA, 5CG, 5CK, 5F1, 5FB, 5FJ, 5GA, 5GG, 5HH, 5IA, 5J5, 5JD, 5JH, 60H, 632, 66H, 66J, 678, 68J, 69G, 6F8, 6FA, 6GJ, 6H8, 6HK, 6IH, 6J4, 6JG, 6KH, 711, 71B, 71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, GGJ, GJ4, GK5, H0K, H15, H1B, H1J, H31, H3D, H3H, H48, H51, H55, H5J, H6K, H75, H84, H88, H91, H95, H9D, H9H, HBD, HC4, HC8, HDJ, HF5, HFB, HFH, HHD, HHJ, HI4, HI8, HJ5, HJB, HKK, I0D, I2D, I6H, I78, I8H, I9K, IFA, IFK, IID, J08, J1J, J31, J35, J3J, J44, J5H, J68, J6G, J71, J7B, J7D, J7H, J8G, J95, J9D, JBD, JBH, JBJ, JCG, JDB, JDH, JE4, JE8, JEG, JF5, JFD, JFJ, JG4, JGG, JH1, JH5, JHB, JHH, JI4, JJ1, JJ5, K0H, K5G, K6H, K9K, KG5, KH4, KKH, 101B, 103D, 1044, 1055, 10DH, 10FB, 10H1, 10H5, 10J1, 110J, 11CD, 11EB, 11ID, 130D, 13C1, 13EB, 141B, 1444, 1471, 14FB, 1505, 15CD, 167H, 16FJ, 16HH, 176H, 17C1, 17IH, 190J, 1BF4, 1BID, 1C04, 1C3D, 1CD1, 1D6H, 1DE1, 1DEB, 1E1B, 1E71, 1E95, 1EFB, 1F0B, 1F4B, 1FB4, 1H01, 1H05, 1H41, 1H6H, 1H6J, 1HC1, 1HE1, 1HF4, 1IDH, 1IHH, 1JE5, 1JF4, 20CD, 20D2, 20ID, 2278, 22F8, 239D, 23FD, 2478, 2708, 27FD, 293D, 29DD, 2A78, 2C0D, 2CF8, 2F0A, 2F9D, 2FAA, 2FDD, 2GFA, 2IF8, 3015, 301D, 301J, 3031, 3035, 305D, 30F1, 30FB, 30FD, 30HB, 30I2, 3105, 31C1, 329D, 330H, 336J, 33E1, 33EB, 33IH, 35CD, 363H, 390H, 39E5, 39ID, 3B0J, 3B6J, 3BC1, 3BEB, 3CC2, 3CF1, 3CFD, 3D32, 3E3B, 3E91, 3EF1, 3EFB, 3F05, 3FC1, 3FE1, 3FEB, 3H01, 3H0B, 3HC1, 3I3H, 3I9D, 404B, 4078, 40E1, 411B, 4131, 4171, 41FB, 4331, 4391, 43BB, 43FB, 43I2, 4432, 444B, 44E1, 477B, 4931, 4971, 499B, 49CG, 4A32, 4A4B, 4A78, 4A9G, 4B1B, 4B31, 4B3B, 4BFB, 4C78, 4E32, 4F1B, 4FB1, 5011, 509D, 50BD, 50DB, 50DD, 50EK, 50FD, 50H1, 50HJ, 50IK, 50JJ, 5101, 5105, 51E1, 530B, 53EB, 53IH, 550D, 550H, 55AB, 55BG, 55DG, 55E5, 55KH, 563H, 56JJ, 5701, 570H, 57EB, 57ID, 57IH, 590D, 59C1, 59CD, 5A3B, 5A7B, 5AGK, 5AIG, 5AKG, 5B0D, 5B9G, 5BCD, 5BGJ, 5BIH, 5C91, 5CD1, 5D0D, 5D0J, 5D6H, 5D6J, 5DC1, 5DG5, 5DIH, 5DJG, 5E0A, 5E11, 5E3B, 5E71, 5EAK, 5EEA, 5EKK, 5F0H, 5F9K, 5G0K, 5G9J, 5GF5, 5GIK, 5H0B, 5H5K, 5H9K, 5HE5, 5HID, 5I3H, 5I7D, 5IKG, 5J6J, 5JE1, 5KAG, 5KEK, 5KFH, 5KH5, 60FK, 63FH, 67JH, 69JH, 6E9K, 6F0K, 6F64, 6FJH, 6H04, 6HE4, 6HFJ, 6IF4, 6J9H, 6J9J, 7015, 7031, 703B, 7055, 705J, 7088, 709B, 709D, 709H, 70DJ, 70I8, 710J, 716J, 730B, 73C1, 73E5, 7501, 769H, 76FJ, 76JJ, 770D, 77E1, 77E5, 77EB, 7888, 78FH, 78H1, 78JH, 790H, 79CD, 79IH, 7B0J, 7B6H, 7C31, 7C91, 7C9D, 7E35, 7E75, 7E91, 7EF1, 7F01, 7F78, 7FCD, 7FH8, 7FIH, 7FJ8, 7H81, 7HC1, 7HF8, 7J6J, 7J78, 7JC1, 7JF8, 7JH8, 7JJ8, 809K, 80H8, 81H1, 81HH, 81J1, 839H, 83FH, 869K, 871H, 8731, 8771, 8788, 879J, 87F1, 87FJ, 87JH, 888H, 8911, 8931, 896K, 8A78, 8A9K, 8CC1, 8E9K, 8F3H, 8F9H, 8FF1, 8FHH, 8FJH, 8G9K, 8GJ8, 8H3J, 8H7H, 8H8K, 8HEK, 8HF1, 8HFJ, 8HH1, 8HJJ, 8HK8, 8J7J, 8J91, 8JC8, 8JFH, 8JJH, 8K78, 8K8H, 8KH8, 901J, 9035, 903B, 9071, 90CG, 90GG, 90JD, 9101, 91E5, 9301, 93E1, 93EB, 95CD, 95DG, 95FK, 967J, 969H, 96HH, 96HJ, 96JJ, 9705, 97C1, 9905, 995K, 999G, 99CD, 99EB, 99K5, 9B5G, 9BE5, 9BID, 9C0G, 9C11, 9C71, 9C9D, 9CEG, 9CGG, 9D05, 9D5G, 9DDG, 9DE5, 9E35, 9E3B, 9E71, 9EB5, 9EEG, 9EF1, 9EFB, 9F0J, 9F6H, 9F9K, 9FC1, 9FIH, 9FKH, 9G35, 9G95, 9GIK, 9GJJ, 9GKG, 9H0H, 9H5K, 9H6H, 9HKH, 9I6K, 9I9H, 9IDD, 9J6J, 9JEB, 9JGJ, 9JID, 9JJG, 9K35, 9K95, 9KEG, A33B, A3F5, A432, A478, A53B, A5IK, A788, A795, A878, A975, A99B, A9CG, A9GG, AAAB, AB44, AB55, ABFB, AC32, AC78, ACFA, AE32, AE5A, AE9K, AEEB, AEFA, AEFK, AF75, AFCA, AFCK, AFEA, AFFB, AG9G, AGFA, AI32, B064, B091, B09H, B0B1, B0D5, B0DH, B0EG, B0FB, B0FH, B0H5, B0HD, B114, B141, B14B, B1B4, B1E1, B411, B431, B444, B4B1, B59G, B5G5, B5JG, B67H, B67J, B69H, B6FH, B6FJ, B6HH, B6JH, B96H, B9DG, B9G5, BB14, BBB4, BC1D, BC31, BCDD, BD05, BD0H, BD6H, BDC1, BDE5, BDIH, BE0G, BE31, BEBB, BEE4, BEIG, BF0B, BF0H, BF6J, BFEB, BFIH, BG0G, BG55, BG95, BGG4, BGGG, BGIG, BGJ5, BH0B, BHEB, BIEG, BIGG, BIHH, BJ01, BJ0J, BJC1, BJF4, BJID, C041, C081, C09G, C0E1, C0FK, C104, C131, C171, C1C4, C1D1, C2F8, C2FA, C39D, C3F1, C478, C6FK, C7FD, C8C1, C90G, C911, C971, C99D, CAFK, CC78, CC81, CCE1, CCF8, CD02, CD31, CDFD, CE01, CE32, CF0A, CF48, CF64, CF68, CF6A, CF91, CFAK, CFDD, CFEA, CFEK, CIF4, CK32, CK9G, CKFA, D01D, D01H, D07B, D0DB, D0F1, D0FD, D0HD, D0HH, D10D, D10H, D16H, D1EB, D1IH, D23D, D3E1, D56H, D63J, D6CG, D6IG, D70B, D70D, D70J, D99G, D9GJ, D9ID, D9JG, DBC1, DBCD, DBE1, DBJG, DC11, DC9D, DCF1, DCFD, DCGG, DCIG, DD0H, DD6J, DDGJ, DDID, DDJG, DE31, DE75, DFC1, DFID, DFIH, DG22, DG6G, DG75, DG9J, DGCG, DGIG, DH6H, DHE5, DI6G, DICG, DIDH, DIFD, DIFH, DIGG, DJCD, DJDG, DJG5, E032, E05A, E0EB, E11B, E2F8, E31B, E331, E375, E395, E3C2, E3F1, E3F5, E3FB, E3I2, E53B, E555, E591, E59B, E5EG, E6F4, E755, E775, E89K, E915, E935, E93B, E96K, E9FB, EA32, EAEB, EAFK, EB44, EB95, ECFA, ECFK, EE32, EEEB, EEFK, EF08, EF35, EF48, EF55, EF64, EF68, EF75, EF95, EFB1, EFCA, EFCK, EFEA, EFIA, EFKK, EGF4, EIF4, EKF4, EKF8, EKFA, F008, F051, F075, F095, F09H, F0CA, F0CK, F0DD, F0FB, F0H1, F0IA, F0JJ, F105, F10B, F10D, F16J, F1B4, F1EB, F1ID, F1J4, F36H, F468, F47B, F491, F4C8, F501, F50B, F648, F66A, F69H, F705, F778, F864, F8F1, F90D, F99K, F9C1, F9HK, FA6A, FA75, FA7B, FA95, FAEA, FAFB, FB0D, FB14, FBCD, FBE1, FBJ4, FC1D, FC68, FC9D, FCAA, FCCK, FCD1, FCEA, FD6H, FDIH, FE0K, FE1B, FE64, FE95, 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====Additional known quasi-minimal primes (not necessarily the next)====

40&lt;sub&gt;47333&lt;/sub&gt;9G, CF&lt;sub&gt;479147&lt;/sub&gt;0K

===Base 22===

11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, IHH, IIJ, IJ7, IL7, J07, J55, J69, J8J, J99, J9J, JA5, JAH, JE7, JEH, JHH, JI9, JJ5, JJ9, JJH, JK7, K2J, K51, K5D, K75, K81, KA1, KB1, KB7, KBJ, KD1, KEJ, KG5, KIF, KJ5, KKD, KKJ, L0D, L47, L7F, L97, LAF, LD3, LD9, LDD, LEF, LGD, LI7, LJ7, LJJ, LLD, 104H, 10D5, 1205, 12B5, 140H, 1433, 144H, 14AH, 14B3, 16ED, 1AIH, 1B43, 1DD5, 1DDD, 1E6D, 1EGD, 1G05, 1GDD, 1GED, 1GGD, 1HB3, 1HHH, 1IAH, 200L, 2015, 2051, 20A1, 20DJ, 20GD, 20IL, 21B5, 21DD, 220D, 226D, 228D, 22B5, 22G5, 22K5, 22KD, 2555, 2557, 2581, 25C1, 26A1, 26B1, 2725, 2755, 2801, 2861, 288D, 28B1, 28KD, 2AA1, 2B25, 2B51, 2BB1, 2C81, 2D6D, 2DA1, 2DDJ, 2DGD, 2G0J, 2GB5, 2GDD, 2GGJ, 2I0L, 2I6L, 2ICL, 2J05, 2JK5, 2K07, 2K2D, 2K55, 2K6D, 2KB5, 2KI7, 2L2D, 2L8D, 2LK7, 302L, 30G3, 320L, 32IL, 332L, 33G3, 33G9, 36AF, 36EF, 382J, 388J, 39EL, 3AAF, 3BBJ, 3CG9, 3E2J, 3E6F, 3E6L, 3EEF, 3FAF, 3G69, 3GI3, 3GI9, 3IG9, 3LEL, 3LG3, 4025, 404H, 4063, 4075, 40AH, 40B5, 40B7, 40GH, 4225, 4363, 4447, 444H, 447H, 4487, 44B7, 44H7, 4525, 4555, 45B5, 4663, 4777, 47GH, 4807, 4B0H, 4BB5, 4BG3, 4EBH, 4G4H, 4GB3, 4HH3, 4I03, 4I63, 4IGH, 5069, 5077, 50KD, 5255, 52IJ, 5455, 5477, 5507, 5527, 556F, 557F, 5587, 56EF, 56FF, 56GD, 56KD, 5771, 57C1, 57EF, 5807, 580J, 589J, 58E7, 58IJ, 58K7, 5905, 5945, 5957, 5969, 5989, 598J, 5999, 59EJ, 59K7, 5AEF, 5B25, 5B55, 5BA5, 5BI7, 5BIJ, 5BK5, 5BK7, 5C21, 5EAF, 5EE7, 5F09, 5F6F, 5F95, 5FKD, 5G6D, 5G89, 5I09, 5I0F, 5I27, 5I6F, 5I89, 5IB7, 5IBJ, 5IFF, 5J77, 5K07, 5K25, 5K27, 5K6F, 5K87, 5KK7, 6013, 60A1, 60AF, 60EL, 6281, 62B1, 62KD, 63G9, 6403, 6643, 66EL, 66G9, 68KD, 69B3, 6A01, 6A0F, 6AAF, 6AFF, 6B21, 6B8D, 6BBD, 6BK1, 6D03, 6D43, 6D69, 6D6F, 6D93, 6D9F, 6E21, 6E81, 6E8D, 6EBD, 6ECL, 6ELL, 6FD9, 6GBD, 6GG9, 6IEF, 6K21, 6K8D, 6KBD, 6KE1, 6L43, 6LB3, 6LDF, 6LEL, 6LG9, 700F, 7027, 7057, 70EH, 70LF, 710H, 71G5, 7207, 7225, 7255, 727L, 7505, 7507, 755F, 75EF, 75F5, 75I7, 766F, 7681, 76CF, 7771, 7781, 77IL, 7861, 7A0H, 7A4H, 7BB5, 7C6F, 7EEH, 7EHL, 7FG5, 7G45, 7HEL, 7I0L, 7I27, 7IE7, 7IGH, 7K07, 7K61, 7K77, 7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, H0D1, H0HJ, H0LL, H1HH, H20D, H22D, H26D, H2I7, H2K7, H30L, H3KJ, H40H, H447, H4G3, H4H3, H4HH, H66L, H6ED, 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5IIIIIIIIIIIIIIIIIIIIIIIIIIF, D555555555555555555555550555, EEAAAAAAAAAAAAAAAAAAAAAAAAAF, HHHHHHHHHHHHHHHHHHHHHHHHEEBH, K66666666666666666666666666F, LLLLLLLLLLLLLLLLLLLLLLLLEEB7, D5555555555555555555555555A55, GGGGGGGGGGGGGGGGGGGGGGGGGGGG3, GIG0000000000000000000000000H, HH00000000000000000000000001H, K0000000000000000000000005KEF, 5BBBBBBBBBBBBBBBBBBBBBBBBBBBBD, HB0000000000000000000000000001, K000000000000000000000000505EF, L7777777777777777777777777772L, 2000000000000000000000000000CB1, C8CCCCCCCCCCCCCCCCCCCCCCCCCCCC9, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKFF, JE0000000000000000000000000000J, K000000000000000000000000000261, A0000000000000000000000000004I4H, HD000000000000000000000000000001, K000000000000000000000000000EC01, K0FFFFFFFFFFFFFFFFFFFFFFFFFFFFCF, D0002222222222222222222222222222D, FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL2L, I700000000000000000000000000000GH, K00000000000000000000000000000E61, 20000000000000000000000000000000JJ, DD5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL0KE7, 44HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, F0000000000000000000000000000000000000000000000000000000L89, J000000000000000000000000000000000000000000000000000000JJIJ, K222222222222222222222222222222222222222222222222222222222DD, KCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCF, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKCF, D55KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, K000000000000000000000000000000000000000000000000000000000000077, 22222222222222222222222222222222222222222222222222222222222222222D, K000000000000000000000000000000000000000000000000000000000000008IJ, DFA00000000000000000000000000000000000000000000000000000000000000005, CC4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJE0J, 777777777777777777777777777777777777777777777777777777777777777777777777EK7, 6000000000000000000000000000000000000000000000000000000000000000000000000000000043, KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKB5, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEEH, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 50000000000000000000000000000000000000000000000000000000000000000000000000000000002C1, K0000000000000000000000000000000000000000000000000000000000000000000000000000000000055EF, H700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000K1, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0I7, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, A400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, E0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000071, 7LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIL, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLK77, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEI7, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIB3, I7G00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 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77EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEK7, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJKJ, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777772L, BKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK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KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK5

===Base 24===

15, 17, 1D, 1H, 1J, 1N, 25, 2B, 2D, 2J, 2N, 31, 37, 3B, 3H, 41, 45, 47, 4B, 4D, 4H, 57, 5B, 5H, 5J, 65, 67, 6D, 6J, 6N, 75, 7B, 7D, 7N, 81, 85, 87, 8J, 97, 9B, 9D, 9H, 9N, A1, AB, AH, AN, B5, B7, BD, BH, BJ, C5, CJ, CN, D1, D5, DJ, E1, EB, ED, EH, EN, F7, FD, FJ, FN, G5, GD, GH, H1, HB, HD, HN, I1, I7, IB, IH, J1, J5, J7, JB, JN, K7, KB, KJ, KN, L5, LH, LJ, MD, MJ, N5, NB, NH, NJ, 101, 10B, 111, 1F1, 1FB, 1GB, 1LB, 201, 221, 22H, 261, 271, 277, 28H, 2A7, 2C7, 2G7, 2H7, 2L1, 2L7, 2MH, 305, 30D, 30J, 33N, 34N, 35D, 35N, 38D, 395, 3A5, 3AJ, 3CD, 3DD, 3DN, 3E5, 3EJ, 3GJ, 3IJ, 3JJ, 3K5, 3KD, 3ND, 43N, 44N, 49J, 4EJ, 4GJ, 4GN, 4NN, 50N, 535, 54N, 551, 55N, 5C1, 5CD, 5E5, 5K1, 5KD, 5LN, 5M5, 5N1, 601, 60B, 61B, 66H, 68B, 691, 6CH, 6FH, 6GB, 6HH, 6MH, 70H, 70J, 711, 761, 771, 77H, 77J, 78H, 7C7, 7CH, 7FH, 7G7, 7H7, 7HH, 7IJ, 7JJ, 7K1, 7M1, 7M7, 80D, 82H, 83N, 88D, 88H, 8AD, 8CD, 8DB, 8DD, 8DH, 8DN, 8GB, 8KD, 8MB, 8MH, 905, 911, 921, 935, 955, 99J, 9AJ, 9G1, 9JJ, 9K5, 9L1, 9M5, A0J, A3J, A95, AA7, AD7, AE5, AG7, AGJ, AI5, AIJ, AJD, AL7, ALD, B01, B0N, B11, B61, B6B, B8N, B91, BIN, BL1, BLN, BNN, C1B, C21, C27, C2H, C3D, C61, C8H, C91, CA7, CB1, CBB, CC7, CCB, CCD, CDD, CFB, CG1, CGB, CK1, CL1, CMB, CMH, D0B, D3D, D3N, D4N, D6B, D6H, D7H, D8B, D8N, DAD, DCD, DCH, DDH, DDN, DG7, DGB, DID, DMN, DND, E05, E4J, EA7, EEJ, EF5, EGJ, EI5, EJJ, EM5, EM7, F01, F21, F51, F8H, F95, FC1, FF1, FFB, FKH, FM5, G0B, G0N, G11, G3N, G6B, G77, G7J, G8B, G8N, G91, GA7, GBB, GC7, GFB, GG1, GGJ, GGN, GK1, GL1, GLN, GMN, GN1, GNN, H0J, H2H, H3J, H4J, H77, HA5, HA7, HE5, HFH, HIJ, HJJ, HKH, HL7, HMH, I0N, I3D, I3J, I3N, I4N, I5D, I95, IA5, IAJ, IE5, IEJ, IF5, IGJ, IJD, IK5, IKD, J0D, J4J, J8D, JAD, JDH, JEJ, JFH, JHH, JKD, JMH, K35, K6H, KCD, KFH, KH5, KLD, KM1, L01, L0B, L0D, L0N, L61, L6B, L8D, LA7, LC7, LDD, LF1, LG7, LGB, LGN, LID, LK1, LKD, LL1, LLB, LLD, LMN, LNN, M0H, M11, M21, M4N, M71, M91, M95, MA5, MA7, MBN, MC7, MF1, MF5, MFB, MFH, MG7, MI5, MIN, ML1, ML7, MLB, MMH, N01, N21, N4N, N71, N8N, NC1, ND7, NE7, NG1, NID, NK1, NL7, NMN, NN7, 11CB, 11MB, 1291, 12G1, 16C1, 16CB, 16K1, 186B, 18CB, 19K1, 1BK1, 1C8B, 1K91, 1KC1, 1KL1, 1L21, 1LC1, 1LM1, 1M61, 1M8B, 1MG1, 206H, 20CH, 20M7, 21C1, 21M1, 2207, 260H, 26KH, 2991, 2C6H, 2CC1, 2CM1, 2F11, 2FHH, 2MC1, 2MK1, 2MM1, 308N, 30GN, 30IN, 30LN, 30MN, 333J, 33JD, 33LD, 343J, 344J, 35I5, 380N, 393J, 394J, 3A3D, 3FI5, 3IMN, 3J3D, 3JID, 3L3D, 3L8N, 3M0N, 3M55, 3NGN, 404J, 408N, 40LN, 434J, 44AJ, 4ILN, 4JAJ, 4L8N, 5091, 5095, 50F1, 50I5, 51L1, 5211, 5291, 52G1, 53ID, 53MN, 5595, 55AD, 56F1, 588N, 58MN, 58ND, 5961, 5991, 5A5D, 5AAD, 5F91, 5GF1, 5GIN, 5I05, 5I55, 5I8D, 5IDD, 5IDN, 5IIN, 5IMN, 5KI5, 5M61, 5M8N, 5N3N, 602H, 6211, 62F1, 62G1, 66C1, 66FB, 66M1, 66MB, 6B21, 6BM1, 6BMB, 6C6B, 6CF1, 6CLB, 6FG1, 6K21, 6K2H, 6KG1, 6KKH, 6L21, 6LCB, 6LM1, 6MB1, 6MBB, 6MG1, 6MK1, 7001, 7027, 7207, 726H, 739J, 793J, 79C1, 7A4J, 7A9J, 7AE7, 7C01, 7CC1, 7FL1, 7G21, 7G9J, 7GAJ, 7GC1, 7HGJ, 7J2H, 7J6H, 7MKH, 800B, 800H, 804N, 806H, 808N, 80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, NK3D, NK8D, 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5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, 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EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEG7, 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===Base 30===

11, 17, 1B, 1D, 1H, 1N, 1T, 21, 27, 2B, 2D, 2J, 2N, 2T, 37, 3B, 3D, 3H, 3J, 3N, 47, 4B, 4H, 4J, 4T, 51, 57, 5D, 5H, 5N, 5T, 61, 6B, 6D, 6H, 6J, 71, 7D, 7H, 7J, 7N, 7T, 81, 8B, 8H, 8N, 8T, 91, 97, 9B, 9D, 9N, A7, AB, AD, AH, B1, B7, BH, BJ, BN, BT, C7, CD, CJ, CN, CT, D7, DB, DJ, DT, E1, EB, ED, EJ, EN, ET, F7, FB, FD, FH, FT, G7, GB, GJ, GN, GT, HB, HD, I1, I7, IH, IN, IT, J1, J7, JH, JN, JT, K1, K7, KD, KH, KJ, L1, LB, LD, LH, LN, LT, M1, MD, MH, MN, N1, NB, NJ, NT, O7, OD, OJ, ON, P1, P7, PB, PJ, PN, Q7, QH, QT, R1, RB, RD, RH, RJ, RT, SD, SH, SJ, SN, T7, TB, TD, TH, 10J, 15J, 1IJ, 1JJ, 1LJ, 1MJ, 1QJ, 22H, 29H, 2EH, 2GH, 30T, 331, 33T, 36T, 39T, 3A1, 3C1, 3G1, 3KT, 3MT, 3OT, 3S1, 3T1, 40D, 40N, 431, 44D, 46N, 48D, 4AN, 4DD, 4DN, 4F1, 4FN, 4GD, 4ID, 4PD, 4S1, 50J, 58J, 59J, 5CB, 5FJ, 5IB, 5IJ, 5JB, 5MB, 5MJ, 5OB, 5SB, 607, 63T, 687, 6E7, 6KT, 6L7, 6M7, 6MT, 6NN, 6QN, 6R7, 6S7, 6ST, 6TT, 70B, 77B, 787, 7KB, 7M7, 7MB, 7SB, 807, 80D, 80J, 84D, 85J, 877, 88J, 89J, 8DD, 8FJ, 8ID, 8IJ, 8JJ, 8M7, 8MJ, 8PD, 8QD, 8R7, 90H, 93T, 95J, 99H, 99J, 9AJ, 9AT, 9EH, 9GH, 9HH, 9HJ, 9JJ, 9MJ, 9OH, 9OT, 9PH, 9ST, 9TT, A01, A0T, A1J, A31, A6N, A6T, AAJ, AAN, AFN, AKN, AKT, ALJ, AMJ, AMT, AO1, AOT, AQ1, AQN, ARN, AT1, B5B, BBD, BCB, BDD, BID, BOB, BPD, BQB, C31, C9H, CC1, CCB, CCH, CF1, CH1, CIB, CKB, CMB, D01, D0H, D2H, D41, D4D, D6N, D8D, D9H, DDH, DDN, DGH, DH1, DHH, DID, DKN, DNN, DO1, DOH, DQN, DS1, EEH, EHH, EM7, EOH, EPH, ER7, F0N, F31, F5J, F8J, F9J, FLJ, FO1, FQ1, FQN, FS1, G01, G4D, G8D, GF1, GGH, GO1, GOH, GQD, GS1, H07, H0J, H0T, H1J, H2H, H31, H4N, H8J, HA1, HAJ, HAT, HC1, HE7, HEH, HFN, HGH, HH7, HIJ, HJJ, HKN, HL7, HNH, HPH, HQN, HS1, HTJ, HTN, I4D, I5B, I5J, I8D, IAJ, IDD, IGD, IIJ, IKB, IMB, IMJ, IOB, IPD, IQD, J8J, J9J, JAJ, JBD, JCB, JFJ, JIJ, JJB, JJD, JLJ, JPD, JQB, K3T, K4N, KAT, KBB, KCB, KMB, KNN, KOB, KOT, KQN, KST, KTT, L0J, L5J, L67, LAJ, LJJ, LQJ, LR7, M0J, M3T, M5B, M7B, M87, M9J, MAT, MFJ, MIJ, MJJ, MKB, MMJ, MOT, MQJ, N07, N0H, N67, N6N, N87, NAN, NDH, NGD, NHH, NKN, NN7, NNH, NPH, NQD, NQN, NR7, O01, O0B, O0H, OAT, OC1, OCH, OEH, OF1, OH1, OKT, OMB, OOT, OPH, OQ1, OQB, OS1, OST, P3T, P6T, P9H, PCH, PDH, PEH, PGD, PHH, PHT, PID, PMT, PPT, PQD, PST, PTT, Q5B, Q6N, Q9J, QAJ, QBB, QBD, QC1, QDN, QFJ, QFN, QGD, QJB, QKN, QLJ, QMB, QND, QNN, QO1, QQ1, QQN, QSB, R4N, R77, R87, RAN, RKN, RM7, RNN, RR7, RS7, S0T, S41, S6T, S87, SBB, SC1, SM7, SMT, SOB, SQ1, SR7, SS1, STT, T0J, T6T, T9T, TAN, TFN, TKN, TKT, TNN, TO1, TOT, TPT, TQ1, TQN, TTN, 18AJ, 19FJ, 1AFJ, 1FAJ, 20CH, 20PH, 2C0H, 2CHH, 2COH, 2H0H, 2HCH, 2HHH, 2POH, 2PPH, 3001, 34Q1, 3F41, 3FF1, 3QF1, 3SPT, 3SST, 3TTT, 40A1, 4441, 44KN, 44QN, 4AC1, 4C01, 4CA1, 4DG1, 4GA1, 4GC1, 4NND, 4NRN, 4OA1, 4Q4N, 4QA1, 4QRN, 4RQN, 550B, 555B, 55KB, 5A5J, 5AJJ, 5BKB, 5JQJ, 5KKB, 5QQB, 5QQJ, 604N, 606T, 60FN, 60KN, 60OT, 60PT, 660N, 660T, 664N, 66AT, 66FN, 66KN, 66TN, 6A0N, 6A9T, 6AAT, 6FKN, 6FRN, 6K0N, 6KFN, 6KKN, 6O6T, 6PAT, 6RFN, 6RRN, 6T6N, 6TRN, 7067, 70R7, 75BB, 77R7, 7C5B, 7CQB, 7E67, 7IQB, 7O5B, 7OIB, 7QOB, 7R07, 7RE7, 8667, 88L7, 88S7, 8E87, 8EE7, 8EL7, 8J8D, 8LL7, 8LS7, 906T, 908J, 90FJ, 90QJ, 90TJ, 92CH, 966T, 99MT, 99PT, 9I0J, 9I8J, 9LFJ, 9LIJ, 9M6T, 9MKT, 9MMT, 9PKT, 9QQJ, 9TLJ, A04N, A0QJ, A3AT, A3ST, A4A1, A4G1, A5QJ, A8QJ, A90J, A9PT, AA3T, AA41, AAAT, AAF1, AAG1, AAPT, AC41, ACS1, AF0J, AFC1, AFIJ, AG41, AGA1, AGG1, AI8J, AIJJ, AIQJ, AJ5J, AJJJ, AQ8J, AQJJ, AQQJ, AS3T, AT4N, AT5J, ATST, B04D, B0QD, BBIB, BBKB, BI0B, BK0B, BKIB, BKSB, BMIB, BMSB, BQ4D, BQQD, BSMB, C00B, C05B, C0A1, C0BB, C0EH, C0GH, C0Q1, C0QB, C2OH, C2PH, C441, C4Q1, C50B, CB0B, CBSB, CEGH, CG2H, CGHH, CHHH, CHOH, COGH, COO1, COOB, COOH, CQ01, CQ0B, CQQB, CQS1, CS0B, D00N, D0PD, D44N, DA0N, DCEH, DCG1, DCQ1, DDC1, DDF1, DDPD, DDQ1, DECH, DF4N, DFA1, DFRN, DG0D, DGD1, DGQ1, DNEH, DPDD, DQA1, DQPD, DRFN, DRRN, E20H, E667, E767, E8L7, E8S7, ECGH, EG2H, EH67, EH77, ES67, F001, F01J, F0MJ, F0QJ, F44N, F4G1, F4RN, F6RN, FA0J, FAC1, FAF1, FAIJ, FC01, FCG1, FFAJ, FFC1, FFG1, FFIJ, FGA1, FJJJ, FJQJ, FKAN, FR6N, FRFN, G09H, G0EH, G0HH, G20H, G341, G3Q1, G4G1, G92H, GA41, GAC1, GC0H, GCEH, GCG1, GCPH, GD31, GDA1, GDG1, GDPD, GE2H, GE9H, GGG1, GGGD, GHCH, GHHH, GP0D, GP2H, GPPD, GQ31, GQA1, GQG1, H00N, H0CH, H3ST, H4O1, H667, H66N, H677, H69T, H99T, H9CH, H9KT, H9LJ, H9PT, H9QJ, HH6N, HHF1, HHFJ, HHH1, HHHT, HHLJ, HHOH, HHQ1, HHST, HK6T, HL9J, HLMJ, HMKT, HMMT, HNM7, HNRN, HOPT, HQ01, HQ5J, HR0N, HR6N, HS77, HSKT, HT01, I0BB, I0JD, I0JJ, I98J, I9LJ, I9QJ, ICBB, IF0J, IFFJ, IFQJ, IIBD, IJBB, IJID, IL8J, ILFJ, IQ8J, IQJJ, IQQJ, IS0B, ISQB, J00D, J05J, J0BB, J0MJ, J0OB, J4QD, J50B, J5QJ, JBKB, JBMB, JDGD, JDQD, JGDD, JI0D, JIIB, JISB, JKIB, JKKB, JM0B, JOIB, JQ4D, JQDD, JQID, JQJJ, JQMJ, JSIB, JSMB, K06N, K0KB, K0KT, K0MT, K0PT, K0SB, K5QB, K60T, K66T, K6AN, K6FN, K6PT, K6RN, K96T, K99T, KA0N, KF6N, KFKN, KI0B, KK0T, KK6N, KK6T, KKFN, KKKB, KKKT, KKQB, KKTN, KMMT, KQ0B, KQQB, KR0N, KS5B, L0M7, L8E7, L98J, LFFJ, LI8J, LIFJ, LL87, LL9J, LLIJ, LLM7, LM77, LML7, LMM7, M09T, M0E7, M0IB, M0L7, M0M7, M0TT, M55J, M5LJ, M60T, M69T, M707, M767, M777, M7E7, M7S7, M8LJ, M96T, M9KT, MC0B, MCOB, ME07, MIBB, MJ0B, MJBB, MJSB, MLLJ, MLS7, MM07, MM0T, MML7, MMOB, MMR7, MOBB, MOCB, MP0T, MP9T, MPKT, MQ0B, MQIB, MQOB, MS07, MSPT, MSSB, MT8J, MTTT, N00D, N04D, N04N, N0FN, N0ND, N0RN, N2OH, N92H, NCGH, ND0D, ND4N, NDPD, NEGH, NEL7, NF4N, NFRN, NGCH, NGEH, NHM7, NME7, NML7, NMM7, NN0D, NN4D, NN4N, NNDN, NNID, NNND, NNRN, NP4D, NSL7, NSS7, O00T, O03T, O0MT, O0PT, O2HH, O2OH, O341, O4A1, O4G1, O4O1, O5KB, O90T, OA41, OBSB, OC5B, OCOB, OG2H, OG31, OH9H, OH9T, OHHH, OHTT, OICB, OISB, OM0T, OM6T, OMPT, OMTT, OO2H, OO9H, OOA1, OOCB, OOGH, OOKB, OOO1, OSKB, OTMT, P00T, P04D, P08D, P09T, P0KT, P90T, PAAT, PKKT, PO0T, PP0D, PPPD, Q00B, Q0AN, Q0D1, Q0F1, Q0IB, Q0JJ, Q0MJ, Q0OB, Q55J, Q5QJ, Q88D, QA0N, QA41, QA4N, QDF1, QDPD, QF41, QFA1, QG31, QIQJ, QJ0J, QJ8D, QJDD, QJJJ, QJMJ, QJQD, QKKB, QKQB, QOKB, QOOB, QP0D, QPDD, QQ8D, QQID, QQJD, QQPD, QQQB, QS01, QSA1, R00N, R067, R06N, R0FN, R0L7, R60N, RE67, RFRN, RRFN, RRRN, S00B, S3AT, S3ST, S50B, S5QB, S99T, SAAT, SC0B, SE67, SGG1, SICB, SIQB, SK5B, SKIB, SKKT, SKPT, SKQB, SM0B, SMMB, SMSB, SOA1, SOG1, SPOT, SQCB, SQQB, SS0B, SS67, ST31, STG1, T00N, T03T, T041, T04N, T0AT, T0C1, T0MT, T0ST, T1AJ, T3F1, T4A1, T4RN, T5JJ, T66N, T8AJ, T9FJ, T9QJ, TA3T, TAA1, TAG1, TAIJ, TCS1, TFA1, TFAJ, TFF1, TFFJ, TFG1, TGA1, TI9J, TIJJ, TIQJ, TL8J, TL9J, TM0T, TM5J, TM8J, TMAJ, TMLJ, TMST, TSF1, TSG1, TT01, TT5J, TT8J, TTA1, TTC1, TTFJ, TTG1, TTLJ, TTMJ, 30441, 30O41, 30OO1, 34O41, 3AATT, 3ASAT, 3O441, 3Q041, 40G41, 40GQ1, 40O41, 40OG1, 40OO1, 44001, 440C1, 440G1, 4444N, 44CG1, 44GQ1, 4AAA1, 4COG1, 4DAA1, 4GQ41, 4KRRN, 4OGG1, 4Q041, 4QGG1, 5005B, 500BB, 500KB, 500QB, 50K0B, 50Q0B, 50QKB, 555AJ, 555LJ, 555QJ, 55J5J, 5B00B, 5B0BB, 5J55J, 5J5JJ, 5JJ5J, 5JJJJ, 5K00B, 5QJ5J, 5QK0B, 6006N, 6009T, 600AT, 6066N, 60R0N, 666N7, 6696T, 66OPT, 67767, 6900T, 696PT, 69P0T, 6F6AN, 6O9PT, 6OP0T, 6OP9T, 6P99T, 755QB, 77767, 7CBBB, 7IBIB, 7IIBB, 7IICB, 7QIIB, 8888D, 88E67, 8L887, 8SSE7, 9099T, 90KKT, 90KMT, 90MPT, 9690T, 96P9T, 990KT, 9990T, 9999T, 99K6T, 9FFFJ, 9HKMT, 9HMPT, 9ILLJ, 9KKMT, 9KKPT, 9M00T, 9T8QJ, 9TFIJ, A008J, A00NN, A055J, A0FJJ, A0I0J, A0I9J, A0J0J, A44C1, A555J, A9FFJ, A9QIJ, A9T8J, AFFA1, AFFFJ, AFFJJ, AFFQJ, AJ00J, AP99T, ASA9T, ASFF1, ASP9T, ASSAT, AT3TT, ATFQJ, ATQIJ, ATT9J, ATTQJ, B088D, B08GD, B0GGD, B0SKB, B8GGD, BG00D, BIISB, BISIB, BS0IB, BS0SB, BSSKB, BSSSB, C0041, C0G41, C0H0H, C0PPH, C0S01, C4001, C4AG1, C4OG1, C5BBB, CG00H, CGE0H, CGGA1, CGQ41, COSSB, CP20H, CPGPH, CPP2H, CPPOH, CPPPH, CS001, CS55B, CSSQB, D00GD, D0A4N, D0FAN, D0GDD, D0N0D, DAAA1, DAAC1, DAFG1, DDAA1, DDGGD, DFFAN, DFGC1, DGDGD, DGG31, DGGA1, DH0AN, DPPPH, DQDD1, DQGG1, E0L87, E88E7, EC00H, EC02H, EE867, EE887, EEL87, ELE87, F00IJ, F0441, F0AG1, F0F0J, F0F41, F0FF1, F0GG1, F4041, F44A1, F44C1, F4A41, F64KN, F6K6N, FAAA1, FAFQJ, FCAA1, FF041, FF0F1, FF64N, FF6KN, FFA41, FFA4N, FFF4N, FFFF1, FFFFN, FFKRN, FFQMJ, FIJ0J, FJ00J, FKKKN, FNNFN, FQQ0J, FQQMJ, G00DD, G00DH, G00PD, G02PH, G0DDD, G0PDD, GAAA1, GC2HH, GDDDD, GDGDD, GDPPH, GG0PD, GGCA1, GGCQ1, GGDD1, GH4Q1, GHHG1, GII0D, GQ441, H009H, H00G1, H00H1, H04Q1, H0ANN, H0H01, H0H9H, H0HG1, H0HO1, H0O41, H0OHH, H0QG1, H40G1, H4G41, H4GG1, H60AN, HAN0N, HF0G1, HFF41, HFFMJ, HFFQJ, HGQ41, HH3TT, HH401, HH441, HH4G1, HH55J, HH66T, HH6OT, HH96T, HHA0N, HHANN, HHC0H, HHCHH, HHH0N, HHHCH, HHHHN, HHHNN, HHKMT, HHO41, HHP9T, HHPKT, HHTMT, HHTT1, HK9MT, HKKMT, HLLFJ, HM66T, HM7R7, HMM67, HMM77, HMMM7, HMTST, HO9HT, HOO41, HOOOH, HOT3T, HOTT1, HPO9T, HSO3T, HTGG1, I000J, I009J, I00BD, I00QJ, I00SB, I08QJ, I0I0D, I0IQB, I0JIB, I0Q0B, I0SCB, I0SIB, I0SSB, IB00D, IBBSB, IBISB, ICSSB, II0CB, II0ID, II0SB, IIIBB, IIQ0B, IIQIB, IISIB, IJ0SB, IJJQJ, ILLLJ, IQ00J, IQC0B, IQIQB, IQQIB, ISSCB, J00MB, J08GD, J0IID, J0QQD, J55BB, J55JJ, J5J5J, J5JJJ, J88GD, JB0SB, JBB0B, JBBSB, JDDDD, JG0ID, JGG0D, JIIID, JJ0QJ, JJ5JJ, JQ08D, JQ0QJ, JQQ0D, JQQ5J, JS55B, JSK0B, JSS5B, K000T, K006T, K00TN, K0FAN, K505B, K6T0N, K9KMT, K9KPT, K9P0T, KIQIB, KK00N, KK05B, KK0AN, KK0KN, KKM9T, KS0IB, KS0QB, KSQIB, KTR6N, KTRRN, L0087, L08S7, LE087, LEE87, LL8LJ, LMEE7, LMSE7, LMSS7, M00CB, M00PT, M066T, M0CQB, M0K6T, M0MQB, M0MSB, M0QQB, M0SMB, M0SST, M666T, M900T, MBBSB, MEE77, MEL77, MELE7, MES77, MESE7, MESS7, MI00B, MIICB, MIISB, MIQCB, MK9PT, ML7L7, MLEL7, MLLE7, MLME7, MM677, MMBIB, MMBSB, MMCQB, MME77, MMEE7, MMICB, MMISB, MMK6T, MMKKT, MMM9T, MMMMT, MMMTT, MMQQB, MMSCB, MMSKT, MMSMB, MMSS7, MMTST, MOIIB, MOOIB, MOSIB, MQCQB, MR007, MR667, MRL07, MS7L7, MSEL7, MSK9T, MSL77, MSSL7, MT00T, MTMMT, N0DDD, N4NNN, N7LE7, N7S77, NE2CH, NE9CH, NEC2H, NEE77, NFFNN, NFNFN, NII0D, NL777, NL7L7, NLES7, NLLL7, NLS77, NOG9H, NRR0N, O6P9T, O9H6T, O9HPT, O9P9T, OCBBB, OG441, OGAG1, OGGA1, OHH6T, OHOOH, OKIIB, OKK5B, OKKIB, OO5BB, OOBIB, OOIIB, OSCSB, OT3TT, OT441, OTG41, OTGG1, P00PH, P0D0D, P0G2H, P0OGH, P0PGH, PA99T, PGP0H, POOOH, PP20H, PP88D, PPP0H, Q0001, Q000N, Q001J, Q00ID, Q00PD, Q044N, Q04G1, Q0I0D, Q0PPD, Q40G1, Q444N, Q44RN, Q4AA1, Q8QQJ, QAAS1, QAFF1, QAFG1, QASG1, QDGG1, QFGG1, QI00D, QIICB, QIIQB, QIQCB, QOIIB, QPP4D, QQ08J, QQ0CB, QQC0B, QQI0J, QQJ5J, QSFG1, R00E7, R0E07, R0NE7, R6F6N, R6FFN, REE07, RELE7, RFF6N, RL0E7, RLE07, RLEE7, RLLE7, RQR0N, RR0QN, RRQ0N, S03O1, S0AF1, S0AG1, S0O31, S0QIB, S0SIB, S30F1, S30O1, S7QIB, SA3PT, SAAA1, SAFG1, SASST, SCSQB, SF0G1, SFFF1, SI0SB, SISSB, SKSSB, SSCQB, SSCSB, SSMIB, SSPAT, SSSKT, SSSMB, SSSSB, STAF1, STF01, T0001, T0031, T0AF1, T0AS1, T0G31, T0R6N, T0T31, T0TF1, T3AAT, T40G1, T4CG1, T4G41, T5LLJ, T8LLJ, TA441, TA98J, TAFJJ, TAQ5J, TASST, TATAT, TC401, TCGG1, TFQIJ, TFQJJ, TFQMJ, TG441, TGC41, TI8LJ, TLLMJ, TMMMT, TMTMT, TQ8QJ, TSS3T, TT3AT, TT9IJ, TTAAT, TTQIJ, TTS31, TTS3T, TTT9J, TTTST, 20000H, 200OOH, 3440O1, 3TAAST, 404CQ1, 4KKKKN, 4KKKRN, 4QQQQD, 505BBB, 50BB0B, 6000AN, 6444RN, 66666N, 666O9T, 66999T, 669P9T, 66N777, 6A444N, 6FF66N, 6FFF6N, 6R666N, 766767, 77S677, 7IBBBB, 8888E7, 8LLLLJ, 8SSSL7, 9000MT, 90K90T, 90KP0T, 90M90T, 99000T, 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===Base 36===

11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 2UJ, 2WN, 2XN, 30Z, 337, 33B, 33H, 33P, 34H, 34P, 37D, 38Z, 39H, 3AB, 3AP, 3AZ, 3D7, 3DH, 3FD, 3FZ, 3HD, 3HH, 3KZ, 3L7, 3LZ, 3MB, 3O7, 3OZ, 3PD, 3PP, 3Q7, 3RH, 3S7, 3SZ, 3UP, 3UZ, 3XB, 40P, 43B, 43H, 46V, 48B, 48V, 49B, 4AP, 4BB, 4E1, 4FP, 4HB, 4HH, 4HP, 4IH, 4MB, 4OP, 4OV, 4PH, 4RH, 4VB, 4VH, 4WH, 4WP, 4WV, 4X1, 4XH, 55T, 565, 58Z, 5C5, 5E7, 5ET, 5EZ, 5IN, 5KT, 5L7, 5MP, 5NN, 5O5, 5O7, 5OP, 5R7, 5RT, 5RZ, 5ST, 5SZ, 5TN, 5TZ, 5W7, 5XN, 5YN, 61T, 625, 62J, 64V, 661, 66J, 691, 6AV, 6EV, 6IJ, 6J1, 6K5, 6O1, 6OT, 6QJ, 6QT, 6RV, 6SJ, 6T1, 6TT, 6U5, 6UV, 6VV, 6W5, 6Y1, 6YT, 72D, 737, 77D, 791, 7A1, 7AZ, 7D7, 7GD, 7I1, 7IN, 7LN, 7M7, 7MN, 7N7, 7NN, 7RN, 7RZ, 7WN, 7X7, 7Z1, 801, 80V, 82D, 83B, 841, 84H, 877, 881, 887, 88B, 88V, 88Z, 89H, 89V, 8A1, 8AB, 8B7, 8BH, 8D1, 8DH, 8EB, 8EV, 8GD, 8GZ, 8HD, 8IV, 8L7, 8LZ, 8M1, 8MB, 8MD, 8O7, 8OB, 8Q7, 8QD, 8RD, 8S7, 8SH, 8SZ, 8UZ, 8W7, 8WV, 8XV, 8Y1, 90H, 91J, 93B, 93H, 94V, 981, 98H, 99J, 9AH, 9BB, 9C5, 9IH, 9IV, 9JV, 9KH, 9M1, 9OB, 9QJ, 9R1, 9RH, 9SH, 9UJ, 9VB, 9VJ, 9W5, 9X1, 9Y1, 9YB, 9YJ, 9YV, A25, A3P, A3Z, A4H, A61, A81, A8B, AAB, AAH, ABB, AC5, AEZ, AHP, AK1, AKB, AKH, ALZ, AMB, AO5, AOZ, APH, AQP, AR1, ARV, ARZ, AUV, AV5, AVB, AVV, AWV, AX1, AXB, AZ1, B2J, B4B, B4V, B6V, B87, B8H, B9B, BAB, BAH, BBH, BBV, BE7, BEB, BGJ, BHB, BIJ, BJB, BJT, BLJ, BOB, BOT, BQ7, BRV, BS7, BTT, BTV, BVB, BVJ, BVT, BWV, BX7, BXH, BYH, BYV, C25, C2J, C2N, C55, C65, C6J, C95, CCN, CDD, CED, CF5, CFJ, CGD, CIN, CJD, CO5, CRN, CUJ, CXJ, CXN, D37, D3P, D41, D55, D5P, D6T, D77, D81, D9H, DA1, DCD, DD1, DD5, DDH, DDP, DE7, DEP, DF5, DFD, DG7, DHH, DI1, DK1, DK5, DKT, DO1, DOP, DP1, DPD, DQ7, DR7, DRH, DS7, DTT, DU1, DWD, DX7, DYH, E0P, E2D, E3Z, E41, E4P, E71, E81, E87, E8B, EAZ, EDP, EED, EEN, EEV, EFZ, EGT, EI1, EKZ, ELB, EMB, EMN, EN7, EO1, EOT, EPP, EPZ, EQ7, ERN, EST, ETN, ETV, EUP, EUZ, EVT, EWD, EWN, EX1, EYB, EYN, EZD, EZP, F2J, F3B, F3Z, F4P, F65, F8B, F8Z, FCJ, FD5, FEZ, FFB, FFD, FG5, FGD, FGZ, FIJ, FJ5, FJJ, FLZ, FPD, FQD, FSZ, FXB, G07, G0D, G0Z, G3D, G3Z, G55, G6T, G7D, G7Z, G87, G8Z, GA5, GCJ, GD7, GE7, GET, GG5, GG7, GGT, GGZ, GI7, GJD, GKZ, GL7, GLT, GQT, GRT, GS7, GST, GU5, GUZ, GW5, GXJ, GZ7, GZZ, H3D, H3H, H4H, H9B, HAH, HED, HGD, HHP, HIH, HKH, HKP, HLN, HNH, HOB, HOP, HQP, HRD, HRH, HRN, HSH, HWD, HWH, HWP, HYN, I41, I6J, I71, I7N, I87, I8H, I9H, I9J, IA1, IAV, IE1, IFJ, IG7, IHH, II1, IIH, IIV, IJV, IK1, IL7, ILJ, ILN, IM1, INN, IOV, IRH, IUV, IX1, IXH, IXV, IYJ, J2D, J4V, J55, J61, J6J, J8B, J8V, J95, J9J, J9V, JA5, JBB, JD5, JDT, JF5, JFJ, JGJ, JGT, JIV, JJ1, JJD, JJV, JK5, JKT, JLB, JLT, JMN, JN5, JNJ, JQJ, JQN, JQT, JRD, JTB, JVJ, JVN, JVV, JWN, JY1, JYJ, K0B, K25, K3P, K65, K81, K95, K9H, KAH, KC5, KEP, KEZ, KG5, KHP, KK1, KKT, KLB, KLP, KLZ, KM1, KMB, KMP, KOT, KP1, KQP, KR1, KRT, KRZ, KTT, KW5, KX1, KZB, L0N, L0P, L8Z, L9B, LAZ, LBJ, LEN, LET, LFB, LFZ, LG7, LGZ, LIJ, LJJ, LKB, LKP, LLB, LLP, LLT, LMB, LMN, LN7, LO7, LOT, LPZ, LQT, LRN, LTN, LTT, LWP, LX7, LYN, M01, M0P, M2D, M2N, M37, M3B, M41, M61, M77, M87, M8D, M91, ME1, ME7, MEB, MFB, MFP, MGD, MI7, MKB, MM7, MMN, MN7, MNN, MO7, MOB, MOP, MPP, MQP, MS7, MW7, MXN, MYN, N5N, N65, N87, N8D, N8H, N95, NCD, NCJ, NDH, NE7, NFJ, NG5, NG7, NGJ, NJ5, NN7, NND, NO5, NQD, NQJ, NRN, NU5, NWH, NWN, NXH, NXN, NYJ, O0Z, O25, O37, O3B, O3P, O3Z, O41, O4B, O5Z, O61, O71, O7Z, O81, OA5, OAP, OBB, OBV, OC5, OEZ, OF5, OG7, OKP, OKZ, OMB, OO5, OOZ, OP5, OPP, OQT, OR1, OR7, OS7, OST, OTB, OTZ, OU5, OVV, OW5, OX7, OXB, OXV, OYV, OZ5, OZ7, P01, P0D, P0N, P3P, P4H, P4P, P5N, P65, P6V, P8V, PD1, PED, PEZ, PHH, PHP, PI1, PIN, PLN, PLP, PLZ, POP, PP1, PPH, PPV, PQD, PQN, PRV, PV5, PVH, PVV, PWH, PWP, PX1, PXV, PYH, PYN, Q07, Q0P, Q2N, Q37, Q3P, Q6J, Q6T, Q7D, Q8D, Q9J, QCD, QCJ, QD7, QED, QFP, QGT, QI7, QIN, QJN, QLJ, QMD, QMN, QMP, QND, QNJ, QOT, QQJ, QTN, QWN, QYJ, QYT, R0V, R0Z, R37, R3H, R4H, R77, R7D, R7N, R7Z, R81, R8V, R91, RA1, RCD, RCN, RD1, RED, REV, REZ, RGT, RGZ, RHD, RIV, RKH, RKZ, RLN, RMD, RNH, RO7, RQN, RRD, RRT, RRZ, RS7, RSH, RT1, RTV, RU1, RUZ, RVN, RVT, RW7, RWH, RX7, RY1, S0J, S6J, S87, S9H, SAZ, SCJ, SET, SFJ, SG7, SGZ, SI7, SM7, SO7, SOT, SQJ, SQT, SRH, SSH, STT, SW7, SX7, SXH, SYH, T0N, T0Z, T1D, T1T, T6V, T8Z, TBB, TBV, TCN, TD1, TEV, TGT, TKT, TLB, TM1, TO1, TOB, TQN, TR1, TRD, TT1, TTB, TTN, TUZ, TVB, TVN, TVT, TWD, TWV, TXB, TXV, TYV, U2J, U3Z, U61, U8V, U95, UA1, UC5, UEP, UEZ, UFJ, UG5, UJ5, ULZ, UO5, UOP, UOZ, UQP, URV, URZ, UUJ, UW5, UWV, UXJ, UXV, UYJ, V0H, V25, V4V, V55, V6V, V9J, V9V, VBB, VBJ, VCJ, VET, VGJ, VHP, VIH, VIN, VJJ, VJN, VLT, VMP, VNJ, VOB, VP5, VQJ, VQT, VRT, VRV, VSH, VSJ, VST, VTB, VTN, VUP, VW5, VWN, VXH, VXN, VYB, W07, W25, W3D, W3H, W4P, W4V, W57, W6V, W7D, W8H, W95, W9H, WAH, WAV, WCN, WD7, WDD, WDH, WE7, WEN, WF5, WGD, WHH, WK5, WKH, WN7, WNN, WPP, WPV, WQP, WR7, WRD, WRH, WRN, WS7, WU5, WUP, WVN, WWH, WWP, WX7, WXH, WXN, WYN, WYV, X0J, X2J, X2N, X4B, X4H, X4V, X6J, X87, X8B, X91, X9B, XAV, XCN, XEB, XFB, XHB, XHH, XHN, XKB, XLJ, XNH, XO1, XQ7, XS7, XSH, XUJ, XWN, XWV, XX7, XXH, XXV, Y0N, Y1J, Y1T, Y2N, Y3H, Y61, Y8V, Y91, YBV, YCN, YEB, YFJ, YHB, YHN, YIH, YJN, YJT, YLJ, YLN, YMB, YMN, YNH, YOB, YOV, YRH, YTB, YTN, YTT, YTV, YVB, YVH, YWH, YWV, YXB, YY1, YYJ, YYT, YYV, Z01, Z3D, Z3Z, Z71, Z8B, ZB7, ZBB, ZD5, ZDD, ZDP, ZG7, ZGD, ZGZ, ZKB, ZLP, ZM1, ZO5, ZP1, ZPD, ZQD, ZU1, ZX1, ZXB, ZZD, 102J, 106T, 10ET, 10GJ, 10LJ, 10TD, 12FJ, 12LJ, 13ED, 13GD, 13MD, 192J, 1D0T, 1DET, 1E3D, 1EFD, 1F0D, 1FWD, 1G0J, 1GGJ, 1GIJ, 1GUJ, 1I0J, 1ISJ, 1JOT, 1JTT, 1M0D, 1M3D, 1MFD, 1MWD, 1OKT, 1OOT, 1Q3D, 1QDT, 1QKT, 1QTD, 1QWD, 1SLJ, 1TDT, 1TFD, 1TOT, 1TTD, 1TTT, 1UGJ, 1ULJ, 1WFD, 1WWD, 20CJ, 20FJ, 20G5, 20IN, 20JD, 20MD, 20RN, 20WD, 20YN, 2295, 25RN, 260J, 26LJ, 288D, 28DD, 28FD, 296J, 29G5, 29K5, 29LJ, 2C05, 2C0J, 2C0N, 2CJJ, 2CK5, 2CLJ, 2CSJ, 2D25, 2DC5, 2DDD, 2DW5, 2F0J, 2FDD, 2FMD, 2FSJ, 2G65, 2G95, 2GK5, 2I0N, 2JC5, 2JCJ, 2JWD, 2K05, 2L6J, 2L9J, 2N0N, 2NIN, 2NN5, 2NSJ, 2NW5, 2NYN, 2Q0N, 2QNN, 2R0N, 2RFD, 2RWD, 2SJJ, 2SXJ, 2SYJ, 2W05, 2WC5, 2WD5, 2WMD, 2XCJ, 2YCJ, 2YJJ, 2YRN, 2YSJ, 300D, 302D, 304B, 3077, 3087, 308D, 30BH, 30DD, 30FP, 30G7, 30GD, 30MD, 30QP, 30YH, 320D, 333D, 333Z, 33WD, 344B, 34FB, 373Z, 37E7, 37EZ, 37G7, 3807, 38BB, 38E7, 38G7, 38HB, 38I7, 38X7, 390B, 394B, 399B, 39EB, 39FB, 39KB, 3ASH, 3AWH, 3B8B, 3BG7, 3BLB, 3BR7, 3BW7, 3C8D, 3D0P, 3D2D, 3DKP, 3E3D, 3E77, 3EBB, 3EFB, 3EKB, 3EQP, 3ERD, 3ERZ, 3EZZ, 3F4B, 3FEP, 3G2D, 3GCD, 3GED, 3GM7, 3GR7, 3GWD, 3GZD, 3H0B, 3HEP, 3HKB, 3IAH, 3ISH, 3K4B, 3KBH, 3KEB, 3KKB, 3L4B, 3L8B, 3M3D, 3MEP, 3MG7, 3MWP, 3OFP, 3OKB, 3OMP, 3OOB, 3OYB, 3P8H, 3PXH, 3QDP, 3QQD, 3QRD, 3QWD, 3R3D, 3R3Z, 3R8D, 3RG7, 3RM7, 3RQD, 3RR7, 3RZ7, 3SIH, 3W0H, 3WED, 3WM7, 3WQD, 3WW7, 3WYH, 3X8H, 3XE7, 3XG7, 3XKH, 3Y0B, 3Y4B, 3Y8H, 3YAH, 3YSH, 3YYB, 3Z4B, 3ZFP, 404V, 4061, 40EB, 40I1, 40IV, 40KH, 40M1, 40R1, 40XB, 40XV, 43QP, 4441, 444H, 444P, 4481, 448H, 4491, 449H, 44EB, 44EV, 44K1, 44KB, 44KP, 44M1, 44R1, 44RV, 44SH, 44U1, 44VP, 44YB, 46I1, 46K1, 46M1, 480H, 488H, 48AH, 48I1, 48K1, 48U1, 4991, 49EV, 49I1, 49O1, 49UV, 4A0H, 4A41, 4A4B, 4A8H, 4A91, 4ALB, 4ASH, 4BXV, 4E4B, 4EEP, 4EFB, 4ELP, 4EUV, 4EXB, 4FOB, 4I01, 4I0V, 4I4V, 4I61, 4I81, 4I91, 4IEV, 4IY1, 4K0H, 4K91, 4KI1, 4KKH, 4KKP, 4KUP, 4L3P, 4L4P, 4LAB, 4LMP, 4MA1, 4MK1, 4MMP, 4MP1, 4MU1, 4O0B, 4OFB, 4OKB, 4OM1, 4OU1, 4OYB, 4P41, 4P81, 4PO1, 4PYV, 4QLP, 4QQP, 4RAV, 4RK1, 4RRV, 4S8H, 4U3P, 4U91, 4U9V, 4UIV, 4UUP, 4VAV, 4VLP, 4VVV, 4X0B, 4X0V, 4XAB, 4XBV, 4XVV, 4XYV, 4Y0V, 4Y41, 4Y4H, 4Y4V, 4Y81, 4YAV, 4YEV, 4YIV, 4YK1, 4YKB, 4YM1, 4YR1, 4YUV, 4YYB, 5025, 503Z, 5095, 50G5, 50PN, 50QN, 50WP, 50Z7, 5205, 52RN, 53FP, 53M7, 53PZ, 53X7, 53ZZ, 54PP, 550N, 553P, 554P, 5557, 555N, 557Z, 5595, 55EN, 55G5, 55GZ, 55LZ, 55OZ, 55QN, 55Z7, 572N, 577Z, 5787, 57CN, 57FZ, 57QN, 57S7, 57ZZ, 58I7, 58X7, 5955, 5A95, 5AEP, 5CMN, 5CWN, 5E2N, 5EAP, 5ECN, 5EKP, 5EQN, 5F95, 5FLP, 5FQP, 5G05, 5GK5, 5GQ7, 5I37, 5II7, 5IM7, 5L3P, 5LCN, 5LFP, 5LGT, 5LKZ, 5LUP, 5LZZ, 5M07, 5N55, 5NQ7, 5NX7, 5OAZ, 5OFZ, 5OLT, 5OOT, 5P25, 5P95, 5PG5, 5PGZ, 5PPN, 5PQP, 5PW5, 5Q0N, 5Q87, 5QCN, 5QKP, 5QN7, 5QPN, 5QPP, 5QQT, 5QTT, 5QWP, 5QX7, 5S37, 5SS7, 5T6T, 5TQT, 5U4P, 5UGZ, 5W0P, 5W2N, 5W5P, 5WLN, 5WP5, 5WQN, 5XM7, 5Y0T, 5YOT, 5Z25, 5Z4P, 5Z57, 5ZG5, 5ZLZ, 5ZQ7, 5ZWP, 6005, 600V, 601J, 6055, 606V, 60A1, 60C5, 60GT, 60JJ, 60LJ, 60M1, 60R1, 60VT, 60YV, 610J, 61FJ, 6401, 64K1, 6505, 665T, 6665, 66A5, 66F5, 66G5, 66GT, 66KT, 66V5, 66XV, 68OV, 68YV, 690J, 69G5, 69O5, 69XJ, 69XV, 6A55, 6AA1, 6AA5, 6AF5, 6C9J, 6CA5, 6CGJ, 6CYJ, 6EK1, 6ER1, 6FLJ, 6FXJ, 6G0J, 6G65, 6GGJ, 6GJ5, 6GLJ, 6I81, 6I9V, 6J65, 6J6V, 6JET, 6JST, 6JWV, 6K01, 6KET, 6KST, 6L0J, 6L6T, 6LFJ, 6LRT, 6LXJ, 6MA1, 6MM1, 6MX1, 6O8V, 6OV5, 6R0T, 6RET, 6RKT, 6S6T, 6T8V, 6U01, 6U9J, 6UE1, 6UGJ, 6UJJ, 6ULJ, 6UU1, 6V0T, 6VA5, 6VF5, 6VG5, 6VKT, 6VXJ, 6WWV, 6WXV, 6X41, 6X6V, 6XIV, 6XJV, 6XOV, 6Y0V, 6YCJ, 6YXV, 7001, 702N, 7041, 707N, 7081, 70GZ, 70I7, 70KZ, 70ND, 70O7, 70OZ, 70R7, 70U1, 70UZ, 70WD, 70YN, 71QD, 71WD, 720N, 72CN, 72YN, 733Z, 73ED, 73ZD, 7401, 7441, 74M1, 74O1, 74R1, 7641, 76K1, 76M1, 76R1, 76U1, 76X1, 773Z, 7741, 7771, 7781, 77FZ, 77G7, 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8KRH, 8KU1, 8KYH, 8KZ1, 8L4B, 8LBB, 8MG7, 8MR7, 8O4V, 8OI1, 8OK1, 8OOV, 8OX1, 8RAV, 8RHH, 8RUV, 8RX1, 8U4V, 8UK1, 8V4B, 8V9B, 8VHH, 8VRH, 8VVV, 8VYV, 8W0H, 8WFD, 8X07, 8XAH, 8XR1, 8XU1, 8XYB, 8Y6V, 8Y8H, 8YBB, 8YFB, 8YHH, 8YKB, 8YLB, 8YUV, 8YVV, 8YYH, 8Z1D, 8Z4B, 8Z8D, 8Z91, 8ZCD, 8ZEZ, 8ZFD, 8ZFZ, 8ZK1, 8ZO1, 8ZX7, 90A5, 90BJ, 90BV, 90EB, 90EV, 90GJ, 90IJ, 90J5, 90JB, 90LB, 90O1, 90OV, 90XB, 90XV, 9225, 92K5, 9441, 94BH, 94EB, 94KB, 94XB, 9505, 9601, 960V, 96A1, 96J5, 96K1, 96V5, 96WV, 96XV, 984B, 98AV, 98BV, 98FB, 994H, 998B, 998V, 999B, 99A1, 99A5, 99E1, 99G5, 99O1, 99O5, 99WH, 9A05, 9A55, 9A95, 9AA5, 9ABV, 9AE1, 9AEV, 9AI1, 9AO1, 9AOV, 9B0J, 9B0V, 9B8V, 9B9H, 9BCJ, 9BFJ, 9BSJ, 9BVV, 9CCJ, 9CLJ, 9CSJ, 9E01, 9E4B, 9EA1, 9EAB, 9EAV, 9EK1, 9F05, 9F25, 9FA5, 9FBJ, 9FFJ, 9FJB, 9FSJ, 9FXJ, 9G25, 9G2J, 9GIJ, 9GSJ, 9HFB, 9HLB, 9IIJ, 9IJ1, 9J01, 9J0B, 9J0J, 9J9B, 9JCJ, 9JE1, 9K05, 9K41, 9K55, 9K61, 9K8B, 9KA5, 9KI1, 9KK5, 9KU5, 9L2J, 9LCJ, 9O05, 9O0V, 9O95, 9OI1, 9OO1, 9OOV, 9OWV, 9RRV, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ

====Additional known quasi-minimal primes (not necessarily the next)====

P&lt;sub&gt;81993&lt;/sub&gt;SZ

==Unsolved families==

Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')) by covering congruence, algebraic factorization, or combine of them.

{|class=wikitable
|base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family
|-
|13||9{5}||(113×13&lt;sup&gt;''n''&lt;/sup&gt;−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|13||A{3}A||(41×13&lt;sup&gt;''n''+1&lt;/sup&gt;+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|16||{3}AF||(16&lt;sup&gt;''n''+2&lt;/sup&gt;+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1{7}||(23×17&lt;sup&gt;''n''&lt;/sup&gt;−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1F{0}7||32×17&lt;sup&gt;''n''+1&lt;/sup&gt;+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||4{7}A||(71×17&lt;sup&gt;''n''+1&lt;/sup&gt;+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||51{0}D||86×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||70F{0}D||2038×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||8{B}9||(139×17&lt;sup&gt;''n''+1&lt;/sup&gt;−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||9{5}9||(149×17&lt;sup&gt;''n''+1&lt;/sup&gt;+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||95{F}||(2543×17&lt;sup&gt;''n''&lt;/sup&gt;−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||A{D}F||(173×17&lt;sup&gt;''n''+1&lt;/sup&gt;+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}B3||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+190||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}DB||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+232||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2BE||(11×17&lt;sup&gt;''n''+3&lt;/sup&gt;−41579)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2E||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;−2411)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}E9||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+773)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}EE||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+853)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||D0G{D}||(60381×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2860381*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||E9{B}||(3963×17&lt;sup&gt;''n''&lt;/sup&gt;−11)/16||29000||[http://factordb.com/index.php?query=%283963*17%5En-11%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||F1{9}||(4105×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16||1000000||[http://factordb.com/index.php?query=%284105*17%5En-9%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||FD0{D}||(72909×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2872909*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||G{7}F||(263×17&lt;sup&gt;''n''&lt;/sup&gt;+121)/16||29000||[http://factordb.com/index.php?query=%28263*17%5E%28n%2B1%29%2B121%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||2{7}9D||(47×21&lt;sup&gt;''n''&lt;/sup&gt;+953)/20||20000||[http://factordb.com/index.php?query=%2847*21%5E%28n%2B2%29%2B953%29%2F20&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||2F{C}A||(288×21&lt;sup&gt;''n''&lt;/sup&gt;−13)/5||20000||[http://factordb.com/index.php?query=%28288*21%5E%28n%2B1%29-13%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||7{K}Z||(53×36&lt;sup&gt;''n''+1&lt;/sup&gt;+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||B{0}EUV||11×36&lt;sup&gt;''n''+3&lt;/sup&gt;+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||HM{0}N||634×36&lt;sup&gt;''n''+1&lt;/sup&gt;+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||N{0}YYN||23×36&lt;sup&gt;''n''+3&lt;/sup&gt;+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||O{L}Z||(123×36&lt;sup&gt;''n''+1&lt;/sup&gt;+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||S{0}8H||28×36&lt;sup&gt;''n''+2&lt;/sup&gt;+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|}

For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E

For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D}

=== Base 17 ===

All searched to length 29000

* 1{7}
* 1F{0}7
* 4{7}A
* 51{0}D
* 70F{0}D
* 8{B}9
* 9{5}9
* 95{F}
* A{D}F
* B{0}B3
* B{0}DB
* {B}2BE
* {B}2E
* {B}E9
* {B}EE
* D0G{D}
* E9{B}
* F1{9}
* FD0{D}
* G{7}F

=== Base 21 ===

All searched to length 20000

* 2{7}9D
* 2F{C}A
* 4{3}B
* 5{0}DJ
* {5}FEK
* {7}ID
* 99{0}99H
* {9}0D
* {9}D
* B0{H}6H
* B3{0}EB
* B9{0}E5
* B{D}B
* B{H}6H
* DH{D}
* F{9}D
* {F}35
* G{0}FK
* H{0}7771
* H{D}
* {J}6J

=== Base 36 ===

All searched to length 20000

* 7{K}Z
* B{0}EUV
* HM{0}N
* N{0}YYN
* O{L}Z
* S{0}8H

==Primality certificates for the proven primes &gt; 10&lt;sup&gt;299&lt;/sup&gt;==

See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]]

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime
|-
|9||149||76&lt;sub&gt;329&lt;/sub&gt;2||(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642]
|-
|9||150||27&lt;sub&gt;686&lt;/sub&gt;07||(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486]
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||3×9&lt;sup&gt;1160&lt;/sup&gt;+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423]
|-
|11||1065||A&lt;sub&gt;713&lt;/sub&gt;58||11&lt;sup&gt;715&lt;/sup&gt;−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487]
|-
|11||1066||7&lt;sub&gt;759&lt;/sub&gt;44||(7×11&lt;sup&gt;761&lt;/sup&gt;−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840]
|-
|11||1067||557&lt;sub&gt;1011&lt;/sub&gt;||(607×11&lt;sup&gt;1011&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522]
|-
|13||3165||50&lt;sub&gt;270&lt;/sub&gt;44||5×13&lt;sup&gt;272&lt;/sup&gt;+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005]
|-
|13||3166||9&lt;sub&gt;271&lt;/sub&gt;095||(3×13&lt;sup&gt;274&lt;/sup&gt;−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654]
|-
|13||3167||10&lt;sub&gt;286&lt;/sub&gt;7771||13&lt;sup&gt;290&lt;/sup&gt;+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633]
|-
|13||3168||9&lt;sub&gt;308&lt;/sub&gt;1||(3×13&lt;sup&gt;309&lt;/sup&gt;−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=308&amp;c0=-&amp;EN= 13&lt;sup&gt;308&lt;/sup&gt;−1]
|-
|13||3169||B&lt;sub&gt;341&lt;/sub&gt;C4||(11×13&lt;sup&gt;343&lt;/sup&gt;+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618]
|-
|13||3170||8B&lt;sub&gt;343&lt;/sub&gt;||(107×13&lt;sup&gt;343&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736]
|-
|13||3171||710&lt;sub&gt;371&lt;/sub&gt;111||92×13&lt;sup&gt;374&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609]
|-
|13||3172||75&lt;sub&gt;375&lt;/sub&gt;7||(89×13&lt;sup&gt;376&lt;/sup&gt;+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596]
|-
|13||3173||9B0&lt;sub&gt;391&lt;/sub&gt;9||128×13&lt;sup&gt;392&lt;/sup&gt;+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790]
|-
|13||3174||7B0B&lt;sub&gt;397&lt;/sub&gt;||(15923×13&lt;sup&gt;397&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574]
|-
|13||3175||10&lt;sub&gt;414&lt;/sub&gt;93||13&lt;sup&gt;416&lt;/sup&gt;+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240]
|-
|13||3176||81010&lt;sub&gt;415&lt;/sub&gt;1||17746×13&lt;sup&gt;416&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3177||8110&lt;sub&gt;435&lt;/sub&gt;1||1366×13&lt;sup&gt;436&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3178||B7&lt;sub&gt;486&lt;/sub&gt;||(139×13&lt;sup&gt;486&lt;/sup&gt;−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892]
|-
|13||3179||B&lt;sub&gt;563&lt;/sub&gt;C||(11×13&lt;sup&gt;564&lt;/sup&gt;+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=564&amp;c0=-&amp;EN= 13&lt;sup&gt;564&lt;/sup&gt;−1]
|-
|13||3180||1B&lt;sub&gt;576&lt;/sub&gt;||(23×13&lt;sup&gt;576&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=576&amp;c0=-&amp;EN= 13&lt;sup&gt;576&lt;/sup&gt;−1]
|-
|13||3181||80&lt;sub&gt;693&lt;/sub&gt;87||8×13&lt;sup&gt;695&lt;/sup&gt;+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532]
|-
|13||3182||CC5&lt;sub&gt;713&lt;/sub&gt;||(2021×13&lt;sup&gt;713&lt;/sup&gt;−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353]
|-
|13||3183||B&lt;sub&gt;834&lt;/sub&gt;74||(11×13&lt;sup&gt;836&lt;/sup&gt;−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871]
|-
|13||3184||9&lt;sub&gt;968&lt;/sub&gt;B||(3×13&lt;sup&gt;969&lt;/sup&gt;+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244]
|-
|13||3185||10&lt;sub&gt;1295&lt;/sub&gt;181||13&lt;sup&gt;1298&lt;/sup&gt;+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013]
|-
|13||3186||9&lt;sub&gt;1362&lt;/sub&gt;5||(3×13&lt;sup&gt;1363&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776]
|-
|13||3187||7&lt;sub&gt;1504&lt;/sub&gt;1||(7×13&lt;sup&gt;1505&lt;/sup&gt;−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755]
|-
|13||3188||930&lt;sub&gt;1551&lt;/sub&gt;1||120×13&lt;sup&gt;1552&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3189||720&lt;sub&gt;2297&lt;/sub&gt;2||93×13&lt;sup&gt;2298&lt;/sup&gt;+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910]
|-
|13||3190||1770&lt;sub&gt;2703&lt;/sub&gt;17||267×13&lt;sup&gt;2705&lt;/sup&gt;+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825]
|-
|13||3191||390&lt;sub&gt;6266&lt;/sub&gt;1||48×13&lt;sup&gt;6267&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3192||B0&lt;sub&gt;6540&lt;/sub&gt;BBA||11×13&lt;sup&gt;6543&lt;/sup&gt;+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906]
|-
|13||3193||C&lt;sub&gt;10631&lt;/sub&gt;92||13&lt;sup&gt;10633&lt;/sup&gt;−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750]
|-
|14||649||34D&lt;sub&gt;708&lt;/sub&gt;||47×14&lt;sup&gt;708&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||5×14&lt;sup&gt;19698&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2328||880&lt;sub&gt;246&lt;/sub&gt;7||136×16&lt;sup&gt;247&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is &lt; 10&lt;sup&gt;299&lt;/sup&gt;
|-
|16||2329||D4&lt;sub&gt;263&lt;/sub&gt;D||(199×16&lt;sup&gt;264&lt;/sup&gt;+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238]
|-
|16||2330||E0&lt;sub&gt;261&lt;/sub&gt;4DD||14×16&lt;sup&gt;264&lt;/sup&gt;+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352]
|-
|16||2331||8C0&lt;sub&gt;290&lt;/sub&gt;ED||140×16&lt;sup&gt;292&lt;/sup&gt;+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307]
|-
|16||2332||DA&lt;sub&gt;305&lt;/sub&gt;5||(41×16&lt;sup&gt;306&lt;/sup&gt;−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284]
|-
|16||2333||CE80&lt;sub&gt;422&lt;/sub&gt;D||3304×16&lt;sup&gt;423&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257]
|-
|16||2334||5F&lt;sub&gt;544&lt;/sub&gt;6F||6×16&lt;sup&gt;546&lt;/sup&gt;−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967]
|-
|16||2335||88F&lt;sub&gt;545&lt;/sub&gt;||137×16&lt;sup&gt;545&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2336||BE0&lt;sub&gt;792&lt;/sub&gt;BB||190×16&lt;sup&gt;794&lt;/sup&gt;+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938]
|-
|16||2337||D9&lt;sub&gt;1052&lt;/sub&gt;||(68×16&lt;sup&gt;1052&lt;/sup&gt;−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020]
|-
|16||2338||FAF&lt;sub&gt;1062&lt;/sub&gt;45||251×16&lt;sup&gt;1064&lt;/sup&gt;−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610]
|-
|16||2339||F8&lt;sub&gt;1517&lt;/sub&gt;F||(233×16&lt;sup&gt;1518&lt;/sup&gt;+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824]
|-
|16||2340||20&lt;sub&gt;1713&lt;/sub&gt;321||2×16&lt;sup&gt;1716&lt;/sup&gt;+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735]
|-
|16||2341||300F&lt;sub&gt;1960&lt;/sub&gt;AF||769×16&lt;sup&gt;1962&lt;/sup&gt;−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750]
|-
|16||2342||90&lt;sub&gt;3542&lt;/sub&gt;91||9×16&lt;sup&gt;3544&lt;/sup&gt;+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191]
|-
|16||2343||5BC&lt;sub&gt;3700&lt;/sub&gt;D||(459×16&lt;sup&gt;3701&lt;/sup&gt;+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322]
|-
|16||2344||D0B&lt;sub&gt;17804&lt;/sub&gt;||(3131×16&lt;sup&gt;17804&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511]
|-
|18||547||80&lt;sub&gt;298&lt;/sub&gt;B||8×18&lt;sup&gt;299&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is an algebraic factor of ''N''+1, factordb entry of 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is [http://factordb.com/index.php?id=1100000000900149167]
|-
|18||548||H&lt;sub&gt;766&lt;/sub&gt;FH||18&lt;sup&gt;768&lt;/sup&gt;−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490]
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||12×18&lt;sup&gt;6270&lt;/sup&gt;+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437]
|-
|20||3301||H&lt;sub&gt;247&lt;/sub&gt;A0H||(17×20&lt;sup&gt;250&lt;/sup&gt;−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619]
|-
|20||3302||7&lt;sub&gt;249&lt;/sub&gt;A7||(7×20&lt;sup&gt;251&lt;/sup&gt;+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602]
|-
|20||3303||J7&lt;sub&gt;270&lt;/sub&gt;||(368×20&lt;sup&gt;270&lt;/sup&gt;−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462]
|-
|20||3304||J&lt;sub&gt;330&lt;/sub&gt;CCC7||20&lt;sup&gt;334&lt;/sup&gt;−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572]
|-
|20||3305||40&lt;sub&gt;387&lt;/sub&gt;404B||4×20&lt;sup&gt;391&lt;/sup&gt;+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563]
|-
|20||3306||EC0&lt;sub&gt;429&lt;/sub&gt;7||292×20&lt;sup&gt;430&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702]
|-
|20||3307||G&lt;sub&gt;447&lt;/sub&gt;99||(16×20&lt;sup&gt;449&lt;/sup&gt;−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753]
|-
|20||3308||3A&lt;sub&gt;527&lt;/sub&gt;3||(67×20&lt;sup&gt;528&lt;/sup&gt;−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531]
|-
|20||3309||E&lt;sub&gt;566&lt;/sub&gt;C7||(14×20&lt;sup&gt;568&lt;/sup&gt;−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516]
|-
|20||3310||JCJ&lt;sub&gt;629&lt;/sub&gt;||393×20&lt;sup&gt;629&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|20||3311||J&lt;sub&gt;655&lt;/sub&gt;05J||20&lt;sup&gt;658&lt;/sup&gt;−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052]
|-
|20||3312||50&lt;sub&gt;1163&lt;/sub&gt;AJ||5×20&lt;sup&gt;1165&lt;/sup&gt;+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412]
|-
|20||3313||CD&lt;sub&gt;2449&lt;/sub&gt;||(241×20&lt;sup&gt;2449&lt;/sup&gt;−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915]
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||16×20&lt;sup&gt;6270&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457]
|-
|22||7984||I7G0&lt;sub&gt;254&lt;/sub&gt;H||8882×22&lt;sup&gt;255&lt;/sup&gt;+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788]
|-
|22||7985||D0&lt;sub&gt;255&lt;/sub&gt;5EEF||13×22&lt;sup&gt;259&lt;/sup&gt;+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932]
|-
|22||7986||IK&lt;sub&gt;322&lt;/sub&gt;F||(398×22&lt;sup&gt;323&lt;/sup&gt;−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145]
|-
|22||7987||C0&lt;sub&gt;340&lt;/sub&gt;G9||12×22&lt;sup&gt;342&lt;/sup&gt;+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159]
|-
|22||7988||77E&lt;sub&gt;348&lt;/sub&gt;K7||(485×22&lt;sup&gt;350&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779]
|-
|22||7989||J&lt;sub&gt;379&lt;/sub&gt;KJ||(19×22&lt;sup&gt;381&lt;/sup&gt;+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027]
|-
|22||7990||J&lt;sub&gt;388&lt;/sub&gt;EJ||(19×22&lt;sup&gt;390&lt;/sup&gt;−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729]
|-
|22||7991||DJ&lt;sub&gt;400&lt;/sub&gt;||(292×22&lt;sup&gt;400&lt;/sup&gt;−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110]
|-
|22||7992||E&lt;sub&gt;404&lt;/sub&gt;K7||(2×22&lt;sup&gt;406&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298]
|-
|22||7993||66F&lt;sub&gt;453&lt;/sub&gt;B3||(971×22&lt;sup&gt;455&lt;/sup&gt;−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809]
|-
|22||7994||L0&lt;sub&gt;454&lt;/sub&gt;B63||21×22&lt;sup&gt;457&lt;/sup&gt;+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331]
|-
|22||7995||L&lt;sub&gt;483&lt;/sub&gt;G3||22&lt;sup&gt;485&lt;/sup&gt;−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730]
|-
|22||7996||E60&lt;sub&gt;496&lt;/sub&gt;L||314×22&lt;sup&gt;497&lt;/sup&gt;+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239]
|-
|22||7997||I&lt;sub&gt;626&lt;/sub&gt;AF||(6×22&lt;sup&gt;628&lt;/sup&gt;−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334]
|-
|22||7998||K0&lt;sub&gt;760&lt;/sub&gt;EC1||20×22&lt;sup&gt;763&lt;/sup&gt;+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415]
|-
|22||7999||J0&lt;sub&gt;767&lt;/sub&gt;IGGJ||19×22&lt;sup&gt;771&lt;/sup&gt;+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567]
|-
|22||8000||7&lt;sub&gt;959&lt;/sub&gt;K7||(22&lt;sup&gt;961&lt;/sup&gt;+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817]
|-
|22||8001||L&lt;sub&gt;2385&lt;/sub&gt;KE7||22&lt;sup&gt;2388&lt;/sup&gt;−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774]
|-
|22||8002||7&lt;sub&gt;3815&lt;/sub&gt;2L||(22&lt;sup&gt;3817&lt;/sup&gt;−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839]
|-
|24||3400||I0&lt;sub&gt;241&lt;/sub&gt;I5||18×24&lt;sup&gt;243&lt;/sup&gt;+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037]
|-
|24||3401||D0&lt;sub&gt;259&lt;/sub&gt;KKD||13×24&lt;sup&gt;262&lt;/sup&gt;+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725]
|-
|24||3402||C7&lt;sub&gt;298&lt;/sub&gt;||(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235]
|-
|24||3403||20&lt;sub&gt;313&lt;/sub&gt;7||2×24&lt;sup&gt;314&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241]
|-
|24||3404||BC0&lt;sub&gt;331&lt;/sub&gt;B||276×24&lt;sup&gt;332&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842]
|-
|24||3405||N&lt;sub&gt;2644&lt;/sub&gt;LLN||24&lt;sup&gt;2647&lt;/sup&gt;−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089]
|-
|24||3406||D&lt;sub&gt;2698&lt;/sub&gt;LD||(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876]
|-
|24||3407||A0&lt;sub&gt;2951&lt;/sub&gt;8ID||10×24&lt;sup&gt;2954&lt;/sup&gt;+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654]
|-
|24||3408||88N&lt;sub&gt;5951&lt;/sub&gt;||201×24&lt;sup&gt;5951&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606]
|-
|30||2613||AN&lt;sub&gt;206&lt;/sub&gt;||(313×30&lt;sup&gt;206&lt;/sup&gt;−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073]
|-
|30||2614||M&lt;sub&gt;241&lt;/sub&gt;QB||(22×30&lt;sup&gt;243&lt;/sup&gt;+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295]
|-
|30||2615||M0&lt;sub&gt;547&lt;/sub&gt;SS7||22×30&lt;sup&gt;550&lt;/sup&gt;+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988]
|-
|30||2616||C0&lt;sub&gt;1022&lt;/sub&gt;1||12×30&lt;sup&gt;1023&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|30||2617||5&lt;sub&gt;4882&lt;/sub&gt;J||(5×30&lt;sup&gt;4883&lt;/sup&gt;+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423]
|-
|30||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||25×30&lt;sup&gt;34205&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|}

==Unproven PRPs==

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this PRP
|-
|11||1068||57&lt;sub&gt;62668&lt;/sub&gt;||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000003573679860]
|-
|13||3194||C5&lt;sub&gt;23755&lt;/sub&gt;C||(149×13&lt;sup&gt;23756&lt;/sup&gt;+79)/12||[http://factordb.com/index.php?id=1100000003590647776]
|-
|13||3195||80&lt;sub&gt;32017&lt;/sub&gt;111||8×13&lt;sup&gt;32020&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000000490878060]
|-
|16||2345||DB&lt;sub&gt;32234&lt;/sub&gt;||(206×16&lt;sup&gt;32234&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000002383583629]
|-
|16||2346||4&lt;sub&gt;72785&lt;/sub&gt;DD||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15||[http://factordb.com/index.php?id=1100000003615909841]
|-
|22||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21||[http://factordb.com/index.php?id=1100000003594696838]
|-
|30||2618||I0&lt;sub&gt;24608&lt;/sub&gt;D||18×30&lt;sup&gt;24609&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003593967511]
|}

All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10&lt;sup&gt;16&lt;/sup&gt;. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]])

==Proof==

===Base 2===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.

===Base 3===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (2,1), (2,2)

* Case (1,1):
** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes)
*** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime.

===Base 4===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (2,1), (2,3), (3,1), (3,3)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.

===Base 5===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4)

* Case (1,1):
** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes)
*** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes)
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes)
***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb])
***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes)
**** The smallest prime of the form 1{4}4 is 14444.
**** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes)
*** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes)
**** For the 3{0,3}1 family, we can separate this family to four families:
***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01.
****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime.
***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031.
***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001.
***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31.
****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime.
**** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.
* Case (3,4):
** 34 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes)
**** The smallest prime of the form 4{4}1 is 44441.
**** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime.
* Case (4,2):
** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime.

===Base 6===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes)
*** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes)
**** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 4{0}41 is 40041
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes)
*** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.

===Base 7===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

* Case (1,1):
** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes)
*** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {}
**** If there are &gt;=3 number of 1 and 5 in the {}:
***** If there is 111 in the {}, then we have the prime 11111
***** If there is 115 in the {}, then the prime 115 is a subsequence
***** If there is 151 in the {}, then the prime 115 is a subsequence
***** If there is 155 in the {}, then the prime 155 is a subsequence
***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime
***** If there is 515 in the {}, then the prime 515 is a subsequence
***** If there is 551 in the {}, then the prime 551 is a subsequence
***** If there is 555 in the {}, then the prime 551 is a subsequence
**** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1
***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1
****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1
******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21
******** The smallest prime of the form 11{0}21 is 1100021
******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime
******** The smallest prime of the form 1001{0}21 is 100121
********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families
***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1
****** Since 1051 is prime, we only need to consider the family 15{0}1
******* The smallest prime of the form 15{0}1 is 150001
* Case (1,2):
** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes)
*** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {}
**** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime.
**** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime.
**** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2
***** Since 1022 and 1202 are primes, we only need to consider the number 122
****** 122 is not prime.
**** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime.
* Case (1,3):
** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** 14 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime.
* Case (1,6):
** 16 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime.
* Case (2,2):
** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,6):
** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes)
*** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes)
**** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's)
***** For the 3{0,1}1 family:
****** If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
****** If there are no 1's between (3,1), then the form will be 3{0}1
******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime.
****** If there are no 0's between (3,1), then the form will be 3{1}1
******* The smallest prime of the form 3{1}1 is 31111
****** If there are exactly 1 0's between (3,1), then there must be &lt;3 1's between (3,1), or 31111 will be a subsequence.
******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime.
******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011
******** Neither 3101 nor 3011 is prime.
******* If there are no 1's between (3,1), then the number must be 301
******** 301 is not prime.
***** For the 3{3}3{0,1}1 family:
****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence.
****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1
******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence.
******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime.
******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime.
******* If there are no 1's between (33,1), then the form will be 33{0}1
******** The smallest prime of the form 33{0}1 is 33001
****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1
******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence.
******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301
******** The smallest prime of the form 3{3}301 is 33333301
******* If there are no 0's between (3{3}3,1), then the form is 3{3}31
******** The smallest prime of the form 3{3}31 is 33333333333333331
**** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1
***** Numbers containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1,3}3{0,1,3}1
******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes)
******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes)
********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes)
********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence.
********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime.
********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1
*********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime.
********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime.
***** Numbers not containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1}1
******* If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1
******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime.
******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1
******** If there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111
********* None of 31151, 31511, 35111 are primes.
******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 351
********* 351 is not prime.
******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111
********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes.
********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime.
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1
********* Neither 3051 nor 3501 is prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** If there are &gt;=2 5's in {}, then 553 will be a subsequence.
*** If there are no 5's in {}, then the family will be 3{0,3,6}3
**** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
*** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3
**** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes)
***** The smallest prime of the form 3{0}53 is 300053
* Case (3,4):
** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes)
*** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes)
**** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes)
**** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes)
* Case (3,5):
** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes)
*** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence.
*** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence.
*** If there are at least two 1's in {}, then 115 will be a subsequence.
*** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5
**** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime.
*** If there is no 1's in {}, then the family will be 3{0,4,5}5
**** If there are at least to 4's in {}, then 344 and 445 will be subsequences.
**** If there is no 4's in {}, then the family will be 3{0,5}5
***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5
****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime.
****** The smallest prime of the form 3{5}5 is 35555
**** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5
***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes)
****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime.
* Case (3,6):
** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,2):
** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes)
*** If there is no 5's in {}, then the family will be 4{0,2,4,6}4
**** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4
**** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes)
***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence.
***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4
****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes)
***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4
****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4
******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes)
* Case (4,5):
** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** If there are at least two 5's in {}, then 4555 will be a subsequence.
*** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there is no 5's in {}, then the family will be 4{0}5
**** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime.
* Case (4,6):
** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (5,1):
** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 3's in {}, then 533 will be a subsequence.
*** If there is no 3's in {}, then the family will be 5{0,1}1
**** Since 5011 is prime, we only need to consider the family 5{1}{0}1
***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes)
****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime.
****** The smallest prime of the form 51{0}1 is 5100000001
****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime.
****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime.
*** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1
**** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence.
***** Thus we only need to consider the family 5{0}3{0,1}1
****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime.
****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime.
****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence.
****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1
******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0)
******** The smallest prime of the form 5311{0}1 is 531101
* Case (5,2):
** 52 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 1's in {}, then 113 will be a subsequence.
*** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
*** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime.
* Case (5,4):
** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes)
*** If there are no 5's in {}, then the family will be 5{0,3}4
**** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime.
*** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence.
*** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there are at least two 5's in {}, then 5554 will be a subsequence.
* Case (5,5):
** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes)
*** If there are no 4's in {}, then the family will be 5{0,5}5
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
*** If there are no 5's in {}, then the family will be 5{0,4}5
**** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime.
*** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence.
* Case (5,6):
** 56 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,2):
** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (6,3):
** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,4):
** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are at least three 3's in {}, then 3334 will be a subsequence.
*** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4
**** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence.
**** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence.
**** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence.
**** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence.
**** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence.
**** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence.
**** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence.
**** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence.
**** Thus the number can only be 634
***** 634 is not prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,6):
** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}6
**** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime.
*** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6
**** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime.
*** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence.
*** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence.

===Base 8===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7)

* Case (1,1):
** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes)
**** 171 is not prime.
**** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 &gt; 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 &gt; 1, this factorization is nontrivial), thus cannot be prime.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes)
*** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime)
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes)
*** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes)
*** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes)
**** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime.
**** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes)
***** The smallest prime of the form 33{4}1 is 3344441
***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime.
**** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes)
***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime.
***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes)
****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes.
* Case (3,3):
** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes)
*** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes)
**** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime)
**** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461
***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes)
****** The smallest prime of the form 4{4}41 is 444444441
****** The smallest prime of the form 4{4}641 is 444641
***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes)
****** The smallest prime of the form 4{4}411 is 444444441
****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes)
***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461
****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime)
**** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes)
***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime)
***** The smallest prime of the form 4{4}641 is 444641
* Case (4,3):
** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes)
*** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes)
**** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime.
**** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime.
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes)
*** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes)
**** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb])
**** The smallest prime of the form 4{4}77 is 4444477
**** The smallest prime of the form 4{7}7 is 47777
**** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime)
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes)
*** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes)
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
**** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes)
***** 500525 is not prime.
***** The smallest prime of the form 5{5}25 is 555555555555525
***** The smallest prime of the form 5{5}025 is 55555025
***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes)
***** The smallest prime of the form 5{5}0525 is 5550525
***** The smallest prime of the form 5{5}00525 is 5500525
***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes)
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence)
**** The form is 6{0,1,7}4{0,1,7}1
***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1
****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411
******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41
******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41
********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.
********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime.
********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime.
******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411
******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime)
*** Numbers not containing 4:
**** The form is 6{0,1,7}1
***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1
****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime.
****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1
******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1
******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.
******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime)
* Case (6,3):
** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,7):
** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them
**** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17
***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime.
**** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77
***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime.
**** If there is neither 1 nor 7 between them, then the form is 6{0,4}7
***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes)
****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime.
****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime.
* Case (7,1):
** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes)
*** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes)
**** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes)
***** The smallest prime of the form 7{7}1 is 7777777777771
***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime)
***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime.
***** The smallest prime of the form 7{7}4444441 is 77774444441
****** Since this prime has just 4 7's, we only need to consider the families with &lt;=3 7's
******* The smallest prime of the form 7{4}1 is 744444441
******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime.
******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes)
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.

===Base 10===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,9):
** 19 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 2{0}21 is 20021
**** The smallest prime of the form 22{0}1 is 22000001
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes)
*** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes)
**** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime.
**** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime.
* Case (2,9):
** 29 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,9):
** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** 47 is prime, and thus the only minimal prime in this family.
* Case (4,9):
** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime.
* Case (5,1):
** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes)
**** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes)
***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 5{5}1 is 555555555551
**** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581
***** 581 is not prime.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,7):
** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027
**** The smallest prime of the form 52{0}7 is 5200007
* Case (5,9):
** 59 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,3):
** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,9):
** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49
***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49
****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes)
******* The smallest prime of the form 6{6}49 is 666649
******** Since this prime has just 4 6's, we only need to consider the families with &lt;=3 6's
********* The smallest prime of the form 6{0}49 is 60000049
********* The smallest prime of the form 66{0}49 is 66000049
********* The smallest prime of the form 666{0}49 is 66600049
* Case (7,1):
** 71 is prime, and thus the only minimal prime in this family.
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,9):
** 79 is prime, and thus the only minimal prime in this family.
* Case (8,1):
** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes)
**** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes)
***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime)
***** The smallest prime of the form 80{5}1 is 80555551
* Case (8,3):
** 83 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime.
* Case (8,9):
** 89 is prime, and thus the only minimal prime in this family.
* Case (9,1):
** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes)
**** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021
***** None of 921, 9201, 9021 are primes.
**** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801
***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
* Case (9,3):
** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (9,7):
** 97 is prime, and thus the only minimal prime in this family.
* Case (9,9):
** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9
***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes)
****** The smallest prime of the form 94{6}9 is 946669

===Base 12===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,B):
** 1B is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes)
*** The smallest prime of the form 2{0}1 is 2001
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (2,B):
** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes)
**** None of 20B, 22B, 29B, 209B, 220B are primes.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,B):
** 3B is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes)
**** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 44{A}1 is 44AAA1
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes)
**** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077
* Case (4,B):
** 4B is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes)
*** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime.
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (5,B):
** 5B is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,5):
** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes)
**** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 6{0}A5 is 600A5
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,B):
** 6B is prime, and thus the only minimal prime in this family.
* Case (7,1):
** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes)
*** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1
**** None of 741, 7B1, 74B1 are primes.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,B):
** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes)
**** The smallest prime of the form 7{9}B is 7999B
**** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes)
**** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime)
* Case (8,1):
** 81 is prime, and thus the only minimal prime in this family.
* Case (8,5):
** 85 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** 87 is prime, and thus the only minimal prime in this family.
* Case (8,B):
** 8B is prime, and thus the only minimal prime in this family.
* Case (9,1):
** 91 is prime, and thus the only minimal prime in this family.
* Case (9,5):
** 95 is prime, and thus the only minimal prime in this family.
* Case (9,7):
** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes)
**** None of 947, 977, 997, 9477, 9977 are primes.
* Case (9,B):
** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes)
**** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes)
***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime)
***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime)
**** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes)
***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes)
****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime.
****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime)
****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime.
**** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B
***** None of 99B, 999B, 979B, 9799B, 9979B are primes.
* Case (A,1):
** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes)
**** For the A{A}{0}1 family:
***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form AA{0}1 is AA000001
***** The smallest prime of the form AAA{0}1 is AAA0001
***** The smallest prime of the form AAAA{0}1 is AAAA1
****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc.
**** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime.
* Case (A,5):
** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes)
*** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes)
**** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form A{0}65 is A00065
* Case (A,7):
** A7 is prime, and thus the only minimal prime in this family.
* Case (A,B):
** AB is prime, and thus the only minimal prime in this family.
* Case (B,1):
** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes)
**** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1
***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1
****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime)
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1
****** The smallest prime of the form B{A}01 is BAA01
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
**** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1
***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1
****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1
******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1
******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes.
****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1
******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime)
***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes)
****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B04{A}1 is B04A1
****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime)
**** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1
***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1
****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1
******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime.
******* The smallest prime of the form B{B}AA1 is BBBAA1
****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1
******* The smallest prime of the form B{B}AA1 is BBBAA1
**** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes)
***** The smallest prime of the form B{B}B1 is BBBB1
***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form B{B}BAA1 is BBBAA1
* Case (B,5):
** B5 is prime, and thus the only minimal prime in this family.
* Case (B,7):
** B7 is prime, and thus the only minimal prime in this family.
* Case (B,B):
** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes)
*** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B
**** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B
***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime.
***** For the B{0,B}9B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes)
******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B
******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 &gt; 1, 10^n+5 ≥ 10^1+5 = 15 &gt; 1, this factorization is nontrivial), thus cannot be prime.
***** For the B{0,B}99B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes)
******* The smallest prime of the form B{0}99B is B00099B
******* The smallest prime of the form B{B}99B is BBBBBB99B
***** For the B{0,B}999B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes)
******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes)
******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime)

== Examples of families which can be ruled out as contain no primes &gt; ''b'' ==

It is not known if this problem is solvable:

Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''?

It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime &gt; ''b'' or not. We use two different heuristic strategies to show that such families contain no primes &gt; ''b''.

In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;.

Examples of first strategy: (we can show that the corresponding numbers are &gt; all elements in ''S'', if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial)

* In base 10, all numbers of the form 4{6}9 are divisible by 7
* In base 6, all numbers of the form 4{0}1 are divisible by 5
* In base 15, all numbers of the form 9{6}8 are divisible by 11
* In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5}
* In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3}
* In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5}
* In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13}
* In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17}
* In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13}
* In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13}

Examples of second strategy: (we can show that both factors are &gt; 1, if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial)

* In base 9, all numbers of the form {1} factored as difference of squares
* In base 8, all numbers of the form 1{0}1 factored as sum of cubes
* In base 9, all numbers of the form 3{8} factored as difference of squares
* In base 16, all numbers of the form 8{F} factored as difference of squares
* In base 16, all numbers of the form {F}7 factored as difference of squares
* In base 9, all numbers of the form 3{1} factored as difference of squares
* In base 16, all numbers of the form {4}1 factored as difference of squares
* In base 16, all numbers of the form 1{5} factored as difference of squares
* In base 16, all numbers of the from {C}D factored as ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;
* In base 16, all numbers of the form B{4}1 factored as difference of squares

Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are &gt; all elements in S, and for the part of the second strategy, both factors are &gt; 1, if n makes corresponding numbers &gt; b, thus these factorizations are nontrivial)

* In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd
* In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even

== Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes &gt; ''b'' ==

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-powers factorization

=== 1{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 1{0}3 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== 1{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 1{0}5 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 5: Finite covering set {5}

=== 1{0}6 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== 1{0}7 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 7: Finite covering set {7}

=== 1{0}z ===

(none)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

(none)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {1} ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with ''r''&gt;1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2))

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' == 0 mod 2: Finite covering set {2}

=== 1{2} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 1{3} ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361)
* ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865)

=== 1{4} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization

=== 1{z} ===

(none)

=== 2{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 2{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {2}1 ===

* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 2{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 3{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {3}1 ===

* ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121)

=== 3{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 4{0}1 ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 4{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {4}1 ===

* ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225)

=== 4{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 5{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 5{z} ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}
* ''b'' = 6''m''&lt;sup&gt;2&lt;/sup&gt; with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014)

=== 6{0}1 ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== 6{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 7{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 7{z} ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993)
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Difference-of-cubes factorization

=== 8{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022)
* ''b'' = 467: Finite covering set {3, 5, 7, 19, 37}
* ''b'' = 722: Finite covering set {3, 5, 13, 73, 109}
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Sum-of-cubes factorization
* ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2

=== 8{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 9{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 9{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{0}1 ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{z} ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 307: Finite covering set {5, 11, 29}
* ''b'' = 901: Finite covering set {7, 11, 13, 19}

=== C{0}1 ===

* ''b'' == 1 mod 13: Finite covering set {13}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 296, 901: Finite covering set {7, 11, 13, 19}
* ''b'' = 562, 828, 900: Finite covering set {7, 13, 19}
* ''b'' = 563: Finite covering set {5, 7, 13, 19, 29}
* ''b'' = 597: Finite covering set {5, 13, 29}

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

(none)

=== {y}z ===

(none)

=== z{0}1 ===

(none)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3))
* ''b'' = 4''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt; (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}t ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 7: Finite covering set {7}

=== {z}u ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== {z}v ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 5: Finite covering set {5}

=== {z}w ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== {z}x ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== {z}y ===

* ''b'' == 0 mod 2: Finite covering set {2}

== Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) ==

Format: base (length)

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

(none)

=== 1{0}2 ===

(none)

=== 1{0}3 ===

(none)

=== 1{0}4 ===

53 (13403)

113 (10647)

=== 1{0}z ===

113 (20089)

123 (64371)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

(none)

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

208 (26682)

607 (11032)

828 (19659)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

201 (31276)

222 (52727)

227 (36323)

327 (135983)

425 (11231)

710 (24112)

717 (37508)

719 (13420)

=== {1} ===

152 (270217)

184 (16703)

200 (17807)

311 (36497)

326 (26713)

331 (25033)

371 (15527)

485 (99523)

629 (32233)

649 (43987)

670 (18617)

684 (22573)

691 (62903)

693 (41189)

731 (15427)

752 (32833)

872 (10093)

932 (20431)

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

(none)

=== 1{z} ===

107 (21911)

170 (166429)

278 (43909)

303 (40175)

383 (20957)

515 (58467)

522 (62289)

578 (129469)

590 (15527)

647 (21577)

662 (16591)

698 (127559)

704 (62035)

845 (39407)

938 (40423)

969 (24097)

989 (26869)

=== 2{0}1 ===

101 (192276)

206 (46206)

218 (333926)

236 (161230)

257 (12184)

305 (16808)

467 (126776)

578 (44166)

626 (174204)

695 (94626)

752 (26164)

788 (72918)

869 (49150)

887 (27772)

899 (15732)

932 (13644)

=== 2{z} ===

432 (16003)

=== 3{0}1 ===

912 (132174)

=== 3{z} ===

72 (1119850)

212 (34414)

218 (23050)

270 (89662)

303 (198358)

312 (51566)

422 (21738)

480 (93610)

513 (38032)

527 (46074)

566 (23874)

650 (498102)

686 (16584)

758 (15574)

783 (12508)

800 (33838)

921 (98668)

947 (10056)

=== 4{0}1 ===

107 (32587)

227 (13347)

257 (160423)

355 (10990)

410 (144079)

440 (56087)

452 (14155)

482 (30691)

542 (15983)

579 (67776)

608 (20707)

635 (11723)

650 (96223)

679 (69450)

737 (269303)

740 (58043)

789 (149140)

797 (468703)

920 (103687)

934 (101404)

962 (84235)

=== 4{z} ===

14 (19699)

68 (13575)

254 (15451)

800 (20509)

=== 5{0}1 ===

326 (400786)

350 (20392)

554 (10630)

662 (13390)

926 (40036)

=== 5{z} ===

258 (212135)

272 (148427)

299 (64898)

307 (26263)

354 (25566)

433 (283919)

635 (36163)

678 (40859)

692 (45447)

719 (20552)

768 (70214)

857 (23083)

867 (61411)

972 (36703)

=== 6{0}1 ===

108 (16318)

129 (16797)

409 (369833)

522 (52604)

587 (24120)

643 (164916)

762 (11152)

789 (27297)

986 (21634)

=== 6{z} ===

68 (25396)

332 (15222)

338 (42868)

362 (146342)

488 (33164)

566 (164828)

980 (50878)

986 (12506)

1016 (23336)

=== 7{0}1 ===

398 (17473)

1004 (54849)

=== 7{z} ===

97 (192336)

170 (15423)

194 (38361)

202 (155772)

282 (21413)

283 (164769)

332 (13205)

412 (29792)

560 (19905)

639 (10668)

655 (53009)

811 (31784)

814 (17366)

866 (108591)

908 (61797)

962 (31841)

992 (10605)

997 (15815)

=== 8{0}1 ===

23 (119216)

53 (227184)

158 (123476)

254 (67716)

320 (52004)

410 (279992)

425 (94662)

513 (19076)

518 (11768)

596 (148446)

641 (87702)

684 (23387)

695 (39626)

785 (900326)

788 (11408)

893 (86772)

908 (243440)

920 (107822)

962 (47222)

998 (81240)

1013 (43872)

=== 8{z} ===

138 (35686)

412 (12154)

788 (11326)

990 (23032)

=== 9{0}1 ===

248 (39511)

592 (96870)

=== 9{z} ===

431 (43574)

446 (152028)

458 (126262)

599 (11776)

846 (12781)

=== A{0}1 ===

173 (264235)

198 (47665)

311 (314807)

341 (106009)

449 (18507)

492 (42843)

605 (12395)

708 (17563)

710 (31039)

743 (285479)

744 (137056)

786 (68169)

800 (15105)

802 (149320)

879 (25004)

929 (13065)

977 (125873)

986 (48279)

1004 (10645)

=== A{z} ===

368 (10867)

488 (10231)

534 (80328)

662 (13307)

978 (14066)

=== B{0}1 ===

710 (15272)

740 (33520)

878 (227482)

=== B{z} ===

153 (21660)

186 (112718)

439 (18752)

593 (16064)

602 (36518)

707 (10573)

717 (67707)

=== C{0}1 ===

68 (656922)

219 (29231)

230 (94751)

312 (21163)

334 (83334)

353 (20262)

359 (61295)

457 (10024)

481 (45941)

501 (20140)

593 (42779)

600 (11242)

604 (17371)

641 (26422)

700 (91953)

887 (13961)

919 (45359)

923 (64365)

992 (10300)

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

(none)

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

38 (136212)

83 (21496)

113 (286644)

188 (13508)

401 (103670)

417 (21003)

458 (46900)

494 (21580)

518 (129372)

527 (65822)

602 (17644)

608 (36228)

638 (74528)

663 (47557)

723 (24536)

758 (50564)

833 (12220)

904 (13430)

938 (50008)

950 (16248)

=== z{0}1 ===

202 (46774)

251 (102979)

272 (16681)

297 (14314)

298 (60671)

326 (64757)

347 (69661)

363 (142877)

452 (71941)

543 (10042)

564 (38065)

634 (84823)

788 (13541)

869 (12289)

890 (37377)

953 (60995)

1004 (29685)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

53 (21942)

124 (16426)

175 (31626)

188 (22036)

316 (48538)

365 (25578)

373 (24006)

434 (10090)

530 (11086)

545 (12346)

560 (15072)

596 (12762)

701 (12576)

706 (10656)

821 (13536)

833 (17116)

966 (14820)

983 (11272)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}y ===

317 (13896)

== Bases 2≤''b''≤1024 which have these families as unsolved families ==

Unsolved families are families which are neither primes (&gt;''b'') found nor can be ruled out as contain no primes &gt; ''b''

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608)

1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000)

1{0}3: 646, 718, 998 (length limit: 2000)

1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000)

1{0}5

1{0}6

1{0}7

1{0}8

1{0}9

1{0}A

1{0}B

1{0}C

1{0}D

1{0}E

1{0}F

1{0}G

1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000)

1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000)

10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000)

11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000)

{1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000)

{1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000)

11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z})

{1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000)

{1}z

1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000)

1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995

1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000)

1{z}: 581, 992, 1019 (length limit: ≥100000)

2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000)

2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000)

2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000)

{2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000)

2{z}: 588, 972 (length limit: ≥100000)

3{0}1: 718 (length limit: ≥100000)

3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000)

3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000)

{3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000)

3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000)

4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000)

4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000)

{4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000)

4{z}: 338, 998 (length limit: ≥100000)

5{0}1: 308, 512, 824 (length limit: ≥100000)

5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000)

6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000)

6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000)

7{0}1: (none)

7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000)

8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000)

8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000)

9{0}1: 724, 884 (length limit: ≥100000)

9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000)

A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000)

A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000)

B{0}1: 560, 770, 968 (length limit: ≥100000)

B{z}: 263, 615, 912, 978 (length limit: ≥100000)

C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000)

C{z}

D{0}1

D{z}

E{0}1

E{z}

F{0}1

F{z}

G{0}1

{&amp;#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000)

{&amp;#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143)

&amp;#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000)

${&amp;#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2)

x{z}

y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000)

z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000)

{y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000)

{z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326)

zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z})

{z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000)

{z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1)

{z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000)

{z}k

{z}l

{z}m

{z}n

{z}o

{z}p

{z}q

{z}r

{z}s

{z}t

{z}u

{z}v

{z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000)

{z}x: (none)

{z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000)

== List of lengths for quasi-minimal primes in some simple families ==

[https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024]

NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases &lt;=7, family {z}x in bases &lt;=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime &gt; base) in this base)

RC: this family can be proven to only contain composite numbers (only count numbers &gt; base)

unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers &gt; base)

Background color: red for title (bases or families), green for length &gt; 10000, orange for 2500 &lt; length ≤ 10000, white for length ≤ 2500, cyan for &quot;RC&quot;, pink for &quot;NB&quot;, yellow for &quot;unknown&quot;.

Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families.

== References ==

* [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem]

* [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are &gt; ''b'' for bases ''b''≤16]

* [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary]

* [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia]

* [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime&gt;base is not required) in The Prime Puzzles &amp; Problems Connection]

* [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles &amp; Problems Connection]

* [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16]

* [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime&gt;base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29]

* [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime&gt;base is not required, search limits of lengths: 10000 for all b]

* [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime&gt;base is not required]

* [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime&gt;base is not required]

* [http://www.prothsearch.com/sierp.html the Sierpinski problem]

* [http://www.prothsearch.com/rieselprob.html the Riesel problem]

* [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem]

* [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030]

* [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100]

* [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem]

* [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024]

* [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project]

* [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project]

* [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count]

* [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty]

* [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers]

* [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares]

* [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software]

* [https://sourceforge.net/projects/openpfgw/ pfgw software]

* [http://jpenne.free.fr/index2.html llr software]

* [http://www.ellipsa.eu/public/primo/primo.html PRIMO software]

* [https://primes.utm.edu/prove/index.html website for primality proving]

* [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!]

* [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime&gt;base is not required]

* [https://primes.utm.edu/primes/lists/all.txt top proven primes]

* [http://www.primenumbers.net/prptop/prptop.php top PRPs]

* [http://factordb.com online factor database, including many primes which are minimal primes in a small base]</text>
      <sha1>85oizxsnifmfzami0jqgrompsa3m2ty</sha1>
    </revision>
    <revision>
      <id>2412025</id>
      <parentid>2412024</parentid>
      <timestamp>2022-08-05T03:41:44Z</timestamp>
      <contributor>
        <ip>2402:7500:901:2B90:BD69:8142:3D96:F08F</ip>
      </contributor>
      <comment>/* Unsolved families */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1960605" xml:space="preserve">A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime &gt; ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime &gt; 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 &gt; 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 &gt; 10 by changing the order.

(using A−Z to represent digit values 10 to 35)

For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')).

I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E&lt;sub&gt;800873&lt;/sub&gt;, and base 36 has quasi-minimal prime P&lt;sub&gt;81993&lt;/sub&gt;SZ).

Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to:

* Prove that all elements in ''S'', when read as base ''b'' representation, are primes &gt; ''b''.

* Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are &gt; ''b'', are composite.

* Prove that all primes &gt; ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are &gt; ''b'', are composite).

e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to:

* Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes &gt; 10.

* Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are &gt; 10 are composite.

* Prove that all primes &gt; 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers &gt; 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite).

==Condensed table==

{|class=wikitable
|''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form of largest known quasi-minimal prime base ''b''
|-
|2||1||11||2||3
|-
|3||3||111||3||13
|-
|4||5||221||3||41
|-
|5||22||10&lt;sub&gt;93&lt;/sub&gt;13||96||5&lt;sup&gt;95&lt;/sup&gt;+8
|-
|6||11||40041||5||5209
|-
|7||71||3&lt;sub&gt;16&lt;/sub&gt;1||17||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2
|-
|8||75||4&lt;sub&gt;220&lt;/sub&gt;7||221||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||1161||3×9&lt;sup&gt;1160&lt;/sup&gt;+10
|-
|10||77||50&lt;sub&gt;28&lt;/sub&gt;27||31||5×10&lt;sup&gt;30&lt;/sup&gt;+27
|-
|11&lt;sup&gt;*&lt;/sup&gt;||1068||57&lt;sub&gt;62668&lt;/sub&gt;||62669||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10
|-
|12||106||40&lt;sub&gt;39&lt;/sub&gt;77||42||4×12&lt;sup&gt;41&lt;/sup&gt;+91
|-
|13&lt;sup&gt;*&lt;/sup&gt;||3195~3197||80&lt;sub&gt;32017&lt;/sub&gt;111||32021||8×13&lt;sup&gt;32020&lt;/sup&gt;+183
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||19699||5×14&lt;sup&gt;19698&lt;/sup&gt;−1
|-
|15||1284||7&lt;sub&gt;155&lt;/sub&gt;97||157||(15&lt;sup&gt;157&lt;/sup&gt;+59)/2
|-
|16&lt;sup&gt;*&lt;/sup&gt;||2346~2347||4&lt;sub&gt;72785&lt;/sub&gt;DD||72787||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15
|-
|17&lt;sup&gt;*&lt;/sup&gt;||10408~10428||F70&lt;sub&gt;186767&lt;/sub&gt;1||186770||262×17&lt;sup&gt;186768&lt;/sup&gt;+1
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||6271||12×18&lt;sup&gt;6270&lt;/sup&gt;+221
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||6271||16×20&lt;sup&gt;6270&lt;/sup&gt;+13
|-
|21&lt;sup&gt;*&lt;/sup&gt;||13375~13396||CF&lt;sub&gt;479147&lt;/sub&gt;0K||479150||(51×21&lt;sup&gt;479149&lt;/sup&gt;−1243)/4
|-
|22&lt;sup&gt;*&lt;/sup&gt;||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||22003||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||8134||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49
|-
|30&lt;sup&gt;*&lt;/sup&gt;||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||34206||25×30&lt;sup&gt;34205&lt;/sup&gt;−1
|-
|36&lt;sup&gt;*&lt;/sup&gt;||35257~35263||P&lt;sub&gt;81993&lt;/sub&gt;SZ||81995||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7
|}

&lt;sup&gt;*&lt;/sup&gt; Data assumes the primality of the [[w:probable prime|probable prime]]s.

Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes &gt; 10&lt;sup&gt;25000&lt;/sup&gt;), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below.

Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists))

Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists))

==Data for quasi-minimal primes==

===Base 2===

11

===Base 3===

12, 21, 111

===Base 4===

11, 13, 23, 31, 221

===Base 5===

12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013

===Base 6===

11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041

===Base 7===

14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331

===Base 8===

13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447

===Base 9===

12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011

===Base 10===

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027

===Base 11===

12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, 2888882883, 2888888883, 3555555509, 3577777077, 3700000001, 4000000005, 40000005AA, 5377777707, 5555505553, 555555580A, 600000A999, 7000100047, 7000704777, 7007777107, 7057777777, 7070007774, 7077707774, 7077777074, 7100000704, 7470000041, 7701000004, 7707077774, 7770707774, 7777707074, 8888822883, 9555555503, 9900000979, 9999770007, A000144444, A900000066, A999999971, 10000000477, 33333333337, 44444444447, 44444444777, 55A55555552, 60000008883, 68888888306, 68888888883, 70000003999, 70000007717, 70004777777, 70477777777, 77007770004, 77700000477, 77707000704, 77707770074, 77707777774, 77777077774, 77777770004, 83000000006, 97000000999, A0000000001, A0014444444, A4777777771, 100000000057, 305007777777, 305777777777, 370000000007, 377777770007, 377777777107, 700000007474, 707077000074, 707077777774, 707777777717, 770000010004, 771007000007, 777070700004, 777700000704, A95555555555, A99999777777, 1000000003007, 40000000A0041, 58A5555555555, 7004400000004, 7700000000104, 7707000007047, 7707707000004, 7777007000004, 7777700000004, 7777770077704, 7777777710077, 9977777777717, A000000014444, A044444444441, A144444444411, 40000000000401, 45557777777777, 4555AAAAAAAAAA, 59077777777777, 70007777777771, 70077070000074, 70700000004777, 77000007700704, 77700000700047, 77777777770704, 88888888830006, 90900000000799, A0000044444441, 300000000005777, 302000000000002, 55555555A555552, 700000000000174, 770000000000474, 771700000000007, 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7707777777777777777704, 7900000000000000000005, 9777777777777777777707, A999999999999999999999, 10000000000000000000747, 970000000000000000000777, 999900000000000000000007, 3577777777777777777777777, 5555555555555555555555A52, 7000000000000000000777771, 7000000000000000077777771, AAAAAAAAAAAAAAAAAAA000058, 10000000000000000000000044, 77700000000000000000000008, 500777777777777777777777777, 777777777777777777777770774, 855555555555555555555055555, A44444444444444444444444441, 1500000000000000000000000007, 40000000000000000000000000041, 440000000000000000000000000001, 70000000000000000000000000007771, 999999999999999999999999999999991, 95555555555555555555555555555555553, 1900000000000000000000000000000000001, 7777777777777777777777777777777777474, 7777777777777777777777777777777777704, 10000000000000000000000000000000000000307, 50777777777777777777777777777777777777707, 475777777777777777777777777777777777777777, 555555555555555555555555555555555555555A2A, 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===Base 12===

11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077

===Base 13===

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===Base 14===

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===Base 15===

12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797

===Base 16===

11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD

===Base 17===

12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 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====Additional known quasi-minimal primes (not necessarily the next)====

570&lt;sub&gt;51310&lt;/sub&gt;1, 49&lt;sub&gt;111333&lt;/sub&gt;, 970&lt;sub&gt;166047&lt;/sub&gt;1, F70&lt;sub&gt;186767&lt;/sub&gt;1

===Base 18===

11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, 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===Base 20===

13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, GGJ0J, GGJJJ, GJ00J, GJGJJ, GJJJ7, H00EH, H024B, H04B1, H0E4B, H0F41, H0H11, H0HEH, H4E0B, H6I11, HAAA3, HAAG3, HAGA3, HB44B, HBBBB, HBHHH, HGGA3, HGGG3, HH001, HH061, HH3AH, HH6F1, HHG63, HHIA3, HI011, HIII3, I0029, I02I9, I085H, I0A41, I0F29, I2229, I2929, I2I29, I2I99, I2II9, I33A3, I3EHH, I422B, I424B, I4EEB, I4IEB, I80G1, I9IA9, IA099, IA309, IA841, IA939, IA9A9, IAF09, IAF99, IAI99, IAIA9, IB0A1, IB4IB, IB54B, IBB4B, IBBIB, IBE0H, IBH0H, IC929, ICC29, IF841, IFAG1, IGA81, IHB4B, II0A9, II42B, II44B, IIB5B, III4B, IIIA3, IIIIB, J00G1, J00JB, J0601, J06D1, J06I1, J0861, J08B1, J0B01, J0CGJ, J0E0B, J0GA1, J0GG1, J0I2B, J0J2B, J0JIB, J55CJ, J60A1, J60G1, J6D81, JA777, JA7A7, JAJG1, JAJJ1, JAJJ7, JB00B, JB08B, JC00J, JC0GJ, JEECJ, JGGGJ, JGGJJ, JJ0B1, JJ0JB, JJ55J, JJAG1, JJAJ1, JJC0J, JJE0B, JJEEB, JJGG1, JJGGJ, JJGJJ, JJJ01, JJJEB, JJJJB, 17555J, 175EEJ, 1E0007, 1JE55J, 1JJJ5J, 20005H, 2000HH, 222I29, 22E889, 22EE89, 24222B, 2999I9, 29III9, 2A000H, 2B0I0H, 2I2999, 2I9I29, 2III29, 333AF3, 336IA3, 36A3F3, 36I3A3, 388E8H, 3AF333, 3EHHHH, 3HEHHH, 3HHEHH, 40008B, 40054B, 40405B, 40448B, 415E5J, 44440B, 44452B, 44524B, 44E08B, 45444B, 4AAAE7, 4AEEE7, 4E4447, 4GEEE7, 4I544B, 4IEEEB, 500EFJ, 500FCJ, 5055FJ, 50AFFJ, 50EEFJ, 50EF5J, 50F5FJ, 50FFCJ, 52224B, 54EEEB, 5558AH, 555CAJ, 5585AH, 55F55J, 55FFFJ, 5888AH, 588A5H, 5E555J, 5F055J, 5F05CJ, 5F5F5J, 5FFFCJ, 5JEEEB, 60006D, 6000F1, 6000G3, 6001G1, 600A63, 600G1D, 600G6D, 603A33, 606G6D, 608001, 608F89, 608IF9, 60A333, 60A363, 60AAA3, 60CCCD, 60D04D, 60D0DD, 60F8I9, 60FF89, 60GCG3, 636AF3, 63A333, 63A3F3, 63AFF3, 63IIA3, 64AAAD, 660089, 660A33, 660DDD, 660F89, 662289, 666A33, 666D4D, 66A6A3, 66AF33, 66AF63, 66D0DD, 66FF89, 6A33F3, 6AFF33, 6AG6A3, 6AGA63, 6D000D, 6D004D, 6D0D4D, 6F6689, 7000G7, 7060DD, 70AAAD, 70D0CD, 733C33, 7366C3, 73AAA3, 7600DD, 760CDD, 766333, 76664D, 766FC3, 76CCDD, 76DD4D, 770G07, 771JEJ, 7771EJ, 777481, 7777I1, 777F81, 77C7JJ, 77EC77, 77EEC7, 77EJJJ, 77GCG7, 77JECJ, 7A7E77, 7AAAG3, 7AAF63, 7AEEE7, 7C7GG7, 7CCCAD, 7CCCG7, 7CGJJJ, 7D000D, 7J000D, 7JCGGJ, 80005B, 8000G1, 8000I1, 800GA1, 800I01, 80GGA1, 84405B, 84454B, 888EAH, 88E8AH, 8A4441, 8B8EAH, 8E8IF9, 8I0AG1, 902229, 909A09, 909C29, 909FA9, 90IC29, 9222I9, 92I999, 944EE7, 94AEE7, 977GG7, 97AEE7, 97GGG7, 990I29, 999I29, 99I299, 99IIA9, 9A9447, 9A9AF9, 9A9FF9, 9CII29, 9EEEE7, 9FA9A9, 9I9C29, 9I9I29, 9ICI29, 9IIIA9, A1555J, A1E55J, A333C9, A336A3, A33889, A33F63, A36663, A444E7, A4AAE7, A555FJ, A666A3, A6AAA3, A7A7E7, A7AE77, A944E7, A9A9F9, A9AAA9, AA3389, AA9AF9, AAA099, AAA3C9, AAA7E7, AAA939, AAACG9, AAAFF9, AAAI99, AAAIC9, AAE4E7, AAG9A9, AAGAA3, ADE55J, AF6663, AF9FF9, AGA963, AGG6A3, AH4441, AI000H, AI99C9, AI9AA9, AII099, B0004B, B0054B, B0080B, B00H0H, B04IIB, B05B8B, B0A6I1, B0BB8B, B0BIBB, B0E0IH, B0HA0H, B0I0HH, B0I4IB, B0IIBB, B888AH, BB058B, BB05BB, BB080B, BB0I4B, BBB04B, BBB08B, BBIB8B, BE88AH, BEHA0H, BEHHAH, BHEA0H, BHEHAH, BHHA0H, BIE00H, C007A7, C00GG7, C00JA7, C00JAJ, C00JC7, C0C7A7, C0JAC7, C0JJC7, C3G999, CAJJ77, CC0629, CC0929, CCC007, CCCCC7, CCCCG3, CCCGG7, CCCI29, CCI029, CD4GGJ, CE0009, CEFF69, CEFFF9, CEI609, CII029, CIII29, CJACC7, CJAJC7, CJJJA7, D000D7, D000J7, D004CD, D0D007, DCCDC7, DDD04D, DDDAI1, DEJJJJ, DG000D, DG5E5J, DGGG5J, DJ00CJ, DJEEE7, DJJJ5J, E00097, E00A0H, E00AA9, E02289, E028I9, E029I9, E055AJ, E0774J, E0777J, E07JCJ, E0AF5J, E0BI8B, E0E4CJ, E0EA5J, E0ECJJ, E0FFF9, E0H0HH, E0HA0H, E0IF29, E0J5AJ, E0JJAJ, E29299, E444E7, E4EEE7, E4IIIB, E66289, E66629, E66909, E66F69, E69609, E69999, E7774J, E77C77, E7C7C7, E8B8AH, E90029, E90229, E90909, E90FF9, E92299, E94EE7, E97EE7, E990F9, E99I09, E9F669, E9I029, EA8AF9, EAAAI9, EAIAA9, EAJJJJ, EBIBIB, EC7CC7, EE08F9, EE408B, EE8IF9, EEAF5J, EEE84B, EEEC07, EEEC77, EEEE0B, EEF55J, EEFFFJ, EEJ5AJ, EFE009, EFEF09, EFF669, EFFFF9, EGGGG9, EIF669, F004AJ, F00A41, F00JA1, F04481, F055CJ, F0A15J, F0EI09, F0G01J, F0J0GJ, F0J0JJ, F0JJ5J, F14441, F68F89, F68IF9, F800G1, F8E8I9, F99299, FA0009, FA1JJJ, FAAIA9, FAII09, FAII99, FCC929, FCE229, FCE609, FCEF29, FE68F9, FE8FA9, FF0E09, FF68F9, FF9FA9, FFC4AJ, FFC929, FFCE09, FFE009, FFEE89, FFEF09, FFFA99, FFFFFJ, FFFGJJ, FG1JJJ, FI99A9, FIA999, FIAAA9, FIAII9, FIIA09, G00093, G000EJ, G000G9, G00781, G007G3, G00C09, G07GC3, G09033, G0C903, G0CGG3, G0G903, G0G933, G0GCC3, G0GGC9, G30003, G36003, G5000J, G5E0EJ, G60303, G6AGG3, G6GAA3, G7AAA3, G7G363, G90AA9, G90C63, G9AAA9, GAAAA3, GAAGG3, GC0003, GCCC93, GCCCC3, GCG903, GDEEE7, GE0007, GG0CG3, GGA8G1, GGC6G3, GGCCG3, GGCGC9, GGG0I9, GGG363, GGG6G3, GGG75J, GGG8G1, GGG963, GGGAA3, GGGCG3, GGGCG9, H044EB, H0BIBB, H0E00B, H0F011, H0IIEB, H2444B, H400EB, H44IEB, H4EEEB, HBB4IB, HE000B, HEE40B, HEEE4B, HEHHAH, HF0011, HF4441, HH3HEH, HHGAA3, HHGAG3, HHHAG3, HHHEAH, HI0001, HIBIBB, HIEBBB, I0000H, I000A9, I002HH, I00A09, I00A99, I00AG1, I00AI9, I00H3H, I02999, I03HEH, I09AF9, I0A009, I0E0HH, I0I299, I0IIA9, I4445B, I4I45B, I90I29, I99029, I99A09, I9I029, IA3999, IB000H, IB00EH, IB0HEH, IB5B8B, IF2999, II0929, II92I9, II99A9, II9C29, IIAFF9, IIF299, III5BB, III8BB, J0000B, J00081, J000IB, J0010D, J006A1, J00BI1, J00I8B, J00IA1, J05EEJ, J06G81, J0C0JJ, J0E0CJ, J0EEEB, J0J0EB, J5EAEJ, J5EEAJ, J608I1, JE505J, JEEEEB, JJ0001, JJ0JCJ, JJ0JGJ, JJAJC7, JJJAE7, JJJBI1, JJJI8B, 14GGGG1, 1J5555J, 1JGGGG1, 205555H, 20I000H, 222222B, 2929999, 2BI000H, 3333A33, 3333G09, 333A3A3, 333AAA3, 336AAA3, 336AF33, 33II3A3, 363AF33, 3GIIII9, 400445B, 404454B, 404800B, 415555J, 440045B, 440080B, 442222B, 444004B, 444422B, 444444B, 4444G47, 444GGG7, 444I4IB, 44AAA47, 44I44IB, 44I4IIB, 4AAAAA7, 4EEE4E7, 4I4454B, 5000AFJ, 525555H, 555552H, 555555J, 555585H, 58BBBBB, 5BB8BBB, 5E0E55J, 5EBBBBB, 60003A3, 60008I9, 6000DDD, 6006DDD, 60080A1, 60080I1, 600A3A3, 600D8I1, 600F841, 60CCDDD, 60CGGG3, 60DDD0D, 60DDDDD, 60GGG63, 60GGGC3, 66600A3, 66603A3, 66608I9, 6666089, 66666A3, 6666C29, 6666F89, 666AFF3, 66DDDDD, 6800001, 6AGGGG3, 6CCCC29, 6G6AAA3, 6GAGGG3, 6GCCC63, 6GGG633, 700000D, 70000CD, 7000EC7, 70070C7, 700EEC7, 7070EC7, 70EEEC7, 76666CD, 77333A3, 7733A33, 775EE5J, 777A777, 777CGJJ, 777GE5J, 77A7777, 77JCJJJ, 77JJJCJ, 77JJJJJ, 7A77777, 7A777A7, 7AAAA63, 7AAAAAD, 7C00007, 7E77JCJ, 7EE7JCJ, 7EEC007, 7EEEEC7, 7G5E55J, 7GE555J, 7GJJJJJ, 7J6666D, 9000029, 9000299, 9009I29, 900I2I9, 90FAAA9, 90I0299, 90I0I29, 90I9929, 90II029, 90II299, 90II2I9, 94444E7, 97777E7, 9902I99, 99902I9, 9992929, 9992999, 9999929, 99I0C29, 99I9029, 99I92I9, 99III29, 9AAAF09, 9AFFFF9, 9I0AAA9, 9IAAA09, 9II0I29, 9II9929, 9III029, 9III2I9, 9IIIC29, A5F555J, A777E77, A994447, A9AAF99, AA00089, AA6A6A3, AAA0009, AAA3999, AAA9A99, AAA9AA9, AAAAA47, AAAAA77, AAAAA99, AAAAG99, AAAAIA9, AAAD447, AAAGAC9, AAGAAAD, AAGAAC9, AF00009, AGAAA99, AGAGG63, AGGGA63, AI0C999, AJ777A7, B000B8B, B000HEH, B000I0H, B00I00H, B0B805B, B0HHHEH, B0IHE0H, B0III8B, B8B000B, BB8BBBB, BBB8B0B, BEH0HHH, BHHHEHH, C00000J, C000C07, C066629, C0AJJJJ, C0C0A47, C0G000J, C0JAJJJ, C0JJAJJ, C0JJJJJ, C6CCG33, CC004A7, CC00A47, CC6CG33, CC92229, CCCC7G7, CCCGGG3, CE66609, CEF6669, D00007D, D00007J, D00071J, D000C0J, D000I01, D00700D, D00C007, D00C0C7, D00G00D, D0C00C7, D0CDCC7, D444447, D7JJJJJ, DC0000J, DDD7DDD, DDDCCC7, DJ0000J, DJ000EJ, DJJJJ0J, DJJJJJJ, E000009, E00084B, E000889, E000909, E000HAH, E008AF9, E04000B, E0AA8F9, E0CJJJJ, E0HHHHH, E40400B, E6660F9, E666609, E669969, E699669, E7CCC07, E996669, E99I669, E9FAAA9, EB8BBBB, EBBII8B, EBH0HHH, EE092I9, EE44IIB, EE4E4E7, EEBB8BB, EEC0007, EEE0IBB, EEE0IIB, EEEBB8B, EEEE44B, EEJCJJJ, EF00F09, EF0555J, EFAAAA9, EFF0009, EFF00F9, EFF60F9, EI0AAA9, EI660F9, EIIBB8B, EJ0CJJJ, EJ5055J, EJJ0CJJ, F0E00F9, F0E88I9, F0JJJJ1, F299929, F4G555J, F5555CJ, F5555FJ, F6668I9, F8GGGG1, F929999, F999C29, FA84441, FAAA9A9, FCE00F9, FCEF009, FCFEE29, FE00F09, FEE2889, FEF0009, FFAA9A9, FFCEE29, FFFF4AJ, FIII9A9, FIIIAI9, FJJJJ81, G000009, G0000J7, G000303, G000AA9, G000CC3, G000GG3, G003003, G0090C3, G00CG03, G00D007, G00G0G3, G00GG03, G00GG93, G080001, G0G00I9, G0G09C3, G0G0C93, G0G0GG9, G0GG093, G0GGC63, G0GGGG9, G44GG47, G6AAGA3, G999999, GAA9AA9, GCCCGG9, GCGGCC9, GCGGG03, GEE4447, GEEEE47, GG0C007, GG0GC09, GG0GGG3, GG6AAA3, GG6AGA3, GG6GAG3, GGEEE47, GGG00C9, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 21===

12, 18, 1A, 1G, 1K, 21, 25, 2B, 2H, 2J, 34, 38, 3A, 3G, 3K, 45, 4D, 4H, 4J, 52, 54, 58, 61, 65, 6B, 6D, 72, 74, 7A, 7G, 7K, 85, 8B, 8D, 92, 94, 98, 9A, A1, AD, AH, AJ, B2, B8, BA, BK, C5, CB, CH, CJ, D4, D8, DA, DK, ED, EH, EJ, F2, FG, G1, GB, GD, GH, H2, HA, HG, I1, I5, IB, IJ, J2, JA, JK, K1, KB, KD, KJ, 111, 115, 11H, 135, 13H, 13J, 151, 15B, 15H, 164, 175, 17B, 17D, 17J, 191, 19B, 19D, 19H, 1B1, 1B5, 1BB, 1BJ, 1D5, 1DD, 1DJ, 1E4, 1F1, 1F5, 1FD, 1FH, 1HB, 1HD, 1I4, 1JD, 1JH, 1JJ, 22D, 232, 29G, 29K, 2DG, 2F4, 2FK, 31H, 322, 33D, 351, 355, 35B, 35J, 362, 371, 37B, 37D, 37H, 37J, 39B, 39J, 3B5, 3BD, 3BH, 3D1, 3D5, 3DB, 3DD, 3DH, 3E2, 3FJ, 3HD, 3HH, 3HJ, 3J1, 3JB, 3JJ, 414, 481, 49K, 4B4, 4BG, 4C1, 4EB, 4F4, 4F8, 4FA, 4FK, 50G, 51B, 51D, 51H, 531, 535, 53D, 53J, 551, 56A, 56G, 56K, 575, 57J, 595, 59H, 5B1, 5B5, 5BB, 5CA, 5CG, 5CK, 5F1, 5FB, 5FJ, 5GA, 5GG, 5HH, 5IA, 5J5, 5JD, 5JH, 60H, 632, 66H, 66J, 678, 68J, 69G, 6F8, 6FA, 6GJ, 6H8, 6HK, 6IH, 6J4, 6JG, 6KH, 711, 71B, 71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, GGJ, GJ4, GK5, H0K, H15, H1B, H1J, H31, H3D, H3H, H48, H51, H55, H5J, H6K, H75, H84, H88, H91, H95, H9D, H9H, HBD, HC4, HC8, HDJ, HF5, HFB, HFH, HHD, HHJ, HI4, HI8, HJ5, HJB, HKK, I0D, I2D, I6H, I78, I8H, I9K, IFA, IFK, IID, J08, J1J, J31, J35, J3J, J44, J5H, J68, J6G, J71, J7B, J7D, J7H, J8G, J95, J9D, JBD, JBH, JBJ, JCG, JDB, JDH, JE4, JE8, JEG, JF5, JFD, JFJ, JG4, JGG, JH1, JH5, JHB, JHH, JI4, JJ1, JJ5, K0H, K5G, K6H, K9K, KG5, KH4, KKH, 101B, 103D, 1044, 1055, 10DH, 10FB, 10H1, 10H5, 10J1, 110J, 11CD, 11EB, 11ID, 130D, 13C1, 13EB, 141B, 1444, 1471, 14FB, 1505, 15CD, 167H, 16FJ, 16HH, 176H, 17C1, 17IH, 190J, 1BF4, 1BID, 1C04, 1C3D, 1CD1, 1D6H, 1DE1, 1DEB, 1E1B, 1E71, 1E95, 1EFB, 1F0B, 1F4B, 1FB4, 1H01, 1H05, 1H41, 1H6H, 1H6J, 1HC1, 1HE1, 1HF4, 1IDH, 1IHH, 1JE5, 1JF4, 20CD, 20D2, 20ID, 2278, 22F8, 239D, 23FD, 2478, 2708, 27FD, 293D, 29DD, 2A78, 2C0D, 2CF8, 2F0A, 2F9D, 2FAA, 2FDD, 2GFA, 2IF8, 3015, 301D, 301J, 3031, 3035, 305D, 30F1, 30FB, 30FD, 30HB, 30I2, 3105, 31C1, 329D, 330H, 336J, 33E1, 33EB, 33IH, 35CD, 363H, 390H, 39E5, 39ID, 3B0J, 3B6J, 3BC1, 3BEB, 3CC2, 3CF1, 3CFD, 3D32, 3E3B, 3E91, 3EF1, 3EFB, 3F05, 3FC1, 3FE1, 3FEB, 3H01, 3H0B, 3HC1, 3I3H, 3I9D, 404B, 4078, 40E1, 411B, 4131, 4171, 41FB, 4331, 4391, 43BB, 43FB, 43I2, 4432, 444B, 44E1, 477B, 4931, 4971, 499B, 49CG, 4A32, 4A4B, 4A78, 4A9G, 4B1B, 4B31, 4B3B, 4BFB, 4C78, 4E32, 4F1B, 4FB1, 5011, 509D, 50BD, 50DB, 50DD, 50EK, 50FD, 50H1, 50HJ, 50IK, 50JJ, 5101, 5105, 51E1, 530B, 53EB, 53IH, 550D, 550H, 55AB, 55BG, 55DG, 55E5, 55KH, 563H, 56JJ, 5701, 570H, 57EB, 57ID, 57IH, 590D, 59C1, 59CD, 5A3B, 5A7B, 5AGK, 5AIG, 5AKG, 5B0D, 5B9G, 5BCD, 5BGJ, 5BIH, 5C91, 5CD1, 5D0D, 5D0J, 5D6H, 5D6J, 5DC1, 5DG5, 5DIH, 5DJG, 5E0A, 5E11, 5E3B, 5E71, 5EAK, 5EEA, 5EKK, 5F0H, 5F9K, 5G0K, 5G9J, 5GF5, 5GIK, 5H0B, 5H5K, 5H9K, 5HE5, 5HID, 5I3H, 5I7D, 5IKG, 5J6J, 5JE1, 5KAG, 5KEK, 5KFH, 5KH5, 60FK, 63FH, 67JH, 69JH, 6E9K, 6F0K, 6F64, 6FJH, 6H04, 6HE4, 6HFJ, 6IF4, 6J9H, 6J9J, 7015, 7031, 703B, 7055, 705J, 7088, 709B, 709D, 709H, 70DJ, 70I8, 710J, 716J, 730B, 73C1, 73E5, 7501, 769H, 76FJ, 76JJ, 770D, 77E1, 77E5, 77EB, 7888, 78FH, 78H1, 78JH, 790H, 79CD, 79IH, 7B0J, 7B6H, 7C31, 7C91, 7C9D, 7E35, 7E75, 7E91, 7EF1, 7F01, 7F78, 7FCD, 7FH8, 7FIH, 7FJ8, 7H81, 7HC1, 7HF8, 7J6J, 7J78, 7JC1, 7JF8, 7JH8, 7JJ8, 809K, 80H8, 81H1, 81HH, 81J1, 839H, 83FH, 869K, 871H, 8731, 8771, 8788, 879J, 87F1, 87FJ, 87JH, 888H, 8911, 8931, 896K, 8A78, 8A9K, 8CC1, 8E9K, 8F3H, 8F9H, 8FF1, 8FHH, 8FJH, 8G9K, 8GJ8, 8H3J, 8H7H, 8H8K, 8HEK, 8HF1, 8HFJ, 8HH1, 8HJJ, 8HK8, 8J7J, 8J91, 8JC8, 8JFH, 8JJH, 8K78, 8K8H, 8KH8, 901J, 9035, 903B, 9071, 90CG, 90GG, 90JD, 9101, 91E5, 9301, 93E1, 93EB, 95CD, 95DG, 95FK, 967J, 969H, 96HH, 96HJ, 96JJ, 9705, 97C1, 9905, 995K, 999G, 99CD, 99EB, 99K5, 9B5G, 9BE5, 9BID, 9C0G, 9C11, 9C71, 9C9D, 9CEG, 9CGG, 9D05, 9D5G, 9DDG, 9DE5, 9E35, 9E3B, 9E71, 9EB5, 9EEG, 9EF1, 9EFB, 9F0J, 9F6H, 9F9K, 9FC1, 9FIH, 9FKH, 9G35, 9G95, 9GIK, 9GJJ, 9GKG, 9H0H, 9H5K, 9H6H, 9HKH, 9I6K, 9I9H, 9IDD, 9J6J, 9JEB, 9JGJ, 9JID, 9JJG, 9K35, 9K95, 9KEG, A33B, A3F5, A432, A478, A53B, A5IK, A788, A795, A878, A975, A99B, A9CG, A9GG, AAAB, AB44, AB55, ABFB, AC32, AC78, ACFA, AE32, AE5A, AE9K, AEEB, AEFA, AEFK, AF75, AFCA, AFCK, AFEA, AFFB, AG9G, AGFA, AI32, B064, B091, B09H, B0B1, B0D5, B0DH, B0EG, B0FB, B0FH, B0H5, B0HD, B114, B141, B14B, B1B4, B1E1, B411, B431, B444, B4B1, B59G, B5G5, B5JG, B67H, B67J, B69H, B6FH, B6FJ, B6HH, B6JH, B96H, B9DG, B9G5, BB14, BBB4, BC1D, BC31, BCDD, BD05, BD0H, BD6H, BDC1, BDE5, BDIH, BE0G, BE31, BEBB, BEE4, BEIG, BF0B, BF0H, BF6J, BFEB, BFIH, BG0G, BG55, BG95, BGG4, BGGG, BGIG, BGJ5, BH0B, BHEB, BIEG, BIGG, BIHH, BJ01, BJ0J, BJC1, BJF4, BJID, C041, C081, C09G, C0E1, C0FK, C104, C131, C171, C1C4, C1D1, C2F8, C2FA, C39D, C3F1, C478, C6FK, C7FD, C8C1, C90G, C911, C971, C99D, CAFK, CC78, CC81, CCE1, CCF8, CD02, CD31, CDFD, CE01, CE32, CF0A, CF48, CF64, CF68, CF6A, CF91, CFAK, CFDD, CFEA, CFEK, CIF4, CK32, CK9G, CKFA, D01D, D01H, D07B, D0DB, D0F1, D0FD, D0HD, D0HH, D10D, D10H, D16H, D1EB, D1IH, D23D, D3E1, D56H, D63J, D6CG, D6IG, D70B, D70D, D70J, D99G, D9GJ, D9ID, D9JG, DBC1, DBCD, DBE1, DBJG, DC11, DC9D, DCF1, DCFD, DCGG, DCIG, DD0H, DD6J, DDGJ, DDID, DDJG, DE31, DE75, DFC1, DFID, DFIH, DG22, DG6G, DG75, DG9J, DGCG, DGIG, DH6H, DHE5, DI6G, DICG, DIDH, DIFD, DIFH, DIGG, DJCD, DJDG, DJG5, E032, E05A, E0EB, E11B, E2F8, E31B, E331, E375, E395, E3C2, E3F1, E3F5, E3FB, E3I2, E53B, E555, E591, E59B, E5EG, E6F4, E755, E775, E89K, E915, E935, E93B, E96K, E9FB, EA32, EAEB, EAFK, EB44, EB95, ECFA, ECFK, EE32, EEEB, EEFK, EF08, EF35, EF48, EF55, EF64, EF68, EF75, EF95, EFB1, EFCA, EFCK, EFEA, EFIA, EFKK, EGF4, EIF4, EKF4, EKF8, EKFA, F008, F051, F075, F095, F09H, F0CA, F0CK, F0DD, F0FB, F0H1, F0IA, F0JJ, F105, F10B, F10D, F16J, F1B4, F1EB, F1ID, F1J4, F36H, F468, F47B, F491, F4C8, F501, F50B, F648, F66A, F69H, F705, F778, F864, F8F1, F90D, F99K, F9C1, F9HK, FA6A, FA75, FA7B, FA95, FAEA, FAFB, FB0D, FB14, FBCD, FBE1, FBJ4, FC1D, FC68, FC9D, FCAA, FCCK, FCD1, FCEA, FD6H, FDIH, FE0K, FE1B, FE64, FE95, 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====Additional known quasi-minimal primes (not necessarily the next)====

40&lt;sub&gt;47333&lt;/sub&gt;9G, CF&lt;sub&gt;479147&lt;/sub&gt;0K

===Base 22===

11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, IHH, IIJ, IJ7, IL7, J07, J55, J69, J8J, J99, J9J, JA5, JAH, JE7, JEH, JHH, JI9, JJ5, JJ9, JJH, JK7, K2J, K51, K5D, K75, K81, KA1, KB1, KB7, KBJ, KD1, KEJ, KG5, KIF, KJ5, KKD, KKJ, L0D, L47, L7F, L97, LAF, LD3, LD9, LDD, LEF, LGD, LI7, LJ7, LJJ, LLD, 104H, 10D5, 1205, 12B5, 140H, 1433, 144H, 14AH, 14B3, 16ED, 1AIH, 1B43, 1DD5, 1DDD, 1E6D, 1EGD, 1G05, 1GDD, 1GED, 1GGD, 1HB3, 1HHH, 1IAH, 200L, 2015, 2051, 20A1, 20DJ, 20GD, 20IL, 21B5, 21DD, 220D, 226D, 228D, 22B5, 22G5, 22K5, 22KD, 2555, 2557, 2581, 25C1, 26A1, 26B1, 2725, 2755, 2801, 2861, 288D, 28B1, 28KD, 2AA1, 2B25, 2B51, 2BB1, 2C81, 2D6D, 2DA1, 2DDJ, 2DGD, 2G0J, 2GB5, 2GDD, 2GGJ, 2I0L, 2I6L, 2ICL, 2J05, 2JK5, 2K07, 2K2D, 2K55, 2K6D, 2KB5, 2KI7, 2L2D, 2L8D, 2LK7, 302L, 30G3, 320L, 32IL, 332L, 33G3, 33G9, 36AF, 36EF, 382J, 388J, 39EL, 3AAF, 3BBJ, 3CG9, 3E2J, 3E6F, 3E6L, 3EEF, 3FAF, 3G69, 3GI3, 3GI9, 3IG9, 3LEL, 3LG3, 4025, 404H, 4063, 4075, 40AH, 40B5, 40B7, 40GH, 4225, 4363, 4447, 444H, 447H, 4487, 44B7, 44H7, 4525, 4555, 45B5, 4663, 4777, 47GH, 4807, 4B0H, 4BB5, 4BG3, 4EBH, 4G4H, 4GB3, 4HH3, 4I03, 4I63, 4IGH, 5069, 5077, 50KD, 5255, 52IJ, 5455, 5477, 5507, 5527, 556F, 557F, 5587, 56EF, 56FF, 56GD, 56KD, 5771, 57C1, 57EF, 5807, 580J, 589J, 58E7, 58IJ, 58K7, 5905, 5945, 5957, 5969, 5989, 598J, 5999, 59EJ, 59K7, 5AEF, 5B25, 5B55, 5BA5, 5BI7, 5BIJ, 5BK5, 5BK7, 5C21, 5EAF, 5EE7, 5F09, 5F6F, 5F95, 5FKD, 5G6D, 5G89, 5I09, 5I0F, 5I27, 5I6F, 5I89, 5IB7, 5IBJ, 5IFF, 5J77, 5K07, 5K25, 5K27, 5K6F, 5K87, 5KK7, 6013, 60A1, 60AF, 60EL, 6281, 62B1, 62KD, 63G9, 6403, 6643, 66EL, 66G9, 68KD, 69B3, 6A01, 6A0F, 6AAF, 6AFF, 6B21, 6B8D, 6BBD, 6BK1, 6D03, 6D43, 6D69, 6D6F, 6D93, 6D9F, 6E21, 6E81, 6E8D, 6EBD, 6ECL, 6ELL, 6FD9, 6GBD, 6GG9, 6IEF, 6K21, 6K8D, 6KBD, 6KE1, 6L43, 6LB3, 6LDF, 6LEL, 6LG9, 700F, 7027, 7057, 70EH, 70LF, 710H, 71G5, 7207, 7225, 7255, 727L, 7505, 7507, 755F, 75EF, 75F5, 75I7, 766F, 7681, 76CF, 7771, 7781, 77IL, 7861, 7A0H, 7A4H, 7BB5, 7C6F, 7EEH, 7EHL, 7FG5, 7G45, 7HEL, 7I0L, 7I27, 7IE7, 7IGH, 7K07, 7K61, 7K77, 7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, H0D1, H0HJ, H0LL, H1HH, H20D, H22D, H26D, H2I7, H2K7, H30L, H3KJ, H40H, H447, H4G3, H4H3, H4HH, H66L, H6ED, 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5IIIIIIIIIIIIIIIIIIIIIIIIIIF, D555555555555555555555550555, EEAAAAAAAAAAAAAAAAAAAAAAAAAF, HHHHHHHHHHHHHHHHHHHHHHHHEEBH, K66666666666666666666666666F, LLLLLLLLLLLLLLLLLLLLLLLLEEB7, D5555555555555555555555555A55, GGGGGGGGGGGGGGGGGGGGGGGGGGGG3, GIG0000000000000000000000000H, HH00000000000000000000000001H, K0000000000000000000000005KEF, 5BBBBBBBBBBBBBBBBBBBBBBBBBBBBD, HB0000000000000000000000000001, K000000000000000000000000505EF, L7777777777777777777777777772L, 2000000000000000000000000000CB1, C8CCCCCCCCCCCCCCCCCCCCCCCCCCCC9, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKFF, JE0000000000000000000000000000J, K000000000000000000000000000261, A0000000000000000000000000004I4H, HD000000000000000000000000000001, K000000000000000000000000000EC01, K0FFFFFFFFFFFFFFFFFFFFFFFFFFFFCF, D0002222222222222222222222222222D, FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL2L, I700000000000000000000000000000GH, K00000000000000000000000000000E61, 20000000000000000000000000000000JJ, DD5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL0KE7, 44HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, F0000000000000000000000000000000000000000000000000000000L89, J000000000000000000000000000000000000000000000000000000JJIJ, K222222222222222222222222222222222222222222222222222222222DD, KCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCF, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKCF, D55KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, K000000000000000000000000000000000000000000000000000000000000077, 22222222222222222222222222222222222222222222222222222222222222222D, K000000000000000000000000000000000000000000000000000000000000008IJ, DFA00000000000000000000000000000000000000000000000000000000000000005, CC4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJE0J, 777777777777777777777777777777777777777777777777777777777777777777777777EK7, 6000000000000000000000000000000000000000000000000000000000000000000000000000000043, KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKB5, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEEH, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 50000000000000000000000000000000000000000000000000000000000000000000000000000000002C1, K0000000000000000000000000000000000000000000000000000000000000000000000000000000000055EF, H700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000K1, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0I7, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, A400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, E0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000071, 7LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIL, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLK77, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEI7, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIB3, I7G00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 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77EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEK7, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJKJ, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777772L, BKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK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KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK5

===Base 24===

15, 17, 1D, 1H, 1J, 1N, 25, 2B, 2D, 2J, 2N, 31, 37, 3B, 3H, 41, 45, 47, 4B, 4D, 4H, 57, 5B, 5H, 5J, 65, 67, 6D, 6J, 6N, 75, 7B, 7D, 7N, 81, 85, 87, 8J, 97, 9B, 9D, 9H, 9N, A1, AB, AH, AN, B5, B7, BD, BH, BJ, C5, CJ, CN, D1, D5, DJ, E1, EB, ED, EH, EN, F7, FD, FJ, FN, G5, GD, GH, H1, HB, HD, HN, I1, I7, IB, IH, J1, J5, J7, JB, JN, K7, KB, KJ, KN, L5, LH, LJ, MD, MJ, N5, NB, NH, NJ, 101, 10B, 111, 1F1, 1FB, 1GB, 1LB, 201, 221, 22H, 261, 271, 277, 28H, 2A7, 2C7, 2G7, 2H7, 2L1, 2L7, 2MH, 305, 30D, 30J, 33N, 34N, 35D, 35N, 38D, 395, 3A5, 3AJ, 3CD, 3DD, 3DN, 3E5, 3EJ, 3GJ, 3IJ, 3JJ, 3K5, 3KD, 3ND, 43N, 44N, 49J, 4EJ, 4GJ, 4GN, 4NN, 50N, 535, 54N, 551, 55N, 5C1, 5CD, 5E5, 5K1, 5KD, 5LN, 5M5, 5N1, 601, 60B, 61B, 66H, 68B, 691, 6CH, 6FH, 6GB, 6HH, 6MH, 70H, 70J, 711, 761, 771, 77H, 77J, 78H, 7C7, 7CH, 7FH, 7G7, 7H7, 7HH, 7IJ, 7JJ, 7K1, 7M1, 7M7, 80D, 82H, 83N, 88D, 88H, 8AD, 8CD, 8DB, 8DD, 8DH, 8DN, 8GB, 8KD, 8MB, 8MH, 905, 911, 921, 935, 955, 99J, 9AJ, 9G1, 9JJ, 9K5, 9L1, 9M5, A0J, A3J, A95, AA7, AD7, AE5, AG7, AGJ, AI5, AIJ, AJD, AL7, ALD, B01, B0N, B11, B61, B6B, B8N, B91, BIN, BL1, BLN, BNN, C1B, C21, C27, C2H, C3D, C61, C8H, C91, CA7, CB1, CBB, CC7, CCB, CCD, CDD, CFB, CG1, CGB, CK1, CL1, CMB, CMH, D0B, D3D, D3N, D4N, D6B, D6H, D7H, D8B, D8N, DAD, DCD, DCH, DDH, DDN, DG7, DGB, DID, DMN, DND, E05, E4J, EA7, EEJ, EF5, EGJ, EI5, EJJ, EM5, EM7, F01, F21, F51, F8H, F95, FC1, FF1, FFB, FKH, FM5, G0B, G0N, G11, G3N, G6B, G77, G7J, G8B, G8N, G91, GA7, GBB, GC7, GFB, GG1, GGJ, GGN, GK1, GL1, GLN, GMN, GN1, GNN, H0J, H2H, H3J, H4J, H77, HA5, HA7, HE5, HFH, HIJ, HJJ, HKH, HL7, HMH, I0N, I3D, I3J, I3N, I4N, I5D, I95, IA5, IAJ, IE5, IEJ, IF5, IGJ, IJD, IK5, IKD, J0D, J4J, J8D, JAD, JDH, JEJ, JFH, JHH, JKD, JMH, K35, K6H, KCD, KFH, KH5, KLD, KM1, L01, L0B, L0D, L0N, L61, L6B, L8D, LA7, LC7, LDD, LF1, LG7, LGB, LGN, LID, LK1, LKD, LL1, LLB, LLD, LMN, LNN, M0H, M11, M21, M4N, M71, M91, M95, MA5, MA7, MBN, MC7, MF1, MF5, MFB, MFH, MG7, MI5, MIN, ML1, ML7, MLB, MMH, N01, N21, N4N, N71, N8N, NC1, ND7, NE7, NG1, NID, NK1, NL7, NMN, NN7, 11CB, 11MB, 1291, 12G1, 16C1, 16CB, 16K1, 186B, 18CB, 19K1, 1BK1, 1C8B, 1K91, 1KC1, 1KL1, 1L21, 1LC1, 1LM1, 1M61, 1M8B, 1MG1, 206H, 20CH, 20M7, 21C1, 21M1, 2207, 260H, 26KH, 2991, 2C6H, 2CC1, 2CM1, 2F11, 2FHH, 2MC1, 2MK1, 2MM1, 308N, 30GN, 30IN, 30LN, 30MN, 333J, 33JD, 33LD, 343J, 344J, 35I5, 380N, 393J, 394J, 3A3D, 3FI5, 3IMN, 3J3D, 3JID, 3L3D, 3L8N, 3M0N, 3M55, 3NGN, 404J, 408N, 40LN, 434J, 44AJ, 4ILN, 4JAJ, 4L8N, 5091, 5095, 50F1, 50I5, 51L1, 5211, 5291, 52G1, 53ID, 53MN, 5595, 55AD, 56F1, 588N, 58MN, 58ND, 5961, 5991, 5A5D, 5AAD, 5F91, 5GF1, 5GIN, 5I05, 5I55, 5I8D, 5IDD, 5IDN, 5IIN, 5IMN, 5KI5, 5M61, 5M8N, 5N3N, 602H, 6211, 62F1, 62G1, 66C1, 66FB, 66M1, 66MB, 6B21, 6BM1, 6BMB, 6C6B, 6CF1, 6CLB, 6FG1, 6K21, 6K2H, 6KG1, 6KKH, 6L21, 6LCB, 6LM1, 6MB1, 6MBB, 6MG1, 6MK1, 7001, 7027, 7207, 726H, 739J, 793J, 79C1, 7A4J, 7A9J, 7AE7, 7C01, 7CC1, 7FL1, 7G21, 7G9J, 7GAJ, 7GC1, 7HGJ, 7J2H, 7J6H, 7MKH, 800B, 800H, 804N, 806H, 808N, 80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, NK3D, NK8D, 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5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, 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EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEG7, 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===Base 30===

11, 17, 1B, 1D, 1H, 1N, 1T, 21, 27, 2B, 2D, 2J, 2N, 2T, 37, 3B, 3D, 3H, 3J, 3N, 47, 4B, 4H, 4J, 4T, 51, 57, 5D, 5H, 5N, 5T, 61, 6B, 6D, 6H, 6J, 71, 7D, 7H, 7J, 7N, 7T, 81, 8B, 8H, 8N, 8T, 91, 97, 9B, 9D, 9N, A7, AB, AD, AH, B1, B7, BH, BJ, BN, BT, C7, CD, CJ, CN, CT, D7, DB, DJ, DT, E1, EB, ED, EJ, EN, ET, F7, FB, FD, FH, FT, G7, GB, GJ, GN, GT, HB, HD, I1, I7, IH, IN, IT, J1, J7, JH, JN, JT, K1, K7, KD, KH, KJ, L1, LB, LD, LH, LN, LT, M1, MD, MH, MN, N1, NB, NJ, NT, O7, OD, OJ, ON, P1, P7, PB, PJ, PN, Q7, QH, QT, R1, RB, RD, RH, RJ, RT, SD, SH, SJ, SN, T7, TB, TD, TH, 10J, 15J, 1IJ, 1JJ, 1LJ, 1MJ, 1QJ, 22H, 29H, 2EH, 2GH, 30T, 331, 33T, 36T, 39T, 3A1, 3C1, 3G1, 3KT, 3MT, 3OT, 3S1, 3T1, 40D, 40N, 431, 44D, 46N, 48D, 4AN, 4DD, 4DN, 4F1, 4FN, 4GD, 4ID, 4PD, 4S1, 50J, 58J, 59J, 5CB, 5FJ, 5IB, 5IJ, 5JB, 5MB, 5MJ, 5OB, 5SB, 607, 63T, 687, 6E7, 6KT, 6L7, 6M7, 6MT, 6NN, 6QN, 6R7, 6S7, 6ST, 6TT, 70B, 77B, 787, 7KB, 7M7, 7MB, 7SB, 807, 80D, 80J, 84D, 85J, 877, 88J, 89J, 8DD, 8FJ, 8ID, 8IJ, 8JJ, 8M7, 8MJ, 8PD, 8QD, 8R7, 90H, 93T, 95J, 99H, 99J, 9AJ, 9AT, 9EH, 9GH, 9HH, 9HJ, 9JJ, 9MJ, 9OH, 9OT, 9PH, 9ST, 9TT, A01, A0T, A1J, A31, A6N, A6T, AAJ, AAN, AFN, AKN, AKT, ALJ, AMJ, AMT, AO1, AOT, AQ1, AQN, ARN, AT1, B5B, BBD, BCB, BDD, BID, BOB, BPD, BQB, C31, C9H, CC1, CCB, CCH, CF1, CH1, CIB, CKB, CMB, D01, D0H, D2H, D41, D4D, D6N, D8D, D9H, DDH, DDN, DGH, DH1, DHH, DID, DKN, DNN, DO1, DOH, DQN, DS1, EEH, EHH, EM7, EOH, EPH, ER7, F0N, F31, F5J, F8J, F9J, FLJ, FO1, FQ1, FQN, FS1, G01, G4D, G8D, GF1, GGH, GO1, GOH, GQD, GS1, H07, H0J, H0T, H1J, H2H, H31, H4N, H8J, HA1, HAJ, HAT, HC1, HE7, HEH, HFN, HGH, HH7, HIJ, HJJ, HKN, HL7, HNH, HPH, HQN, HS1, HTJ, HTN, I4D, I5B, I5J, I8D, IAJ, IDD, IGD, IIJ, IKB, IMB, IMJ, IOB, IPD, IQD, J8J, J9J, JAJ, JBD, JCB, JFJ, JIJ, JJB, JJD, JLJ, JPD, JQB, K3T, K4N, KAT, KBB, KCB, KMB, KNN, KOB, KOT, KQN, KST, KTT, L0J, L5J, L67, LAJ, LJJ, LQJ, LR7, M0J, M3T, M5B, M7B, M87, M9J, MAT, MFJ, MIJ, MJJ, MKB, MMJ, MOT, MQJ, N07, N0H, N67, N6N, N87, NAN, NDH, NGD, NHH, NKN, NN7, NNH, NPH, NQD, NQN, NR7, O01, O0B, O0H, OAT, OC1, OCH, OEH, OF1, OH1, OKT, OMB, OOT, OPH, OQ1, OQB, OS1, OST, P3T, P6T, P9H, PCH, PDH, PEH, PGD, PHH, PHT, PID, PMT, PPT, PQD, PST, PTT, Q5B, Q6N, Q9J, QAJ, QBB, QBD, QC1, QDN, QFJ, QFN, QGD, QJB, QKN, QLJ, QMB, QND, QNN, QO1, QQ1, QQN, QSB, R4N, R77, R87, RAN, RKN, RM7, RNN, RR7, RS7, S0T, S41, S6T, S87, SBB, SC1, SM7, SMT, SOB, SQ1, SR7, SS1, STT, T0J, T6T, T9T, TAN, TFN, TKN, TKT, TNN, TO1, TOT, TPT, TQ1, TQN, TTN, 18AJ, 19FJ, 1AFJ, 1FAJ, 20CH, 20PH, 2C0H, 2CHH, 2COH, 2H0H, 2HCH, 2HHH, 2POH, 2PPH, 3001, 34Q1, 3F41, 3FF1, 3QF1, 3SPT, 3SST, 3TTT, 40A1, 4441, 44KN, 44QN, 4AC1, 4C01, 4CA1, 4DG1, 4GA1, 4GC1, 4NND, 4NRN, 4OA1, 4Q4N, 4QA1, 4QRN, 4RQN, 550B, 555B, 55KB, 5A5J, 5AJJ, 5BKB, 5JQJ, 5KKB, 5QQB, 5QQJ, 604N, 606T, 60FN, 60KN, 60OT, 60PT, 660N, 660T, 664N, 66AT, 66FN, 66KN, 66TN, 6A0N, 6A9T, 6AAT, 6FKN, 6FRN, 6K0N, 6KFN, 6KKN, 6O6T, 6PAT, 6RFN, 6RRN, 6T6N, 6TRN, 7067, 70R7, 75BB, 77R7, 7C5B, 7CQB, 7E67, 7IQB, 7O5B, 7OIB, 7QOB, 7R07, 7RE7, 8667, 88L7, 88S7, 8E87, 8EE7, 8EL7, 8J8D, 8LL7, 8LS7, 906T, 908J, 90FJ, 90QJ, 90TJ, 92CH, 966T, 99MT, 99PT, 9I0J, 9I8J, 9LFJ, 9LIJ, 9M6T, 9MKT, 9MMT, 9PKT, 9QQJ, 9TLJ, A04N, A0QJ, A3AT, A3ST, A4A1, A4G1, A5QJ, A8QJ, A90J, A9PT, AA3T, AA41, AAAT, AAF1, AAG1, AAPT, AC41, ACS1, AF0J, AFC1, AFIJ, AG41, AGA1, AGG1, AI8J, AIJJ, AIQJ, AJ5J, AJJJ, AQ8J, AQJJ, AQQJ, AS3T, AT4N, AT5J, ATST, B04D, B0QD, BBIB, BBKB, BI0B, BK0B, BKIB, BKSB, BMIB, BMSB, BQ4D, BQQD, BSMB, C00B, C05B, C0A1, C0BB, C0EH, C0GH, C0Q1, C0QB, C2OH, C2PH, C441, C4Q1, C50B, CB0B, CBSB, CEGH, CG2H, CGHH, CHHH, CHOH, COGH, COO1, COOB, COOH, CQ01, CQ0B, CQQB, CQS1, CS0B, D00N, D0PD, D44N, DA0N, DCEH, DCG1, DCQ1, DDC1, DDF1, DDPD, DDQ1, DECH, DF4N, DFA1, DFRN, DG0D, DGD1, DGQ1, DNEH, DPDD, DQA1, DQPD, DRFN, DRRN, E20H, E667, E767, E8L7, E8S7, ECGH, EG2H, EH67, EH77, ES67, F001, F01J, F0MJ, F0QJ, F44N, F4G1, F4RN, F6RN, FA0J, FAC1, FAF1, FAIJ, FC01, FCG1, FFAJ, FFC1, FFG1, FFIJ, FGA1, FJJJ, FJQJ, FKAN, FR6N, FRFN, G09H, G0EH, G0HH, G20H, G341, G3Q1, G4G1, G92H, GA41, GAC1, GC0H, GCEH, GCG1, GCPH, GD31, GDA1, GDG1, GDPD, GE2H, GE9H, GGG1, GGGD, GHCH, GHHH, GP0D, GP2H, GPPD, GQ31, GQA1, GQG1, H00N, H0CH, H3ST, H4O1, H667, H66N, H677, H69T, H99T, H9CH, H9KT, H9LJ, H9PT, H9QJ, HH6N, HHF1, HHFJ, HHH1, HHHT, HHLJ, HHOH, HHQ1, HHST, HK6T, HL9J, HLMJ, HMKT, HMMT, HNM7, HNRN, HOPT, HQ01, HQ5J, HR0N, HR6N, HS77, HSKT, HT01, I0BB, I0JD, I0JJ, I98J, I9LJ, I9QJ, ICBB, IF0J, IFFJ, IFQJ, IIBD, IJBB, IJID, IL8J, ILFJ, IQ8J, IQJJ, IQQJ, IS0B, ISQB, J00D, J05J, J0BB, J0MJ, J0OB, J4QD, J50B, J5QJ, JBKB, JBMB, JDGD, JDQD, JGDD, JI0D, JIIB, JISB, JKIB, JKKB, JM0B, JOIB, JQ4D, JQDD, JQID, JQJJ, JQMJ, JSIB, JSMB, K06N, K0KB, K0KT, K0MT, K0PT, K0SB, K5QB, K60T, K66T, K6AN, K6FN, K6PT, K6RN, K96T, K99T, KA0N, KF6N, KFKN, KI0B, KK0T, KK6N, KK6T, KKFN, KKKB, KKKT, KKQB, KKTN, KMMT, KQ0B, KQQB, KR0N, KS5B, L0M7, L8E7, L98J, LFFJ, LI8J, LIFJ, LL87, LL9J, LLIJ, LLM7, LM77, LML7, LMM7, M09T, M0E7, M0IB, M0L7, M0M7, M0TT, M55J, M5LJ, M60T, M69T, M707, M767, M777, M7E7, M7S7, M8LJ, M96T, M9KT, MC0B, MCOB, ME07, MIBB, MJ0B, MJBB, MJSB, MLLJ, MLS7, MM07, MM0T, MML7, MMOB, MMR7, MOBB, MOCB, MP0T, MP9T, MPKT, MQ0B, MQIB, MQOB, MS07, MSPT, MSSB, MT8J, MTTT, N00D, N04D, N04N, N0FN, N0ND, N0RN, N2OH, N92H, NCGH, ND0D, ND4N, NDPD, NEGH, NEL7, NF4N, NFRN, NGCH, NGEH, NHM7, NME7, NML7, NMM7, NN0D, NN4D, NN4N, NNDN, NNID, NNND, NNRN, NP4D, NSL7, NSS7, O00T, O03T, O0MT, O0PT, O2HH, O2OH, O341, O4A1, O4G1, O4O1, O5KB, O90T, OA41, OBSB, OC5B, OCOB, OG2H, OG31, OH9H, OH9T, OHHH, OHTT, OICB, OISB, OM0T, OM6T, OMPT, OMTT, OO2H, OO9H, OOA1, OOCB, OOGH, OOKB, OOO1, OSKB, OTMT, P00T, P04D, P08D, P09T, P0KT, P90T, PAAT, PKKT, PO0T, PP0D, PPPD, Q00B, Q0AN, Q0D1, Q0F1, Q0IB, Q0JJ, Q0MJ, Q0OB, Q55J, Q5QJ, Q88D, QA0N, QA41, QA4N, QDF1, QDPD, QF41, QFA1, QG31, QIQJ, QJ0J, QJ8D, QJDD, QJJJ, QJMJ, QJQD, QKKB, QKQB, QOKB, QOOB, QP0D, QPDD, QQ8D, QQID, QQJD, QQPD, QQQB, QS01, QSA1, R00N, R067, R06N, R0FN, R0L7, R60N, RE67, RFRN, RRFN, RRRN, S00B, S3AT, S3ST, S50B, S5QB, S99T, SAAT, SC0B, SE67, SGG1, SICB, SIQB, SK5B, SKIB, SKKT, SKPT, SKQB, SM0B, SMMB, SMSB, SOA1, SOG1, SPOT, SQCB, SQQB, SS0B, SS67, ST31, STG1, T00N, T03T, T041, T04N, T0AT, T0C1, T0MT, T0ST, T1AJ, T3F1, T4A1, T4RN, T5JJ, T66N, T8AJ, T9FJ, T9QJ, TA3T, TAA1, TAG1, TAIJ, TCS1, TFA1, TFAJ, TFF1, TFFJ, TFG1, TGA1, TI9J, TIJJ, TIQJ, TL8J, TL9J, TM0T, TM5J, TM8J, TMAJ, TMLJ, TMST, TSF1, TSG1, TT01, TT5J, TT8J, TTA1, TTC1, TTFJ, TTG1, TTLJ, TTMJ, 30441, 30O41, 30OO1, 34O41, 3AATT, 3ASAT, 3O441, 3Q041, 40G41, 40GQ1, 40O41, 40OG1, 40OO1, 44001, 440C1, 440G1, 4444N, 44CG1, 44GQ1, 4AAA1, 4COG1, 4DAA1, 4GQ41, 4KRRN, 4OGG1, 4Q041, 4QGG1, 5005B, 500BB, 500KB, 500QB, 50K0B, 50Q0B, 50QKB, 555AJ, 555LJ, 555QJ, 55J5J, 5B00B, 5B0BB, 5J55J, 5J5JJ, 5JJ5J, 5JJJJ, 5K00B, 5QJ5J, 5QK0B, 6006N, 6009T, 600AT, 6066N, 60R0N, 666N7, 6696T, 66OPT, 67767, 6900T, 696PT, 69P0T, 6F6AN, 6O9PT, 6OP0T, 6OP9T, 6P99T, 755QB, 77767, 7CBBB, 7IBIB, 7IIBB, 7IICB, 7QIIB, 8888D, 88E67, 8L887, 8SSE7, 9099T, 90KKT, 90KMT, 90MPT, 9690T, 96P9T, 990KT, 9990T, 9999T, 99K6T, 9FFFJ, 9HKMT, 9HMPT, 9ILLJ, 9KKMT, 9KKPT, 9M00T, 9T8QJ, 9TFIJ, A008J, A00NN, A055J, A0FJJ, A0I0J, A0I9J, A0J0J, A44C1, A555J, A9FFJ, A9QIJ, A9T8J, AFFA1, AFFFJ, AFFJJ, AFFQJ, AJ00J, AP99T, ASA9T, ASFF1, ASP9T, ASSAT, AT3TT, ATFQJ, ATQIJ, ATT9J, ATTQJ, B088D, B08GD, B0GGD, B0SKB, B8GGD, BG00D, BIISB, BISIB, BS0IB, BS0SB, BSSKB, BSSSB, C0041, C0G41, C0H0H, C0PPH, C0S01, C4001, C4AG1, C4OG1, C5BBB, CG00H, CGE0H, CGGA1, CGQ41, COSSB, CP20H, CPGPH, CPP2H, CPPOH, CPPPH, CS001, CS55B, CSSQB, D00GD, D0A4N, D0FAN, D0GDD, D0N0D, DAAA1, DAAC1, DAFG1, DDAA1, DDGGD, DFFAN, DFGC1, DGDGD, DGG31, DGGA1, DH0AN, DPPPH, DQDD1, DQGG1, E0L87, E88E7, EC00H, EC02H, EE867, EE887, EEL87, ELE87, F00IJ, F0441, F0AG1, F0F0J, F0F41, F0FF1, F0GG1, F4041, F44A1, F44C1, F4A41, F64KN, F6K6N, FAAA1, FAFQJ, FCAA1, FF041, FF0F1, FF64N, FF6KN, FFA41, FFA4N, FFF4N, FFFF1, FFFFN, FFKRN, FFQMJ, FIJ0J, FJ00J, FKKKN, FNNFN, FQQ0J, FQQMJ, G00DD, G00DH, G00PD, G02PH, G0DDD, G0PDD, GAAA1, GC2HH, GDDDD, GDGDD, GDPPH, GG0PD, GGCA1, GGCQ1, GGDD1, GH4Q1, GHHG1, GII0D, GQ441, H009H, H00G1, H00H1, H04Q1, H0ANN, H0H01, H0H9H, H0HG1, H0HO1, H0O41, H0OHH, H0QG1, H40G1, H4G41, H4GG1, H60AN, HAN0N, HF0G1, HFF41, HFFMJ, HFFQJ, HGQ41, HH3TT, HH401, HH441, HH4G1, HH55J, HH66T, HH6OT, HH96T, HHA0N, HHANN, HHC0H, HHCHH, HHH0N, HHHCH, HHHHN, HHHNN, HHKMT, HHO41, HHP9T, HHPKT, HHTMT, HHTT1, HK9MT, HKKMT, HLLFJ, HM66T, HM7R7, HMM67, HMM77, HMMM7, HMTST, HO9HT, HOO41, HOOOH, HOT3T, HOTT1, HPO9T, HSO3T, HTGG1, I000J, I009J, I00BD, I00QJ, I00SB, I08QJ, I0I0D, I0IQB, I0JIB, I0Q0B, I0SCB, I0SIB, I0SSB, IB00D, IBBSB, IBISB, ICSSB, II0CB, II0ID, II0SB, IIIBB, IIQ0B, IIQIB, IISIB, IJ0SB, IJJQJ, ILLLJ, IQ00J, IQC0B, IQIQB, IQQIB, ISSCB, J00MB, J08GD, J0IID, J0QQD, J55BB, J55JJ, J5J5J, J5JJJ, J88GD, JB0SB, JBB0B, JBBSB, JDDDD, JG0ID, JGG0D, JIIID, JJ0QJ, JJ5JJ, JQ08D, JQ0QJ, JQQ0D, JQQ5J, JS55B, JSK0B, JSS5B, K000T, K006T, K00TN, K0FAN, K505B, K6T0N, K9KMT, K9KPT, K9P0T, KIQIB, KK00N, KK05B, KK0AN, KK0KN, KKM9T, KS0IB, KS0QB, KSQIB, KTR6N, KTRRN, L0087, L08S7, LE087, LEE87, LL8LJ, LMEE7, LMSE7, LMSS7, M00CB, M00PT, M066T, M0CQB, M0K6T, M0MQB, M0MSB, M0QQB, M0SMB, M0SST, M666T, M900T, MBBSB, MEE77, MEL77, MELE7, MES77, MESE7, MESS7, MI00B, MIICB, MIISB, MIQCB, MK9PT, ML7L7, MLEL7, MLLE7, MLME7, MM677, MMBIB, MMBSB, MMCQB, MME77, MMEE7, MMICB, MMISB, MMK6T, MMKKT, MMM9T, MMMMT, MMMTT, MMQQB, MMSCB, MMSKT, MMSMB, MMSS7, MMTST, MOIIB, MOOIB, MOSIB, MQCQB, MR007, MR667, MRL07, MS7L7, MSEL7, MSK9T, MSL77, MSSL7, MT00T, MTMMT, N0DDD, N4NNN, N7LE7, N7S77, NE2CH, NE9CH, NEC2H, NEE77, NFFNN, NFNFN, NII0D, NL777, NL7L7, NLES7, NLLL7, NLS77, NOG9H, NRR0N, O6P9T, O9H6T, O9HPT, O9P9T, OCBBB, OG441, OGAG1, OGGA1, OHH6T, OHOOH, OKIIB, OKK5B, OKKIB, OO5BB, OOBIB, OOIIB, OSCSB, OT3TT, OT441, OTG41, OTGG1, P00PH, P0D0D, P0G2H, P0OGH, P0PGH, PA99T, PGP0H, POOOH, PP20H, PP88D, PPP0H, Q0001, Q000N, Q001J, Q00ID, Q00PD, Q044N, Q04G1, Q0I0D, Q0PPD, Q40G1, Q444N, Q44RN, Q4AA1, Q8QQJ, QAAS1, QAFF1, QAFG1, QASG1, QDGG1, QFGG1, QI00D, QIICB, QIIQB, QIQCB, QOIIB, QPP4D, QQ08J, QQ0CB, QQC0B, QQI0J, QQJ5J, QSFG1, R00E7, R0E07, R0NE7, R6F6N, R6FFN, REE07, RELE7, RFF6N, RL0E7, RLE07, RLEE7, RLLE7, RQR0N, RR0QN, RRQ0N, S03O1, S0AF1, S0AG1, S0O31, S0QIB, S0SIB, S30F1, S30O1, S7QIB, SA3PT, SAAA1, SAFG1, SASST, SCSQB, SF0G1, SFFF1, SI0SB, SISSB, SKSSB, SSCQB, SSCSB, SSMIB, SSPAT, SSSKT, SSSMB, SSSSB, STAF1, STF01, T0001, T0031, T0AF1, T0AS1, T0G31, T0R6N, T0T31, T0TF1, T3AAT, T40G1, T4CG1, T4G41, T5LLJ, T8LLJ, TA441, TA98J, TAFJJ, TAQ5J, TASST, TATAT, TC401, TCGG1, TFQIJ, TFQJJ, TFQMJ, TG441, TGC41, TI8LJ, TLLMJ, TMMMT, TMTMT, TQ8QJ, TSS3T, TT3AT, TT9IJ, TTAAT, TTQIJ, TTS31, TTS3T, TTT9J, TTTST, 20000H, 200OOH, 3440O1, 3TAAST, 404CQ1, 4KKKKN, 4KKKRN, 4QQQQD, 505BBB, 50BB0B, 6000AN, 6444RN, 66666N, 666O9T, 66999T, 669P9T, 66N777, 6A444N, 6FF66N, 6FFF6N, 6R666N, 766767, 77S677, 7IBBBB, 8888E7, 8LLLLJ, 8SSSL7, 9000MT, 90K90T, 90KP0T, 90M90T, 99000T, 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===Base 36===

11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 2UJ, 2WN, 2XN, 30Z, 337, 33B, 33H, 33P, 34H, 34P, 37D, 38Z, 39H, 3AB, 3AP, 3AZ, 3D7, 3DH, 3FD, 3FZ, 3HD, 3HH, 3KZ, 3L7, 3LZ, 3MB, 3O7, 3OZ, 3PD, 3PP, 3Q7, 3RH, 3S7, 3SZ, 3UP, 3UZ, 3XB, 40P, 43B, 43H, 46V, 48B, 48V, 49B, 4AP, 4BB, 4E1, 4FP, 4HB, 4HH, 4HP, 4IH, 4MB, 4OP, 4OV, 4PH, 4RH, 4VB, 4VH, 4WH, 4WP, 4WV, 4X1, 4XH, 55T, 565, 58Z, 5C5, 5E7, 5ET, 5EZ, 5IN, 5KT, 5L7, 5MP, 5NN, 5O5, 5O7, 5OP, 5R7, 5RT, 5RZ, 5ST, 5SZ, 5TN, 5TZ, 5W7, 5XN, 5YN, 61T, 625, 62J, 64V, 661, 66J, 691, 6AV, 6EV, 6IJ, 6J1, 6K5, 6O1, 6OT, 6QJ, 6QT, 6RV, 6SJ, 6T1, 6TT, 6U5, 6UV, 6VV, 6W5, 6Y1, 6YT, 72D, 737, 77D, 791, 7A1, 7AZ, 7D7, 7GD, 7I1, 7IN, 7LN, 7M7, 7MN, 7N7, 7NN, 7RN, 7RZ, 7WN, 7X7, 7Z1, 801, 80V, 82D, 83B, 841, 84H, 877, 881, 887, 88B, 88V, 88Z, 89H, 89V, 8A1, 8AB, 8B7, 8BH, 8D1, 8DH, 8EB, 8EV, 8GD, 8GZ, 8HD, 8IV, 8L7, 8LZ, 8M1, 8MB, 8MD, 8O7, 8OB, 8Q7, 8QD, 8RD, 8S7, 8SH, 8SZ, 8UZ, 8W7, 8WV, 8XV, 8Y1, 90H, 91J, 93B, 93H, 94V, 981, 98H, 99J, 9AH, 9BB, 9C5, 9IH, 9IV, 9JV, 9KH, 9M1, 9OB, 9QJ, 9R1, 9RH, 9SH, 9UJ, 9VB, 9VJ, 9W5, 9X1, 9Y1, 9YB, 9YJ, 9YV, A25, A3P, A3Z, A4H, A61, A81, A8B, AAB, AAH, ABB, AC5, AEZ, AHP, AK1, AKB, AKH, ALZ, AMB, AO5, AOZ, APH, AQP, AR1, ARV, ARZ, AUV, AV5, AVB, AVV, AWV, AX1, AXB, AZ1, B2J, B4B, B4V, B6V, B87, B8H, B9B, BAB, BAH, BBH, BBV, BE7, BEB, BGJ, BHB, BIJ, BJB, BJT, BLJ, BOB, BOT, BQ7, BRV, BS7, BTT, BTV, BVB, BVJ, BVT, BWV, BX7, BXH, BYH, BYV, C25, C2J, C2N, C55, C65, C6J, C95, CCN, CDD, CED, CF5, CFJ, CGD, CIN, CJD, CO5, CRN, CUJ, CXJ, CXN, D37, D3P, D41, D55, D5P, D6T, D77, D81, D9H, DA1, DCD, DD1, DD5, DDH, DDP, DE7, DEP, DF5, DFD, DG7, DHH, DI1, DK1, DK5, DKT, DO1, DOP, DP1, DPD, DQ7, DR7, DRH, DS7, DTT, DU1, DWD, DX7, DYH, E0P, E2D, E3Z, E41, E4P, E71, E81, E87, E8B, EAZ, EDP, EED, EEN, EEV, EFZ, EGT, EI1, EKZ, ELB, EMB, EMN, EN7, EO1, EOT, EPP, EPZ, EQ7, ERN, EST, ETN, ETV, EUP, EUZ, EVT, EWD, EWN, EX1, EYB, EYN, EZD, EZP, F2J, F3B, F3Z, F4P, F65, F8B, F8Z, FCJ, FD5, FEZ, FFB, FFD, FG5, FGD, FGZ, FIJ, FJ5, FJJ, FLZ, FPD, FQD, FSZ, FXB, G07, G0D, G0Z, G3D, G3Z, G55, G6T, G7D, G7Z, G87, G8Z, GA5, GCJ, GD7, GE7, GET, GG5, GG7, GGT, GGZ, GI7, GJD, GKZ, GL7, GLT, GQT, GRT, GS7, GST, GU5, GUZ, GW5, GXJ, GZ7, GZZ, H3D, H3H, H4H, H9B, HAH, HED, HGD, HHP, HIH, HKH, HKP, HLN, HNH, HOB, HOP, HQP, HRD, HRH, HRN, HSH, HWD, HWH, HWP, HYN, I41, I6J, I71, I7N, I87, I8H, I9H, I9J, IA1, IAV, IE1, IFJ, IG7, IHH, II1, IIH, IIV, IJV, IK1, IL7, ILJ, ILN, IM1, INN, IOV, IRH, IUV, IX1, IXH, IXV, IYJ, J2D, J4V, J55, J61, J6J, J8B, J8V, J95, J9J, J9V, JA5, JBB, JD5, JDT, JF5, JFJ, JGJ, JGT, JIV, JJ1, JJD, JJV, JK5, JKT, JLB, JLT, JMN, JN5, JNJ, JQJ, JQN, JQT, JRD, JTB, JVJ, JVN, JVV, JWN, JY1, JYJ, K0B, K25, K3P, K65, K81, K95, K9H, KAH, KC5, KEP, KEZ, KG5, KHP, KK1, KKT, KLB, KLP, KLZ, KM1, KMB, KMP, KOT, KP1, KQP, KR1, KRT, KRZ, KTT, KW5, KX1, KZB, L0N, L0P, L8Z, L9B, LAZ, LBJ, LEN, LET, LFB, LFZ, LG7, LGZ, LIJ, LJJ, LKB, LKP, LLB, LLP, LLT, LMB, LMN, LN7, LO7, LOT, LPZ, LQT, LRN, LTN, LTT, LWP, LX7, LYN, M01, M0P, M2D, M2N, M37, M3B, M41, M61, M77, M87, M8D, M91, ME1, ME7, MEB, MFB, MFP, MGD, MI7, MKB, MM7, MMN, MN7, MNN, MO7, MOB, MOP, MPP, MQP, MS7, MW7, MXN, MYN, N5N, N65, N87, N8D, N8H, N95, NCD, NCJ, NDH, NE7, NFJ, NG5, NG7, NGJ, NJ5, NN7, NND, NO5, NQD, NQJ, NRN, NU5, NWH, NWN, NXH, NXN, NYJ, O0Z, O25, O37, O3B, O3P, O3Z, O41, O4B, O5Z, O61, O71, O7Z, O81, OA5, OAP, OBB, OBV, OC5, OEZ, OF5, OG7, OKP, OKZ, OMB, OO5, OOZ, OP5, OPP, OQT, OR1, OR7, OS7, OST, OTB, OTZ, OU5, OVV, OW5, OX7, OXB, OXV, OYV, OZ5, OZ7, P01, P0D, P0N, P3P, P4H, P4P, P5N, P65, P6V, P8V, PD1, PED, PEZ, PHH, PHP, PI1, PIN, PLN, PLP, PLZ, POP, PP1, PPH, PPV, PQD, PQN, PRV, PV5, PVH, PVV, PWH, PWP, PX1, PXV, PYH, PYN, Q07, Q0P, Q2N, Q37, Q3P, Q6J, Q6T, Q7D, Q8D, Q9J, QCD, QCJ, QD7, QED, QFP, QGT, QI7, QIN, QJN, QLJ, QMD, QMN, QMP, QND, QNJ, QOT, QQJ, QTN, QWN, QYJ, QYT, R0V, R0Z, R37, R3H, R4H, R77, R7D, R7N, R7Z, R81, R8V, R91, RA1, RCD, RCN, RD1, RED, REV, REZ, RGT, RGZ, RHD, RIV, RKH, RKZ, RLN, RMD, RNH, RO7, RQN, RRD, RRT, RRZ, RS7, RSH, RT1, RTV, RU1, RUZ, RVN, RVT, RW7, RWH, RX7, RY1, S0J, S6J, S87, S9H, SAZ, SCJ, SET, SFJ, SG7, SGZ, SI7, SM7, SO7, SOT, SQJ, SQT, SRH, SSH, STT, SW7, SX7, SXH, SYH, T0N, T0Z, T1D, T1T, T6V, T8Z, TBB, TBV, TCN, TD1, TEV, TGT, TKT, TLB, TM1, TO1, TOB, TQN, TR1, TRD, TT1, TTB, TTN, TUZ, TVB, TVN, TVT, TWD, TWV, TXB, TXV, TYV, U2J, U3Z, U61, U8V, U95, UA1, UC5, UEP, UEZ, UFJ, UG5, UJ5, ULZ, UO5, UOP, UOZ, UQP, URV, URZ, UUJ, UW5, UWV, UXJ, UXV, UYJ, V0H, V25, V4V, V55, V6V, V9J, V9V, VBB, VBJ, VCJ, VET, VGJ, VHP, VIH, VIN, VJJ, VJN, VLT, VMP, VNJ, VOB, VP5, VQJ, VQT, VRT, VRV, VSH, VSJ, VST, VTB, VTN, VUP, VW5, VWN, VXH, VXN, VYB, W07, W25, W3D, W3H, W4P, W4V, W57, W6V, W7D, W8H, W95, W9H, WAH, WAV, WCN, WD7, WDD, WDH, WE7, WEN, WF5, WGD, WHH, WK5, WKH, WN7, WNN, WPP, WPV, WQP, WR7, WRD, WRH, WRN, WS7, WU5, WUP, WVN, WWH, WWP, WX7, WXH, WXN, WYN, WYV, X0J, X2J, X2N, X4B, X4H, X4V, X6J, X87, X8B, X91, X9B, XAV, XCN, XEB, XFB, XHB, XHH, XHN, XKB, XLJ, XNH, XO1, XQ7, XS7, XSH, XUJ, XWN, XWV, XX7, XXH, XXV, Y0N, Y1J, Y1T, Y2N, Y3H, Y61, Y8V, Y91, YBV, YCN, YEB, YFJ, YHB, YHN, YIH, YJN, YJT, YLJ, YLN, YMB, YMN, YNH, YOB, YOV, YRH, YTB, YTN, YTT, YTV, YVB, YVH, YWH, YWV, YXB, YY1, YYJ, YYT, YYV, Z01, Z3D, Z3Z, Z71, Z8B, ZB7, ZBB, ZD5, ZDD, ZDP, ZG7, ZGD, ZGZ, ZKB, ZLP, ZM1, ZO5, ZP1, ZPD, ZQD, ZU1, ZX1, ZXB, ZZD, 102J, 106T, 10ET, 10GJ, 10LJ, 10TD, 12FJ, 12LJ, 13ED, 13GD, 13MD, 192J, 1D0T, 1DET, 1E3D, 1EFD, 1F0D, 1FWD, 1G0J, 1GGJ, 1GIJ, 1GUJ, 1I0J, 1ISJ, 1JOT, 1JTT, 1M0D, 1M3D, 1MFD, 1MWD, 1OKT, 1OOT, 1Q3D, 1QDT, 1QKT, 1QTD, 1QWD, 1SLJ, 1TDT, 1TFD, 1TOT, 1TTD, 1TTT, 1UGJ, 1ULJ, 1WFD, 1WWD, 20CJ, 20FJ, 20G5, 20IN, 20JD, 20MD, 20RN, 20WD, 20YN, 2295, 25RN, 260J, 26LJ, 288D, 28DD, 28FD, 296J, 29G5, 29K5, 29LJ, 2C05, 2C0J, 2C0N, 2CJJ, 2CK5, 2CLJ, 2CSJ, 2D25, 2DC5, 2DDD, 2DW5, 2F0J, 2FDD, 2FMD, 2FSJ, 2G65, 2G95, 2GK5, 2I0N, 2JC5, 2JCJ, 2JWD, 2K05, 2L6J, 2L9J, 2N0N, 2NIN, 2NN5, 2NSJ, 2NW5, 2NYN, 2Q0N, 2QNN, 2R0N, 2RFD, 2RWD, 2SJJ, 2SXJ, 2SYJ, 2W05, 2WC5, 2WD5, 2WMD, 2XCJ, 2YCJ, 2YJJ, 2YRN, 2YSJ, 300D, 302D, 304B, 3077, 3087, 308D, 30BH, 30DD, 30FP, 30G7, 30GD, 30MD, 30QP, 30YH, 320D, 333D, 333Z, 33WD, 344B, 34FB, 373Z, 37E7, 37EZ, 37G7, 3807, 38BB, 38E7, 38G7, 38HB, 38I7, 38X7, 390B, 394B, 399B, 39EB, 39FB, 39KB, 3ASH, 3AWH, 3B8B, 3BG7, 3BLB, 3BR7, 3BW7, 3C8D, 3D0P, 3D2D, 3DKP, 3E3D, 3E77, 3EBB, 3EFB, 3EKB, 3EQP, 3ERD, 3ERZ, 3EZZ, 3F4B, 3FEP, 3G2D, 3GCD, 3GED, 3GM7, 3GR7, 3GWD, 3GZD, 3H0B, 3HEP, 3HKB, 3IAH, 3ISH, 3K4B, 3KBH, 3KEB, 3KKB, 3L4B, 3L8B, 3M3D, 3MEP, 3MG7, 3MWP, 3OFP, 3OKB, 3OMP, 3OOB, 3OYB, 3P8H, 3PXH, 3QDP, 3QQD, 3QRD, 3QWD, 3R3D, 3R3Z, 3R8D, 3RG7, 3RM7, 3RQD, 3RR7, 3RZ7, 3SIH, 3W0H, 3WED, 3WM7, 3WQD, 3WW7, 3WYH, 3X8H, 3XE7, 3XG7, 3XKH, 3Y0B, 3Y4B, 3Y8H, 3YAH, 3YSH, 3YYB, 3Z4B, 3ZFP, 404V, 4061, 40EB, 40I1, 40IV, 40KH, 40M1, 40R1, 40XB, 40XV, 43QP, 4441, 444H, 444P, 4481, 448H, 4491, 449H, 44EB, 44EV, 44K1, 44KB, 44KP, 44M1, 44R1, 44RV, 44SH, 44U1, 44VP, 44YB, 46I1, 46K1, 46M1, 480H, 488H, 48AH, 48I1, 48K1, 48U1, 4991, 49EV, 49I1, 49O1, 49UV, 4A0H, 4A41, 4A4B, 4A8H, 4A91, 4ALB, 4ASH, 4BXV, 4E4B, 4EEP, 4EFB, 4ELP, 4EUV, 4EXB, 4FOB, 4I01, 4I0V, 4I4V, 4I61, 4I81, 4I91, 4IEV, 4IY1, 4K0H, 4K91, 4KI1, 4KKH, 4KKP, 4KUP, 4L3P, 4L4P, 4LAB, 4LMP, 4MA1, 4MK1, 4MMP, 4MP1, 4MU1, 4O0B, 4OFB, 4OKB, 4OM1, 4OU1, 4OYB, 4P41, 4P81, 4PO1, 4PYV, 4QLP, 4QQP, 4RAV, 4RK1, 4RRV, 4S8H, 4U3P, 4U91, 4U9V, 4UIV, 4UUP, 4VAV, 4VLP, 4VVV, 4X0B, 4X0V, 4XAB, 4XBV, 4XVV, 4XYV, 4Y0V, 4Y41, 4Y4H, 4Y4V, 4Y81, 4YAV, 4YEV, 4YIV, 4YK1, 4YKB, 4YM1, 4YR1, 4YUV, 4YYB, 5025, 503Z, 5095, 50G5, 50PN, 50QN, 50WP, 50Z7, 5205, 52RN, 53FP, 53M7, 53PZ, 53X7, 53ZZ, 54PP, 550N, 553P, 554P, 5557, 555N, 557Z, 5595, 55EN, 55G5, 55GZ, 55LZ, 55OZ, 55QN, 55Z7, 572N, 577Z, 5787, 57CN, 57FZ, 57QN, 57S7, 57ZZ, 58I7, 58X7, 5955, 5A95, 5AEP, 5CMN, 5CWN, 5E2N, 5EAP, 5ECN, 5EKP, 5EQN, 5F95, 5FLP, 5FQP, 5G05, 5GK5, 5GQ7, 5I37, 5II7, 5IM7, 5L3P, 5LCN, 5LFP, 5LGT, 5LKZ, 5LUP, 5LZZ, 5M07, 5N55, 5NQ7, 5NX7, 5OAZ, 5OFZ, 5OLT, 5OOT, 5P25, 5P95, 5PG5, 5PGZ, 5PPN, 5PQP, 5PW5, 5Q0N, 5Q87, 5QCN, 5QKP, 5QN7, 5QPN, 5QPP, 5QQT, 5QTT, 5QWP, 5QX7, 5S37, 5SS7, 5T6T, 5TQT, 5U4P, 5UGZ, 5W0P, 5W2N, 5W5P, 5WLN, 5WP5, 5WQN, 5XM7, 5Y0T, 5YOT, 5Z25, 5Z4P, 5Z57, 5ZG5, 5ZLZ, 5ZQ7, 5ZWP, 6005, 600V, 601J, 6055, 606V, 60A1, 60C5, 60GT, 60JJ, 60LJ, 60M1, 60R1, 60VT, 60YV, 610J, 61FJ, 6401, 64K1, 6505, 665T, 6665, 66A5, 66F5, 66G5, 66GT, 66KT, 66V5, 66XV, 68OV, 68YV, 690J, 69G5, 69O5, 69XJ, 69XV, 6A55, 6AA1, 6AA5, 6AF5, 6C9J, 6CA5, 6CGJ, 6CYJ, 6EK1, 6ER1, 6FLJ, 6FXJ, 6G0J, 6G65, 6GGJ, 6GJ5, 6GLJ, 6I81, 6I9V, 6J65, 6J6V, 6JET, 6JST, 6JWV, 6K01, 6KET, 6KST, 6L0J, 6L6T, 6LFJ, 6LRT, 6LXJ, 6MA1, 6MM1, 6MX1, 6O8V, 6OV5, 6R0T, 6RET, 6RKT, 6S6T, 6T8V, 6U01, 6U9J, 6UE1, 6UGJ, 6UJJ, 6ULJ, 6UU1, 6V0T, 6VA5, 6VF5, 6VG5, 6VKT, 6VXJ, 6WWV, 6WXV, 6X41, 6X6V, 6XIV, 6XJV, 6XOV, 6Y0V, 6YCJ, 6YXV, 7001, 702N, 7041, 707N, 7081, 70GZ, 70I7, 70KZ, 70ND, 70O7, 70OZ, 70R7, 70U1, 70UZ, 70WD, 70YN, 71QD, 71WD, 720N, 72CN, 72YN, 733Z, 73ED, 73ZD, 7401, 7441, 74M1, 74O1, 74R1, 7641, 76K1, 76M1, 76R1, 76U1, 76X1, 773Z, 7741, 7771, 7781, 77FZ, 77G7, 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8KRH, 8KU1, 8KYH, 8KZ1, 8L4B, 8LBB, 8MG7, 8MR7, 8O4V, 8OI1, 8OK1, 8OOV, 8OX1, 8RAV, 8RHH, 8RUV, 8RX1, 8U4V, 8UK1, 8V4B, 8V9B, 8VHH, 8VRH, 8VVV, 8VYV, 8W0H, 8WFD, 8X07, 8XAH, 8XR1, 8XU1, 8XYB, 8Y6V, 8Y8H, 8YBB, 8YFB, 8YHH, 8YKB, 8YLB, 8YUV, 8YVV, 8YYH, 8Z1D, 8Z4B, 8Z8D, 8Z91, 8ZCD, 8ZEZ, 8ZFD, 8ZFZ, 8ZK1, 8ZO1, 8ZX7, 90A5, 90BJ, 90BV, 90EB, 90EV, 90GJ, 90IJ, 90J5, 90JB, 90LB, 90O1, 90OV, 90XB, 90XV, 9225, 92K5, 9441, 94BH, 94EB, 94KB, 94XB, 9505, 9601, 960V, 96A1, 96J5, 96K1, 96V5, 96WV, 96XV, 984B, 98AV, 98BV, 98FB, 994H, 998B, 998V, 999B, 99A1, 99A5, 99E1, 99G5, 99O1, 99O5, 99WH, 9A05, 9A55, 9A95, 9AA5, 9ABV, 9AE1, 9AEV, 9AI1, 9AO1, 9AOV, 9B0J, 9B0V, 9B8V, 9B9H, 9BCJ, 9BFJ, 9BSJ, 9BVV, 9CCJ, 9CLJ, 9CSJ, 9E01, 9E4B, 9EA1, 9EAB, 9EAV, 9EK1, 9F05, 9F25, 9FA5, 9FBJ, 9FFJ, 9FJB, 9FSJ, 9FXJ, 9G25, 9G2J, 9GIJ, 9GSJ, 9HFB, 9HLB, 9IIJ, 9IJ1, 9J01, 9J0B, 9J0J, 9J9B, 9JCJ, 9JE1, 9K05, 9K41, 9K55, 9K61, 9K8B, 9KA5, 9KI1, 9KK5, 9KU5, 9L2J, 9LCJ, 9O05, 9O0V, 9O95, 9OI1, 9OO1, 9OOV, 9OWV, 9RRV, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ

====Additional known quasi-minimal primes (not necessarily the next)====

P&lt;sub&gt;81993&lt;/sub&gt;SZ

==Unsolved families==

Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')) by covering congruence, algebraic factorization, or combine of them.

{|class=wikitable
|base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family
|-
|13||9{5}||(113×13&lt;sup&gt;''n''&lt;/sup&gt;−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|13||A{3}A||(41×13&lt;sup&gt;''n''+1&lt;/sup&gt;+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|16||{3}AF||(16&lt;sup&gt;''n''+2&lt;/sup&gt;+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1{7}||(23×17&lt;sup&gt;''n''&lt;/sup&gt;−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1F{0}7||32×17&lt;sup&gt;''n''+1&lt;/sup&gt;+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||4{7}A||(71×17&lt;sup&gt;''n''+1&lt;/sup&gt;+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||51{0}D||86×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||70F{0}D||2038×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||8{B}9||(139×17&lt;sup&gt;''n''+1&lt;/sup&gt;−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||9{5}9||(149×17&lt;sup&gt;''n''+1&lt;/sup&gt;+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||95{F}||(2543×17&lt;sup&gt;''n''&lt;/sup&gt;−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||A{D}F||(173×17&lt;sup&gt;''n''+1&lt;/sup&gt;+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}B3||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+190||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}DB||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+232||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2BE||(11×17&lt;sup&gt;''n''+3&lt;/sup&gt;−41579)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2E||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;−2411)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}E9||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+773)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}EE||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+853)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||D0G{D}||(60381×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2860381*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||E9{B}||(3963×17&lt;sup&gt;''n''&lt;/sup&gt;−11)/16||29000||[http://factordb.com/index.php?query=%283963*17%5En-11%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||F1{9}||(4105×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16||1000000||[http://factordb.com/index.php?query=%284105*17%5En-9%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||FD0{D}||(72909×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2872909*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||G{7}F||(263×17&lt;sup&gt;''n''+1&lt;/sup&gt;+121)/16||29000||[http://factordb.com/index.php?query=%28263*17%5E%28n%2B1%29%2B121%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||2{7}9D||(47×21&lt;sup&gt;''n''+2&lt;/sup&gt;+953)/20||20000||[http://factordb.com/index.php?query=%2847*21%5E%28n%2B2%29%2B953%29%2F20&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||2F{C}A||(288×21&lt;sup&gt;''n''+1&lt;/sup&gt;−13)/5||20000||[http://factordb.com/index.php?query=%28288*21%5E%28n%2B1%29-13%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||7{K}Z||(53×36&lt;sup&gt;''n''+1&lt;/sup&gt;+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||B{0}EUV||11×36&lt;sup&gt;''n''+3&lt;/sup&gt;+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||HM{0}N||634×36&lt;sup&gt;''n''+1&lt;/sup&gt;+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||N{0}YYN||23×36&lt;sup&gt;''n''+3&lt;/sup&gt;+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||O{L}Z||(123×36&lt;sup&gt;''n''+1&lt;/sup&gt;+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||S{0}8H||28×36&lt;sup&gt;''n''+2&lt;/sup&gt;+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|}

For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E

For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D}

=== Base 17 ===

All searched to length 29000

* 1{7}
* 1F{0}7
* 4{7}A
* 51{0}D
* 70F{0}D
* 8{B}9
* 9{5}9
* 95{F}
* A{D}F
* B{0}B3
* B{0}DB
* {B}2BE
* {B}2E
* {B}E9
* {B}EE
* D0G{D}
* E9{B}
* F1{9}
* FD0{D}
* G{7}F

=== Base 21 ===

All searched to length 20000

* 2{7}9D
* 2F{C}A
* 4{3}B
* 5{0}DJ
* {5}FEK
* {7}ID
* 99{0}99H
* {9}0D
* {9}D
* B0{H}6H
* B3{0}EB
* B9{0}E5
* B{D}B
* B{H}6H
* DH{D}
* F{9}D
* {F}35
* G{0}FK
* H{0}7771
* H{D}
* {J}6J

=== Base 36 ===

All searched to length 20000

* 7{K}Z
* B{0}EUV
* HM{0}N
* N{0}YYN
* O{L}Z
* S{0}8H

==Primality certificates for the proven primes &gt; 10&lt;sup&gt;299&lt;/sup&gt;==

See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]]

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime
|-
|9||149||76&lt;sub&gt;329&lt;/sub&gt;2||(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642]
|-
|9||150||27&lt;sub&gt;686&lt;/sub&gt;07||(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486]
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||3×9&lt;sup&gt;1160&lt;/sup&gt;+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423]
|-
|11||1065||A&lt;sub&gt;713&lt;/sub&gt;58||11&lt;sup&gt;715&lt;/sup&gt;−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487]
|-
|11||1066||7&lt;sub&gt;759&lt;/sub&gt;44||(7×11&lt;sup&gt;761&lt;/sup&gt;−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840]
|-
|11||1067||557&lt;sub&gt;1011&lt;/sub&gt;||(607×11&lt;sup&gt;1011&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522]
|-
|13||3165||50&lt;sub&gt;270&lt;/sub&gt;44||5×13&lt;sup&gt;272&lt;/sup&gt;+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005]
|-
|13||3166||9&lt;sub&gt;271&lt;/sub&gt;095||(3×13&lt;sup&gt;274&lt;/sup&gt;−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654]
|-
|13||3167||10&lt;sub&gt;286&lt;/sub&gt;7771||13&lt;sup&gt;290&lt;/sup&gt;+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633]
|-
|13||3168||9&lt;sub&gt;308&lt;/sub&gt;1||(3×13&lt;sup&gt;309&lt;/sup&gt;−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=308&amp;c0=-&amp;EN= 13&lt;sup&gt;308&lt;/sup&gt;−1]
|-
|13||3169||B&lt;sub&gt;341&lt;/sub&gt;C4||(11×13&lt;sup&gt;343&lt;/sup&gt;+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618]
|-
|13||3170||8B&lt;sub&gt;343&lt;/sub&gt;||(107×13&lt;sup&gt;343&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736]
|-
|13||3171||710&lt;sub&gt;371&lt;/sub&gt;111||92×13&lt;sup&gt;374&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609]
|-
|13||3172||75&lt;sub&gt;375&lt;/sub&gt;7||(89×13&lt;sup&gt;376&lt;/sup&gt;+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596]
|-
|13||3173||9B0&lt;sub&gt;391&lt;/sub&gt;9||128×13&lt;sup&gt;392&lt;/sup&gt;+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790]
|-
|13||3174||7B0B&lt;sub&gt;397&lt;/sub&gt;||(15923×13&lt;sup&gt;397&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574]
|-
|13||3175||10&lt;sub&gt;414&lt;/sub&gt;93||13&lt;sup&gt;416&lt;/sup&gt;+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240]
|-
|13||3176||81010&lt;sub&gt;415&lt;/sub&gt;1||17746×13&lt;sup&gt;416&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3177||8110&lt;sub&gt;435&lt;/sub&gt;1||1366×13&lt;sup&gt;436&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3178||B7&lt;sub&gt;486&lt;/sub&gt;||(139×13&lt;sup&gt;486&lt;/sup&gt;−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892]
|-
|13||3179||B&lt;sub&gt;563&lt;/sub&gt;C||(11×13&lt;sup&gt;564&lt;/sup&gt;+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=564&amp;c0=-&amp;EN= 13&lt;sup&gt;564&lt;/sup&gt;−1]
|-
|13||3180||1B&lt;sub&gt;576&lt;/sub&gt;||(23×13&lt;sup&gt;576&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=576&amp;c0=-&amp;EN= 13&lt;sup&gt;576&lt;/sup&gt;−1]
|-
|13||3181||80&lt;sub&gt;693&lt;/sub&gt;87||8×13&lt;sup&gt;695&lt;/sup&gt;+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532]
|-
|13||3182||CC5&lt;sub&gt;713&lt;/sub&gt;||(2021×13&lt;sup&gt;713&lt;/sup&gt;−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353]
|-
|13||3183||B&lt;sub&gt;834&lt;/sub&gt;74||(11×13&lt;sup&gt;836&lt;/sup&gt;−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871]
|-
|13||3184||9&lt;sub&gt;968&lt;/sub&gt;B||(3×13&lt;sup&gt;969&lt;/sup&gt;+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244]
|-
|13||3185||10&lt;sub&gt;1295&lt;/sub&gt;181||13&lt;sup&gt;1298&lt;/sup&gt;+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013]
|-
|13||3186||9&lt;sub&gt;1362&lt;/sub&gt;5||(3×13&lt;sup&gt;1363&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776]
|-
|13||3187||7&lt;sub&gt;1504&lt;/sub&gt;1||(7×13&lt;sup&gt;1505&lt;/sup&gt;−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755]
|-
|13||3188||930&lt;sub&gt;1551&lt;/sub&gt;1||120×13&lt;sup&gt;1552&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3189||720&lt;sub&gt;2297&lt;/sub&gt;2||93×13&lt;sup&gt;2298&lt;/sup&gt;+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910]
|-
|13||3190||1770&lt;sub&gt;2703&lt;/sub&gt;17||267×13&lt;sup&gt;2705&lt;/sup&gt;+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825]
|-
|13||3191||390&lt;sub&gt;6266&lt;/sub&gt;1||48×13&lt;sup&gt;6267&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3192||B0&lt;sub&gt;6540&lt;/sub&gt;BBA||11×13&lt;sup&gt;6543&lt;/sup&gt;+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906]
|-
|13||3193||C&lt;sub&gt;10631&lt;/sub&gt;92||13&lt;sup&gt;10633&lt;/sup&gt;−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750]
|-
|14||649||34D&lt;sub&gt;708&lt;/sub&gt;||47×14&lt;sup&gt;708&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||5×14&lt;sup&gt;19698&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2328||880&lt;sub&gt;246&lt;/sub&gt;7||136×16&lt;sup&gt;247&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is &lt; 10&lt;sup&gt;299&lt;/sup&gt;
|-
|16||2329||D4&lt;sub&gt;263&lt;/sub&gt;D||(199×16&lt;sup&gt;264&lt;/sup&gt;+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238]
|-
|16||2330||E0&lt;sub&gt;261&lt;/sub&gt;4DD||14×16&lt;sup&gt;264&lt;/sup&gt;+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352]
|-
|16||2331||8C0&lt;sub&gt;290&lt;/sub&gt;ED||140×16&lt;sup&gt;292&lt;/sup&gt;+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307]
|-
|16||2332||DA&lt;sub&gt;305&lt;/sub&gt;5||(41×16&lt;sup&gt;306&lt;/sup&gt;−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284]
|-
|16||2333||CE80&lt;sub&gt;422&lt;/sub&gt;D||3304×16&lt;sup&gt;423&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257]
|-
|16||2334||5F&lt;sub&gt;544&lt;/sub&gt;6F||6×16&lt;sup&gt;546&lt;/sup&gt;−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967]
|-
|16||2335||88F&lt;sub&gt;545&lt;/sub&gt;||137×16&lt;sup&gt;545&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2336||BE0&lt;sub&gt;792&lt;/sub&gt;BB||190×16&lt;sup&gt;794&lt;/sup&gt;+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938]
|-
|16||2337||D9&lt;sub&gt;1052&lt;/sub&gt;||(68×16&lt;sup&gt;1052&lt;/sup&gt;−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020]
|-
|16||2338||FAF&lt;sub&gt;1062&lt;/sub&gt;45||251×16&lt;sup&gt;1064&lt;/sup&gt;−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610]
|-
|16||2339||F8&lt;sub&gt;1517&lt;/sub&gt;F||(233×16&lt;sup&gt;1518&lt;/sup&gt;+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824]
|-
|16||2340||20&lt;sub&gt;1713&lt;/sub&gt;321||2×16&lt;sup&gt;1716&lt;/sup&gt;+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735]
|-
|16||2341||300F&lt;sub&gt;1960&lt;/sub&gt;AF||769×16&lt;sup&gt;1962&lt;/sup&gt;−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750]
|-
|16||2342||90&lt;sub&gt;3542&lt;/sub&gt;91||9×16&lt;sup&gt;3544&lt;/sup&gt;+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191]
|-
|16||2343||5BC&lt;sub&gt;3700&lt;/sub&gt;D||(459×16&lt;sup&gt;3701&lt;/sup&gt;+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322]
|-
|16||2344||D0B&lt;sub&gt;17804&lt;/sub&gt;||(3131×16&lt;sup&gt;17804&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511]
|-
|18||547||80&lt;sub&gt;298&lt;/sub&gt;B||8×18&lt;sup&gt;299&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is an algebraic factor of ''N''+1, factordb entry of 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is [http://factordb.com/index.php?id=1100000000900149167]
|-
|18||548||H&lt;sub&gt;766&lt;/sub&gt;FH||18&lt;sup&gt;768&lt;/sup&gt;−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490]
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||12×18&lt;sup&gt;6270&lt;/sup&gt;+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437]
|-
|20||3301||H&lt;sub&gt;247&lt;/sub&gt;A0H||(17×20&lt;sup&gt;250&lt;/sup&gt;−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619]
|-
|20||3302||7&lt;sub&gt;249&lt;/sub&gt;A7||(7×20&lt;sup&gt;251&lt;/sup&gt;+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602]
|-
|20||3303||J7&lt;sub&gt;270&lt;/sub&gt;||(368×20&lt;sup&gt;270&lt;/sup&gt;−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462]
|-
|20||3304||J&lt;sub&gt;330&lt;/sub&gt;CCC7||20&lt;sup&gt;334&lt;/sup&gt;−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572]
|-
|20||3305||40&lt;sub&gt;387&lt;/sub&gt;404B||4×20&lt;sup&gt;391&lt;/sup&gt;+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563]
|-
|20||3306||EC0&lt;sub&gt;429&lt;/sub&gt;7||292×20&lt;sup&gt;430&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702]
|-
|20||3307||G&lt;sub&gt;447&lt;/sub&gt;99||(16×20&lt;sup&gt;449&lt;/sup&gt;−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753]
|-
|20||3308||3A&lt;sub&gt;527&lt;/sub&gt;3||(67×20&lt;sup&gt;528&lt;/sup&gt;−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531]
|-
|20||3309||E&lt;sub&gt;566&lt;/sub&gt;C7||(14×20&lt;sup&gt;568&lt;/sup&gt;−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516]
|-
|20||3310||JCJ&lt;sub&gt;629&lt;/sub&gt;||393×20&lt;sup&gt;629&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|20||3311||J&lt;sub&gt;655&lt;/sub&gt;05J||20&lt;sup&gt;658&lt;/sup&gt;−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052]
|-
|20||3312||50&lt;sub&gt;1163&lt;/sub&gt;AJ||5×20&lt;sup&gt;1165&lt;/sup&gt;+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412]
|-
|20||3313||CD&lt;sub&gt;2449&lt;/sub&gt;||(241×20&lt;sup&gt;2449&lt;/sup&gt;−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915]
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||16×20&lt;sup&gt;6270&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457]
|-
|22||7984||I7G0&lt;sub&gt;254&lt;/sub&gt;H||8882×22&lt;sup&gt;255&lt;/sup&gt;+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788]
|-
|22||7985||D0&lt;sub&gt;255&lt;/sub&gt;5EEF||13×22&lt;sup&gt;259&lt;/sup&gt;+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932]
|-
|22||7986||IK&lt;sub&gt;322&lt;/sub&gt;F||(398×22&lt;sup&gt;323&lt;/sup&gt;−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145]
|-
|22||7987||C0&lt;sub&gt;340&lt;/sub&gt;G9||12×22&lt;sup&gt;342&lt;/sup&gt;+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159]
|-
|22||7988||77E&lt;sub&gt;348&lt;/sub&gt;K7||(485×22&lt;sup&gt;350&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779]
|-
|22||7989||J&lt;sub&gt;379&lt;/sub&gt;KJ||(19×22&lt;sup&gt;381&lt;/sup&gt;+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027]
|-
|22||7990||J&lt;sub&gt;388&lt;/sub&gt;EJ||(19×22&lt;sup&gt;390&lt;/sup&gt;−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729]
|-
|22||7991||DJ&lt;sub&gt;400&lt;/sub&gt;||(292×22&lt;sup&gt;400&lt;/sup&gt;−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110]
|-
|22||7992||E&lt;sub&gt;404&lt;/sub&gt;K7||(2×22&lt;sup&gt;406&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298]
|-
|22||7993||66F&lt;sub&gt;453&lt;/sub&gt;B3||(971×22&lt;sup&gt;455&lt;/sup&gt;−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809]
|-
|22||7994||L0&lt;sub&gt;454&lt;/sub&gt;B63||21×22&lt;sup&gt;457&lt;/sup&gt;+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331]
|-
|22||7995||L&lt;sub&gt;483&lt;/sub&gt;G3||22&lt;sup&gt;485&lt;/sup&gt;−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730]
|-
|22||7996||E60&lt;sub&gt;496&lt;/sub&gt;L||314×22&lt;sup&gt;497&lt;/sup&gt;+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239]
|-
|22||7997||I&lt;sub&gt;626&lt;/sub&gt;AF||(6×22&lt;sup&gt;628&lt;/sup&gt;−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334]
|-
|22||7998||K0&lt;sub&gt;760&lt;/sub&gt;EC1||20×22&lt;sup&gt;763&lt;/sup&gt;+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415]
|-
|22||7999||J0&lt;sub&gt;767&lt;/sub&gt;IGGJ||19×22&lt;sup&gt;771&lt;/sup&gt;+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567]
|-
|22||8000||7&lt;sub&gt;959&lt;/sub&gt;K7||(22&lt;sup&gt;961&lt;/sup&gt;+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817]
|-
|22||8001||L&lt;sub&gt;2385&lt;/sub&gt;KE7||22&lt;sup&gt;2388&lt;/sup&gt;−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774]
|-
|22||8002||7&lt;sub&gt;3815&lt;/sub&gt;2L||(22&lt;sup&gt;3817&lt;/sup&gt;−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839]
|-
|24||3400||I0&lt;sub&gt;241&lt;/sub&gt;I5||18×24&lt;sup&gt;243&lt;/sup&gt;+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037]
|-
|24||3401||D0&lt;sub&gt;259&lt;/sub&gt;KKD||13×24&lt;sup&gt;262&lt;/sup&gt;+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725]
|-
|24||3402||C7&lt;sub&gt;298&lt;/sub&gt;||(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235]
|-
|24||3403||20&lt;sub&gt;313&lt;/sub&gt;7||2×24&lt;sup&gt;314&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241]
|-
|24||3404||BC0&lt;sub&gt;331&lt;/sub&gt;B||276×24&lt;sup&gt;332&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842]
|-
|24||3405||N&lt;sub&gt;2644&lt;/sub&gt;LLN||24&lt;sup&gt;2647&lt;/sup&gt;−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089]
|-
|24||3406||D&lt;sub&gt;2698&lt;/sub&gt;LD||(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876]
|-
|24||3407||A0&lt;sub&gt;2951&lt;/sub&gt;8ID||10×24&lt;sup&gt;2954&lt;/sup&gt;+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654]
|-
|24||3408||88N&lt;sub&gt;5951&lt;/sub&gt;||201×24&lt;sup&gt;5951&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606]
|-
|30||2613||AN&lt;sub&gt;206&lt;/sub&gt;||(313×30&lt;sup&gt;206&lt;/sup&gt;−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073]
|-
|30||2614||M&lt;sub&gt;241&lt;/sub&gt;QB||(22×30&lt;sup&gt;243&lt;/sup&gt;+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295]
|-
|30||2615||M0&lt;sub&gt;547&lt;/sub&gt;SS7||22×30&lt;sup&gt;550&lt;/sup&gt;+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988]
|-
|30||2616||C0&lt;sub&gt;1022&lt;/sub&gt;1||12×30&lt;sup&gt;1023&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|30||2617||5&lt;sub&gt;4882&lt;/sub&gt;J||(5×30&lt;sup&gt;4883&lt;/sup&gt;+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423]
|-
|30||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||25×30&lt;sup&gt;34205&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|}

==Unproven PRPs==

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this PRP
|-
|11||1068||57&lt;sub&gt;62668&lt;/sub&gt;||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000003573679860]
|-
|13||3194||C5&lt;sub&gt;23755&lt;/sub&gt;C||(149×13&lt;sup&gt;23756&lt;/sup&gt;+79)/12||[http://factordb.com/index.php?id=1100000003590647776]
|-
|13||3195||80&lt;sub&gt;32017&lt;/sub&gt;111||8×13&lt;sup&gt;32020&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000000490878060]
|-
|16||2345||DB&lt;sub&gt;32234&lt;/sub&gt;||(206×16&lt;sup&gt;32234&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000002383583629]
|-
|16||2346||4&lt;sub&gt;72785&lt;/sub&gt;DD||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15||[http://factordb.com/index.php?id=1100000003615909841]
|-
|22||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21||[http://factordb.com/index.php?id=1100000003594696838]
|-
|30||2618||I0&lt;sub&gt;24608&lt;/sub&gt;D||18×30&lt;sup&gt;24609&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003593967511]
|}

All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10&lt;sup&gt;16&lt;/sup&gt;. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]])

==Proof==

===Base 2===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.

===Base 3===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (2,1), (2,2)

* Case (1,1):
** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes)
*** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime.

===Base 4===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (2,1), (2,3), (3,1), (3,3)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.

===Base 5===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4)

* Case (1,1):
** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes)
*** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes)
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes)
***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb])
***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes)
**** The smallest prime of the form 1{4}4 is 14444.
**** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes)
*** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes)
**** For the 3{0,3}1 family, we can separate this family to four families:
***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01.
****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime.
***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031.
***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001.
***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31.
****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime.
**** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.
* Case (3,4):
** 34 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes)
**** The smallest prime of the form 4{4}1 is 44441.
**** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime.
* Case (4,2):
** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime.

===Base 6===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes)
*** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes)
**** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 4{0}41 is 40041
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes)
*** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.

===Base 7===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

* Case (1,1):
** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes)
*** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {}
**** If there are &gt;=3 number of 1 and 5 in the {}:
***** If there is 111 in the {}, then we have the prime 11111
***** If there is 115 in the {}, then the prime 115 is a subsequence
***** If there is 151 in the {}, then the prime 115 is a subsequence
***** If there is 155 in the {}, then the prime 155 is a subsequence
***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime
***** If there is 515 in the {}, then the prime 515 is a subsequence
***** If there is 551 in the {}, then the prime 551 is a subsequence
***** If there is 555 in the {}, then the prime 551 is a subsequence
**** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1
***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1
****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1
******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21
******** The smallest prime of the form 11{0}21 is 1100021
******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime
******** The smallest prime of the form 1001{0}21 is 100121
********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families
***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1
****** Since 1051 is prime, we only need to consider the family 15{0}1
******* The smallest prime of the form 15{0}1 is 150001
* Case (1,2):
** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes)
*** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {}
**** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime.
**** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime.
**** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2
***** Since 1022 and 1202 are primes, we only need to consider the number 122
****** 122 is not prime.
**** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime.
* Case (1,3):
** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** 14 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime.
* Case (1,6):
** 16 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime.
* Case (2,2):
** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,6):
** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes)
*** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes)
**** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's)
***** For the 3{0,1}1 family:
****** If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
****** If there are no 1's between (3,1), then the form will be 3{0}1
******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime.
****** If there are no 0's between (3,1), then the form will be 3{1}1
******* The smallest prime of the form 3{1}1 is 31111
****** If there are exactly 1 0's between (3,1), then there must be &lt;3 1's between (3,1), or 31111 will be a subsequence.
******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime.
******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011
******** Neither 3101 nor 3011 is prime.
******* If there are no 1's between (3,1), then the number must be 301
******** 301 is not prime.
***** For the 3{3}3{0,1}1 family:
****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence.
****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1
******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence.
******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime.
******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime.
******* If there are no 1's between (33,1), then the form will be 33{0}1
******** The smallest prime of the form 33{0}1 is 33001
****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1
******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence.
******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301
******** The smallest prime of the form 3{3}301 is 33333301
******* If there are no 0's between (3{3}3,1), then the form is 3{3}31
******** The smallest prime of the form 3{3}31 is 33333333333333331
**** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1
***** Numbers containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1,3}3{0,1,3}1
******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes)
******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes)
********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes)
********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence.
********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime.
********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1
*********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime.
********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime.
***** Numbers not containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1}1
******* If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1
******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime.
******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1
******** If there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111
********* None of 31151, 31511, 35111 are primes.
******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 351
********* 351 is not prime.
******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111
********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes.
********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime.
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1
********* Neither 3051 nor 3501 is prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** If there are &gt;=2 5's in {}, then 553 will be a subsequence.
*** If there are no 5's in {}, then the family will be 3{0,3,6}3
**** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
*** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3
**** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes)
***** The smallest prime of the form 3{0}53 is 300053
* Case (3,4):
** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes)
*** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes)
**** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes)
**** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes)
* Case (3,5):
** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes)
*** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence.
*** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence.
*** If there are at least two 1's in {}, then 115 will be a subsequence.
*** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5
**** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime.
*** If there is no 1's in {}, then the family will be 3{0,4,5}5
**** If there are at least to 4's in {}, then 344 and 445 will be subsequences.
**** If there is no 4's in {}, then the family will be 3{0,5}5
***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5
****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime.
****** The smallest prime of the form 3{5}5 is 35555
**** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5
***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes)
****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime.
* Case (3,6):
** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,2):
** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes)
*** If there is no 5's in {}, then the family will be 4{0,2,4,6}4
**** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4
**** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes)
***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence.
***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4
****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes)
***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4
****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4
******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes)
* Case (4,5):
** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** If there are at least two 5's in {}, then 4555 will be a subsequence.
*** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there is no 5's in {}, then the family will be 4{0}5
**** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime.
* Case (4,6):
** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (5,1):
** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 3's in {}, then 533 will be a subsequence.
*** If there is no 3's in {}, then the family will be 5{0,1}1
**** Since 5011 is prime, we only need to consider the family 5{1}{0}1
***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes)
****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime.
****** The smallest prime of the form 51{0}1 is 5100000001
****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime.
****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime.
*** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1
**** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence.
***** Thus we only need to consider the family 5{0}3{0,1}1
****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime.
****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime.
****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence.
****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1
******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0)
******** The smallest prime of the form 5311{0}1 is 531101
* Case (5,2):
** 52 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 1's in {}, then 113 will be a subsequence.
*** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
*** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime.
* Case (5,4):
** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes)
*** If there are no 5's in {}, then the family will be 5{0,3}4
**** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime.
*** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence.
*** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there are at least two 5's in {}, then 5554 will be a subsequence.
* Case (5,5):
** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes)
*** If there are no 4's in {}, then the family will be 5{0,5}5
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
*** If there are no 5's in {}, then the family will be 5{0,4}5
**** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime.
*** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence.
* Case (5,6):
** 56 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,2):
** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (6,3):
** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,4):
** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are at least three 3's in {}, then 3334 will be a subsequence.
*** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4
**** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence.
**** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence.
**** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence.
**** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence.
**** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence.
**** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence.
**** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence.
**** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence.
**** Thus the number can only be 634
***** 634 is not prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,6):
** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}6
**** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime.
*** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6
**** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime.
*** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence.
*** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence.

===Base 8===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7)

* Case (1,1):
** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes)
**** 171 is not prime.
**** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 &gt; 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 &gt; 1, this factorization is nontrivial), thus cannot be prime.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes)
*** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime)
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes)
*** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes)
*** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes)
**** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime.
**** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes)
***** The smallest prime of the form 33{4}1 is 3344441
***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime.
**** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes)
***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime.
***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes)
****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes.
* Case (3,3):
** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes)
*** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes)
**** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime)
**** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461
***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes)
****** The smallest prime of the form 4{4}41 is 444444441
****** The smallest prime of the form 4{4}641 is 444641
***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes)
****** The smallest prime of the form 4{4}411 is 444444441
****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes)
***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461
****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime)
**** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes)
***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime)
***** The smallest prime of the form 4{4}641 is 444641
* Case (4,3):
** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes)
*** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes)
**** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime.
**** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime.
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes)
*** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes)
**** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb])
**** The smallest prime of the form 4{4}77 is 4444477
**** The smallest prime of the form 4{7}7 is 47777
**** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime)
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes)
*** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes)
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
**** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes)
***** 500525 is not prime.
***** The smallest prime of the form 5{5}25 is 555555555555525
***** The smallest prime of the form 5{5}025 is 55555025
***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes)
***** The smallest prime of the form 5{5}0525 is 5550525
***** The smallest prime of the form 5{5}00525 is 5500525
***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes)
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence)
**** The form is 6{0,1,7}4{0,1,7}1
***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1
****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411
******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41
******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41
********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.
********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime.
********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime.
******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411
******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime)
*** Numbers not containing 4:
**** The form is 6{0,1,7}1
***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1
****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime.
****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1
******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1
******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.
******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime)
* Case (6,3):
** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,7):
** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them
**** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17
***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime.
**** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77
***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime.
**** If there is neither 1 nor 7 between them, then the form is 6{0,4}7
***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes)
****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime.
****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime.
* Case (7,1):
** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes)
*** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes)
**** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes)
***** The smallest prime of the form 7{7}1 is 7777777777771
***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime)
***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime.
***** The smallest prime of the form 7{7}4444441 is 77774444441
****** Since this prime has just 4 7's, we only need to consider the families with &lt;=3 7's
******* The smallest prime of the form 7{4}1 is 744444441
******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime.
******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes)
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.

===Base 10===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,9):
** 19 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 2{0}21 is 20021
**** The smallest prime of the form 22{0}1 is 22000001
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes)
*** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes)
**** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime.
**** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime.
* Case (2,9):
** 29 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,9):
** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** 47 is prime, and thus the only minimal prime in this family.
* Case (4,9):
** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime.
* Case (5,1):
** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes)
**** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes)
***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 5{5}1 is 555555555551
**** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581
***** 581 is not prime.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,7):
** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027
**** The smallest prime of the form 52{0}7 is 5200007
* Case (5,9):
** 59 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,3):
** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,9):
** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49
***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49
****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes)
******* The smallest prime of the form 6{6}49 is 666649
******** Since this prime has just 4 6's, we only need to consider the families with &lt;=3 6's
********* The smallest prime of the form 6{0}49 is 60000049
********* The smallest prime of the form 66{0}49 is 66000049
********* The smallest prime of the form 666{0}49 is 66600049
* Case (7,1):
** 71 is prime, and thus the only minimal prime in this family.
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,9):
** 79 is prime, and thus the only minimal prime in this family.
* Case (8,1):
** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes)
**** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes)
***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime)
***** The smallest prime of the form 80{5}1 is 80555551
* Case (8,3):
** 83 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime.
* Case (8,9):
** 89 is prime, and thus the only minimal prime in this family.
* Case (9,1):
** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes)
**** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021
***** None of 921, 9201, 9021 are primes.
**** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801
***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
* Case (9,3):
** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (9,7):
** 97 is prime, and thus the only minimal prime in this family.
* Case (9,9):
** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9
***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes)
****** The smallest prime of the form 94{6}9 is 946669

===Base 12===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,B):
** 1B is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes)
*** The smallest prime of the form 2{0}1 is 2001
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (2,B):
** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes)
**** None of 20B, 22B, 29B, 209B, 220B are primes.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,B):
** 3B is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes)
**** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 44{A}1 is 44AAA1
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes)
**** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077
* Case (4,B):
** 4B is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes)
*** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime.
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (5,B):
** 5B is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,5):
** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes)
**** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 6{0}A5 is 600A5
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,B):
** 6B is prime, and thus the only minimal prime in this family.
* Case (7,1):
** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes)
*** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1
**** None of 741, 7B1, 74B1 are primes.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,B):
** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes)
**** The smallest prime of the form 7{9}B is 7999B
**** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes)
**** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime)
* Case (8,1):
** 81 is prime, and thus the only minimal prime in this family.
* Case (8,5):
** 85 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** 87 is prime, and thus the only minimal prime in this family.
* Case (8,B):
** 8B is prime, and thus the only minimal prime in this family.
* Case (9,1):
** 91 is prime, and thus the only minimal prime in this family.
* Case (9,5):
** 95 is prime, and thus the only minimal prime in this family.
* Case (9,7):
** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes)
**** None of 947, 977, 997, 9477, 9977 are primes.
* Case (9,B):
** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes)
**** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes)
***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime)
***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime)
**** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes)
***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes)
****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime.
****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime)
****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime.
**** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B
***** None of 99B, 999B, 979B, 9799B, 9979B are primes.
* Case (A,1):
** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes)
**** For the A{A}{0}1 family:
***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form AA{0}1 is AA000001
***** The smallest prime of the form AAA{0}1 is AAA0001
***** The smallest prime of the form AAAA{0}1 is AAAA1
****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc.
**** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime.
* Case (A,5):
** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes)
*** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes)
**** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form A{0}65 is A00065
* Case (A,7):
** A7 is prime, and thus the only minimal prime in this family.
* Case (A,B):
** AB is prime, and thus the only minimal prime in this family.
* Case (B,1):
** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes)
**** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1
***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1
****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime)
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1
****** The smallest prime of the form B{A}01 is BAA01
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
**** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1
***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1
****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1
******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1
******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes.
****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1
******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime)
***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes)
****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B04{A}1 is B04A1
****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime)
**** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1
***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1
****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1
******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime.
******* The smallest prime of the form B{B}AA1 is BBBAA1
****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1
******* The smallest prime of the form B{B}AA1 is BBBAA1
**** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes)
***** The smallest prime of the form B{B}B1 is BBBB1
***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form B{B}BAA1 is BBBAA1
* Case (B,5):
** B5 is prime, and thus the only minimal prime in this family.
* Case (B,7):
** B7 is prime, and thus the only minimal prime in this family.
* Case (B,B):
** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes)
*** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B
**** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B
***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime.
***** For the B{0,B}9B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes)
******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B
******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 &gt; 1, 10^n+5 ≥ 10^1+5 = 15 &gt; 1, this factorization is nontrivial), thus cannot be prime.
***** For the B{0,B}99B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes)
******* The smallest prime of the form B{0}99B is B00099B
******* The smallest prime of the form B{B}99B is BBBBBB99B
***** For the B{0,B}999B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes)
******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes)
******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime)

== Examples of families which can be ruled out as contain no primes &gt; ''b'' ==

It is not known if this problem is solvable:

Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''?

It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime &gt; ''b'' or not. We use two different heuristic strategies to show that such families contain no primes &gt; ''b''.

In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;.

Examples of first strategy: (we can show that the corresponding numbers are &gt; all elements in ''S'', if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial)

* In base 10, all numbers of the form 4{6}9 are divisible by 7
* In base 6, all numbers of the form 4{0}1 are divisible by 5
* In base 15, all numbers of the form 9{6}8 are divisible by 11
* In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5}
* In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3}
* In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5}
* In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13}
* In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17}
* In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13}
* In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13}

Examples of second strategy: (we can show that both factors are &gt; 1, if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial)

* In base 9, all numbers of the form {1} factored as difference of squares
* In base 8, all numbers of the form 1{0}1 factored as sum of cubes
* In base 9, all numbers of the form 3{8} factored as difference of squares
* In base 16, all numbers of the form 8{F} factored as difference of squares
* In base 16, all numbers of the form {F}7 factored as difference of squares
* In base 9, all numbers of the form 3{1} factored as difference of squares
* In base 16, all numbers of the form {4}1 factored as difference of squares
* In base 16, all numbers of the form 1{5} factored as difference of squares
* In base 16, all numbers of the from {C}D factored as ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;
* In base 16, all numbers of the form B{4}1 factored as difference of squares

Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are &gt; all elements in S, and for the part of the second strategy, both factors are &gt; 1, if n makes corresponding numbers &gt; b, thus these factorizations are nontrivial)

* In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd
* In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even

== Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes &gt; ''b'' ==

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-powers factorization

=== 1{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 1{0}3 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== 1{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 1{0}5 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 5: Finite covering set {5}

=== 1{0}6 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== 1{0}7 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 7: Finite covering set {7}

=== 1{0}z ===

(none)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

(none)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {1} ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with ''r''&gt;1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2))

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' == 0 mod 2: Finite covering set {2}

=== 1{2} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 1{3} ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361)
* ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865)

=== 1{4} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization

=== 1{z} ===

(none)

=== 2{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 2{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {2}1 ===

* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 2{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 3{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {3}1 ===

* ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121)

=== 3{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 4{0}1 ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 4{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {4}1 ===

* ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225)

=== 4{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 5{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 5{z} ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}
* ''b'' = 6''m''&lt;sup&gt;2&lt;/sup&gt; with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014)

=== 6{0}1 ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== 6{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 7{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 7{z} ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993)
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Difference-of-cubes factorization

=== 8{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022)
* ''b'' = 467: Finite covering set {3, 5, 7, 19, 37}
* ''b'' = 722: Finite covering set {3, 5, 13, 73, 109}
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Sum-of-cubes factorization
* ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2

=== 8{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 9{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 9{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{0}1 ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{z} ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 307: Finite covering set {5, 11, 29}
* ''b'' = 901: Finite covering set {7, 11, 13, 19}

=== C{0}1 ===

* ''b'' == 1 mod 13: Finite covering set {13}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 296, 901: Finite covering set {7, 11, 13, 19}
* ''b'' = 562, 828, 900: Finite covering set {7, 13, 19}
* ''b'' = 563: Finite covering set {5, 7, 13, 19, 29}
* ''b'' = 597: Finite covering set {5, 13, 29}

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

(none)

=== {y}z ===

(none)

=== z{0}1 ===

(none)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3))
* ''b'' = 4''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt; (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}t ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 7: Finite covering set {7}

=== {z}u ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== {z}v ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 5: Finite covering set {5}

=== {z}w ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== {z}x ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== {z}y ===

* ''b'' == 0 mod 2: Finite covering set {2}

== Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) ==

Format: base (length)

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

(none)

=== 1{0}2 ===

(none)

=== 1{0}3 ===

(none)

=== 1{0}4 ===

53 (13403)

113 (10647)

=== 1{0}z ===

113 (20089)

123 (64371)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

(none)

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

208 (26682)

607 (11032)

828 (19659)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

201 (31276)

222 (52727)

227 (36323)

327 (135983)

425 (11231)

710 (24112)

717 (37508)

719 (13420)

=== {1} ===

152 (270217)

184 (16703)

200 (17807)

311 (36497)

326 (26713)

331 (25033)

371 (15527)

485 (99523)

629 (32233)

649 (43987)

670 (18617)

684 (22573)

691 (62903)

693 (41189)

731 (15427)

752 (32833)

872 (10093)

932 (20431)

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

(none)

=== 1{z} ===

107 (21911)

170 (166429)

278 (43909)

303 (40175)

383 (20957)

515 (58467)

522 (62289)

578 (129469)

590 (15527)

647 (21577)

662 (16591)

698 (127559)

704 (62035)

845 (39407)

938 (40423)

969 (24097)

989 (26869)

=== 2{0}1 ===

101 (192276)

206 (46206)

218 (333926)

236 (161230)

257 (12184)

305 (16808)

467 (126776)

578 (44166)

626 (174204)

695 (94626)

752 (26164)

788 (72918)

869 (49150)

887 (27772)

899 (15732)

932 (13644)

=== 2{z} ===

432 (16003)

=== 3{0}1 ===

912 (132174)

=== 3{z} ===

72 (1119850)

212 (34414)

218 (23050)

270 (89662)

303 (198358)

312 (51566)

422 (21738)

480 (93610)

513 (38032)

527 (46074)

566 (23874)

650 (498102)

686 (16584)

758 (15574)

783 (12508)

800 (33838)

921 (98668)

947 (10056)

=== 4{0}1 ===

107 (32587)

227 (13347)

257 (160423)

355 (10990)

410 (144079)

440 (56087)

452 (14155)

482 (30691)

542 (15983)

579 (67776)

608 (20707)

635 (11723)

650 (96223)

679 (69450)

737 (269303)

740 (58043)

789 (149140)

797 (468703)

920 (103687)

934 (101404)

962 (84235)

=== 4{z} ===

14 (19699)

68 (13575)

254 (15451)

800 (20509)

=== 5{0}1 ===

326 (400786)

350 (20392)

554 (10630)

662 (13390)

926 (40036)

=== 5{z} ===

258 (212135)

272 (148427)

299 (64898)

307 (26263)

354 (25566)

433 (283919)

635 (36163)

678 (40859)

692 (45447)

719 (20552)

768 (70214)

857 (23083)

867 (61411)

972 (36703)

=== 6{0}1 ===

108 (16318)

129 (16797)

409 (369833)

522 (52604)

587 (24120)

643 (164916)

762 (11152)

789 (27297)

986 (21634)

=== 6{z} ===

68 (25396)

332 (15222)

338 (42868)

362 (146342)

488 (33164)

566 (164828)

980 (50878)

986 (12506)

1016 (23336)

=== 7{0}1 ===

398 (17473)

1004 (54849)

=== 7{z} ===

97 (192336)

170 (15423)

194 (38361)

202 (155772)

282 (21413)

283 (164769)

332 (13205)

412 (29792)

560 (19905)

639 (10668)

655 (53009)

811 (31784)

814 (17366)

866 (108591)

908 (61797)

962 (31841)

992 (10605)

997 (15815)

=== 8{0}1 ===

23 (119216)

53 (227184)

158 (123476)

254 (67716)

320 (52004)

410 (279992)

425 (94662)

513 (19076)

518 (11768)

596 (148446)

641 (87702)

684 (23387)

695 (39626)

785 (900326)

788 (11408)

893 (86772)

908 (243440)

920 (107822)

962 (47222)

998 (81240)

1013 (43872)

=== 8{z} ===

138 (35686)

412 (12154)

788 (11326)

990 (23032)

=== 9{0}1 ===

248 (39511)

592 (96870)

=== 9{z} ===

431 (43574)

446 (152028)

458 (126262)

599 (11776)

846 (12781)

=== A{0}1 ===

173 (264235)

198 (47665)

311 (314807)

341 (106009)

449 (18507)

492 (42843)

605 (12395)

708 (17563)

710 (31039)

743 (285479)

744 (137056)

786 (68169)

800 (15105)

802 (149320)

879 (25004)

929 (13065)

977 (125873)

986 (48279)

1004 (10645)

=== A{z} ===

368 (10867)

488 (10231)

534 (80328)

662 (13307)

978 (14066)

=== B{0}1 ===

710 (15272)

740 (33520)

878 (227482)

=== B{z} ===

153 (21660)

186 (112718)

439 (18752)

593 (16064)

602 (36518)

707 (10573)

717 (67707)

=== C{0}1 ===

68 (656922)

219 (29231)

230 (94751)

312 (21163)

334 (83334)

353 (20262)

359 (61295)

457 (10024)

481 (45941)

501 (20140)

593 (42779)

600 (11242)

604 (17371)

641 (26422)

700 (91953)

887 (13961)

919 (45359)

923 (64365)

992 (10300)

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

(none)

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

38 (136212)

83 (21496)

113 (286644)

188 (13508)

401 (103670)

417 (21003)

458 (46900)

494 (21580)

518 (129372)

527 (65822)

602 (17644)

608 (36228)

638 (74528)

663 (47557)

723 (24536)

758 (50564)

833 (12220)

904 (13430)

938 (50008)

950 (16248)

=== z{0}1 ===

202 (46774)

251 (102979)

272 (16681)

297 (14314)

298 (60671)

326 (64757)

347 (69661)

363 (142877)

452 (71941)

543 (10042)

564 (38065)

634 (84823)

788 (13541)

869 (12289)

890 (37377)

953 (60995)

1004 (29685)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

53 (21942)

124 (16426)

175 (31626)

188 (22036)

316 (48538)

365 (25578)

373 (24006)

434 (10090)

530 (11086)

545 (12346)

560 (15072)

596 (12762)

701 (12576)

706 (10656)

821 (13536)

833 (17116)

966 (14820)

983 (11272)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}y ===

317 (13896)

== Bases 2≤''b''≤1024 which have these families as unsolved families ==

Unsolved families are families which are neither primes (&gt;''b'') found nor can be ruled out as contain no primes &gt; ''b''

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608)

1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000)

1{0}3: 646, 718, 998 (length limit: 2000)

1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000)

1{0}5

1{0}6

1{0}7

1{0}8

1{0}9

1{0}A

1{0}B

1{0}C

1{0}D

1{0}E

1{0}F

1{0}G

1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000)

1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000)

10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000)

11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000)

{1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000)

{1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000)

11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z})

{1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000)

{1}z

1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000)

1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995

1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000)

1{z}: 581, 992, 1019 (length limit: ≥100000)

2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000)

2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000)

2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000)

{2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000)

2{z}: 588, 972 (length limit: ≥100000)

3{0}1: 718 (length limit: ≥100000)

3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000)

3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000)

{3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000)

3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000)

4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000)

4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000)

{4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000)

4{z}: 338, 998 (length limit: ≥100000)

5{0}1: 308, 512, 824 (length limit: ≥100000)

5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000)

6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000)

6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000)

7{0}1: (none)

7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000)

8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000)

8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000)

9{0}1: 724, 884 (length limit: ≥100000)

9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000)

A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000)

A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000)

B{0}1: 560, 770, 968 (length limit: ≥100000)

B{z}: 263, 615, 912, 978 (length limit: ≥100000)

C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000)

C{z}

D{0}1

D{z}

E{0}1

E{z}

F{0}1

F{z}

G{0}1

{&amp;#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000)

{&amp;#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143)

&amp;#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000)

${&amp;#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2)

x{z}

y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000)

z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000)

{y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000)

{z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326)

zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z})

{z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000)

{z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1)

{z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000)

{z}k

{z}l

{z}m

{z}n

{z}o

{z}p

{z}q

{z}r

{z}s

{z}t

{z}u

{z}v

{z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000)

{z}x: (none)

{z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000)

== List of lengths for quasi-minimal primes in some simple families ==

[https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024]

NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases &lt;=7, family {z}x in bases &lt;=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime &gt; base) in this base)

RC: this family can be proven to only contain composite numbers (only count numbers &gt; base)

unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers &gt; base)

Background color: red for title (bases or families), green for length &gt; 10000, orange for 2500 &lt; length ≤ 10000, white for length ≤ 2500, cyan for &quot;RC&quot;, pink for &quot;NB&quot;, yellow for &quot;unknown&quot;.

Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families.

== References ==

* [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem]

* [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are &gt; ''b'' for bases ''b''≤16]

* [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary]

* [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia]

* [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime&gt;base is not required) in The Prime Puzzles &amp; Problems Connection]

* [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles &amp; Problems Connection]

* [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16]

* [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime&gt;base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29]

* [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime&gt;base is not required, search limits of lengths: 10000 for all b]

* [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime&gt;base is not required]

* [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime&gt;base is not required]

* [http://www.prothsearch.com/sierp.html the Sierpinski problem]

* [http://www.prothsearch.com/rieselprob.html the Riesel problem]

* [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem]

* [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030]

* [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100]

* [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem]

* [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024]

* [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project]

* [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project]

* [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count]

* [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty]

* [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers]

* [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares]

* [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software]

* [https://sourceforge.net/projects/openpfgw/ pfgw software]

* [http://jpenne.free.fr/index2.html llr software]

* [http://www.ellipsa.eu/public/primo/primo.html PRIMO software]

* [https://primes.utm.edu/prove/index.html website for primality proving]

* [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!]

* [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime&gt;base is not required]

* [https://primes.utm.edu/primes/lists/all.txt top proven primes]

* [http://www.primenumbers.net/prptop/prptop.php top PRPs]

* [http://factordb.com online factor database, including many primes which are minimal primes in a small base]</text>
      <sha1>azpcwivj97bvqyrqtm168a4zkcu5x0g</sha1>
    </revision>
    <revision>
      <id>2412026</id>
      <parentid>2412025</parentid>
      <timestamp>2022-08-05T03:45:25Z</timestamp>
      <contributor>
        <ip>2402:7500:901:2B90:BD69:8142:3D96:F08F</ip>
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      <comment>/* Unsolved families */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1960726" xml:space="preserve">A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime &gt; ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime &gt; 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 &gt; 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 &gt; 10 by changing the order.

(using A−Z to represent digit values 10 to 35)

For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')).

I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E&lt;sub&gt;800873&lt;/sub&gt;, and base 36 has quasi-minimal prime P&lt;sub&gt;81993&lt;/sub&gt;SZ).

Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to:

* Prove that all elements in ''S'', when read as base ''b'' representation, are primes &gt; ''b''.

* Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are &gt; ''b'', are composite.

* Prove that all primes &gt; ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are &gt; ''b'', are composite).

e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to:

* Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes &gt; 10.

* Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are &gt; 10 are composite.

* Prove that all primes &gt; 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers &gt; 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite).

==Condensed table==

{|class=wikitable
|''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form of largest known quasi-minimal prime base ''b''
|-
|2||1||11||2||3
|-
|3||3||111||3||13
|-
|4||5||221||3||41
|-
|5||22||10&lt;sub&gt;93&lt;/sub&gt;13||96||5&lt;sup&gt;95&lt;/sup&gt;+8
|-
|6||11||40041||5||5209
|-
|7||71||3&lt;sub&gt;16&lt;/sub&gt;1||17||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2
|-
|8||75||4&lt;sub&gt;220&lt;/sub&gt;7||221||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||1161||3×9&lt;sup&gt;1160&lt;/sup&gt;+10
|-
|10||77||50&lt;sub&gt;28&lt;/sub&gt;27||31||5×10&lt;sup&gt;30&lt;/sup&gt;+27
|-
|11&lt;sup&gt;*&lt;/sup&gt;||1068||57&lt;sub&gt;62668&lt;/sub&gt;||62669||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10
|-
|12||106||40&lt;sub&gt;39&lt;/sub&gt;77||42||4×12&lt;sup&gt;41&lt;/sup&gt;+91
|-
|13&lt;sup&gt;*&lt;/sup&gt;||3195~3197||80&lt;sub&gt;32017&lt;/sub&gt;111||32021||8×13&lt;sup&gt;32020&lt;/sup&gt;+183
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||19699||5×14&lt;sup&gt;19698&lt;/sup&gt;−1
|-
|15||1284||7&lt;sub&gt;155&lt;/sub&gt;97||157||(15&lt;sup&gt;157&lt;/sup&gt;+59)/2
|-
|16&lt;sup&gt;*&lt;/sup&gt;||2346~2347||4&lt;sub&gt;72785&lt;/sub&gt;DD||72787||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15
|-
|17&lt;sup&gt;*&lt;/sup&gt;||10408~10428||F70&lt;sub&gt;186767&lt;/sub&gt;1||186770||262×17&lt;sup&gt;186768&lt;/sup&gt;+1
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||6271||12×18&lt;sup&gt;6270&lt;/sup&gt;+221
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||6271||16×20&lt;sup&gt;6270&lt;/sup&gt;+13
|-
|21&lt;sup&gt;*&lt;/sup&gt;||13375~13396||CF&lt;sub&gt;479147&lt;/sub&gt;0K||479150||(51×21&lt;sup&gt;479149&lt;/sup&gt;−1243)/4
|-
|22&lt;sup&gt;*&lt;/sup&gt;||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||22003||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||8134||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49
|-
|30&lt;sup&gt;*&lt;/sup&gt;||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||34206||25×30&lt;sup&gt;34205&lt;/sup&gt;−1
|-
|36&lt;sup&gt;*&lt;/sup&gt;||35257~35263||P&lt;sub&gt;81993&lt;/sub&gt;SZ||81995||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7
|}

&lt;sup&gt;*&lt;/sup&gt; Data assumes the primality of the [[w:probable prime|probable prime]]s.

Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes &gt; 10&lt;sup&gt;25000&lt;/sup&gt;), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below.

Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists))

Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists))

==Data for quasi-minimal primes==

===Base 2===

11

===Base 3===

12, 21, 111

===Base 4===

11, 13, 23, 31, 221

===Base 5===

12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013

===Base 6===

11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041

===Base 7===

14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331

===Base 8===

13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447

===Base 9===

12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011

===Base 10===

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027

===Base 11===

12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, 2888882883, 2888888883, 3555555509, 3577777077, 3700000001, 4000000005, 40000005AA, 5377777707, 5555505553, 555555580A, 600000A999, 7000100047, 7000704777, 7007777107, 7057777777, 7070007774, 7077707774, 7077777074, 7100000704, 7470000041, 7701000004, 7707077774, 7770707774, 7777707074, 8888822883, 9555555503, 9900000979, 9999770007, A000144444, A900000066, A999999971, 10000000477, 33333333337, 44444444447, 44444444777, 55A55555552, 60000008883, 68888888306, 68888888883, 70000003999, 70000007717, 70004777777, 70477777777, 77007770004, 77700000477, 77707000704, 77707770074, 77707777774, 77777077774, 77777770004, 83000000006, 97000000999, A0000000001, A0014444444, A4777777771, 100000000057, 305007777777, 305777777777, 370000000007, 377777770007, 377777777107, 700000007474, 707077000074, 707077777774, 707777777717, 770000010004, 771007000007, 777070700004, 777700000704, A95555555555, A99999777777, 1000000003007, 40000000A0041, 58A5555555555, 7004400000004, 7700000000104, 7707000007047, 7707707000004, 7777007000004, 7777700000004, 7777770077704, 7777777710077, 9977777777717, A000000014444, A044444444441, A144444444411, 40000000000401, 45557777777777, 4555AAAAAAAAAA, 59077777777777, 70007777777771, 70077070000074, 70700000004777, 77000007700704, 77700000700047, 77777777770704, 88888888830006, 90900000000799, A0000044444441, 300000000005777, 302000000000002, 55555555A555552, 700000000000174, 770000000000474, 771700000000007, 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7707777777777777777704, 7900000000000000000005, 9777777777777777777707, A999999999999999999999, 10000000000000000000747, 970000000000000000000777, 999900000000000000000007, 3577777777777777777777777, 5555555555555555555555A52, 7000000000000000000777771, 7000000000000000077777771, AAAAAAAAAAAAAAAAAAA000058, 10000000000000000000000044, 77700000000000000000000008, 500777777777777777777777777, 777777777777777777777770774, 855555555555555555555055555, A44444444444444444444444441, 1500000000000000000000000007, 40000000000000000000000000041, 440000000000000000000000000001, 70000000000000000000000000007771, 999999999999999999999999999999991, 95555555555555555555555555555555553, 1900000000000000000000000000000000001, 7777777777777777777777777777777777474, 7777777777777777777777777777777777704, 10000000000000000000000000000000000000307, 50777777777777777777777777777777777777707, 475777777777777777777777777777777777777777, 555555555555555555555555555555555555555A2A, 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===Base 12===

11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077

===Base 13===

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===Base 14===

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===Base 15===

12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797

===Base 16===

11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD

===Base 17===

12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 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====Additional known quasi-minimal primes (not necessarily the next)====

570&lt;sub&gt;51310&lt;/sub&gt;1, 49&lt;sub&gt;111333&lt;/sub&gt;, 970&lt;sub&gt;166047&lt;/sub&gt;1, F70&lt;sub&gt;186767&lt;/sub&gt;1

===Base 18===

11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, 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===Base 20===

13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, GGJ0J, GGJJJ, GJ00J, GJGJJ, GJJJ7, H00EH, H024B, H04B1, H0E4B, H0F41, H0H11, H0HEH, H4E0B, H6I11, HAAA3, HAAG3, HAGA3, HB44B, HBBBB, HBHHH, HGGA3, HGGG3, HH001, HH061, HH3AH, HH6F1, HHG63, HHIA3, HI011, HIII3, I0029, I02I9, I085H, I0A41, I0F29, I2229, I2929, I2I29, I2I99, I2II9, I33A3, I3EHH, I422B, I424B, I4EEB, I4IEB, I80G1, I9IA9, IA099, IA309, IA841, IA939, IA9A9, IAF09, IAF99, IAI99, IAIA9, IB0A1, IB4IB, IB54B, IBB4B, IBBIB, IBE0H, IBH0H, IC929, ICC29, IF841, IFAG1, IGA81, IHB4B, II0A9, II42B, II44B, IIB5B, III4B, IIIA3, IIIIB, J00G1, J00JB, J0601, J06D1, J06I1, J0861, J08B1, J0B01, J0CGJ, J0E0B, J0GA1, J0GG1, J0I2B, J0J2B, J0JIB, J55CJ, J60A1, J60G1, J6D81, JA777, JA7A7, JAJG1, JAJJ1, JAJJ7, JB00B, JB08B, JC00J, JC0GJ, JEECJ, JGGGJ, JGGJJ, JJ0B1, JJ0JB, JJ55J, JJAG1, JJAJ1, JJC0J, JJE0B, JJEEB, JJGG1, JJGGJ, JJGJJ, JJJ01, JJJEB, JJJJB, 17555J, 175EEJ, 1E0007, 1JE55J, 1JJJ5J, 20005H, 2000HH, 222I29, 22E889, 22EE89, 24222B, 2999I9, 29III9, 2A000H, 2B0I0H, 2I2999, 2I9I29, 2III29, 333AF3, 336IA3, 36A3F3, 36I3A3, 388E8H, 3AF333, 3EHHHH, 3HEHHH, 3HHEHH, 40008B, 40054B, 40405B, 40448B, 415E5J, 44440B, 44452B, 44524B, 44E08B, 45444B, 4AAAE7, 4AEEE7, 4E4447, 4GEEE7, 4I544B, 4IEEEB, 500EFJ, 500FCJ, 5055FJ, 50AFFJ, 50EEFJ, 50EF5J, 50F5FJ, 50FFCJ, 52224B, 54EEEB, 5558AH, 555CAJ, 5585AH, 55F55J, 55FFFJ, 5888AH, 588A5H, 5E555J, 5F055J, 5F05CJ, 5F5F5J, 5FFFCJ, 5JEEEB, 60006D, 6000F1, 6000G3, 6001G1, 600A63, 600G1D, 600G6D, 603A33, 606G6D, 608001, 608F89, 608IF9, 60A333, 60A363, 60AAA3, 60CCCD, 60D04D, 60D0DD, 60F8I9, 60FF89, 60GCG3, 636AF3, 63A333, 63A3F3, 63AFF3, 63IIA3, 64AAAD, 660089, 660A33, 660DDD, 660F89, 662289, 666A33, 666D4D, 66A6A3, 66AF33, 66AF63, 66D0DD, 66FF89, 6A33F3, 6AFF33, 6AG6A3, 6AGA63, 6D000D, 6D004D, 6D0D4D, 6F6689, 7000G7, 7060DD, 70AAAD, 70D0CD, 733C33, 7366C3, 73AAA3, 7600DD, 760CDD, 766333, 76664D, 766FC3, 76CCDD, 76DD4D, 770G07, 771JEJ, 7771EJ, 777481, 7777I1, 777F81, 77C7JJ, 77EC77, 77EEC7, 77EJJJ, 77GCG7, 77JECJ, 7A7E77, 7AAAG3, 7AAF63, 7AEEE7, 7C7GG7, 7CCCAD, 7CCCG7, 7CGJJJ, 7D000D, 7J000D, 7JCGGJ, 80005B, 8000G1, 8000I1, 800GA1, 800I01, 80GGA1, 84405B, 84454B, 888EAH, 88E8AH, 8A4441, 8B8EAH, 8E8IF9, 8I0AG1, 902229, 909A09, 909C29, 909FA9, 90IC29, 9222I9, 92I999, 944EE7, 94AEE7, 977GG7, 97AEE7, 97GGG7, 990I29, 999I29, 99I299, 99IIA9, 9A9447, 9A9AF9, 9A9FF9, 9CII29, 9EEEE7, 9FA9A9, 9I9C29, 9I9I29, 9ICI29, 9IIIA9, A1555J, A1E55J, A333C9, A336A3, A33889, A33F63, A36663, A444E7, A4AAE7, A555FJ, A666A3, A6AAA3, A7A7E7, A7AE77, A944E7, A9A9F9, A9AAA9, AA3389, AA9AF9, AAA099, AAA3C9, AAA7E7, AAA939, AAACG9, AAAFF9, AAAI99, AAAIC9, AAE4E7, AAG9A9, AAGAA3, ADE55J, AF6663, AF9FF9, AGA963, AGG6A3, AH4441, AI000H, AI99C9, AI9AA9, AII099, B0004B, B0054B, B0080B, B00H0H, B04IIB, B05B8B, B0A6I1, B0BB8B, B0BIBB, B0E0IH, B0HA0H, B0I0HH, B0I4IB, B0IIBB, B888AH, BB058B, BB05BB, BB080B, BB0I4B, BBB04B, BBB08B, BBIB8B, BE88AH, BEHA0H, BEHHAH, BHEA0H, BHEHAH, BHHA0H, BIE00H, C007A7, C00GG7, C00JA7, C00JAJ, C00JC7, C0C7A7, C0JAC7, C0JJC7, C3G999, CAJJ77, CC0629, CC0929, CCC007, CCCCC7, CCCCG3, CCCGG7, CCCI29, CCI029, CD4GGJ, CE0009, CEFF69, CEFFF9, CEI609, CII029, CIII29, CJACC7, CJAJC7, CJJJA7, D000D7, D000J7, D004CD, D0D007, DCCDC7, DDD04D, DDDAI1, DEJJJJ, DG000D, DG5E5J, DGGG5J, DJ00CJ, DJEEE7, DJJJ5J, E00097, E00A0H, E00AA9, E02289, E028I9, E029I9, E055AJ, E0774J, E0777J, E07JCJ, E0AF5J, E0BI8B, E0E4CJ, E0EA5J, E0ECJJ, E0FFF9, E0H0HH, E0HA0H, E0IF29, E0J5AJ, E0JJAJ, E29299, E444E7, E4EEE7, E4IIIB, E66289, E66629, E66909, E66F69, E69609, E69999, E7774J, E77C77, E7C7C7, E8B8AH, E90029, E90229, E90909, E90FF9, E92299, E94EE7, E97EE7, E990F9, E99I09, E9F669, E9I029, EA8AF9, EAAAI9, EAIAA9, EAJJJJ, EBIBIB, EC7CC7, EE08F9, EE408B, EE8IF9, EEAF5J, EEE84B, EEEC07, EEEC77, EEEE0B, EEF55J, EEFFFJ, EEJ5AJ, EFE009, EFEF09, EFF669, EFFFF9, EGGGG9, EIF669, F004AJ, F00A41, F00JA1, F04481, F055CJ, F0A15J, F0EI09, F0G01J, F0J0GJ, F0J0JJ, F0JJ5J, F14441, F68F89, F68IF9, F800G1, F8E8I9, F99299, FA0009, FA1JJJ, FAAIA9, FAII09, FAII99, FCC929, FCE229, FCE609, FCEF29, FE68F9, FE8FA9, FF0E09, FF68F9, FF9FA9, FFC4AJ, FFC929, FFCE09, FFE009, FFEE89, FFEF09, FFFA99, FFFFFJ, FFFGJJ, FG1JJJ, FI99A9, FIA999, FIAAA9, FIAII9, FIIA09, G00093, G000EJ, G000G9, G00781, G007G3, G00C09, G07GC3, G09033, G0C903, G0CGG3, G0G903, G0G933, G0GCC3, G0GGC9, G30003, G36003, G5000J, G5E0EJ, G60303, G6AGG3, G6GAA3, G7AAA3, G7G363, G90AA9, G90C63, G9AAA9, GAAAA3, GAAGG3, GC0003, GCCC93, GCCCC3, GCG903, GDEEE7, GE0007, GG0CG3, GGA8G1, GGC6G3, GGCCG3, GGCGC9, GGG0I9, GGG363, GGG6G3, GGG75J, GGG8G1, GGG963, GGGAA3, GGGCG3, GGGCG9, H044EB, H0BIBB, H0E00B, H0F011, H0IIEB, H2444B, H400EB, H44IEB, H4EEEB, HBB4IB, HE000B, HEE40B, HEEE4B, HEHHAH, HF0011, HF4441, HH3HEH, HHGAA3, HHGAG3, HHHAG3, HHHEAH, HI0001, HIBIBB, HIEBBB, I0000H, I000A9, I002HH, I00A09, I00A99, I00AG1, I00AI9, I00H3H, I02999, I03HEH, I09AF9, I0A009, I0E0HH, I0I299, I0IIA9, I4445B, I4I45B, I90I29, I99029, I99A09, I9I029, IA3999, IB000H, IB00EH, IB0HEH, IB5B8B, IF2999, II0929, II92I9, II99A9, II9C29, IIAFF9, IIF299, III5BB, III8BB, J0000B, J00081, J000IB, J0010D, J006A1, J00BI1, J00I8B, J00IA1, J05EEJ, J06G81, J0C0JJ, J0E0CJ, J0EEEB, J0J0EB, J5EAEJ, J5EEAJ, J608I1, JE505J, JEEEEB, JJ0001, JJ0JCJ, JJ0JGJ, JJAJC7, JJJAE7, JJJBI1, JJJI8B, 14GGGG1, 1J5555J, 1JGGGG1, 205555H, 20I000H, 222222B, 2929999, 2BI000H, 3333A33, 3333G09, 333A3A3, 333AAA3, 336AAA3, 336AF33, 33II3A3, 363AF33, 3GIIII9, 400445B, 404454B, 404800B, 415555J, 440045B, 440080B, 442222B, 444004B, 444422B, 444444B, 4444G47, 444GGG7, 444I4IB, 44AAA47, 44I44IB, 44I4IIB, 4AAAAA7, 4EEE4E7, 4I4454B, 5000AFJ, 525555H, 555552H, 555555J, 555585H, 58BBBBB, 5BB8BBB, 5E0E55J, 5EBBBBB, 60003A3, 60008I9, 6000DDD, 6006DDD, 60080A1, 60080I1, 600A3A3, 600D8I1, 600F841, 60CCDDD, 60CGGG3, 60DDD0D, 60DDDDD, 60GGG63, 60GGGC3, 66600A3, 66603A3, 66608I9, 6666089, 66666A3, 6666C29, 6666F89, 666AFF3, 66DDDDD, 6800001, 6AGGGG3, 6CCCC29, 6G6AAA3, 6GAGGG3, 6GCCC63, 6GGG633, 700000D, 70000CD, 7000EC7, 70070C7, 700EEC7, 7070EC7, 70EEEC7, 76666CD, 77333A3, 7733A33, 775EE5J, 777A777, 777CGJJ, 777GE5J, 77A7777, 77JCJJJ, 77JJJCJ, 77JJJJJ, 7A77777, 7A777A7, 7AAAA63, 7AAAAAD, 7C00007, 7E77JCJ, 7EE7JCJ, 7EEC007, 7EEEEC7, 7G5E55J, 7GE555J, 7GJJJJJ, 7J6666D, 9000029, 9000299, 9009I29, 900I2I9, 90FAAA9, 90I0299, 90I0I29, 90I9929, 90II029, 90II299, 90II2I9, 94444E7, 97777E7, 9902I99, 99902I9, 9992929, 9992999, 9999929, 99I0C29, 99I9029, 99I92I9, 99III29, 9AAAF09, 9AFFFF9, 9I0AAA9, 9IAAA09, 9II0I29, 9II9929, 9III029, 9III2I9, 9IIIC29, A5F555J, A777E77, A994447, A9AAF99, AA00089, AA6A6A3, AAA0009, AAA3999, AAA9A99, AAA9AA9, AAAAA47, AAAAA77, AAAAA99, AAAAG99, AAAAIA9, AAAD447, AAAGAC9, AAGAAAD, AAGAAC9, AF00009, AGAAA99, AGAGG63, AGGGA63, AI0C999, AJ777A7, B000B8B, B000HEH, B000I0H, B00I00H, B0B805B, B0HHHEH, B0IHE0H, B0III8B, B8B000B, BB8BBBB, BBB8B0B, BEH0HHH, BHHHEHH, C00000J, C000C07, C066629, C0AJJJJ, C0C0A47, C0G000J, C0JAJJJ, C0JJAJJ, C0JJJJJ, C6CCG33, CC004A7, CC00A47, CC6CG33, CC92229, CCCC7G7, CCCGGG3, CE66609, CEF6669, D00007D, D00007J, D00071J, D000C0J, D000I01, D00700D, D00C007, D00C0C7, D00G00D, D0C00C7, D0CDCC7, D444447, D7JJJJJ, DC0000J, DDD7DDD, DDDCCC7, DJ0000J, DJ000EJ, DJJJJ0J, DJJJJJJ, E000009, E00084B, E000889, E000909, E000HAH, E008AF9, E04000B, E0AA8F9, E0CJJJJ, E0HHHHH, E40400B, E6660F9, E666609, E669969, E699669, E7CCC07, E996669, E99I669, E9FAAA9, EB8BBBB, EBBII8B, EBH0HHH, EE092I9, EE44IIB, EE4E4E7, EEBB8BB, EEC0007, EEE0IBB, EEE0IIB, EEEBB8B, EEEE44B, EEJCJJJ, EF00F09, EF0555J, EFAAAA9, EFF0009, EFF00F9, EFF60F9, EI0AAA9, EI660F9, EIIBB8B, EJ0CJJJ, EJ5055J, EJJ0CJJ, F0E00F9, F0E88I9, F0JJJJ1, F299929, F4G555J, F5555CJ, F5555FJ, F6668I9, F8GGGG1, F929999, F999C29, FA84441, FAAA9A9, FCE00F9, FCEF009, FCFEE29, FE00F09, FEE2889, FEF0009, FFAA9A9, FFCEE29, FFFF4AJ, FIII9A9, FIIIAI9, FJJJJ81, G000009, G0000J7, G000303, G000AA9, G000CC3, G000GG3, G003003, G0090C3, G00CG03, G00D007, G00G0G3, G00GG03, G00GG93, G080001, G0G00I9, G0G09C3, G0G0C93, G0G0GG9, G0GG093, G0GGC63, G0GGGG9, G44GG47, G6AAGA3, G999999, GAA9AA9, GCCCGG9, GCGGCC9, GCGGG03, GEE4447, GEEEE47, GG0C007, GG0GC09, GG0GGG3, GG6AAA3, GG6AGA3, GG6GAG3, GGEEE47, GGG00C9, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 21===

12, 18, 1A, 1G, 1K, 21, 25, 2B, 2H, 2J, 34, 38, 3A, 3G, 3K, 45, 4D, 4H, 4J, 52, 54, 58, 61, 65, 6B, 6D, 72, 74, 7A, 7G, 7K, 85, 8B, 8D, 92, 94, 98, 9A, A1, AD, AH, AJ, B2, B8, BA, BK, C5, CB, CH, CJ, D4, D8, DA, DK, ED, EH, EJ, F2, FG, G1, GB, GD, GH, H2, HA, HG, I1, I5, IB, IJ, J2, JA, JK, K1, KB, KD, KJ, 111, 115, 11H, 135, 13H, 13J, 151, 15B, 15H, 164, 175, 17B, 17D, 17J, 191, 19B, 19D, 19H, 1B1, 1B5, 1BB, 1BJ, 1D5, 1DD, 1DJ, 1E4, 1F1, 1F5, 1FD, 1FH, 1HB, 1HD, 1I4, 1JD, 1JH, 1JJ, 22D, 232, 29G, 29K, 2DG, 2F4, 2FK, 31H, 322, 33D, 351, 355, 35B, 35J, 362, 371, 37B, 37D, 37H, 37J, 39B, 39J, 3B5, 3BD, 3BH, 3D1, 3D5, 3DB, 3DD, 3DH, 3E2, 3FJ, 3HD, 3HH, 3HJ, 3J1, 3JB, 3JJ, 414, 481, 49K, 4B4, 4BG, 4C1, 4EB, 4F4, 4F8, 4FA, 4FK, 50G, 51B, 51D, 51H, 531, 535, 53D, 53J, 551, 56A, 56G, 56K, 575, 57J, 595, 59H, 5B1, 5B5, 5BB, 5CA, 5CG, 5CK, 5F1, 5FB, 5FJ, 5GA, 5GG, 5HH, 5IA, 5J5, 5JD, 5JH, 60H, 632, 66H, 66J, 678, 68J, 69G, 6F8, 6FA, 6GJ, 6H8, 6HK, 6IH, 6J4, 6JG, 6KH, 711, 71B, 71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, GGJ, GJ4, GK5, H0K, H15, H1B, H1J, H31, H3D, H3H, H48, H51, H55, H5J, H6K, H75, H84, H88, H91, H95, H9D, H9H, HBD, HC4, HC8, HDJ, HF5, HFB, HFH, HHD, HHJ, HI4, HI8, HJ5, HJB, HKK, I0D, I2D, I6H, I78, I8H, I9K, IFA, IFK, IID, J08, J1J, J31, J35, J3J, J44, J5H, J68, J6G, J71, J7B, J7D, J7H, J8G, J95, J9D, JBD, JBH, JBJ, JCG, JDB, JDH, JE4, JE8, JEG, JF5, JFD, JFJ, JG4, JGG, JH1, JH5, JHB, JHH, JI4, JJ1, JJ5, K0H, K5G, K6H, K9K, KG5, KH4, KKH, 101B, 103D, 1044, 1055, 10DH, 10FB, 10H1, 10H5, 10J1, 110J, 11CD, 11EB, 11ID, 130D, 13C1, 13EB, 141B, 1444, 1471, 14FB, 1505, 15CD, 167H, 16FJ, 16HH, 176H, 17C1, 17IH, 190J, 1BF4, 1BID, 1C04, 1C3D, 1CD1, 1D6H, 1DE1, 1DEB, 1E1B, 1E71, 1E95, 1EFB, 1F0B, 1F4B, 1FB4, 1H01, 1H05, 1H41, 1H6H, 1H6J, 1HC1, 1HE1, 1HF4, 1IDH, 1IHH, 1JE5, 1JF4, 20CD, 20D2, 20ID, 2278, 22F8, 239D, 23FD, 2478, 2708, 27FD, 293D, 29DD, 2A78, 2C0D, 2CF8, 2F0A, 2F9D, 2FAA, 2FDD, 2GFA, 2IF8, 3015, 301D, 301J, 3031, 3035, 305D, 30F1, 30FB, 30FD, 30HB, 30I2, 3105, 31C1, 329D, 330H, 336J, 33E1, 33EB, 33IH, 35CD, 363H, 390H, 39E5, 39ID, 3B0J, 3B6J, 3BC1, 3BEB, 3CC2, 3CF1, 3CFD, 3D32, 3E3B, 3E91, 3EF1, 3EFB, 3F05, 3FC1, 3FE1, 3FEB, 3H01, 3H0B, 3HC1, 3I3H, 3I9D, 404B, 4078, 40E1, 411B, 4131, 4171, 41FB, 4331, 4391, 43BB, 43FB, 43I2, 4432, 444B, 44E1, 477B, 4931, 4971, 499B, 49CG, 4A32, 4A4B, 4A78, 4A9G, 4B1B, 4B31, 4B3B, 4BFB, 4C78, 4E32, 4F1B, 4FB1, 5011, 509D, 50BD, 50DB, 50DD, 50EK, 50FD, 50H1, 50HJ, 50IK, 50JJ, 5101, 5105, 51E1, 530B, 53EB, 53IH, 550D, 550H, 55AB, 55BG, 55DG, 55E5, 55KH, 563H, 56JJ, 5701, 570H, 57EB, 57ID, 57IH, 590D, 59C1, 59CD, 5A3B, 5A7B, 5AGK, 5AIG, 5AKG, 5B0D, 5B9G, 5BCD, 5BGJ, 5BIH, 5C91, 5CD1, 5D0D, 5D0J, 5D6H, 5D6J, 5DC1, 5DG5, 5DIH, 5DJG, 5E0A, 5E11, 5E3B, 5E71, 5EAK, 5EEA, 5EKK, 5F0H, 5F9K, 5G0K, 5G9J, 5GF5, 5GIK, 5H0B, 5H5K, 5H9K, 5HE5, 5HID, 5I3H, 5I7D, 5IKG, 5J6J, 5JE1, 5KAG, 5KEK, 5KFH, 5KH5, 60FK, 63FH, 67JH, 69JH, 6E9K, 6F0K, 6F64, 6FJH, 6H04, 6HE4, 6HFJ, 6IF4, 6J9H, 6J9J, 7015, 7031, 703B, 7055, 705J, 7088, 709B, 709D, 709H, 70DJ, 70I8, 710J, 716J, 730B, 73C1, 73E5, 7501, 769H, 76FJ, 76JJ, 770D, 77E1, 77E5, 77EB, 7888, 78FH, 78H1, 78JH, 790H, 79CD, 79IH, 7B0J, 7B6H, 7C31, 7C91, 7C9D, 7E35, 7E75, 7E91, 7EF1, 7F01, 7F78, 7FCD, 7FH8, 7FIH, 7FJ8, 7H81, 7HC1, 7HF8, 7J6J, 7J78, 7JC1, 7JF8, 7JH8, 7JJ8, 809K, 80H8, 81H1, 81HH, 81J1, 839H, 83FH, 869K, 871H, 8731, 8771, 8788, 879J, 87F1, 87FJ, 87JH, 888H, 8911, 8931, 896K, 8A78, 8A9K, 8CC1, 8E9K, 8F3H, 8F9H, 8FF1, 8FHH, 8FJH, 8G9K, 8GJ8, 8H3J, 8H7H, 8H8K, 8HEK, 8HF1, 8HFJ, 8HH1, 8HJJ, 8HK8, 8J7J, 8J91, 8JC8, 8JFH, 8JJH, 8K78, 8K8H, 8KH8, 901J, 9035, 903B, 9071, 90CG, 90GG, 90JD, 9101, 91E5, 9301, 93E1, 93EB, 95CD, 95DG, 95FK, 967J, 969H, 96HH, 96HJ, 96JJ, 9705, 97C1, 9905, 995K, 999G, 99CD, 99EB, 99K5, 9B5G, 9BE5, 9BID, 9C0G, 9C11, 9C71, 9C9D, 9CEG, 9CGG, 9D05, 9D5G, 9DDG, 9DE5, 9E35, 9E3B, 9E71, 9EB5, 9EEG, 9EF1, 9EFB, 9F0J, 9F6H, 9F9K, 9FC1, 9FIH, 9FKH, 9G35, 9G95, 9GIK, 9GJJ, 9GKG, 9H0H, 9H5K, 9H6H, 9HKH, 9I6K, 9I9H, 9IDD, 9J6J, 9JEB, 9JGJ, 9JID, 9JJG, 9K35, 9K95, 9KEG, A33B, A3F5, A432, A478, A53B, A5IK, A788, A795, A878, A975, A99B, A9CG, A9GG, AAAB, AB44, AB55, ABFB, AC32, AC78, ACFA, AE32, AE5A, AE9K, AEEB, AEFA, AEFK, AF75, AFCA, AFCK, AFEA, AFFB, AG9G, AGFA, AI32, B064, B091, B09H, B0B1, B0D5, B0DH, B0EG, B0FB, B0FH, B0H5, B0HD, B114, B141, B14B, B1B4, B1E1, B411, B431, B444, B4B1, B59G, B5G5, B5JG, B67H, B67J, B69H, B6FH, B6FJ, B6HH, B6JH, B96H, B9DG, B9G5, BB14, BBB4, BC1D, BC31, BCDD, BD05, BD0H, BD6H, BDC1, BDE5, BDIH, BE0G, BE31, BEBB, BEE4, BEIG, BF0B, BF0H, BF6J, BFEB, BFIH, BG0G, BG55, BG95, BGG4, BGGG, BGIG, BGJ5, BH0B, BHEB, BIEG, BIGG, BIHH, BJ01, BJ0J, BJC1, BJF4, BJID, C041, C081, C09G, C0E1, C0FK, C104, C131, C171, C1C4, C1D1, C2F8, C2FA, C39D, C3F1, C478, C6FK, C7FD, C8C1, C90G, C911, C971, C99D, CAFK, CC78, CC81, CCE1, CCF8, CD02, CD31, CDFD, CE01, CE32, CF0A, CF48, CF64, CF68, CF6A, CF91, CFAK, CFDD, CFEA, CFEK, CIF4, CK32, CK9G, CKFA, D01D, D01H, D07B, D0DB, D0F1, D0FD, D0HD, D0HH, D10D, D10H, D16H, D1EB, D1IH, D23D, D3E1, D56H, D63J, D6CG, D6IG, D70B, D70D, D70J, D99G, D9GJ, D9ID, D9JG, DBC1, DBCD, DBE1, DBJG, DC11, DC9D, DCF1, DCFD, DCGG, DCIG, DD0H, DD6J, DDGJ, DDID, DDJG, DE31, DE75, DFC1, DFID, DFIH, DG22, DG6G, DG75, DG9J, DGCG, DGIG, DH6H, DHE5, DI6G, DICG, DIDH, DIFD, DIFH, DIGG, DJCD, DJDG, DJG5, E032, E05A, E0EB, E11B, E2F8, E31B, E331, E375, E395, E3C2, E3F1, E3F5, E3FB, E3I2, E53B, E555, E591, E59B, E5EG, E6F4, E755, E775, E89K, E915, E935, E93B, E96K, E9FB, EA32, EAEB, EAFK, EB44, EB95, ECFA, ECFK, EE32, EEEB, EEFK, EF08, EF35, EF48, EF55, EF64, EF68, EF75, EF95, EFB1, EFCA, EFCK, EFEA, EFIA, EFKK, EGF4, EIF4, EKF4, EKF8, EKFA, F008, F051, F075, F095, F09H, F0CA, F0CK, F0DD, F0FB, F0H1, F0IA, F0JJ, F105, F10B, F10D, F16J, F1B4, F1EB, F1ID, F1J4, F36H, F468, F47B, F491, F4C8, F501, F50B, F648, F66A, F69H, F705, F778, F864, F8F1, F90D, F99K, F9C1, F9HK, FA6A, FA75, FA7B, FA95, FAEA, FAFB, FB0D, FB14, FBCD, FBE1, FBJ4, FC1D, FC68, FC9D, FCAA, FCCK, FCD1, FCEA, FD6H, FDIH, FE0K, FE1B, FE64, FE95, 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====Additional known quasi-minimal primes (not necessarily the next)====

40&lt;sub&gt;47333&lt;/sub&gt;9G, CF&lt;sub&gt;479147&lt;/sub&gt;0K

===Base 22===

11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, IHH, IIJ, IJ7, IL7, J07, J55, J69, J8J, J99, J9J, JA5, JAH, JE7, JEH, JHH, JI9, JJ5, JJ9, JJH, JK7, K2J, K51, K5D, K75, K81, KA1, KB1, KB7, KBJ, KD1, KEJ, KG5, KIF, KJ5, KKD, KKJ, L0D, L47, L7F, L97, LAF, LD3, LD9, LDD, LEF, LGD, LI7, LJ7, LJJ, LLD, 104H, 10D5, 1205, 12B5, 140H, 1433, 144H, 14AH, 14B3, 16ED, 1AIH, 1B43, 1DD5, 1DDD, 1E6D, 1EGD, 1G05, 1GDD, 1GED, 1GGD, 1HB3, 1HHH, 1IAH, 200L, 2015, 2051, 20A1, 20DJ, 20GD, 20IL, 21B5, 21DD, 220D, 226D, 228D, 22B5, 22G5, 22K5, 22KD, 2555, 2557, 2581, 25C1, 26A1, 26B1, 2725, 2755, 2801, 2861, 288D, 28B1, 28KD, 2AA1, 2B25, 2B51, 2BB1, 2C81, 2D6D, 2DA1, 2DDJ, 2DGD, 2G0J, 2GB5, 2GDD, 2GGJ, 2I0L, 2I6L, 2ICL, 2J05, 2JK5, 2K07, 2K2D, 2K55, 2K6D, 2KB5, 2KI7, 2L2D, 2L8D, 2LK7, 302L, 30G3, 320L, 32IL, 332L, 33G3, 33G9, 36AF, 36EF, 382J, 388J, 39EL, 3AAF, 3BBJ, 3CG9, 3E2J, 3E6F, 3E6L, 3EEF, 3FAF, 3G69, 3GI3, 3GI9, 3IG9, 3LEL, 3LG3, 4025, 404H, 4063, 4075, 40AH, 40B5, 40B7, 40GH, 4225, 4363, 4447, 444H, 447H, 4487, 44B7, 44H7, 4525, 4555, 45B5, 4663, 4777, 47GH, 4807, 4B0H, 4BB5, 4BG3, 4EBH, 4G4H, 4GB3, 4HH3, 4I03, 4I63, 4IGH, 5069, 5077, 50KD, 5255, 52IJ, 5455, 5477, 5507, 5527, 556F, 557F, 5587, 56EF, 56FF, 56GD, 56KD, 5771, 57C1, 57EF, 5807, 580J, 589J, 58E7, 58IJ, 58K7, 5905, 5945, 5957, 5969, 5989, 598J, 5999, 59EJ, 59K7, 5AEF, 5B25, 5B55, 5BA5, 5BI7, 5BIJ, 5BK5, 5BK7, 5C21, 5EAF, 5EE7, 5F09, 5F6F, 5F95, 5FKD, 5G6D, 5G89, 5I09, 5I0F, 5I27, 5I6F, 5I89, 5IB7, 5IBJ, 5IFF, 5J77, 5K07, 5K25, 5K27, 5K6F, 5K87, 5KK7, 6013, 60A1, 60AF, 60EL, 6281, 62B1, 62KD, 63G9, 6403, 6643, 66EL, 66G9, 68KD, 69B3, 6A01, 6A0F, 6AAF, 6AFF, 6B21, 6B8D, 6BBD, 6BK1, 6D03, 6D43, 6D69, 6D6F, 6D93, 6D9F, 6E21, 6E81, 6E8D, 6EBD, 6ECL, 6ELL, 6FD9, 6GBD, 6GG9, 6IEF, 6K21, 6K8D, 6KBD, 6KE1, 6L43, 6LB3, 6LDF, 6LEL, 6LG9, 700F, 7027, 7057, 70EH, 70LF, 710H, 71G5, 7207, 7225, 7255, 727L, 7505, 7507, 755F, 75EF, 75F5, 75I7, 766F, 7681, 76CF, 7771, 7781, 77IL, 7861, 7A0H, 7A4H, 7BB5, 7C6F, 7EEH, 7EHL, 7FG5, 7G45, 7HEL, 7I0L, 7I27, 7IE7, 7IGH, 7K07, 7K61, 7K77, 7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, H0D1, H0HJ, H0LL, H1HH, H20D, H22D, H26D, H2I7, H2K7, H30L, H3KJ, H40H, H447, H4G3, H4H3, H4HH, H66L, H6ED, 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5IIIIIIIIIIIIIIIIIIIIIIIIIIF, D555555555555555555555550555, EEAAAAAAAAAAAAAAAAAAAAAAAAAF, HHHHHHHHHHHHHHHHHHHHHHHHEEBH, K66666666666666666666666666F, LLLLLLLLLLLLLLLLLLLLLLLLEEB7, D5555555555555555555555555A55, GGGGGGGGGGGGGGGGGGGGGGGGGGGG3, GIG0000000000000000000000000H, HH00000000000000000000000001H, K0000000000000000000000005KEF, 5BBBBBBBBBBBBBBBBBBBBBBBBBBBBD, HB0000000000000000000000000001, K000000000000000000000000505EF, L7777777777777777777777777772L, 2000000000000000000000000000CB1, C8CCCCCCCCCCCCCCCCCCCCCCCCCCCC9, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKFF, JE0000000000000000000000000000J, K000000000000000000000000000261, A0000000000000000000000000004I4H, HD000000000000000000000000000001, K000000000000000000000000000EC01, K0FFFFFFFFFFFFFFFFFFFFFFFFFFFFCF, D0002222222222222222222222222222D, FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL2L, I700000000000000000000000000000GH, K00000000000000000000000000000E61, 20000000000000000000000000000000JJ, DD5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL0KE7, 44HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, F0000000000000000000000000000000000000000000000000000000L89, J000000000000000000000000000000000000000000000000000000JJIJ, K222222222222222222222222222222222222222222222222222222222DD, KCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCF, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKCF, D55KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, K000000000000000000000000000000000000000000000000000000000000077, 22222222222222222222222222222222222222222222222222222222222222222D, K000000000000000000000000000000000000000000000000000000000000008IJ, DFA00000000000000000000000000000000000000000000000000000000000000005, CC4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJE0J, 777777777777777777777777777777777777777777777777777777777777777777777777EK7, 6000000000000000000000000000000000000000000000000000000000000000000000000000000043, KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKB5, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEEH, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 50000000000000000000000000000000000000000000000000000000000000000000000000000000002C1, K0000000000000000000000000000000000000000000000000000000000000000000000000000000000055EF, H700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000K1, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0I7, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, A400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, E0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000071, 7LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIL, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLK77, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEI7, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIB3, I7G00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 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77EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEK7, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJKJ, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777772L, BKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK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KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK5

===Base 24===

15, 17, 1D, 1H, 1J, 1N, 25, 2B, 2D, 2J, 2N, 31, 37, 3B, 3H, 41, 45, 47, 4B, 4D, 4H, 57, 5B, 5H, 5J, 65, 67, 6D, 6J, 6N, 75, 7B, 7D, 7N, 81, 85, 87, 8J, 97, 9B, 9D, 9H, 9N, A1, AB, AH, AN, B5, B7, BD, BH, BJ, C5, CJ, CN, D1, D5, DJ, E1, EB, ED, EH, EN, F7, FD, FJ, FN, G5, GD, GH, H1, HB, HD, HN, I1, I7, IB, IH, J1, J5, J7, JB, JN, K7, KB, KJ, KN, L5, LH, LJ, MD, MJ, N5, NB, NH, NJ, 101, 10B, 111, 1F1, 1FB, 1GB, 1LB, 201, 221, 22H, 261, 271, 277, 28H, 2A7, 2C7, 2G7, 2H7, 2L1, 2L7, 2MH, 305, 30D, 30J, 33N, 34N, 35D, 35N, 38D, 395, 3A5, 3AJ, 3CD, 3DD, 3DN, 3E5, 3EJ, 3GJ, 3IJ, 3JJ, 3K5, 3KD, 3ND, 43N, 44N, 49J, 4EJ, 4GJ, 4GN, 4NN, 50N, 535, 54N, 551, 55N, 5C1, 5CD, 5E5, 5K1, 5KD, 5LN, 5M5, 5N1, 601, 60B, 61B, 66H, 68B, 691, 6CH, 6FH, 6GB, 6HH, 6MH, 70H, 70J, 711, 761, 771, 77H, 77J, 78H, 7C7, 7CH, 7FH, 7G7, 7H7, 7HH, 7IJ, 7JJ, 7K1, 7M1, 7M7, 80D, 82H, 83N, 88D, 88H, 8AD, 8CD, 8DB, 8DD, 8DH, 8DN, 8GB, 8KD, 8MB, 8MH, 905, 911, 921, 935, 955, 99J, 9AJ, 9G1, 9JJ, 9K5, 9L1, 9M5, A0J, A3J, A95, AA7, AD7, AE5, AG7, AGJ, AI5, AIJ, AJD, AL7, ALD, B01, B0N, B11, B61, B6B, B8N, B91, BIN, BL1, BLN, BNN, C1B, C21, C27, C2H, C3D, C61, C8H, C91, CA7, CB1, CBB, CC7, CCB, CCD, CDD, CFB, CG1, CGB, CK1, CL1, CMB, CMH, D0B, D3D, D3N, D4N, D6B, D6H, D7H, D8B, D8N, DAD, DCD, DCH, DDH, DDN, DG7, DGB, DID, DMN, DND, E05, E4J, EA7, EEJ, EF5, EGJ, EI5, EJJ, EM5, EM7, F01, F21, F51, F8H, F95, FC1, FF1, FFB, FKH, FM5, G0B, G0N, G11, G3N, G6B, G77, G7J, G8B, G8N, G91, GA7, GBB, GC7, GFB, GG1, GGJ, GGN, GK1, GL1, GLN, GMN, GN1, GNN, H0J, H2H, H3J, H4J, H77, HA5, HA7, HE5, HFH, HIJ, HJJ, HKH, HL7, HMH, I0N, I3D, I3J, I3N, I4N, I5D, I95, IA5, IAJ, IE5, IEJ, IF5, IGJ, IJD, IK5, IKD, J0D, J4J, J8D, JAD, JDH, JEJ, JFH, JHH, JKD, JMH, K35, K6H, KCD, KFH, KH5, KLD, KM1, L01, L0B, L0D, L0N, L61, L6B, L8D, LA7, LC7, LDD, LF1, LG7, LGB, LGN, LID, LK1, LKD, LL1, LLB, LLD, LMN, LNN, M0H, M11, M21, M4N, M71, M91, M95, MA5, MA7, MBN, MC7, MF1, MF5, MFB, MFH, MG7, MI5, MIN, ML1, ML7, MLB, MMH, N01, N21, N4N, N71, N8N, NC1, ND7, NE7, NG1, NID, NK1, NL7, NMN, NN7, 11CB, 11MB, 1291, 12G1, 16C1, 16CB, 16K1, 186B, 18CB, 19K1, 1BK1, 1C8B, 1K91, 1KC1, 1KL1, 1L21, 1LC1, 1LM1, 1M61, 1M8B, 1MG1, 206H, 20CH, 20M7, 21C1, 21M1, 2207, 260H, 26KH, 2991, 2C6H, 2CC1, 2CM1, 2F11, 2FHH, 2MC1, 2MK1, 2MM1, 308N, 30GN, 30IN, 30LN, 30MN, 333J, 33JD, 33LD, 343J, 344J, 35I5, 380N, 393J, 394J, 3A3D, 3FI5, 3IMN, 3J3D, 3JID, 3L3D, 3L8N, 3M0N, 3M55, 3NGN, 404J, 408N, 40LN, 434J, 44AJ, 4ILN, 4JAJ, 4L8N, 5091, 5095, 50F1, 50I5, 51L1, 5211, 5291, 52G1, 53ID, 53MN, 5595, 55AD, 56F1, 588N, 58MN, 58ND, 5961, 5991, 5A5D, 5AAD, 5F91, 5GF1, 5GIN, 5I05, 5I55, 5I8D, 5IDD, 5IDN, 5IIN, 5IMN, 5KI5, 5M61, 5M8N, 5N3N, 602H, 6211, 62F1, 62G1, 66C1, 66FB, 66M1, 66MB, 6B21, 6BM1, 6BMB, 6C6B, 6CF1, 6CLB, 6FG1, 6K21, 6K2H, 6KG1, 6KKH, 6L21, 6LCB, 6LM1, 6MB1, 6MBB, 6MG1, 6MK1, 7001, 7027, 7207, 726H, 739J, 793J, 79C1, 7A4J, 7A9J, 7AE7, 7C01, 7CC1, 7FL1, 7G21, 7G9J, 7GAJ, 7GC1, 7HGJ, 7J2H, 7J6H, 7MKH, 800B, 800H, 804N, 806H, 808N, 80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, NK3D, NK8D, 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5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, 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EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEG7, 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===Base 30===

11, 17, 1B, 1D, 1H, 1N, 1T, 21, 27, 2B, 2D, 2J, 2N, 2T, 37, 3B, 3D, 3H, 3J, 3N, 47, 4B, 4H, 4J, 4T, 51, 57, 5D, 5H, 5N, 5T, 61, 6B, 6D, 6H, 6J, 71, 7D, 7H, 7J, 7N, 7T, 81, 8B, 8H, 8N, 8T, 91, 97, 9B, 9D, 9N, A7, AB, AD, AH, B1, B7, BH, BJ, BN, BT, C7, CD, CJ, CN, CT, D7, DB, DJ, DT, E1, EB, ED, EJ, EN, ET, F7, FB, FD, FH, FT, G7, GB, GJ, GN, GT, HB, HD, I1, I7, IH, IN, IT, J1, J7, JH, JN, JT, K1, K7, KD, KH, KJ, L1, LB, LD, LH, LN, LT, M1, MD, MH, MN, N1, NB, NJ, NT, O7, OD, OJ, ON, P1, P7, PB, PJ, PN, Q7, QH, QT, R1, RB, RD, RH, RJ, RT, SD, SH, SJ, SN, T7, TB, TD, TH, 10J, 15J, 1IJ, 1JJ, 1LJ, 1MJ, 1QJ, 22H, 29H, 2EH, 2GH, 30T, 331, 33T, 36T, 39T, 3A1, 3C1, 3G1, 3KT, 3MT, 3OT, 3S1, 3T1, 40D, 40N, 431, 44D, 46N, 48D, 4AN, 4DD, 4DN, 4F1, 4FN, 4GD, 4ID, 4PD, 4S1, 50J, 58J, 59J, 5CB, 5FJ, 5IB, 5IJ, 5JB, 5MB, 5MJ, 5OB, 5SB, 607, 63T, 687, 6E7, 6KT, 6L7, 6M7, 6MT, 6NN, 6QN, 6R7, 6S7, 6ST, 6TT, 70B, 77B, 787, 7KB, 7M7, 7MB, 7SB, 807, 80D, 80J, 84D, 85J, 877, 88J, 89J, 8DD, 8FJ, 8ID, 8IJ, 8JJ, 8M7, 8MJ, 8PD, 8QD, 8R7, 90H, 93T, 95J, 99H, 99J, 9AJ, 9AT, 9EH, 9GH, 9HH, 9HJ, 9JJ, 9MJ, 9OH, 9OT, 9PH, 9ST, 9TT, A01, A0T, A1J, A31, A6N, A6T, AAJ, AAN, AFN, AKN, AKT, ALJ, AMJ, AMT, AO1, AOT, AQ1, AQN, ARN, AT1, B5B, BBD, BCB, BDD, BID, BOB, BPD, BQB, C31, C9H, CC1, CCB, CCH, CF1, CH1, CIB, CKB, CMB, D01, D0H, D2H, D41, D4D, D6N, D8D, D9H, DDH, DDN, DGH, DH1, DHH, DID, DKN, DNN, DO1, DOH, DQN, DS1, EEH, EHH, EM7, EOH, EPH, ER7, F0N, F31, F5J, F8J, F9J, FLJ, FO1, FQ1, FQN, FS1, G01, G4D, G8D, GF1, GGH, GO1, GOH, GQD, GS1, H07, H0J, H0T, H1J, H2H, H31, H4N, H8J, HA1, HAJ, HAT, HC1, HE7, HEH, HFN, HGH, HH7, HIJ, HJJ, HKN, HL7, HNH, HPH, HQN, HS1, HTJ, HTN, I4D, I5B, I5J, I8D, IAJ, IDD, IGD, IIJ, IKB, IMB, IMJ, IOB, IPD, IQD, J8J, J9J, JAJ, JBD, JCB, JFJ, JIJ, JJB, JJD, JLJ, JPD, JQB, K3T, K4N, KAT, KBB, KCB, KMB, KNN, KOB, KOT, KQN, KST, KTT, L0J, L5J, L67, LAJ, LJJ, LQJ, LR7, M0J, M3T, M5B, M7B, M87, M9J, MAT, MFJ, MIJ, MJJ, MKB, MMJ, MOT, MQJ, N07, N0H, N67, N6N, N87, NAN, NDH, NGD, NHH, NKN, NN7, NNH, NPH, NQD, NQN, NR7, O01, O0B, O0H, OAT, OC1, OCH, OEH, OF1, OH1, OKT, OMB, OOT, OPH, OQ1, OQB, OS1, OST, P3T, P6T, P9H, PCH, PDH, PEH, PGD, PHH, PHT, PID, PMT, PPT, PQD, PST, PTT, Q5B, Q6N, Q9J, QAJ, QBB, QBD, QC1, QDN, QFJ, QFN, QGD, QJB, QKN, QLJ, QMB, QND, QNN, QO1, QQ1, QQN, QSB, R4N, R77, R87, RAN, RKN, RM7, RNN, RR7, RS7, S0T, S41, S6T, S87, SBB, SC1, SM7, SMT, SOB, SQ1, SR7, SS1, STT, T0J, T6T, T9T, TAN, TFN, TKN, TKT, TNN, TO1, TOT, TPT, TQ1, TQN, TTN, 18AJ, 19FJ, 1AFJ, 1FAJ, 20CH, 20PH, 2C0H, 2CHH, 2COH, 2H0H, 2HCH, 2HHH, 2POH, 2PPH, 3001, 34Q1, 3F41, 3FF1, 3QF1, 3SPT, 3SST, 3TTT, 40A1, 4441, 44KN, 44QN, 4AC1, 4C01, 4CA1, 4DG1, 4GA1, 4GC1, 4NND, 4NRN, 4OA1, 4Q4N, 4QA1, 4QRN, 4RQN, 550B, 555B, 55KB, 5A5J, 5AJJ, 5BKB, 5JQJ, 5KKB, 5QQB, 5QQJ, 604N, 606T, 60FN, 60KN, 60OT, 60PT, 660N, 660T, 664N, 66AT, 66FN, 66KN, 66TN, 6A0N, 6A9T, 6AAT, 6FKN, 6FRN, 6K0N, 6KFN, 6KKN, 6O6T, 6PAT, 6RFN, 6RRN, 6T6N, 6TRN, 7067, 70R7, 75BB, 77R7, 7C5B, 7CQB, 7E67, 7IQB, 7O5B, 7OIB, 7QOB, 7R07, 7RE7, 8667, 88L7, 88S7, 8E87, 8EE7, 8EL7, 8J8D, 8LL7, 8LS7, 906T, 908J, 90FJ, 90QJ, 90TJ, 92CH, 966T, 99MT, 99PT, 9I0J, 9I8J, 9LFJ, 9LIJ, 9M6T, 9MKT, 9MMT, 9PKT, 9QQJ, 9TLJ, A04N, A0QJ, A3AT, A3ST, A4A1, A4G1, A5QJ, A8QJ, A90J, A9PT, AA3T, AA41, AAAT, AAF1, AAG1, AAPT, AC41, ACS1, AF0J, AFC1, AFIJ, AG41, AGA1, AGG1, AI8J, AIJJ, AIQJ, AJ5J, AJJJ, AQ8J, AQJJ, AQQJ, AS3T, AT4N, AT5J, ATST, B04D, B0QD, BBIB, BBKB, BI0B, BK0B, BKIB, BKSB, BMIB, BMSB, BQ4D, BQQD, BSMB, C00B, C05B, C0A1, C0BB, C0EH, C0GH, C0Q1, C0QB, C2OH, C2PH, C441, C4Q1, C50B, CB0B, CBSB, CEGH, CG2H, CGHH, CHHH, CHOH, COGH, COO1, COOB, COOH, CQ01, CQ0B, CQQB, CQS1, CS0B, D00N, D0PD, D44N, DA0N, DCEH, DCG1, DCQ1, DDC1, DDF1, DDPD, DDQ1, DECH, DF4N, DFA1, DFRN, DG0D, DGD1, DGQ1, DNEH, DPDD, DQA1, DQPD, DRFN, DRRN, E20H, E667, E767, E8L7, E8S7, ECGH, EG2H, EH67, EH77, ES67, F001, F01J, F0MJ, F0QJ, F44N, F4G1, F4RN, F6RN, FA0J, FAC1, FAF1, FAIJ, FC01, FCG1, FFAJ, FFC1, FFG1, FFIJ, FGA1, FJJJ, FJQJ, FKAN, FR6N, FRFN, G09H, G0EH, G0HH, G20H, G341, G3Q1, G4G1, G92H, GA41, GAC1, GC0H, GCEH, GCG1, GCPH, GD31, GDA1, GDG1, GDPD, GE2H, GE9H, GGG1, GGGD, GHCH, GHHH, GP0D, GP2H, GPPD, GQ31, GQA1, GQG1, H00N, H0CH, H3ST, H4O1, H667, H66N, H677, H69T, H99T, H9CH, H9KT, H9LJ, H9PT, H9QJ, HH6N, HHF1, HHFJ, HHH1, HHHT, HHLJ, HHOH, HHQ1, HHST, HK6T, HL9J, HLMJ, HMKT, HMMT, HNM7, HNRN, HOPT, HQ01, HQ5J, HR0N, HR6N, HS77, HSKT, HT01, I0BB, I0JD, I0JJ, I98J, I9LJ, I9QJ, ICBB, IF0J, IFFJ, IFQJ, IIBD, IJBB, IJID, IL8J, ILFJ, IQ8J, IQJJ, IQQJ, IS0B, ISQB, J00D, J05J, J0BB, J0MJ, J0OB, J4QD, J50B, J5QJ, JBKB, JBMB, JDGD, JDQD, JGDD, JI0D, JIIB, JISB, JKIB, JKKB, JM0B, JOIB, JQ4D, JQDD, JQID, JQJJ, JQMJ, JSIB, JSMB, K06N, K0KB, K0KT, K0MT, K0PT, K0SB, K5QB, K60T, K66T, K6AN, K6FN, K6PT, K6RN, K96T, K99T, KA0N, KF6N, KFKN, KI0B, KK0T, KK6N, KK6T, KKFN, KKKB, KKKT, KKQB, KKTN, KMMT, KQ0B, KQQB, KR0N, KS5B, L0M7, L8E7, L98J, LFFJ, LI8J, LIFJ, LL87, LL9J, LLIJ, LLM7, LM77, LML7, LMM7, M09T, M0E7, M0IB, M0L7, M0M7, M0TT, M55J, M5LJ, M60T, M69T, M707, M767, M777, M7E7, M7S7, M8LJ, M96T, M9KT, MC0B, MCOB, ME07, MIBB, MJ0B, MJBB, MJSB, MLLJ, MLS7, MM07, MM0T, MML7, MMOB, MMR7, MOBB, MOCB, MP0T, MP9T, MPKT, MQ0B, MQIB, MQOB, MS07, MSPT, MSSB, MT8J, MTTT, N00D, N04D, N04N, N0FN, N0ND, N0RN, N2OH, N92H, NCGH, ND0D, ND4N, NDPD, NEGH, NEL7, NF4N, NFRN, NGCH, NGEH, NHM7, NME7, NML7, NMM7, NN0D, NN4D, NN4N, NNDN, NNID, NNND, NNRN, NP4D, NSL7, NSS7, O00T, O03T, O0MT, O0PT, O2HH, O2OH, O341, O4A1, O4G1, O4O1, O5KB, O90T, OA41, OBSB, OC5B, OCOB, OG2H, OG31, OH9H, OH9T, OHHH, OHTT, OICB, OISB, OM0T, OM6T, OMPT, OMTT, OO2H, OO9H, OOA1, OOCB, OOGH, OOKB, OOO1, OSKB, OTMT, P00T, P04D, P08D, P09T, P0KT, P90T, PAAT, PKKT, PO0T, PP0D, PPPD, Q00B, Q0AN, Q0D1, Q0F1, Q0IB, Q0JJ, Q0MJ, Q0OB, Q55J, Q5QJ, Q88D, QA0N, QA41, QA4N, QDF1, QDPD, QF41, QFA1, QG31, QIQJ, QJ0J, QJ8D, QJDD, QJJJ, QJMJ, QJQD, QKKB, QKQB, QOKB, QOOB, QP0D, QPDD, QQ8D, QQID, QQJD, QQPD, QQQB, QS01, QSA1, R00N, R067, R06N, R0FN, R0L7, R60N, RE67, RFRN, RRFN, RRRN, S00B, S3AT, S3ST, S50B, S5QB, S99T, SAAT, SC0B, SE67, SGG1, SICB, SIQB, SK5B, SKIB, SKKT, SKPT, SKQB, SM0B, SMMB, SMSB, SOA1, SOG1, SPOT, SQCB, SQQB, SS0B, SS67, ST31, STG1, T00N, T03T, T041, T04N, T0AT, T0C1, T0MT, T0ST, T1AJ, T3F1, T4A1, T4RN, T5JJ, T66N, T8AJ, T9FJ, T9QJ, TA3T, TAA1, TAG1, TAIJ, TCS1, TFA1, TFAJ, TFF1, TFFJ, TFG1, TGA1, TI9J, TIJJ, TIQJ, TL8J, TL9J, TM0T, TM5J, TM8J, TMAJ, TMLJ, TMST, TSF1, TSG1, TT01, TT5J, TT8J, TTA1, TTC1, TTFJ, TTG1, TTLJ, TTMJ, 30441, 30O41, 30OO1, 34O41, 3AATT, 3ASAT, 3O441, 3Q041, 40G41, 40GQ1, 40O41, 40OG1, 40OO1, 44001, 440C1, 440G1, 4444N, 44CG1, 44GQ1, 4AAA1, 4COG1, 4DAA1, 4GQ41, 4KRRN, 4OGG1, 4Q041, 4QGG1, 5005B, 500BB, 500KB, 500QB, 50K0B, 50Q0B, 50QKB, 555AJ, 555LJ, 555QJ, 55J5J, 5B00B, 5B0BB, 5J55J, 5J5JJ, 5JJ5J, 5JJJJ, 5K00B, 5QJ5J, 5QK0B, 6006N, 6009T, 600AT, 6066N, 60R0N, 666N7, 6696T, 66OPT, 67767, 6900T, 696PT, 69P0T, 6F6AN, 6O9PT, 6OP0T, 6OP9T, 6P99T, 755QB, 77767, 7CBBB, 7IBIB, 7IIBB, 7IICB, 7QIIB, 8888D, 88E67, 8L887, 8SSE7, 9099T, 90KKT, 90KMT, 90MPT, 9690T, 96P9T, 990KT, 9990T, 9999T, 99K6T, 9FFFJ, 9HKMT, 9HMPT, 9ILLJ, 9KKMT, 9KKPT, 9M00T, 9T8QJ, 9TFIJ, A008J, A00NN, A055J, A0FJJ, A0I0J, A0I9J, A0J0J, A44C1, A555J, A9FFJ, A9QIJ, A9T8J, AFFA1, AFFFJ, AFFJJ, AFFQJ, AJ00J, AP99T, ASA9T, ASFF1, ASP9T, ASSAT, AT3TT, ATFQJ, ATQIJ, ATT9J, ATTQJ, B088D, B08GD, B0GGD, B0SKB, B8GGD, BG00D, BIISB, BISIB, BS0IB, BS0SB, BSSKB, BSSSB, C0041, C0G41, C0H0H, C0PPH, C0S01, C4001, C4AG1, C4OG1, C5BBB, CG00H, CGE0H, CGGA1, CGQ41, COSSB, CP20H, CPGPH, CPP2H, CPPOH, CPPPH, CS001, CS55B, CSSQB, D00GD, D0A4N, D0FAN, D0GDD, D0N0D, DAAA1, DAAC1, DAFG1, DDAA1, DDGGD, DFFAN, DFGC1, DGDGD, DGG31, DGGA1, DH0AN, DPPPH, DQDD1, DQGG1, E0L87, E88E7, EC00H, EC02H, EE867, EE887, EEL87, ELE87, F00IJ, F0441, F0AG1, F0F0J, F0F41, F0FF1, F0GG1, F4041, F44A1, F44C1, F4A41, F64KN, F6K6N, FAAA1, FAFQJ, FCAA1, FF041, FF0F1, FF64N, FF6KN, FFA41, FFA4N, FFF4N, FFFF1, FFFFN, FFKRN, FFQMJ, FIJ0J, FJ00J, FKKKN, FNNFN, FQQ0J, FQQMJ, G00DD, G00DH, G00PD, G02PH, G0DDD, G0PDD, GAAA1, GC2HH, GDDDD, GDGDD, GDPPH, GG0PD, GGCA1, GGCQ1, GGDD1, GH4Q1, GHHG1, GII0D, GQ441, H009H, H00G1, H00H1, H04Q1, H0ANN, H0H01, H0H9H, H0HG1, H0HO1, H0O41, H0OHH, H0QG1, H40G1, H4G41, H4GG1, H60AN, HAN0N, HF0G1, HFF41, HFFMJ, HFFQJ, HGQ41, HH3TT, HH401, HH441, HH4G1, HH55J, HH66T, HH6OT, HH96T, HHA0N, HHANN, HHC0H, HHCHH, HHH0N, HHHCH, HHHHN, HHHNN, HHKMT, HHO41, HHP9T, HHPKT, HHTMT, HHTT1, HK9MT, HKKMT, HLLFJ, HM66T, HM7R7, HMM67, HMM77, HMMM7, HMTST, HO9HT, HOO41, HOOOH, HOT3T, HOTT1, HPO9T, HSO3T, HTGG1, I000J, I009J, I00BD, I00QJ, I00SB, I08QJ, I0I0D, I0IQB, I0JIB, I0Q0B, I0SCB, I0SIB, I0SSB, IB00D, IBBSB, IBISB, ICSSB, II0CB, II0ID, II0SB, IIIBB, IIQ0B, IIQIB, IISIB, IJ0SB, IJJQJ, ILLLJ, IQ00J, IQC0B, IQIQB, IQQIB, ISSCB, J00MB, J08GD, J0IID, J0QQD, J55BB, J55JJ, J5J5J, J5JJJ, J88GD, JB0SB, JBB0B, JBBSB, JDDDD, JG0ID, JGG0D, JIIID, JJ0QJ, JJ5JJ, JQ08D, JQ0QJ, JQQ0D, JQQ5J, JS55B, JSK0B, JSS5B, K000T, K006T, K00TN, K0FAN, K505B, K6T0N, K9KMT, K9KPT, K9P0T, KIQIB, KK00N, KK05B, KK0AN, KK0KN, KKM9T, KS0IB, KS0QB, KSQIB, KTR6N, KTRRN, L0087, L08S7, LE087, LEE87, LL8LJ, LMEE7, LMSE7, LMSS7, M00CB, M00PT, M066T, M0CQB, M0K6T, M0MQB, M0MSB, M0QQB, M0SMB, M0SST, M666T, M900T, MBBSB, MEE77, MEL77, MELE7, MES77, MESE7, MESS7, MI00B, MIICB, MIISB, MIQCB, MK9PT, ML7L7, MLEL7, MLLE7, MLME7, MM677, MMBIB, MMBSB, MMCQB, MME77, MMEE7, MMICB, MMISB, MMK6T, MMKKT, MMM9T, MMMMT, MMMTT, MMQQB, MMSCB, MMSKT, MMSMB, MMSS7, MMTST, MOIIB, MOOIB, MOSIB, MQCQB, MR007, MR667, MRL07, MS7L7, MSEL7, MSK9T, MSL77, MSSL7, MT00T, MTMMT, N0DDD, N4NNN, N7LE7, N7S77, NE2CH, NE9CH, NEC2H, NEE77, NFFNN, NFNFN, NII0D, NL777, NL7L7, NLES7, NLLL7, NLS77, NOG9H, NRR0N, O6P9T, O9H6T, O9HPT, O9P9T, OCBBB, OG441, OGAG1, OGGA1, OHH6T, OHOOH, OKIIB, OKK5B, OKKIB, OO5BB, OOBIB, OOIIB, OSCSB, OT3TT, OT441, OTG41, OTGG1, P00PH, P0D0D, P0G2H, P0OGH, P0PGH, PA99T, PGP0H, POOOH, PP20H, PP88D, PPP0H, Q0001, Q000N, Q001J, Q00ID, Q00PD, Q044N, Q04G1, Q0I0D, Q0PPD, Q40G1, Q444N, Q44RN, Q4AA1, Q8QQJ, QAAS1, QAFF1, QAFG1, QASG1, QDGG1, QFGG1, QI00D, QIICB, QIIQB, QIQCB, QOIIB, QPP4D, QQ08J, QQ0CB, QQC0B, QQI0J, QQJ5J, QSFG1, R00E7, R0E07, R0NE7, R6F6N, R6FFN, REE07, RELE7, RFF6N, RL0E7, RLE07, RLEE7, RLLE7, RQR0N, RR0QN, RRQ0N, S03O1, S0AF1, S0AG1, S0O31, S0QIB, S0SIB, S30F1, S30O1, S7QIB, SA3PT, SAAA1, SAFG1, SASST, SCSQB, SF0G1, SFFF1, SI0SB, SISSB, SKSSB, SSCQB, SSCSB, SSMIB, SSPAT, SSSKT, SSSMB, SSSSB, STAF1, STF01, T0001, T0031, T0AF1, T0AS1, T0G31, T0R6N, T0T31, T0TF1, T3AAT, T40G1, T4CG1, T4G41, T5LLJ, T8LLJ, TA441, TA98J, TAFJJ, TAQ5J, TASST, TATAT, TC401, TCGG1, TFQIJ, TFQJJ, TFQMJ, TG441, TGC41, TI8LJ, TLLMJ, TMMMT, TMTMT, TQ8QJ, TSS3T, TT3AT, TT9IJ, TTAAT, TTQIJ, TTS31, TTS3T, TTT9J, TTTST, 20000H, 200OOH, 3440O1, 3TAAST, 404CQ1, 4KKKKN, 4KKKRN, 4QQQQD, 505BBB, 50BB0B, 6000AN, 6444RN, 66666N, 666O9T, 66999T, 669P9T, 66N777, 6A444N, 6FF66N, 6FFF6N, 6R666N, 766767, 77S677, 7IBBBB, 8888E7, 8LLLLJ, 8SSSL7, 9000MT, 90K90T, 90KP0T, 90M90T, 99000T, 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===Base 36===

11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 2UJ, 2WN, 2XN, 30Z, 337, 33B, 33H, 33P, 34H, 34P, 37D, 38Z, 39H, 3AB, 3AP, 3AZ, 3D7, 3DH, 3FD, 3FZ, 3HD, 3HH, 3KZ, 3L7, 3LZ, 3MB, 3O7, 3OZ, 3PD, 3PP, 3Q7, 3RH, 3S7, 3SZ, 3UP, 3UZ, 3XB, 40P, 43B, 43H, 46V, 48B, 48V, 49B, 4AP, 4BB, 4E1, 4FP, 4HB, 4HH, 4HP, 4IH, 4MB, 4OP, 4OV, 4PH, 4RH, 4VB, 4VH, 4WH, 4WP, 4WV, 4X1, 4XH, 55T, 565, 58Z, 5C5, 5E7, 5ET, 5EZ, 5IN, 5KT, 5L7, 5MP, 5NN, 5O5, 5O7, 5OP, 5R7, 5RT, 5RZ, 5ST, 5SZ, 5TN, 5TZ, 5W7, 5XN, 5YN, 61T, 625, 62J, 64V, 661, 66J, 691, 6AV, 6EV, 6IJ, 6J1, 6K5, 6O1, 6OT, 6QJ, 6QT, 6RV, 6SJ, 6T1, 6TT, 6U5, 6UV, 6VV, 6W5, 6Y1, 6YT, 72D, 737, 77D, 791, 7A1, 7AZ, 7D7, 7GD, 7I1, 7IN, 7LN, 7M7, 7MN, 7N7, 7NN, 7RN, 7RZ, 7WN, 7X7, 7Z1, 801, 80V, 82D, 83B, 841, 84H, 877, 881, 887, 88B, 88V, 88Z, 89H, 89V, 8A1, 8AB, 8B7, 8BH, 8D1, 8DH, 8EB, 8EV, 8GD, 8GZ, 8HD, 8IV, 8L7, 8LZ, 8M1, 8MB, 8MD, 8O7, 8OB, 8Q7, 8QD, 8RD, 8S7, 8SH, 8SZ, 8UZ, 8W7, 8WV, 8XV, 8Y1, 90H, 91J, 93B, 93H, 94V, 981, 98H, 99J, 9AH, 9BB, 9C5, 9IH, 9IV, 9JV, 9KH, 9M1, 9OB, 9QJ, 9R1, 9RH, 9SH, 9UJ, 9VB, 9VJ, 9W5, 9X1, 9Y1, 9YB, 9YJ, 9YV, A25, A3P, A3Z, A4H, A61, A81, A8B, AAB, AAH, ABB, AC5, AEZ, AHP, AK1, AKB, AKH, ALZ, AMB, AO5, AOZ, APH, AQP, AR1, ARV, ARZ, AUV, AV5, AVB, AVV, AWV, AX1, AXB, AZ1, B2J, B4B, B4V, B6V, B87, B8H, B9B, BAB, BAH, BBH, BBV, BE7, BEB, BGJ, BHB, BIJ, BJB, BJT, BLJ, BOB, BOT, BQ7, BRV, BS7, BTT, BTV, BVB, BVJ, BVT, BWV, BX7, BXH, BYH, BYV, C25, C2J, C2N, C55, C65, C6J, C95, CCN, CDD, CED, CF5, CFJ, CGD, CIN, CJD, CO5, CRN, CUJ, CXJ, CXN, D37, D3P, D41, D55, D5P, D6T, D77, D81, D9H, DA1, DCD, DD1, DD5, DDH, DDP, DE7, DEP, DF5, DFD, DG7, DHH, DI1, DK1, DK5, DKT, DO1, DOP, DP1, DPD, DQ7, DR7, DRH, DS7, DTT, DU1, DWD, DX7, DYH, E0P, E2D, E3Z, E41, E4P, E71, E81, E87, E8B, EAZ, EDP, EED, EEN, EEV, EFZ, EGT, EI1, EKZ, ELB, EMB, EMN, EN7, EO1, EOT, EPP, EPZ, EQ7, ERN, EST, ETN, ETV, EUP, EUZ, EVT, EWD, EWN, EX1, EYB, EYN, EZD, EZP, F2J, F3B, F3Z, F4P, F65, F8B, F8Z, FCJ, FD5, FEZ, FFB, FFD, FG5, FGD, FGZ, FIJ, FJ5, FJJ, FLZ, FPD, FQD, FSZ, FXB, G07, G0D, G0Z, G3D, G3Z, G55, G6T, G7D, G7Z, G87, G8Z, GA5, GCJ, GD7, GE7, GET, GG5, GG7, GGT, GGZ, GI7, GJD, GKZ, GL7, GLT, GQT, GRT, GS7, GST, GU5, GUZ, GW5, GXJ, GZ7, GZZ, H3D, H3H, H4H, H9B, HAH, HED, HGD, HHP, HIH, HKH, HKP, HLN, HNH, HOB, HOP, HQP, HRD, HRH, HRN, HSH, HWD, HWH, HWP, HYN, I41, I6J, I71, I7N, I87, I8H, I9H, I9J, IA1, IAV, IE1, IFJ, IG7, IHH, II1, IIH, IIV, IJV, IK1, IL7, ILJ, ILN, IM1, INN, IOV, IRH, IUV, IX1, IXH, IXV, IYJ, J2D, J4V, J55, J61, J6J, J8B, J8V, J95, J9J, J9V, JA5, JBB, JD5, JDT, JF5, JFJ, JGJ, JGT, JIV, JJ1, JJD, JJV, JK5, JKT, JLB, JLT, JMN, JN5, JNJ, JQJ, JQN, JQT, JRD, JTB, JVJ, JVN, JVV, JWN, JY1, JYJ, K0B, K25, K3P, K65, K81, K95, K9H, KAH, KC5, KEP, KEZ, KG5, KHP, KK1, KKT, KLB, KLP, KLZ, KM1, KMB, KMP, KOT, KP1, KQP, KR1, KRT, KRZ, KTT, KW5, KX1, KZB, L0N, L0P, L8Z, L9B, LAZ, LBJ, LEN, LET, LFB, LFZ, LG7, LGZ, LIJ, LJJ, LKB, LKP, LLB, LLP, LLT, LMB, LMN, LN7, LO7, LOT, LPZ, LQT, LRN, LTN, LTT, LWP, LX7, LYN, M01, M0P, M2D, M2N, M37, M3B, M41, M61, M77, M87, M8D, M91, ME1, ME7, MEB, MFB, MFP, MGD, MI7, MKB, MM7, MMN, MN7, MNN, MO7, MOB, MOP, MPP, MQP, MS7, MW7, MXN, MYN, N5N, N65, N87, N8D, N8H, N95, NCD, NCJ, NDH, NE7, NFJ, NG5, NG7, NGJ, NJ5, NN7, NND, NO5, NQD, NQJ, NRN, NU5, NWH, NWN, NXH, NXN, NYJ, O0Z, O25, O37, O3B, O3P, O3Z, O41, O4B, O5Z, O61, O71, O7Z, O81, OA5, OAP, OBB, OBV, OC5, OEZ, OF5, OG7, OKP, OKZ, OMB, OO5, OOZ, OP5, OPP, OQT, OR1, OR7, OS7, OST, OTB, OTZ, OU5, OVV, OW5, OX7, OXB, OXV, OYV, OZ5, OZ7, P01, P0D, P0N, P3P, P4H, P4P, P5N, P65, P6V, P8V, PD1, PED, PEZ, PHH, PHP, PI1, PIN, PLN, PLP, PLZ, POP, PP1, PPH, PPV, PQD, PQN, PRV, PV5, PVH, PVV, PWH, PWP, PX1, PXV, PYH, PYN, Q07, Q0P, Q2N, Q37, Q3P, Q6J, Q6T, Q7D, Q8D, Q9J, QCD, QCJ, QD7, QED, QFP, QGT, QI7, QIN, QJN, QLJ, QMD, QMN, QMP, QND, QNJ, QOT, QQJ, QTN, QWN, QYJ, QYT, R0V, R0Z, R37, R3H, R4H, R77, R7D, R7N, R7Z, R81, R8V, R91, RA1, RCD, RCN, RD1, RED, REV, REZ, RGT, RGZ, RHD, RIV, RKH, RKZ, RLN, RMD, RNH, RO7, RQN, RRD, RRT, RRZ, RS7, RSH, RT1, RTV, RU1, RUZ, RVN, RVT, RW7, RWH, RX7, RY1, S0J, S6J, S87, S9H, SAZ, SCJ, SET, SFJ, SG7, SGZ, SI7, SM7, SO7, SOT, SQJ, SQT, SRH, SSH, STT, SW7, SX7, SXH, SYH, T0N, T0Z, T1D, T1T, T6V, T8Z, TBB, TBV, TCN, TD1, TEV, TGT, TKT, TLB, TM1, TO1, TOB, TQN, TR1, TRD, TT1, TTB, TTN, TUZ, TVB, TVN, TVT, TWD, TWV, TXB, TXV, TYV, U2J, U3Z, U61, U8V, U95, UA1, UC5, UEP, UEZ, UFJ, UG5, UJ5, ULZ, UO5, UOP, UOZ, UQP, URV, URZ, UUJ, UW5, UWV, UXJ, UXV, UYJ, V0H, V25, V4V, V55, V6V, V9J, V9V, VBB, VBJ, VCJ, VET, VGJ, VHP, VIH, VIN, VJJ, VJN, VLT, VMP, VNJ, VOB, VP5, VQJ, VQT, VRT, VRV, VSH, VSJ, VST, VTB, VTN, VUP, VW5, VWN, VXH, VXN, VYB, W07, W25, W3D, W3H, W4P, W4V, W57, W6V, W7D, W8H, W95, W9H, WAH, WAV, WCN, WD7, WDD, WDH, WE7, WEN, WF5, WGD, WHH, WK5, WKH, WN7, WNN, WPP, WPV, WQP, WR7, WRD, WRH, WRN, WS7, WU5, WUP, WVN, WWH, WWP, WX7, WXH, WXN, WYN, WYV, X0J, X2J, X2N, X4B, X4H, X4V, X6J, X87, X8B, X91, X9B, XAV, XCN, XEB, XFB, XHB, XHH, XHN, XKB, XLJ, XNH, XO1, XQ7, XS7, XSH, XUJ, XWN, XWV, XX7, XXH, XXV, Y0N, Y1J, Y1T, Y2N, Y3H, Y61, Y8V, Y91, YBV, YCN, YEB, YFJ, YHB, YHN, YIH, YJN, YJT, YLJ, YLN, YMB, YMN, YNH, YOB, YOV, YRH, YTB, YTN, YTT, YTV, YVB, YVH, YWH, YWV, YXB, YY1, YYJ, YYT, YYV, Z01, Z3D, Z3Z, Z71, Z8B, ZB7, ZBB, ZD5, ZDD, ZDP, ZG7, ZGD, ZGZ, ZKB, ZLP, ZM1, ZO5, ZP1, ZPD, ZQD, ZU1, ZX1, ZXB, ZZD, 102J, 106T, 10ET, 10GJ, 10LJ, 10TD, 12FJ, 12LJ, 13ED, 13GD, 13MD, 192J, 1D0T, 1DET, 1E3D, 1EFD, 1F0D, 1FWD, 1G0J, 1GGJ, 1GIJ, 1GUJ, 1I0J, 1ISJ, 1JOT, 1JTT, 1M0D, 1M3D, 1MFD, 1MWD, 1OKT, 1OOT, 1Q3D, 1QDT, 1QKT, 1QTD, 1QWD, 1SLJ, 1TDT, 1TFD, 1TOT, 1TTD, 1TTT, 1UGJ, 1ULJ, 1WFD, 1WWD, 20CJ, 20FJ, 20G5, 20IN, 20JD, 20MD, 20RN, 20WD, 20YN, 2295, 25RN, 260J, 26LJ, 288D, 28DD, 28FD, 296J, 29G5, 29K5, 29LJ, 2C05, 2C0J, 2C0N, 2CJJ, 2CK5, 2CLJ, 2CSJ, 2D25, 2DC5, 2DDD, 2DW5, 2F0J, 2FDD, 2FMD, 2FSJ, 2G65, 2G95, 2GK5, 2I0N, 2JC5, 2JCJ, 2JWD, 2K05, 2L6J, 2L9J, 2N0N, 2NIN, 2NN5, 2NSJ, 2NW5, 2NYN, 2Q0N, 2QNN, 2R0N, 2RFD, 2RWD, 2SJJ, 2SXJ, 2SYJ, 2W05, 2WC5, 2WD5, 2WMD, 2XCJ, 2YCJ, 2YJJ, 2YRN, 2YSJ, 300D, 302D, 304B, 3077, 3087, 308D, 30BH, 30DD, 30FP, 30G7, 30GD, 30MD, 30QP, 30YH, 320D, 333D, 333Z, 33WD, 344B, 34FB, 373Z, 37E7, 37EZ, 37G7, 3807, 38BB, 38E7, 38G7, 38HB, 38I7, 38X7, 390B, 394B, 399B, 39EB, 39FB, 39KB, 3ASH, 3AWH, 3B8B, 3BG7, 3BLB, 3BR7, 3BW7, 3C8D, 3D0P, 3D2D, 3DKP, 3E3D, 3E77, 3EBB, 3EFB, 3EKB, 3EQP, 3ERD, 3ERZ, 3EZZ, 3F4B, 3FEP, 3G2D, 3GCD, 3GED, 3GM7, 3GR7, 3GWD, 3GZD, 3H0B, 3HEP, 3HKB, 3IAH, 3ISH, 3K4B, 3KBH, 3KEB, 3KKB, 3L4B, 3L8B, 3M3D, 3MEP, 3MG7, 3MWP, 3OFP, 3OKB, 3OMP, 3OOB, 3OYB, 3P8H, 3PXH, 3QDP, 3QQD, 3QRD, 3QWD, 3R3D, 3R3Z, 3R8D, 3RG7, 3RM7, 3RQD, 3RR7, 3RZ7, 3SIH, 3W0H, 3WED, 3WM7, 3WQD, 3WW7, 3WYH, 3X8H, 3XE7, 3XG7, 3XKH, 3Y0B, 3Y4B, 3Y8H, 3YAH, 3YSH, 3YYB, 3Z4B, 3ZFP, 404V, 4061, 40EB, 40I1, 40IV, 40KH, 40M1, 40R1, 40XB, 40XV, 43QP, 4441, 444H, 444P, 4481, 448H, 4491, 449H, 44EB, 44EV, 44K1, 44KB, 44KP, 44M1, 44R1, 44RV, 44SH, 44U1, 44VP, 44YB, 46I1, 46K1, 46M1, 480H, 488H, 48AH, 48I1, 48K1, 48U1, 4991, 49EV, 49I1, 49O1, 49UV, 4A0H, 4A41, 4A4B, 4A8H, 4A91, 4ALB, 4ASH, 4BXV, 4E4B, 4EEP, 4EFB, 4ELP, 4EUV, 4EXB, 4FOB, 4I01, 4I0V, 4I4V, 4I61, 4I81, 4I91, 4IEV, 4IY1, 4K0H, 4K91, 4KI1, 4KKH, 4KKP, 4KUP, 4L3P, 4L4P, 4LAB, 4LMP, 4MA1, 4MK1, 4MMP, 4MP1, 4MU1, 4O0B, 4OFB, 4OKB, 4OM1, 4OU1, 4OYB, 4P41, 4P81, 4PO1, 4PYV, 4QLP, 4QQP, 4RAV, 4RK1, 4RRV, 4S8H, 4U3P, 4U91, 4U9V, 4UIV, 4UUP, 4VAV, 4VLP, 4VVV, 4X0B, 4X0V, 4XAB, 4XBV, 4XVV, 4XYV, 4Y0V, 4Y41, 4Y4H, 4Y4V, 4Y81, 4YAV, 4YEV, 4YIV, 4YK1, 4YKB, 4YM1, 4YR1, 4YUV, 4YYB, 5025, 503Z, 5095, 50G5, 50PN, 50QN, 50WP, 50Z7, 5205, 52RN, 53FP, 53M7, 53PZ, 53X7, 53ZZ, 54PP, 550N, 553P, 554P, 5557, 555N, 557Z, 5595, 55EN, 55G5, 55GZ, 55LZ, 55OZ, 55QN, 55Z7, 572N, 577Z, 5787, 57CN, 57FZ, 57QN, 57S7, 57ZZ, 58I7, 58X7, 5955, 5A95, 5AEP, 5CMN, 5CWN, 5E2N, 5EAP, 5ECN, 5EKP, 5EQN, 5F95, 5FLP, 5FQP, 5G05, 5GK5, 5GQ7, 5I37, 5II7, 5IM7, 5L3P, 5LCN, 5LFP, 5LGT, 5LKZ, 5LUP, 5LZZ, 5M07, 5N55, 5NQ7, 5NX7, 5OAZ, 5OFZ, 5OLT, 5OOT, 5P25, 5P95, 5PG5, 5PGZ, 5PPN, 5PQP, 5PW5, 5Q0N, 5Q87, 5QCN, 5QKP, 5QN7, 5QPN, 5QPP, 5QQT, 5QTT, 5QWP, 5QX7, 5S37, 5SS7, 5T6T, 5TQT, 5U4P, 5UGZ, 5W0P, 5W2N, 5W5P, 5WLN, 5WP5, 5WQN, 5XM7, 5Y0T, 5YOT, 5Z25, 5Z4P, 5Z57, 5ZG5, 5ZLZ, 5ZQ7, 5ZWP, 6005, 600V, 601J, 6055, 606V, 60A1, 60C5, 60GT, 60JJ, 60LJ, 60M1, 60R1, 60VT, 60YV, 610J, 61FJ, 6401, 64K1, 6505, 665T, 6665, 66A5, 66F5, 66G5, 66GT, 66KT, 66V5, 66XV, 68OV, 68YV, 690J, 69G5, 69O5, 69XJ, 69XV, 6A55, 6AA1, 6AA5, 6AF5, 6C9J, 6CA5, 6CGJ, 6CYJ, 6EK1, 6ER1, 6FLJ, 6FXJ, 6G0J, 6G65, 6GGJ, 6GJ5, 6GLJ, 6I81, 6I9V, 6J65, 6J6V, 6JET, 6JST, 6JWV, 6K01, 6KET, 6KST, 6L0J, 6L6T, 6LFJ, 6LRT, 6LXJ, 6MA1, 6MM1, 6MX1, 6O8V, 6OV5, 6R0T, 6RET, 6RKT, 6S6T, 6T8V, 6U01, 6U9J, 6UE1, 6UGJ, 6UJJ, 6ULJ, 6UU1, 6V0T, 6VA5, 6VF5, 6VG5, 6VKT, 6VXJ, 6WWV, 6WXV, 6X41, 6X6V, 6XIV, 6XJV, 6XOV, 6Y0V, 6YCJ, 6YXV, 7001, 702N, 7041, 707N, 7081, 70GZ, 70I7, 70KZ, 70ND, 70O7, 70OZ, 70R7, 70U1, 70UZ, 70WD, 70YN, 71QD, 71WD, 720N, 72CN, 72YN, 733Z, 73ED, 73ZD, 7401, 7441, 74M1, 74O1, 74R1, 7641, 76K1, 76M1, 76R1, 76U1, 76X1, 773Z, 7741, 7771, 7781, 77FZ, 77G7, 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8KRH, 8KU1, 8KYH, 8KZ1, 8L4B, 8LBB, 8MG7, 8MR7, 8O4V, 8OI1, 8OK1, 8OOV, 8OX1, 8RAV, 8RHH, 8RUV, 8RX1, 8U4V, 8UK1, 8V4B, 8V9B, 8VHH, 8VRH, 8VVV, 8VYV, 8W0H, 8WFD, 8X07, 8XAH, 8XR1, 8XU1, 8XYB, 8Y6V, 8Y8H, 8YBB, 8YFB, 8YHH, 8YKB, 8YLB, 8YUV, 8YVV, 8YYH, 8Z1D, 8Z4B, 8Z8D, 8Z91, 8ZCD, 8ZEZ, 8ZFD, 8ZFZ, 8ZK1, 8ZO1, 8ZX7, 90A5, 90BJ, 90BV, 90EB, 90EV, 90GJ, 90IJ, 90J5, 90JB, 90LB, 90O1, 90OV, 90XB, 90XV, 9225, 92K5, 9441, 94BH, 94EB, 94KB, 94XB, 9505, 9601, 960V, 96A1, 96J5, 96K1, 96V5, 96WV, 96XV, 984B, 98AV, 98BV, 98FB, 994H, 998B, 998V, 999B, 99A1, 99A5, 99E1, 99G5, 99O1, 99O5, 99WH, 9A05, 9A55, 9A95, 9AA5, 9ABV, 9AE1, 9AEV, 9AI1, 9AO1, 9AOV, 9B0J, 9B0V, 9B8V, 9B9H, 9BCJ, 9BFJ, 9BSJ, 9BVV, 9CCJ, 9CLJ, 9CSJ, 9E01, 9E4B, 9EA1, 9EAB, 9EAV, 9EK1, 9F05, 9F25, 9FA5, 9FBJ, 9FFJ, 9FJB, 9FSJ, 9FXJ, 9G25, 9G2J, 9GIJ, 9GSJ, 9HFB, 9HLB, 9IIJ, 9IJ1, 9J01, 9J0B, 9J0J, 9J9B, 9JCJ, 9JE1, 9K05, 9K41, 9K55, 9K61, 9K8B, 9KA5, 9KI1, 9KK5, 9KU5, 9L2J, 9LCJ, 9O05, 9O0V, 9O95, 9OI1, 9OO1, 9OOV, 9OWV, 9RRV, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ

====Additional known quasi-minimal primes (not necessarily the next)====

P&lt;sub&gt;81993&lt;/sub&gt;SZ

==Unsolved families==

Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')) by covering congruence, algebraic factorization, or combine of them.

{|class=wikitable
|base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family
|-
|13||9{5}||(113×13&lt;sup&gt;''n''&lt;/sup&gt;−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|13||A{3}A||(41×13&lt;sup&gt;''n''+1&lt;/sup&gt;+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|16||{3}AF||(16&lt;sup&gt;''n''+2&lt;/sup&gt;+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1{7}||(23×17&lt;sup&gt;''n''&lt;/sup&gt;−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1F{0}7||32×17&lt;sup&gt;''n''+1&lt;/sup&gt;+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||4{7}A||(71×17&lt;sup&gt;''n''+1&lt;/sup&gt;+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||51{0}D||86×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||70F{0}D||2038×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||8{B}9||(139×17&lt;sup&gt;''n''+1&lt;/sup&gt;−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||9{5}9||(149×17&lt;sup&gt;''n''+1&lt;/sup&gt;+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||95{F}||(2543×17&lt;sup&gt;''n''&lt;/sup&gt;−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||A{D}F||(173×17&lt;sup&gt;''n''+1&lt;/sup&gt;+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}B3||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+190||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}DB||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+232||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2BE||(11×17&lt;sup&gt;''n''+3&lt;/sup&gt;−41579)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2E||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;−2411)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}E9||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+773)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}EE||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+853)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||D0G{D}||(60381×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2860381*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||E9{B}||(3963×17&lt;sup&gt;''n''&lt;/sup&gt;−11)/16||29000||[http://factordb.com/index.php?query=%283963*17%5En-11%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||F1{9}||(4105×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16||1000000||[http://factordb.com/index.php?query=%284105*17%5En-9%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||FD0{D}||(72909×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2872909*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||G{7}F||(263×17&lt;sup&gt;''n''+1&lt;/sup&gt;+121)/16||29000||[http://factordb.com/index.php?query=%28263*17%5E%28n%2B1%29%2B121%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||2{7}9D||(47×21&lt;sup&gt;''n''+2&lt;/sup&gt;+953)/20||20000||[http://factordb.com/index.php?query=%2847*21%5E%28n%2B2%29%2B953%29%2F20&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||2F{C}A||(288×21&lt;sup&gt;''n''+1&lt;/sup&gt;−13)/5||20000||[http://factordb.com/index.php?query=%28288*21%5E%28n%2B1%29-13%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||7{K}Z||(53×36&lt;sup&gt;''n''+1&lt;/sup&gt;+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||B{0}EUV||11×36&lt;sup&gt;''n''+3&lt;/sup&gt;+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||HM{0}N||634×36&lt;sup&gt;''n''+1&lt;/sup&gt;+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||N{0}YYN||23×36&lt;sup&gt;''n''+3&lt;/sup&gt;+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||O{L}Z||(123×36&lt;sup&gt;''n''+1&lt;/sup&gt;+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||S{0}8H||28×36&lt;sup&gt;''n''+2&lt;/sup&gt;+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|}

For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E

For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family B0{H}6H may or may not be quasi-minimal prime, since another unsolved family is B{H}6H, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D}

=== Base 17 ===

All searched to length 29000

* 1{7}
* 1F{0}7
* 4{7}A
* 51{0}D
* 70F{0}D
* 8{B}9
* 9{5}9
* 95{F}
* A{D}F
* B{0}B3
* B{0}DB
* {B}2BE
* {B}2E
* {B}E9
* {B}EE
* D0G{D}
* E9{B}
* F1{9}
* FD0{D}
* G{7}F

=== Base 21 ===

All searched to length 20000

* 2{7}9D
* 2F{C}A
* 4{3}B
* 5{0}DJ
* {5}FEK
* {7}ID
* 99{0}99H
* {9}0D
* {9}D
* B0{H}6H
* B3{0}EB
* B9{0}E5
* B{D}B
* B{H}6H
* DH{D}
* F{9}D
* {F}35
* G{0}FK
* H{0}7771
* H{D}
* {J}6J

=== Base 36 ===

All searched to length 20000

* 7{K}Z
* B{0}EUV
* HM{0}N
* N{0}YYN
* O{L}Z
* S{0}8H

==Primality certificates for the proven primes &gt; 10&lt;sup&gt;299&lt;/sup&gt;==

See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]]

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime
|-
|9||149||76&lt;sub&gt;329&lt;/sub&gt;2||(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642]
|-
|9||150||27&lt;sub&gt;686&lt;/sub&gt;07||(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486]
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||3×9&lt;sup&gt;1160&lt;/sup&gt;+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423]
|-
|11||1065||A&lt;sub&gt;713&lt;/sub&gt;58||11&lt;sup&gt;715&lt;/sup&gt;−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487]
|-
|11||1066||7&lt;sub&gt;759&lt;/sub&gt;44||(7×11&lt;sup&gt;761&lt;/sup&gt;−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840]
|-
|11||1067||557&lt;sub&gt;1011&lt;/sub&gt;||(607×11&lt;sup&gt;1011&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522]
|-
|13||3165||50&lt;sub&gt;270&lt;/sub&gt;44||5×13&lt;sup&gt;272&lt;/sup&gt;+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005]
|-
|13||3166||9&lt;sub&gt;271&lt;/sub&gt;095||(3×13&lt;sup&gt;274&lt;/sup&gt;−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654]
|-
|13||3167||10&lt;sub&gt;286&lt;/sub&gt;7771||13&lt;sup&gt;290&lt;/sup&gt;+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633]
|-
|13||3168||9&lt;sub&gt;308&lt;/sub&gt;1||(3×13&lt;sup&gt;309&lt;/sup&gt;−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=308&amp;c0=-&amp;EN= 13&lt;sup&gt;308&lt;/sup&gt;−1]
|-
|13||3169||B&lt;sub&gt;341&lt;/sub&gt;C4||(11×13&lt;sup&gt;343&lt;/sup&gt;+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618]
|-
|13||3170||8B&lt;sub&gt;343&lt;/sub&gt;||(107×13&lt;sup&gt;343&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736]
|-
|13||3171||710&lt;sub&gt;371&lt;/sub&gt;111||92×13&lt;sup&gt;374&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609]
|-
|13||3172||75&lt;sub&gt;375&lt;/sub&gt;7||(89×13&lt;sup&gt;376&lt;/sup&gt;+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596]
|-
|13||3173||9B0&lt;sub&gt;391&lt;/sub&gt;9||128×13&lt;sup&gt;392&lt;/sup&gt;+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790]
|-
|13||3174||7B0B&lt;sub&gt;397&lt;/sub&gt;||(15923×13&lt;sup&gt;397&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574]
|-
|13||3175||10&lt;sub&gt;414&lt;/sub&gt;93||13&lt;sup&gt;416&lt;/sup&gt;+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240]
|-
|13||3176||81010&lt;sub&gt;415&lt;/sub&gt;1||17746×13&lt;sup&gt;416&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3177||8110&lt;sub&gt;435&lt;/sub&gt;1||1366×13&lt;sup&gt;436&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3178||B7&lt;sub&gt;486&lt;/sub&gt;||(139×13&lt;sup&gt;486&lt;/sup&gt;−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892]
|-
|13||3179||B&lt;sub&gt;563&lt;/sub&gt;C||(11×13&lt;sup&gt;564&lt;/sup&gt;+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=564&amp;c0=-&amp;EN= 13&lt;sup&gt;564&lt;/sup&gt;−1]
|-
|13||3180||1B&lt;sub&gt;576&lt;/sub&gt;||(23×13&lt;sup&gt;576&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=576&amp;c0=-&amp;EN= 13&lt;sup&gt;576&lt;/sup&gt;−1]
|-
|13||3181||80&lt;sub&gt;693&lt;/sub&gt;87||8×13&lt;sup&gt;695&lt;/sup&gt;+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532]
|-
|13||3182||CC5&lt;sub&gt;713&lt;/sub&gt;||(2021×13&lt;sup&gt;713&lt;/sup&gt;−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353]
|-
|13||3183||B&lt;sub&gt;834&lt;/sub&gt;74||(11×13&lt;sup&gt;836&lt;/sup&gt;−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871]
|-
|13||3184||9&lt;sub&gt;968&lt;/sub&gt;B||(3×13&lt;sup&gt;969&lt;/sup&gt;+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244]
|-
|13||3185||10&lt;sub&gt;1295&lt;/sub&gt;181||13&lt;sup&gt;1298&lt;/sup&gt;+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013]
|-
|13||3186||9&lt;sub&gt;1362&lt;/sub&gt;5||(3×13&lt;sup&gt;1363&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776]
|-
|13||3187||7&lt;sub&gt;1504&lt;/sub&gt;1||(7×13&lt;sup&gt;1505&lt;/sup&gt;−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755]
|-
|13||3188||930&lt;sub&gt;1551&lt;/sub&gt;1||120×13&lt;sup&gt;1552&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3189||720&lt;sub&gt;2297&lt;/sub&gt;2||93×13&lt;sup&gt;2298&lt;/sup&gt;+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910]
|-
|13||3190||1770&lt;sub&gt;2703&lt;/sub&gt;17||267×13&lt;sup&gt;2705&lt;/sup&gt;+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825]
|-
|13||3191||390&lt;sub&gt;6266&lt;/sub&gt;1||48×13&lt;sup&gt;6267&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3192||B0&lt;sub&gt;6540&lt;/sub&gt;BBA||11×13&lt;sup&gt;6543&lt;/sup&gt;+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906]
|-
|13||3193||C&lt;sub&gt;10631&lt;/sub&gt;92||13&lt;sup&gt;10633&lt;/sup&gt;−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750]
|-
|14||649||34D&lt;sub&gt;708&lt;/sub&gt;||47×14&lt;sup&gt;708&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||5×14&lt;sup&gt;19698&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2328||880&lt;sub&gt;246&lt;/sub&gt;7||136×16&lt;sup&gt;247&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is &lt; 10&lt;sup&gt;299&lt;/sup&gt;
|-
|16||2329||D4&lt;sub&gt;263&lt;/sub&gt;D||(199×16&lt;sup&gt;264&lt;/sup&gt;+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238]
|-
|16||2330||E0&lt;sub&gt;261&lt;/sub&gt;4DD||14×16&lt;sup&gt;264&lt;/sup&gt;+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352]
|-
|16||2331||8C0&lt;sub&gt;290&lt;/sub&gt;ED||140×16&lt;sup&gt;292&lt;/sup&gt;+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307]
|-
|16||2332||DA&lt;sub&gt;305&lt;/sub&gt;5||(41×16&lt;sup&gt;306&lt;/sup&gt;−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284]
|-
|16||2333||CE80&lt;sub&gt;422&lt;/sub&gt;D||3304×16&lt;sup&gt;423&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257]
|-
|16||2334||5F&lt;sub&gt;544&lt;/sub&gt;6F||6×16&lt;sup&gt;546&lt;/sup&gt;−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967]
|-
|16||2335||88F&lt;sub&gt;545&lt;/sub&gt;||137×16&lt;sup&gt;545&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2336||BE0&lt;sub&gt;792&lt;/sub&gt;BB||190×16&lt;sup&gt;794&lt;/sup&gt;+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938]
|-
|16||2337||D9&lt;sub&gt;1052&lt;/sub&gt;||(68×16&lt;sup&gt;1052&lt;/sup&gt;−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020]
|-
|16||2338||FAF&lt;sub&gt;1062&lt;/sub&gt;45||251×16&lt;sup&gt;1064&lt;/sup&gt;−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610]
|-
|16||2339||F8&lt;sub&gt;1517&lt;/sub&gt;F||(233×16&lt;sup&gt;1518&lt;/sup&gt;+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824]
|-
|16||2340||20&lt;sub&gt;1713&lt;/sub&gt;321||2×16&lt;sup&gt;1716&lt;/sup&gt;+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735]
|-
|16||2341||300F&lt;sub&gt;1960&lt;/sub&gt;AF||769×16&lt;sup&gt;1962&lt;/sup&gt;−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750]
|-
|16||2342||90&lt;sub&gt;3542&lt;/sub&gt;91||9×16&lt;sup&gt;3544&lt;/sup&gt;+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191]
|-
|16||2343||5BC&lt;sub&gt;3700&lt;/sub&gt;D||(459×16&lt;sup&gt;3701&lt;/sup&gt;+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322]
|-
|16||2344||D0B&lt;sub&gt;17804&lt;/sub&gt;||(3131×16&lt;sup&gt;17804&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511]
|-
|18||547||80&lt;sub&gt;298&lt;/sub&gt;B||8×18&lt;sup&gt;299&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is an algebraic factor of ''N''+1, factordb entry of 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is [http://factordb.com/index.php?id=1100000000900149167]
|-
|18||548||H&lt;sub&gt;766&lt;/sub&gt;FH||18&lt;sup&gt;768&lt;/sup&gt;−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490]
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||12×18&lt;sup&gt;6270&lt;/sup&gt;+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437]
|-
|20||3301||H&lt;sub&gt;247&lt;/sub&gt;A0H||(17×20&lt;sup&gt;250&lt;/sup&gt;−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619]
|-
|20||3302||7&lt;sub&gt;249&lt;/sub&gt;A7||(7×20&lt;sup&gt;251&lt;/sup&gt;+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602]
|-
|20||3303||J7&lt;sub&gt;270&lt;/sub&gt;||(368×20&lt;sup&gt;270&lt;/sup&gt;−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462]
|-
|20||3304||J&lt;sub&gt;330&lt;/sub&gt;CCC7||20&lt;sup&gt;334&lt;/sup&gt;−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572]
|-
|20||3305||40&lt;sub&gt;387&lt;/sub&gt;404B||4×20&lt;sup&gt;391&lt;/sup&gt;+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563]
|-
|20||3306||EC0&lt;sub&gt;429&lt;/sub&gt;7||292×20&lt;sup&gt;430&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702]
|-
|20||3307||G&lt;sub&gt;447&lt;/sub&gt;99||(16×20&lt;sup&gt;449&lt;/sup&gt;−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753]
|-
|20||3308||3A&lt;sub&gt;527&lt;/sub&gt;3||(67×20&lt;sup&gt;528&lt;/sup&gt;−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531]
|-
|20||3309||E&lt;sub&gt;566&lt;/sub&gt;C7||(14×20&lt;sup&gt;568&lt;/sup&gt;−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516]
|-
|20||3310||JCJ&lt;sub&gt;629&lt;/sub&gt;||393×20&lt;sup&gt;629&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|20||3311||J&lt;sub&gt;655&lt;/sub&gt;05J||20&lt;sup&gt;658&lt;/sup&gt;−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052]
|-
|20||3312||50&lt;sub&gt;1163&lt;/sub&gt;AJ||5×20&lt;sup&gt;1165&lt;/sup&gt;+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412]
|-
|20||3313||CD&lt;sub&gt;2449&lt;/sub&gt;||(241×20&lt;sup&gt;2449&lt;/sup&gt;−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915]
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||16×20&lt;sup&gt;6270&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457]
|-
|22||7984||I7G0&lt;sub&gt;254&lt;/sub&gt;H||8882×22&lt;sup&gt;255&lt;/sup&gt;+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788]
|-
|22||7985||D0&lt;sub&gt;255&lt;/sub&gt;5EEF||13×22&lt;sup&gt;259&lt;/sup&gt;+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932]
|-
|22||7986||IK&lt;sub&gt;322&lt;/sub&gt;F||(398×22&lt;sup&gt;323&lt;/sup&gt;−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145]
|-
|22||7987||C0&lt;sub&gt;340&lt;/sub&gt;G9||12×22&lt;sup&gt;342&lt;/sup&gt;+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159]
|-
|22||7988||77E&lt;sub&gt;348&lt;/sub&gt;K7||(485×22&lt;sup&gt;350&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779]
|-
|22||7989||J&lt;sub&gt;379&lt;/sub&gt;KJ||(19×22&lt;sup&gt;381&lt;/sup&gt;+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027]
|-
|22||7990||J&lt;sub&gt;388&lt;/sub&gt;EJ||(19×22&lt;sup&gt;390&lt;/sup&gt;−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729]
|-
|22||7991||DJ&lt;sub&gt;400&lt;/sub&gt;||(292×22&lt;sup&gt;400&lt;/sup&gt;−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110]
|-
|22||7992||E&lt;sub&gt;404&lt;/sub&gt;K7||(2×22&lt;sup&gt;406&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298]
|-
|22||7993||66F&lt;sub&gt;453&lt;/sub&gt;B3||(971×22&lt;sup&gt;455&lt;/sup&gt;−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809]
|-
|22||7994||L0&lt;sub&gt;454&lt;/sub&gt;B63||21×22&lt;sup&gt;457&lt;/sup&gt;+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331]
|-
|22||7995||L&lt;sub&gt;483&lt;/sub&gt;G3||22&lt;sup&gt;485&lt;/sup&gt;−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730]
|-
|22||7996||E60&lt;sub&gt;496&lt;/sub&gt;L||314×22&lt;sup&gt;497&lt;/sup&gt;+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239]
|-
|22||7997||I&lt;sub&gt;626&lt;/sub&gt;AF||(6×22&lt;sup&gt;628&lt;/sup&gt;−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334]
|-
|22||7998||K0&lt;sub&gt;760&lt;/sub&gt;EC1||20×22&lt;sup&gt;763&lt;/sup&gt;+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415]
|-
|22||7999||J0&lt;sub&gt;767&lt;/sub&gt;IGGJ||19×22&lt;sup&gt;771&lt;/sup&gt;+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567]
|-
|22||8000||7&lt;sub&gt;959&lt;/sub&gt;K7||(22&lt;sup&gt;961&lt;/sup&gt;+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817]
|-
|22||8001||L&lt;sub&gt;2385&lt;/sub&gt;KE7||22&lt;sup&gt;2388&lt;/sup&gt;−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774]
|-
|22||8002||7&lt;sub&gt;3815&lt;/sub&gt;2L||(22&lt;sup&gt;3817&lt;/sup&gt;−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839]
|-
|24||3400||I0&lt;sub&gt;241&lt;/sub&gt;I5||18×24&lt;sup&gt;243&lt;/sup&gt;+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037]
|-
|24||3401||D0&lt;sub&gt;259&lt;/sub&gt;KKD||13×24&lt;sup&gt;262&lt;/sup&gt;+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725]
|-
|24||3402||C7&lt;sub&gt;298&lt;/sub&gt;||(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235]
|-
|24||3403||20&lt;sub&gt;313&lt;/sub&gt;7||2×24&lt;sup&gt;314&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241]
|-
|24||3404||BC0&lt;sub&gt;331&lt;/sub&gt;B||276×24&lt;sup&gt;332&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842]
|-
|24||3405||N&lt;sub&gt;2644&lt;/sub&gt;LLN||24&lt;sup&gt;2647&lt;/sup&gt;−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089]
|-
|24||3406||D&lt;sub&gt;2698&lt;/sub&gt;LD||(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876]
|-
|24||3407||A0&lt;sub&gt;2951&lt;/sub&gt;8ID||10×24&lt;sup&gt;2954&lt;/sup&gt;+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654]
|-
|24||3408||88N&lt;sub&gt;5951&lt;/sub&gt;||201×24&lt;sup&gt;5951&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606]
|-
|30||2613||AN&lt;sub&gt;206&lt;/sub&gt;||(313×30&lt;sup&gt;206&lt;/sup&gt;−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073]
|-
|30||2614||M&lt;sub&gt;241&lt;/sub&gt;QB||(22×30&lt;sup&gt;243&lt;/sup&gt;+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295]
|-
|30||2615||M0&lt;sub&gt;547&lt;/sub&gt;SS7||22×30&lt;sup&gt;550&lt;/sup&gt;+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988]
|-
|30||2616||C0&lt;sub&gt;1022&lt;/sub&gt;1||12×30&lt;sup&gt;1023&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|30||2617||5&lt;sub&gt;4882&lt;/sub&gt;J||(5×30&lt;sup&gt;4883&lt;/sup&gt;+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423]
|-
|30||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||25×30&lt;sup&gt;34205&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|}

==Unproven PRPs==

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this PRP
|-
|11||1068||57&lt;sub&gt;62668&lt;/sub&gt;||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000003573679860]
|-
|13||3194||C5&lt;sub&gt;23755&lt;/sub&gt;C||(149×13&lt;sup&gt;23756&lt;/sup&gt;+79)/12||[http://factordb.com/index.php?id=1100000003590647776]
|-
|13||3195||80&lt;sub&gt;32017&lt;/sub&gt;111||8×13&lt;sup&gt;32020&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000000490878060]
|-
|16||2345||DB&lt;sub&gt;32234&lt;/sub&gt;||(206×16&lt;sup&gt;32234&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000002383583629]
|-
|16||2346||4&lt;sub&gt;72785&lt;/sub&gt;DD||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15||[http://factordb.com/index.php?id=1100000003615909841]
|-
|22||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21||[http://factordb.com/index.php?id=1100000003594696838]
|-
|30||2618||I0&lt;sub&gt;24608&lt;/sub&gt;D||18×30&lt;sup&gt;24609&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003593967511]
|}

All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10&lt;sup&gt;16&lt;/sup&gt;. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]])

==Proof==

===Base 2===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.

===Base 3===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (2,1), (2,2)

* Case (1,1):
** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes)
*** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime.

===Base 4===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (2,1), (2,3), (3,1), (3,3)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.

===Base 5===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4)

* Case (1,1):
** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes)
*** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes)
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes)
***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb])
***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes)
**** The smallest prime of the form 1{4}4 is 14444.
**** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes)
*** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes)
**** For the 3{0,3}1 family, we can separate this family to four families:
***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01.
****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime.
***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031.
***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001.
***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31.
****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime.
**** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.
* Case (3,4):
** 34 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes)
**** The smallest prime of the form 4{4}1 is 44441.
**** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime.
* Case (4,2):
** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime.

===Base 6===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes)
*** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes)
**** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 4{0}41 is 40041
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes)
*** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.

===Base 7===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

* Case (1,1):
** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes)
*** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {}
**** If there are &gt;=3 number of 1 and 5 in the {}:
***** If there is 111 in the {}, then we have the prime 11111
***** If there is 115 in the {}, then the prime 115 is a subsequence
***** If there is 151 in the {}, then the prime 115 is a subsequence
***** If there is 155 in the {}, then the prime 155 is a subsequence
***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime
***** If there is 515 in the {}, then the prime 515 is a subsequence
***** If there is 551 in the {}, then the prime 551 is a subsequence
***** If there is 555 in the {}, then the prime 551 is a subsequence
**** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1
***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1
****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1
******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21
******** The smallest prime of the form 11{0}21 is 1100021
******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime
******** The smallest prime of the form 1001{0}21 is 100121
********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families
***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1
****** Since 1051 is prime, we only need to consider the family 15{0}1
******* The smallest prime of the form 15{0}1 is 150001
* Case (1,2):
** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes)
*** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {}
**** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime.
**** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime.
**** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2
***** Since 1022 and 1202 are primes, we only need to consider the number 122
****** 122 is not prime.
**** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime.
* Case (1,3):
** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** 14 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime.
* Case (1,6):
** 16 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime.
* Case (2,2):
** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,6):
** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes)
*** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes)
**** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's)
***** For the 3{0,1}1 family:
****** If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
****** If there are no 1's between (3,1), then the form will be 3{0}1
******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime.
****** If there are no 0's between (3,1), then the form will be 3{1}1
******* The smallest prime of the form 3{1}1 is 31111
****** If there are exactly 1 0's between (3,1), then there must be &lt;3 1's between (3,1), or 31111 will be a subsequence.
******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime.
******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011
******** Neither 3101 nor 3011 is prime.
******* If there are no 1's between (3,1), then the number must be 301
******** 301 is not prime.
***** For the 3{3}3{0,1}1 family:
****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence.
****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1
******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence.
******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime.
******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime.
******* If there are no 1's between (33,1), then the form will be 33{0}1
******** The smallest prime of the form 33{0}1 is 33001
****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1
******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence.
******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301
******** The smallest prime of the form 3{3}301 is 33333301
******* If there are no 0's between (3{3}3,1), then the form is 3{3}31
******** The smallest prime of the form 3{3}31 is 33333333333333331
**** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1
***** Numbers containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1,3}3{0,1,3}1
******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes)
******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes)
********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes)
********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence.
********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime.
********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1
*********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime.
********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime.
***** Numbers not containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1}1
******* If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1
******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime.
******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1
******** If there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111
********* None of 31151, 31511, 35111 are primes.
******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 351
********* 351 is not prime.
******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111
********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes.
********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime.
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1
********* Neither 3051 nor 3501 is prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** If there are &gt;=2 5's in {}, then 553 will be a subsequence.
*** If there are no 5's in {}, then the family will be 3{0,3,6}3
**** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
*** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3
**** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes)
***** The smallest prime of the form 3{0}53 is 300053
* Case (3,4):
** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes)
*** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes)
**** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes)
**** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes)
* Case (3,5):
** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes)
*** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence.
*** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence.
*** If there are at least two 1's in {}, then 115 will be a subsequence.
*** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5
**** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime.
*** If there is no 1's in {}, then the family will be 3{0,4,5}5
**** If there are at least to 4's in {}, then 344 and 445 will be subsequences.
**** If there is no 4's in {}, then the family will be 3{0,5}5
***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5
****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime.
****** The smallest prime of the form 3{5}5 is 35555
**** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5
***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes)
****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime.
* Case (3,6):
** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,2):
** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes)
*** If there is no 5's in {}, then the family will be 4{0,2,4,6}4
**** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4
**** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes)
***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence.
***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4
****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes)
***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4
****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4
******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes)
* Case (4,5):
** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** If there are at least two 5's in {}, then 4555 will be a subsequence.
*** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there is no 5's in {}, then the family will be 4{0}5
**** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime.
* Case (4,6):
** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (5,1):
** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 3's in {}, then 533 will be a subsequence.
*** If there is no 3's in {}, then the family will be 5{0,1}1
**** Since 5011 is prime, we only need to consider the family 5{1}{0}1
***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes)
****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime.
****** The smallest prime of the form 51{0}1 is 5100000001
****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime.
****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime.
*** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1
**** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence.
***** Thus we only need to consider the family 5{0}3{0,1}1
****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime.
****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime.
****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence.
****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1
******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0)
******** The smallest prime of the form 5311{0}1 is 531101
* Case (5,2):
** 52 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 1's in {}, then 113 will be a subsequence.
*** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
*** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime.
* Case (5,4):
** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes)
*** If there are no 5's in {}, then the family will be 5{0,3}4
**** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime.
*** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence.
*** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there are at least two 5's in {}, then 5554 will be a subsequence.
* Case (5,5):
** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes)
*** If there are no 4's in {}, then the family will be 5{0,5}5
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
*** If there are no 5's in {}, then the family will be 5{0,4}5
**** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime.
*** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence.
* Case (5,6):
** 56 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,2):
** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (6,3):
** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,4):
** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are at least three 3's in {}, then 3334 will be a subsequence.
*** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4
**** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence.
**** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence.
**** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence.
**** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence.
**** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence.
**** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence.
**** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence.
**** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence.
**** Thus the number can only be 634
***** 634 is not prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,6):
** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}6
**** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime.
*** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6
**** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime.
*** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence.
*** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence.

===Base 8===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7)

* Case (1,1):
** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes)
**** 171 is not prime.
**** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 &gt; 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 &gt; 1, this factorization is nontrivial), thus cannot be prime.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes)
*** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime)
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes)
*** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes)
*** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes)
**** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime.
**** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes)
***** The smallest prime of the form 33{4}1 is 3344441
***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime.
**** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes)
***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime.
***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes)
****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes.
* Case (3,3):
** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes)
*** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes)
**** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime)
**** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461
***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes)
****** The smallest prime of the form 4{4}41 is 444444441
****** The smallest prime of the form 4{4}641 is 444641
***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes)
****** The smallest prime of the form 4{4}411 is 444444441
****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes)
***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461
****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime)
**** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes)
***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime)
***** The smallest prime of the form 4{4}641 is 444641
* Case (4,3):
** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes)
*** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes)
**** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime.
**** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime.
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes)
*** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes)
**** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb])
**** The smallest prime of the form 4{4}77 is 4444477
**** The smallest prime of the form 4{7}7 is 47777
**** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime)
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes)
*** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes)
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
**** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes)
***** 500525 is not prime.
***** The smallest prime of the form 5{5}25 is 555555555555525
***** The smallest prime of the form 5{5}025 is 55555025
***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes)
***** The smallest prime of the form 5{5}0525 is 5550525
***** The smallest prime of the form 5{5}00525 is 5500525
***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes)
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence)
**** The form is 6{0,1,7}4{0,1,7}1
***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1
****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411
******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41
******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41
********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.
********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime.
********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime.
******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411
******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime)
*** Numbers not containing 4:
**** The form is 6{0,1,7}1
***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1
****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime.
****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1
******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1
******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.
******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime)
* Case (6,3):
** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,7):
** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them
**** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17
***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime.
**** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77
***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime.
**** If there is neither 1 nor 7 between them, then the form is 6{0,4}7
***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes)
****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime.
****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime.
* Case (7,1):
** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes)
*** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes)
**** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes)
***** The smallest prime of the form 7{7}1 is 7777777777771
***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime)
***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime.
***** The smallest prime of the form 7{7}4444441 is 77774444441
****** Since this prime has just 4 7's, we only need to consider the families with &lt;=3 7's
******* The smallest prime of the form 7{4}1 is 744444441
******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime.
******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes)
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.

===Base 10===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,9):
** 19 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 2{0}21 is 20021
**** The smallest prime of the form 22{0}1 is 22000001
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes)
*** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes)
**** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime.
**** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime.
* Case (2,9):
** 29 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,9):
** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** 47 is prime, and thus the only minimal prime in this family.
* Case (4,9):
** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime.
* Case (5,1):
** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes)
**** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes)
***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 5{5}1 is 555555555551
**** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581
***** 581 is not prime.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,7):
** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027
**** The smallest prime of the form 52{0}7 is 5200007
* Case (5,9):
** 59 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,3):
** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,9):
** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49
***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49
****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes)
******* The smallest prime of the form 6{6}49 is 666649
******** Since this prime has just 4 6's, we only need to consider the families with &lt;=3 6's
********* The smallest prime of the form 6{0}49 is 60000049
********* The smallest prime of the form 66{0}49 is 66000049
********* The smallest prime of the form 666{0}49 is 66600049
* Case (7,1):
** 71 is prime, and thus the only minimal prime in this family.
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,9):
** 79 is prime, and thus the only minimal prime in this family.
* Case (8,1):
** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes)
**** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes)
***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime)
***** The smallest prime of the form 80{5}1 is 80555551
* Case (8,3):
** 83 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime.
* Case (8,9):
** 89 is prime, and thus the only minimal prime in this family.
* Case (9,1):
** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes)
**** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021
***** None of 921, 9201, 9021 are primes.
**** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801
***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
* Case (9,3):
** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (9,7):
** 97 is prime, and thus the only minimal prime in this family.
* Case (9,9):
** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9
***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes)
****** The smallest prime of the form 94{6}9 is 946669

===Base 12===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,B):
** 1B is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes)
*** The smallest prime of the form 2{0}1 is 2001
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (2,B):
** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes)
**** None of 20B, 22B, 29B, 209B, 220B are primes.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,B):
** 3B is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes)
**** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 44{A}1 is 44AAA1
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes)
**** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077
* Case (4,B):
** 4B is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes)
*** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime.
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (5,B):
** 5B is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,5):
** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes)
**** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 6{0}A5 is 600A5
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,B):
** 6B is prime, and thus the only minimal prime in this family.
* Case (7,1):
** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes)
*** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1
**** None of 741, 7B1, 74B1 are primes.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,B):
** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes)
**** The smallest prime of the form 7{9}B is 7999B
**** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes)
**** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime)
* Case (8,1):
** 81 is prime, and thus the only minimal prime in this family.
* Case (8,5):
** 85 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** 87 is prime, and thus the only minimal prime in this family.
* Case (8,B):
** 8B is prime, and thus the only minimal prime in this family.
* Case (9,1):
** 91 is prime, and thus the only minimal prime in this family.
* Case (9,5):
** 95 is prime, and thus the only minimal prime in this family.
* Case (9,7):
** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes)
**** None of 947, 977, 997, 9477, 9977 are primes.
* Case (9,B):
** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes)
**** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes)
***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime)
***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime)
**** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes)
***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes)
****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime.
****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime)
****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime.
**** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B
***** None of 99B, 999B, 979B, 9799B, 9979B are primes.
* Case (A,1):
** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes)
**** For the A{A}{0}1 family:
***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form AA{0}1 is AA000001
***** The smallest prime of the form AAA{0}1 is AAA0001
***** The smallest prime of the form AAAA{0}1 is AAAA1
****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc.
**** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime.
* Case (A,5):
** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes)
*** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes)
**** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form A{0}65 is A00065
* Case (A,7):
** A7 is prime, and thus the only minimal prime in this family.
* Case (A,B):
** AB is prime, and thus the only minimal prime in this family.
* Case (B,1):
** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes)
**** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1
***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1
****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime)
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1
****** The smallest prime of the form B{A}01 is BAA01
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
**** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1
***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1
****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1
******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1
******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes.
****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1
******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime)
***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes)
****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B04{A}1 is B04A1
****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime)
**** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1
***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1
****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1
******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime.
******* The smallest prime of the form B{B}AA1 is BBBAA1
****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1
******* The smallest prime of the form B{B}AA1 is BBBAA1
**** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes)
***** The smallest prime of the form B{B}B1 is BBBB1
***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form B{B}BAA1 is BBBAA1
* Case (B,5):
** B5 is prime, and thus the only minimal prime in this family.
* Case (B,7):
** B7 is prime, and thus the only minimal prime in this family.
* Case (B,B):
** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes)
*** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B
**** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B
***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime.
***** For the B{0,B}9B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes)
******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B
******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 &gt; 1, 10^n+5 ≥ 10^1+5 = 15 &gt; 1, this factorization is nontrivial), thus cannot be prime.
***** For the B{0,B}99B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes)
******* The smallest prime of the form B{0}99B is B00099B
******* The smallest prime of the form B{B}99B is BBBBBB99B
***** For the B{0,B}999B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes)
******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes)
******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime)

== Examples of families which can be ruled out as contain no primes &gt; ''b'' ==

It is not known if this problem is solvable:

Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''?

It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime &gt; ''b'' or not. We use two different heuristic strategies to show that such families contain no primes &gt; ''b''.

In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;.

Examples of first strategy: (we can show that the corresponding numbers are &gt; all elements in ''S'', if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial)

* In base 10, all numbers of the form 4{6}9 are divisible by 7
* In base 6, all numbers of the form 4{0}1 are divisible by 5
* In base 15, all numbers of the form 9{6}8 are divisible by 11
* In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5}
* In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3}
* In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5}
* In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13}
* In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17}
* In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13}
* In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13}

Examples of second strategy: (we can show that both factors are &gt; 1, if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial)

* In base 9, all numbers of the form {1} factored as difference of squares
* In base 8, all numbers of the form 1{0}1 factored as sum of cubes
* In base 9, all numbers of the form 3{8} factored as difference of squares
* In base 16, all numbers of the form 8{F} factored as difference of squares
* In base 16, all numbers of the form {F}7 factored as difference of squares
* In base 9, all numbers of the form 3{1} factored as difference of squares
* In base 16, all numbers of the form {4}1 factored as difference of squares
* In base 16, all numbers of the form 1{5} factored as difference of squares
* In base 16, all numbers of the from {C}D factored as ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;
* In base 16, all numbers of the form B{4}1 factored as difference of squares

Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are &gt; all elements in S, and for the part of the second strategy, both factors are &gt; 1, if n makes corresponding numbers &gt; b, thus these factorizations are nontrivial)

* In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd
* In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even

== Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes &gt; ''b'' ==

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-powers factorization

=== 1{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 1{0}3 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== 1{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 1{0}5 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 5: Finite covering set {5}

=== 1{0}6 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== 1{0}7 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 7: Finite covering set {7}

=== 1{0}z ===

(none)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

(none)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {1} ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with ''r''&gt;1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2))

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' == 0 mod 2: Finite covering set {2}

=== 1{2} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 1{3} ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361)
* ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865)

=== 1{4} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization

=== 1{z} ===

(none)

=== 2{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 2{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {2}1 ===

* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 2{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 3{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {3}1 ===

* ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121)

=== 3{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 4{0}1 ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 4{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {4}1 ===

* ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225)

=== 4{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 5{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 5{z} ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}
* ''b'' = 6''m''&lt;sup&gt;2&lt;/sup&gt; with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014)

=== 6{0}1 ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== 6{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 7{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 7{z} ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993)
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Difference-of-cubes factorization

=== 8{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022)
* ''b'' = 467: Finite covering set {3, 5, 7, 19, 37}
* ''b'' = 722: Finite covering set {3, 5, 13, 73, 109}
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Sum-of-cubes factorization
* ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2

=== 8{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 9{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 9{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{0}1 ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{z} ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 307: Finite covering set {5, 11, 29}
* ''b'' = 901: Finite covering set {7, 11, 13, 19}

=== C{0}1 ===

* ''b'' == 1 mod 13: Finite covering set {13}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 296, 901: Finite covering set {7, 11, 13, 19}
* ''b'' = 562, 828, 900: Finite covering set {7, 13, 19}
* ''b'' = 563: Finite covering set {5, 7, 13, 19, 29}
* ''b'' = 597: Finite covering set {5, 13, 29}

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

(none)

=== {y}z ===

(none)

=== z{0}1 ===

(none)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3))
* ''b'' = 4''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt; (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}t ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 7: Finite covering set {7}

=== {z}u ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== {z}v ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 5: Finite covering set {5}

=== {z}w ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== {z}x ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== {z}y ===

* ''b'' == 0 mod 2: Finite covering set {2}

== Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) ==

Format: base (length)

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

(none)

=== 1{0}2 ===

(none)

=== 1{0}3 ===

(none)

=== 1{0}4 ===

53 (13403)

113 (10647)

=== 1{0}z ===

113 (20089)

123 (64371)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

(none)

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

208 (26682)

607 (11032)

828 (19659)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

201 (31276)

222 (52727)

227 (36323)

327 (135983)

425 (11231)

710 (24112)

717 (37508)

719 (13420)

=== {1} ===

152 (270217)

184 (16703)

200 (17807)

311 (36497)

326 (26713)

331 (25033)

371 (15527)

485 (99523)

629 (32233)

649 (43987)

670 (18617)

684 (22573)

691 (62903)

693 (41189)

731 (15427)

752 (32833)

872 (10093)

932 (20431)

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

(none)

=== 1{z} ===

107 (21911)

170 (166429)

278 (43909)

303 (40175)

383 (20957)

515 (58467)

522 (62289)

578 (129469)

590 (15527)

647 (21577)

662 (16591)

698 (127559)

704 (62035)

845 (39407)

938 (40423)

969 (24097)

989 (26869)

=== 2{0}1 ===

101 (192276)

206 (46206)

218 (333926)

236 (161230)

257 (12184)

305 (16808)

467 (126776)

578 (44166)

626 (174204)

695 (94626)

752 (26164)

788 (72918)

869 (49150)

887 (27772)

899 (15732)

932 (13644)

=== 2{z} ===

432 (16003)

=== 3{0}1 ===

912 (132174)

=== 3{z} ===

72 (1119850)

212 (34414)

218 (23050)

270 (89662)

303 (198358)

312 (51566)

422 (21738)

480 (93610)

513 (38032)

527 (46074)

566 (23874)

650 (498102)

686 (16584)

758 (15574)

783 (12508)

800 (33838)

921 (98668)

947 (10056)

=== 4{0}1 ===

107 (32587)

227 (13347)

257 (160423)

355 (10990)

410 (144079)

440 (56087)

452 (14155)

482 (30691)

542 (15983)

579 (67776)

608 (20707)

635 (11723)

650 (96223)

679 (69450)

737 (269303)

740 (58043)

789 (149140)

797 (468703)

920 (103687)

934 (101404)

962 (84235)

=== 4{z} ===

14 (19699)

68 (13575)

254 (15451)

800 (20509)

=== 5{0}1 ===

326 (400786)

350 (20392)

554 (10630)

662 (13390)

926 (40036)

=== 5{z} ===

258 (212135)

272 (148427)

299 (64898)

307 (26263)

354 (25566)

433 (283919)

635 (36163)

678 (40859)

692 (45447)

719 (20552)

768 (70214)

857 (23083)

867 (61411)

972 (36703)

=== 6{0}1 ===

108 (16318)

129 (16797)

409 (369833)

522 (52604)

587 (24120)

643 (164916)

762 (11152)

789 (27297)

986 (21634)

=== 6{z} ===

68 (25396)

332 (15222)

338 (42868)

362 (146342)

488 (33164)

566 (164828)

980 (50878)

986 (12506)

1016 (23336)

=== 7{0}1 ===

398 (17473)

1004 (54849)

=== 7{z} ===

97 (192336)

170 (15423)

194 (38361)

202 (155772)

282 (21413)

283 (164769)

332 (13205)

412 (29792)

560 (19905)

639 (10668)

655 (53009)

811 (31784)

814 (17366)

866 (108591)

908 (61797)

962 (31841)

992 (10605)

997 (15815)

=== 8{0}1 ===

23 (119216)

53 (227184)

158 (123476)

254 (67716)

320 (52004)

410 (279992)

425 (94662)

513 (19076)

518 (11768)

596 (148446)

641 (87702)

684 (23387)

695 (39626)

785 (900326)

788 (11408)

893 (86772)

908 (243440)

920 (107822)

962 (47222)

998 (81240)

1013 (43872)

=== 8{z} ===

138 (35686)

412 (12154)

788 (11326)

990 (23032)

=== 9{0}1 ===

248 (39511)

592 (96870)

=== 9{z} ===

431 (43574)

446 (152028)

458 (126262)

599 (11776)

846 (12781)

=== A{0}1 ===

173 (264235)

198 (47665)

311 (314807)

341 (106009)

449 (18507)

492 (42843)

605 (12395)

708 (17563)

710 (31039)

743 (285479)

744 (137056)

786 (68169)

800 (15105)

802 (149320)

879 (25004)

929 (13065)

977 (125873)

986 (48279)

1004 (10645)

=== A{z} ===

368 (10867)

488 (10231)

534 (80328)

662 (13307)

978 (14066)

=== B{0}1 ===

710 (15272)

740 (33520)

878 (227482)

=== B{z} ===

153 (21660)

186 (112718)

439 (18752)

593 (16064)

602 (36518)

707 (10573)

717 (67707)

=== C{0}1 ===

68 (656922)

219 (29231)

230 (94751)

312 (21163)

334 (83334)

353 (20262)

359 (61295)

457 (10024)

481 (45941)

501 (20140)

593 (42779)

600 (11242)

604 (17371)

641 (26422)

700 (91953)

887 (13961)

919 (45359)

923 (64365)

992 (10300)

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

(none)

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

38 (136212)

83 (21496)

113 (286644)

188 (13508)

401 (103670)

417 (21003)

458 (46900)

494 (21580)

518 (129372)

527 (65822)

602 (17644)

608 (36228)

638 (74528)

663 (47557)

723 (24536)

758 (50564)

833 (12220)

904 (13430)

938 (50008)

950 (16248)

=== z{0}1 ===

202 (46774)

251 (102979)

272 (16681)

297 (14314)

298 (60671)

326 (64757)

347 (69661)

363 (142877)

452 (71941)

543 (10042)

564 (38065)

634 (84823)

788 (13541)

869 (12289)

890 (37377)

953 (60995)

1004 (29685)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

53 (21942)

124 (16426)

175 (31626)

188 (22036)

316 (48538)

365 (25578)

373 (24006)

434 (10090)

530 (11086)

545 (12346)

560 (15072)

596 (12762)

701 (12576)

706 (10656)

821 (13536)

833 (17116)

966 (14820)

983 (11272)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}y ===

317 (13896)

== Bases 2≤''b''≤1024 which have these families as unsolved families ==

Unsolved families are families which are neither primes (&gt;''b'') found nor can be ruled out as contain no primes &gt; ''b''

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608)

1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000)

1{0}3: 646, 718, 998 (length limit: 2000)

1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000)

1{0}5

1{0}6

1{0}7

1{0}8

1{0}9

1{0}A

1{0}B

1{0}C

1{0}D

1{0}E

1{0}F

1{0}G

1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000)

1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000)

10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000)

11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000)

{1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000)

{1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000)

11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z})

{1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000)

{1}z

1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000)

1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995

1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000)

1{z}: 581, 992, 1019 (length limit: ≥100000)

2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000)

2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000)

2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000)

{2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000)

2{z}: 588, 972 (length limit: ≥100000)

3{0}1: 718 (length limit: ≥100000)

3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000)

3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000)

{3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000)

3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000)

4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000)

4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000)

{4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000)

4{z}: 338, 998 (length limit: ≥100000)

5{0}1: 308, 512, 824 (length limit: ≥100000)

5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000)

6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000)

6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000)

7{0}1: (none)

7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000)

8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000)

8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000)

9{0}1: 724, 884 (length limit: ≥100000)

9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000)

A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000)

A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000)

B{0}1: 560, 770, 968 (length limit: ≥100000)

B{z}: 263, 615, 912, 978 (length limit: ≥100000)

C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000)

C{z}

D{0}1

D{z}

E{0}1

E{z}

F{0}1

F{z}

G{0}1

{&amp;#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000)

{&amp;#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143)

&amp;#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000)

${&amp;#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2)

x{z}

y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000)

z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000)

{y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000)

{z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326)

zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z})

{z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000)

{z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1)

{z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000)

{z}k

{z}l

{z}m

{z}n

{z}o

{z}p

{z}q

{z}r

{z}s

{z}t

{z}u

{z}v

{z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000)

{z}x: (none)

{z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000)

== List of lengths for quasi-minimal primes in some simple families ==

[https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024]

NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases &lt;=7, family {z}x in bases &lt;=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime &gt; base) in this base)

RC: this family can be proven to only contain composite numbers (only count numbers &gt; base)

unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers &gt; base)

Background color: red for title (bases or families), green for length &gt; 10000, orange for 2500 &lt; length ≤ 10000, white for length ≤ 2500, cyan for &quot;RC&quot;, pink for &quot;NB&quot;, yellow for &quot;unknown&quot;.

Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families.

== References ==

* [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem]

* [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are &gt; ''b'' for bases ''b''≤16]

* [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary]

* [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia]

* [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime&gt;base is not required) in The Prime Puzzles &amp; Problems Connection]

* [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles &amp; Problems Connection]

* [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16]

* [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime&gt;base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29]

* [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime&gt;base is not required, search limits of lengths: 10000 for all b]

* [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime&gt;base is not required]

* [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime&gt;base is not required]

* [http://www.prothsearch.com/sierp.html the Sierpinski problem]

* [http://www.prothsearch.com/rieselprob.html the Riesel problem]

* [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem]

* [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030]

* [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100]

* [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem]

* [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024]

* [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project]

* [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project]

* [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count]

* [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty]

* [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers]

* [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares]

* [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software]

* [https://sourceforge.net/projects/openpfgw/ pfgw software]

* [http://jpenne.free.fr/index2.html llr software]

* [http://www.ellipsa.eu/public/primo/primo.html PRIMO software]

* [https://primes.utm.edu/prove/index.html website for primality proving]

* [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!]

* [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime&gt;base is not required]

* [https://primes.utm.edu/primes/lists/all.txt top proven primes]

* [http://www.primenumbers.net/prptop/prptop.php top PRPs]

* [http://factordb.com online factor database, including many primes which are minimal primes in a small base]</text>
      <sha1>ehtsbqzs5ns8fy8x4ismgezwyo1dvd1</sha1>
    </revision>
    <revision>
      <id>2412027</id>
      <parentid>2412026</parentid>
      <timestamp>2022-08-05T03:49:08Z</timestamp>
      <contributor>
        <ip>2402:7500:901:2B90:BD69:8142:3D96:F08F</ip>
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      <comment>/* Unsolved families */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1961173" xml:space="preserve">A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime &gt; ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime &gt; 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 &gt; 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 &gt; 10 by changing the order.

(using A−Z to represent digit values 10 to 35)

For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')).

I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E&lt;sub&gt;800873&lt;/sub&gt;, and base 36 has quasi-minimal prime P&lt;sub&gt;81993&lt;/sub&gt;SZ).

Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to:

* Prove that all elements in ''S'', when read as base ''b'' representation, are primes &gt; ''b''.

* Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are &gt; ''b'', are composite.

* Prove that all primes &gt; ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are &gt; ''b'', are composite).

e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to:

* Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes &gt; 10.

* Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are &gt; 10 are composite.

* Prove that all primes &gt; 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers &gt; 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite).

==Condensed table==

{|class=wikitable
|''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form of largest known quasi-minimal prime base ''b''
|-
|2||1||11||2||3
|-
|3||3||111||3||13
|-
|4||5||221||3||41
|-
|5||22||10&lt;sub&gt;93&lt;/sub&gt;13||96||5&lt;sup&gt;95&lt;/sup&gt;+8
|-
|6||11||40041||5||5209
|-
|7||71||3&lt;sub&gt;16&lt;/sub&gt;1||17||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2
|-
|8||75||4&lt;sub&gt;220&lt;/sub&gt;7||221||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||1161||3×9&lt;sup&gt;1160&lt;/sup&gt;+10
|-
|10||77||50&lt;sub&gt;28&lt;/sub&gt;27||31||5×10&lt;sup&gt;30&lt;/sup&gt;+27
|-
|11&lt;sup&gt;*&lt;/sup&gt;||1068||57&lt;sub&gt;62668&lt;/sub&gt;||62669||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10
|-
|12||106||40&lt;sub&gt;39&lt;/sub&gt;77||42||4×12&lt;sup&gt;41&lt;/sup&gt;+91
|-
|13&lt;sup&gt;*&lt;/sup&gt;||3195~3197||80&lt;sub&gt;32017&lt;/sub&gt;111||32021||8×13&lt;sup&gt;32020&lt;/sup&gt;+183
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||19699||5×14&lt;sup&gt;19698&lt;/sup&gt;−1
|-
|15||1284||7&lt;sub&gt;155&lt;/sub&gt;97||157||(15&lt;sup&gt;157&lt;/sup&gt;+59)/2
|-
|16&lt;sup&gt;*&lt;/sup&gt;||2346~2347||4&lt;sub&gt;72785&lt;/sub&gt;DD||72787||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15
|-
|17&lt;sup&gt;*&lt;/sup&gt;||10408~10428||F70&lt;sub&gt;186767&lt;/sub&gt;1||186770||262×17&lt;sup&gt;186768&lt;/sup&gt;+1
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||6271||12×18&lt;sup&gt;6270&lt;/sup&gt;+221
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||6271||16×20&lt;sup&gt;6270&lt;/sup&gt;+13
|-
|21&lt;sup&gt;*&lt;/sup&gt;||13375~13396||CF&lt;sub&gt;479147&lt;/sub&gt;0K||479150||(51×21&lt;sup&gt;479149&lt;/sup&gt;−1243)/4
|-
|22&lt;sup&gt;*&lt;/sup&gt;||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||22003||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||8134||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49
|-
|30&lt;sup&gt;*&lt;/sup&gt;||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||34206||25×30&lt;sup&gt;34205&lt;/sup&gt;−1
|-
|36&lt;sup&gt;*&lt;/sup&gt;||35257~35263||P&lt;sub&gt;81993&lt;/sub&gt;SZ||81995||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7
|}

&lt;sup&gt;*&lt;/sup&gt; Data assumes the primality of the [[w:probable prime|probable prime]]s.

Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes &gt; 10&lt;sup&gt;25000&lt;/sup&gt;), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below.

Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists))

Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists))

==Data for quasi-minimal primes==

===Base 2===

11

===Base 3===

12, 21, 111

===Base 4===

11, 13, 23, 31, 221

===Base 5===

12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013

===Base 6===

11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041

===Base 7===

14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331

===Base 8===

13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447

===Base 9===

12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011

===Base 10===

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027

===Base 11===

12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, 2888882883, 2888888883, 3555555509, 3577777077, 3700000001, 4000000005, 40000005AA, 5377777707, 5555505553, 555555580A, 600000A999, 7000100047, 7000704777, 7007777107, 7057777777, 7070007774, 7077707774, 7077777074, 7100000704, 7470000041, 7701000004, 7707077774, 7770707774, 7777707074, 8888822883, 9555555503, 9900000979, 9999770007, A000144444, A900000066, A999999971, 10000000477, 33333333337, 44444444447, 44444444777, 55A55555552, 60000008883, 68888888306, 68888888883, 70000003999, 70000007717, 70004777777, 70477777777, 77007770004, 77700000477, 77707000704, 77707770074, 77707777774, 77777077774, 77777770004, 83000000006, 97000000999, A0000000001, A0014444444, A4777777771, 100000000057, 305007777777, 305777777777, 370000000007, 377777770007, 377777777107, 700000007474, 707077000074, 707077777774, 707777777717, 770000010004, 771007000007, 777070700004, 777700000704, A95555555555, A99999777777, 1000000003007, 40000000A0041, 58A5555555555, 7004400000004, 7700000000104, 7707000007047, 7707707000004, 7777007000004, 7777700000004, 7777770077704, 7777777710077, 9977777777717, A000000014444, A044444444441, A144444444411, 40000000000401, 45557777777777, 4555AAAAAAAAAA, 59077777777777, 70007777777771, 70077070000074, 70700000004777, 77000007700704, 77700000700047, 77777777770704, 88888888830006, 90900000000799, A0000044444441, 300000000005777, 302000000000002, 55555555A555552, 700000000000174, 770000000000474, 771700000000007, 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7707777777777777777704, 7900000000000000000005, 9777777777777777777707, A999999999999999999999, 10000000000000000000747, 970000000000000000000777, 999900000000000000000007, 3577777777777777777777777, 5555555555555555555555A52, 7000000000000000000777771, 7000000000000000077777771, AAAAAAAAAAAAAAAAAAA000058, 10000000000000000000000044, 77700000000000000000000008, 500777777777777777777777777, 777777777777777777777770774, 855555555555555555555055555, A44444444444444444444444441, 1500000000000000000000000007, 40000000000000000000000000041, 440000000000000000000000000001, 70000000000000000000000000007771, 999999999999999999999999999999991, 95555555555555555555555555555555553, 1900000000000000000000000000000000001, 7777777777777777777777777777777777474, 7777777777777777777777777777777777704, 10000000000000000000000000000000000000307, 50777777777777777777777777777777777777707, 475777777777777777777777777777777777777777, 555555555555555555555555555555555555555A2A, 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===Base 12===

11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077

===Base 13===

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===Base 14===

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===Base 15===

12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797

===Base 16===

11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD

===Base 17===

12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 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====Additional known quasi-minimal primes (not necessarily the next)====

570&lt;sub&gt;51310&lt;/sub&gt;1, 49&lt;sub&gt;111333&lt;/sub&gt;, 970&lt;sub&gt;166047&lt;/sub&gt;1, F70&lt;sub&gt;186767&lt;/sub&gt;1

===Base 18===

11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, 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===Base 20===

13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, GGJ0J, GGJJJ, GJ00J, GJGJJ, GJJJ7, H00EH, H024B, H04B1, H0E4B, H0F41, H0H11, H0HEH, H4E0B, H6I11, HAAA3, HAAG3, HAGA3, HB44B, HBBBB, HBHHH, HGGA3, HGGG3, HH001, HH061, HH3AH, HH6F1, HHG63, HHIA3, HI011, HIII3, I0029, I02I9, I085H, I0A41, I0F29, I2229, I2929, I2I29, I2I99, I2II9, I33A3, I3EHH, I422B, I424B, I4EEB, I4IEB, I80G1, I9IA9, IA099, IA309, IA841, IA939, IA9A9, IAF09, IAF99, IAI99, IAIA9, IB0A1, IB4IB, IB54B, IBB4B, IBBIB, IBE0H, IBH0H, IC929, ICC29, IF841, IFAG1, IGA81, IHB4B, II0A9, II42B, II44B, IIB5B, III4B, IIIA3, IIIIB, J00G1, J00JB, J0601, J06D1, J06I1, J0861, J08B1, J0B01, J0CGJ, J0E0B, J0GA1, J0GG1, J0I2B, J0J2B, J0JIB, J55CJ, J60A1, J60G1, J6D81, JA777, JA7A7, JAJG1, JAJJ1, JAJJ7, JB00B, JB08B, JC00J, JC0GJ, JEECJ, JGGGJ, JGGJJ, JJ0B1, JJ0JB, JJ55J, JJAG1, JJAJ1, JJC0J, JJE0B, JJEEB, JJGG1, JJGGJ, JJGJJ, JJJ01, JJJEB, JJJJB, 17555J, 175EEJ, 1E0007, 1JE55J, 1JJJ5J, 20005H, 2000HH, 222I29, 22E889, 22EE89, 24222B, 2999I9, 29III9, 2A000H, 2B0I0H, 2I2999, 2I9I29, 2III29, 333AF3, 336IA3, 36A3F3, 36I3A3, 388E8H, 3AF333, 3EHHHH, 3HEHHH, 3HHEHH, 40008B, 40054B, 40405B, 40448B, 415E5J, 44440B, 44452B, 44524B, 44E08B, 45444B, 4AAAE7, 4AEEE7, 4E4447, 4GEEE7, 4I544B, 4IEEEB, 500EFJ, 500FCJ, 5055FJ, 50AFFJ, 50EEFJ, 50EF5J, 50F5FJ, 50FFCJ, 52224B, 54EEEB, 5558AH, 555CAJ, 5585AH, 55F55J, 55FFFJ, 5888AH, 588A5H, 5E555J, 5F055J, 5F05CJ, 5F5F5J, 5FFFCJ, 5JEEEB, 60006D, 6000F1, 6000G3, 6001G1, 600A63, 600G1D, 600G6D, 603A33, 606G6D, 608001, 608F89, 608IF9, 60A333, 60A363, 60AAA3, 60CCCD, 60D04D, 60D0DD, 60F8I9, 60FF89, 60GCG3, 636AF3, 63A333, 63A3F3, 63AFF3, 63IIA3, 64AAAD, 660089, 660A33, 660DDD, 660F89, 662289, 666A33, 666D4D, 66A6A3, 66AF33, 66AF63, 66D0DD, 66FF89, 6A33F3, 6AFF33, 6AG6A3, 6AGA63, 6D000D, 6D004D, 6D0D4D, 6F6689, 7000G7, 7060DD, 70AAAD, 70D0CD, 733C33, 7366C3, 73AAA3, 7600DD, 760CDD, 766333, 76664D, 766FC3, 76CCDD, 76DD4D, 770G07, 771JEJ, 7771EJ, 777481, 7777I1, 777F81, 77C7JJ, 77EC77, 77EEC7, 77EJJJ, 77GCG7, 77JECJ, 7A7E77, 7AAAG3, 7AAF63, 7AEEE7, 7C7GG7, 7CCCAD, 7CCCG7, 7CGJJJ, 7D000D, 7J000D, 7JCGGJ, 80005B, 8000G1, 8000I1, 800GA1, 800I01, 80GGA1, 84405B, 84454B, 888EAH, 88E8AH, 8A4441, 8B8EAH, 8E8IF9, 8I0AG1, 902229, 909A09, 909C29, 909FA9, 90IC29, 9222I9, 92I999, 944EE7, 94AEE7, 977GG7, 97AEE7, 97GGG7, 990I29, 999I29, 99I299, 99IIA9, 9A9447, 9A9AF9, 9A9FF9, 9CII29, 9EEEE7, 9FA9A9, 9I9C29, 9I9I29, 9ICI29, 9IIIA9, A1555J, A1E55J, A333C9, A336A3, A33889, A33F63, A36663, A444E7, A4AAE7, A555FJ, A666A3, A6AAA3, A7A7E7, A7AE77, A944E7, A9A9F9, A9AAA9, AA3389, AA9AF9, AAA099, AAA3C9, AAA7E7, AAA939, AAACG9, AAAFF9, AAAI99, AAAIC9, AAE4E7, AAG9A9, AAGAA3, ADE55J, AF6663, AF9FF9, AGA963, AGG6A3, AH4441, AI000H, AI99C9, AI9AA9, AII099, B0004B, B0054B, B0080B, B00H0H, B04IIB, B05B8B, B0A6I1, B0BB8B, B0BIBB, B0E0IH, B0HA0H, B0I0HH, B0I4IB, B0IIBB, B888AH, BB058B, BB05BB, BB080B, BB0I4B, BBB04B, BBB08B, BBIB8B, BE88AH, BEHA0H, BEHHAH, BHEA0H, BHEHAH, BHHA0H, BIE00H, C007A7, C00GG7, C00JA7, C00JAJ, C00JC7, C0C7A7, C0JAC7, C0JJC7, C3G999, CAJJ77, CC0629, CC0929, CCC007, CCCCC7, CCCCG3, CCCGG7, CCCI29, CCI029, CD4GGJ, CE0009, CEFF69, CEFFF9, CEI609, CII029, CIII29, CJACC7, CJAJC7, CJJJA7, D000D7, D000J7, D004CD, D0D007, DCCDC7, DDD04D, DDDAI1, DEJJJJ, DG000D, DG5E5J, DGGG5J, DJ00CJ, DJEEE7, DJJJ5J, E00097, E00A0H, E00AA9, E02289, E028I9, E029I9, E055AJ, E0774J, E0777J, E07JCJ, E0AF5J, E0BI8B, E0E4CJ, E0EA5J, E0ECJJ, E0FFF9, E0H0HH, E0HA0H, E0IF29, E0J5AJ, E0JJAJ, E29299, E444E7, E4EEE7, E4IIIB, E66289, E66629, E66909, E66F69, E69609, E69999, E7774J, E77C77, E7C7C7, E8B8AH, E90029, E90229, E90909, E90FF9, E92299, E94EE7, E97EE7, E990F9, E99I09, E9F669, E9I029, EA8AF9, EAAAI9, EAIAA9, EAJJJJ, EBIBIB, EC7CC7, EE08F9, EE408B, EE8IF9, EEAF5J, EEE84B, EEEC07, EEEC77, EEEE0B, EEF55J, EEFFFJ, EEJ5AJ, EFE009, EFEF09, EFF669, EFFFF9, EGGGG9, EIF669, F004AJ, F00A41, F00JA1, F04481, F055CJ, F0A15J, F0EI09, F0G01J, F0J0GJ, F0J0JJ, F0JJ5J, F14441, F68F89, F68IF9, F800G1, F8E8I9, F99299, FA0009, FA1JJJ, FAAIA9, FAII09, FAII99, FCC929, FCE229, FCE609, FCEF29, FE68F9, FE8FA9, FF0E09, FF68F9, FF9FA9, FFC4AJ, FFC929, FFCE09, FFE009, FFEE89, FFEF09, FFFA99, FFFFFJ, FFFGJJ, FG1JJJ, FI99A9, FIA999, FIAAA9, FIAII9, FIIA09, G00093, G000EJ, G000G9, G00781, G007G3, G00C09, G07GC3, G09033, G0C903, G0CGG3, G0G903, G0G933, G0GCC3, G0GGC9, G30003, G36003, G5000J, G5E0EJ, G60303, G6AGG3, G6GAA3, G7AAA3, G7G363, G90AA9, G90C63, G9AAA9, GAAAA3, GAAGG3, GC0003, GCCC93, GCCCC3, GCG903, GDEEE7, GE0007, GG0CG3, GGA8G1, GGC6G3, GGCCG3, GGCGC9, GGG0I9, GGG363, GGG6G3, GGG75J, GGG8G1, GGG963, GGGAA3, GGGCG3, GGGCG9, H044EB, H0BIBB, H0E00B, H0F011, H0IIEB, H2444B, H400EB, H44IEB, H4EEEB, HBB4IB, HE000B, HEE40B, HEEE4B, HEHHAH, HF0011, HF4441, HH3HEH, HHGAA3, HHGAG3, HHHAG3, HHHEAH, HI0001, HIBIBB, HIEBBB, I0000H, I000A9, I002HH, I00A09, I00A99, I00AG1, I00AI9, I00H3H, I02999, I03HEH, I09AF9, I0A009, I0E0HH, I0I299, I0IIA9, I4445B, I4I45B, I90I29, I99029, I99A09, I9I029, IA3999, IB000H, IB00EH, IB0HEH, IB5B8B, IF2999, II0929, II92I9, II99A9, II9C29, IIAFF9, IIF299, III5BB, III8BB, J0000B, J00081, J000IB, J0010D, J006A1, J00BI1, J00I8B, J00IA1, J05EEJ, J06G81, J0C0JJ, J0E0CJ, J0EEEB, J0J0EB, J5EAEJ, J5EEAJ, J608I1, JE505J, JEEEEB, JJ0001, JJ0JCJ, JJ0JGJ, JJAJC7, JJJAE7, JJJBI1, JJJI8B, 14GGGG1, 1J5555J, 1JGGGG1, 205555H, 20I000H, 222222B, 2929999, 2BI000H, 3333A33, 3333G09, 333A3A3, 333AAA3, 336AAA3, 336AF33, 33II3A3, 363AF33, 3GIIII9, 400445B, 404454B, 404800B, 415555J, 440045B, 440080B, 442222B, 444004B, 444422B, 444444B, 4444G47, 444GGG7, 444I4IB, 44AAA47, 44I44IB, 44I4IIB, 4AAAAA7, 4EEE4E7, 4I4454B, 5000AFJ, 525555H, 555552H, 555555J, 555585H, 58BBBBB, 5BB8BBB, 5E0E55J, 5EBBBBB, 60003A3, 60008I9, 6000DDD, 6006DDD, 60080A1, 60080I1, 600A3A3, 600D8I1, 600F841, 60CCDDD, 60CGGG3, 60DDD0D, 60DDDDD, 60GGG63, 60GGGC3, 66600A3, 66603A3, 66608I9, 6666089, 66666A3, 6666C29, 6666F89, 666AFF3, 66DDDDD, 6800001, 6AGGGG3, 6CCCC29, 6G6AAA3, 6GAGGG3, 6GCCC63, 6GGG633, 700000D, 70000CD, 7000EC7, 70070C7, 700EEC7, 7070EC7, 70EEEC7, 76666CD, 77333A3, 7733A33, 775EE5J, 777A777, 777CGJJ, 777GE5J, 77A7777, 77JCJJJ, 77JJJCJ, 77JJJJJ, 7A77777, 7A777A7, 7AAAA63, 7AAAAAD, 7C00007, 7E77JCJ, 7EE7JCJ, 7EEC007, 7EEEEC7, 7G5E55J, 7GE555J, 7GJJJJJ, 7J6666D, 9000029, 9000299, 9009I29, 900I2I9, 90FAAA9, 90I0299, 90I0I29, 90I9929, 90II029, 90II299, 90II2I9, 94444E7, 97777E7, 9902I99, 99902I9, 9992929, 9992999, 9999929, 99I0C29, 99I9029, 99I92I9, 99III29, 9AAAF09, 9AFFFF9, 9I0AAA9, 9IAAA09, 9II0I29, 9II9929, 9III029, 9III2I9, 9IIIC29, A5F555J, A777E77, A994447, A9AAF99, AA00089, AA6A6A3, AAA0009, AAA3999, AAA9A99, AAA9AA9, AAAAA47, AAAAA77, AAAAA99, AAAAG99, AAAAIA9, AAAD447, AAAGAC9, AAGAAAD, AAGAAC9, AF00009, AGAAA99, AGAGG63, AGGGA63, AI0C999, AJ777A7, B000B8B, B000HEH, B000I0H, B00I00H, B0B805B, B0HHHEH, B0IHE0H, B0III8B, B8B000B, BB8BBBB, BBB8B0B, BEH0HHH, BHHHEHH, C00000J, C000C07, C066629, C0AJJJJ, C0C0A47, C0G000J, C0JAJJJ, C0JJAJJ, C0JJJJJ, C6CCG33, CC004A7, CC00A47, CC6CG33, CC92229, CCCC7G7, CCCGGG3, CE66609, CEF6669, D00007D, D00007J, D00071J, D000C0J, D000I01, D00700D, D00C007, D00C0C7, D00G00D, D0C00C7, D0CDCC7, D444447, D7JJJJJ, DC0000J, DDD7DDD, DDDCCC7, DJ0000J, DJ000EJ, DJJJJ0J, DJJJJJJ, E000009, E00084B, E000889, E000909, E000HAH, E008AF9, E04000B, E0AA8F9, E0CJJJJ, E0HHHHH, E40400B, E6660F9, E666609, E669969, E699669, E7CCC07, E996669, E99I669, E9FAAA9, EB8BBBB, EBBII8B, EBH0HHH, EE092I9, EE44IIB, EE4E4E7, EEBB8BB, EEC0007, EEE0IBB, EEE0IIB, EEEBB8B, EEEE44B, EEJCJJJ, EF00F09, EF0555J, EFAAAA9, EFF0009, EFF00F9, EFF60F9, EI0AAA9, EI660F9, EIIBB8B, EJ0CJJJ, EJ5055J, EJJ0CJJ, F0E00F9, F0E88I9, F0JJJJ1, F299929, F4G555J, F5555CJ, F5555FJ, F6668I9, F8GGGG1, F929999, F999C29, FA84441, FAAA9A9, FCE00F9, FCEF009, FCFEE29, FE00F09, FEE2889, FEF0009, FFAA9A9, FFCEE29, FFFF4AJ, FIII9A9, FIIIAI9, FJJJJ81, G000009, G0000J7, G000303, G000AA9, G000CC3, G000GG3, G003003, G0090C3, G00CG03, G00D007, G00G0G3, G00GG03, G00GG93, G080001, G0G00I9, G0G09C3, G0G0C93, G0G0GG9, G0GG093, G0GGC63, G0GGGG9, G44GG47, G6AAGA3, G999999, GAA9AA9, GCCCGG9, GCGGCC9, GCGGG03, GEE4447, GEEEE47, GG0C007, GG0GC09, GG0GGG3, GG6AAA3, GG6AGA3, GG6GAG3, GGEEE47, GGG00C9, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 21===

12, 18, 1A, 1G, 1K, 21, 25, 2B, 2H, 2J, 34, 38, 3A, 3G, 3K, 45, 4D, 4H, 4J, 52, 54, 58, 61, 65, 6B, 6D, 72, 74, 7A, 7G, 7K, 85, 8B, 8D, 92, 94, 98, 9A, A1, AD, AH, AJ, B2, B8, BA, BK, C5, CB, CH, CJ, D4, D8, DA, DK, ED, EH, EJ, F2, FG, G1, GB, GD, GH, H2, HA, HG, I1, I5, IB, IJ, J2, JA, JK, K1, KB, KD, KJ, 111, 115, 11H, 135, 13H, 13J, 151, 15B, 15H, 164, 175, 17B, 17D, 17J, 191, 19B, 19D, 19H, 1B1, 1B5, 1BB, 1BJ, 1D5, 1DD, 1DJ, 1E4, 1F1, 1F5, 1FD, 1FH, 1HB, 1HD, 1I4, 1JD, 1JH, 1JJ, 22D, 232, 29G, 29K, 2DG, 2F4, 2FK, 31H, 322, 33D, 351, 355, 35B, 35J, 362, 371, 37B, 37D, 37H, 37J, 39B, 39J, 3B5, 3BD, 3BH, 3D1, 3D5, 3DB, 3DD, 3DH, 3E2, 3FJ, 3HD, 3HH, 3HJ, 3J1, 3JB, 3JJ, 414, 481, 49K, 4B4, 4BG, 4C1, 4EB, 4F4, 4F8, 4FA, 4FK, 50G, 51B, 51D, 51H, 531, 535, 53D, 53J, 551, 56A, 56G, 56K, 575, 57J, 595, 59H, 5B1, 5B5, 5BB, 5CA, 5CG, 5CK, 5F1, 5FB, 5FJ, 5GA, 5GG, 5HH, 5IA, 5J5, 5JD, 5JH, 60H, 632, 66H, 66J, 678, 68J, 69G, 6F8, 6FA, 6GJ, 6H8, 6HK, 6IH, 6J4, 6JG, 6KH, 711, 71B, 71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, GGJ, GJ4, GK5, H0K, H15, H1B, H1J, H31, H3D, H3H, H48, H51, H55, H5J, H6K, H75, H84, H88, H91, H95, H9D, H9H, HBD, HC4, HC8, HDJ, HF5, HFB, HFH, HHD, HHJ, HI4, HI8, HJ5, HJB, HKK, I0D, I2D, I6H, I78, I8H, I9K, IFA, IFK, IID, J08, J1J, J31, J35, J3J, J44, J5H, J68, J6G, J71, J7B, J7D, J7H, J8G, J95, J9D, JBD, JBH, JBJ, JCG, JDB, JDH, JE4, JE8, JEG, JF5, JFD, JFJ, JG4, JGG, JH1, JH5, JHB, JHH, JI4, JJ1, JJ5, K0H, K5G, K6H, K9K, KG5, KH4, KKH, 101B, 103D, 1044, 1055, 10DH, 10FB, 10H1, 10H5, 10J1, 110J, 11CD, 11EB, 11ID, 130D, 13C1, 13EB, 141B, 1444, 1471, 14FB, 1505, 15CD, 167H, 16FJ, 16HH, 176H, 17C1, 17IH, 190J, 1BF4, 1BID, 1C04, 1C3D, 1CD1, 1D6H, 1DE1, 1DEB, 1E1B, 1E71, 1E95, 1EFB, 1F0B, 1F4B, 1FB4, 1H01, 1H05, 1H41, 1H6H, 1H6J, 1HC1, 1HE1, 1HF4, 1IDH, 1IHH, 1JE5, 1JF4, 20CD, 20D2, 20ID, 2278, 22F8, 239D, 23FD, 2478, 2708, 27FD, 293D, 29DD, 2A78, 2C0D, 2CF8, 2F0A, 2F9D, 2FAA, 2FDD, 2GFA, 2IF8, 3015, 301D, 301J, 3031, 3035, 305D, 30F1, 30FB, 30FD, 30HB, 30I2, 3105, 31C1, 329D, 330H, 336J, 33E1, 33EB, 33IH, 35CD, 363H, 390H, 39E5, 39ID, 3B0J, 3B6J, 3BC1, 3BEB, 3CC2, 3CF1, 3CFD, 3D32, 3E3B, 3E91, 3EF1, 3EFB, 3F05, 3FC1, 3FE1, 3FEB, 3H01, 3H0B, 3HC1, 3I3H, 3I9D, 404B, 4078, 40E1, 411B, 4131, 4171, 41FB, 4331, 4391, 43BB, 43FB, 43I2, 4432, 444B, 44E1, 477B, 4931, 4971, 499B, 49CG, 4A32, 4A4B, 4A78, 4A9G, 4B1B, 4B31, 4B3B, 4BFB, 4C78, 4E32, 4F1B, 4FB1, 5011, 509D, 50BD, 50DB, 50DD, 50EK, 50FD, 50H1, 50HJ, 50IK, 50JJ, 5101, 5105, 51E1, 530B, 53EB, 53IH, 550D, 550H, 55AB, 55BG, 55DG, 55E5, 55KH, 563H, 56JJ, 5701, 570H, 57EB, 57ID, 57IH, 590D, 59C1, 59CD, 5A3B, 5A7B, 5AGK, 5AIG, 5AKG, 5B0D, 5B9G, 5BCD, 5BGJ, 5BIH, 5C91, 5CD1, 5D0D, 5D0J, 5D6H, 5D6J, 5DC1, 5DG5, 5DIH, 5DJG, 5E0A, 5E11, 5E3B, 5E71, 5EAK, 5EEA, 5EKK, 5F0H, 5F9K, 5G0K, 5G9J, 5GF5, 5GIK, 5H0B, 5H5K, 5H9K, 5HE5, 5HID, 5I3H, 5I7D, 5IKG, 5J6J, 5JE1, 5KAG, 5KEK, 5KFH, 5KH5, 60FK, 63FH, 67JH, 69JH, 6E9K, 6F0K, 6F64, 6FJH, 6H04, 6HE4, 6HFJ, 6IF4, 6J9H, 6J9J, 7015, 7031, 703B, 7055, 705J, 7088, 709B, 709D, 709H, 70DJ, 70I8, 710J, 716J, 730B, 73C1, 73E5, 7501, 769H, 76FJ, 76JJ, 770D, 77E1, 77E5, 77EB, 7888, 78FH, 78H1, 78JH, 790H, 79CD, 79IH, 7B0J, 7B6H, 7C31, 7C91, 7C9D, 7E35, 7E75, 7E91, 7EF1, 7F01, 7F78, 7FCD, 7FH8, 7FIH, 7FJ8, 7H81, 7HC1, 7HF8, 7J6J, 7J78, 7JC1, 7JF8, 7JH8, 7JJ8, 809K, 80H8, 81H1, 81HH, 81J1, 839H, 83FH, 869K, 871H, 8731, 8771, 8788, 879J, 87F1, 87FJ, 87JH, 888H, 8911, 8931, 896K, 8A78, 8A9K, 8CC1, 8E9K, 8F3H, 8F9H, 8FF1, 8FHH, 8FJH, 8G9K, 8GJ8, 8H3J, 8H7H, 8H8K, 8HEK, 8HF1, 8HFJ, 8HH1, 8HJJ, 8HK8, 8J7J, 8J91, 8JC8, 8JFH, 8JJH, 8K78, 8K8H, 8KH8, 901J, 9035, 903B, 9071, 90CG, 90GG, 90JD, 9101, 91E5, 9301, 93E1, 93EB, 95CD, 95DG, 95FK, 967J, 969H, 96HH, 96HJ, 96JJ, 9705, 97C1, 9905, 995K, 999G, 99CD, 99EB, 99K5, 9B5G, 9BE5, 9BID, 9C0G, 9C11, 9C71, 9C9D, 9CEG, 9CGG, 9D05, 9D5G, 9DDG, 9DE5, 9E35, 9E3B, 9E71, 9EB5, 9EEG, 9EF1, 9EFB, 9F0J, 9F6H, 9F9K, 9FC1, 9FIH, 9FKH, 9G35, 9G95, 9GIK, 9GJJ, 9GKG, 9H0H, 9H5K, 9H6H, 9HKH, 9I6K, 9I9H, 9IDD, 9J6J, 9JEB, 9JGJ, 9JID, 9JJG, 9K35, 9K95, 9KEG, A33B, A3F5, A432, A478, A53B, A5IK, A788, A795, A878, A975, A99B, A9CG, A9GG, AAAB, AB44, AB55, ABFB, AC32, AC78, ACFA, AE32, AE5A, AE9K, AEEB, AEFA, AEFK, AF75, AFCA, AFCK, AFEA, AFFB, AG9G, AGFA, AI32, B064, B091, B09H, B0B1, B0D5, B0DH, B0EG, B0FB, B0FH, B0H5, B0HD, B114, B141, B14B, B1B4, B1E1, B411, B431, B444, B4B1, B59G, B5G5, B5JG, B67H, B67J, B69H, B6FH, B6FJ, B6HH, B6JH, B96H, B9DG, B9G5, BB14, BBB4, BC1D, BC31, BCDD, BD05, BD0H, BD6H, BDC1, BDE5, BDIH, BE0G, BE31, BEBB, BEE4, BEIG, BF0B, BF0H, BF6J, BFEB, BFIH, BG0G, BG55, BG95, BGG4, BGGG, BGIG, BGJ5, BH0B, BHEB, BIEG, BIGG, BIHH, BJ01, BJ0J, BJC1, BJF4, BJID, C041, C081, C09G, C0E1, C0FK, C104, C131, C171, C1C4, C1D1, C2F8, C2FA, C39D, C3F1, C478, C6FK, C7FD, C8C1, C90G, C911, C971, C99D, CAFK, CC78, CC81, CCE1, CCF8, CD02, CD31, CDFD, CE01, CE32, CF0A, CF48, CF64, CF68, CF6A, CF91, CFAK, CFDD, CFEA, CFEK, CIF4, CK32, CK9G, CKFA, D01D, D01H, D07B, D0DB, D0F1, D0FD, D0HD, D0HH, D10D, D10H, D16H, D1EB, D1IH, D23D, D3E1, D56H, D63J, D6CG, D6IG, D70B, D70D, D70J, D99G, D9GJ, D9ID, D9JG, DBC1, DBCD, DBE1, DBJG, DC11, DC9D, DCF1, DCFD, DCGG, DCIG, DD0H, DD6J, DDGJ, DDID, DDJG, DE31, DE75, DFC1, DFID, DFIH, DG22, DG6G, DG75, DG9J, DGCG, DGIG, DH6H, DHE5, DI6G, DICG, DIDH, DIFD, DIFH, DIGG, DJCD, DJDG, DJG5, E032, E05A, E0EB, E11B, E2F8, E31B, E331, E375, E395, E3C2, E3F1, E3F5, E3FB, E3I2, E53B, E555, E591, E59B, E5EG, E6F4, E755, E775, E89K, E915, E935, E93B, E96K, E9FB, EA32, EAEB, EAFK, EB44, EB95, ECFA, ECFK, EE32, EEEB, EEFK, EF08, EF35, EF48, EF55, EF64, EF68, EF75, EF95, EFB1, EFCA, EFCK, EFEA, EFIA, EFKK, EGF4, EIF4, EKF4, EKF8, EKFA, F008, F051, F075, F095, F09H, F0CA, F0CK, F0DD, F0FB, F0H1, F0IA, F0JJ, F105, F10B, F10D, F16J, F1B4, F1EB, F1ID, F1J4, F36H, F468, F47B, F491, F4C8, F501, F50B, F648, F66A, F69H, F705, F778, F864, F8F1, F90D, F99K, F9C1, F9HK, FA6A, FA75, FA7B, FA95, FAEA, FAFB, FB0D, FB14, FBCD, FBE1, FBJ4, FC1D, FC68, FC9D, FCAA, FCCK, FCD1, FCEA, FD6H, FDIH, FE0K, FE1B, FE64, FE95, 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====Additional known quasi-minimal primes (not necessarily the next)====

40&lt;sub&gt;47333&lt;/sub&gt;9G, CF&lt;sub&gt;479147&lt;/sub&gt;0K

===Base 22===

11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, IHH, IIJ, IJ7, IL7, J07, J55, J69, J8J, J99, J9J, JA5, JAH, JE7, JEH, JHH, JI9, JJ5, JJ9, JJH, JK7, K2J, K51, K5D, K75, K81, KA1, KB1, KB7, KBJ, KD1, KEJ, KG5, KIF, KJ5, KKD, KKJ, L0D, L47, L7F, L97, LAF, LD3, LD9, LDD, LEF, LGD, LI7, LJ7, LJJ, LLD, 104H, 10D5, 1205, 12B5, 140H, 1433, 144H, 14AH, 14B3, 16ED, 1AIH, 1B43, 1DD5, 1DDD, 1E6D, 1EGD, 1G05, 1GDD, 1GED, 1GGD, 1HB3, 1HHH, 1IAH, 200L, 2015, 2051, 20A1, 20DJ, 20GD, 20IL, 21B5, 21DD, 220D, 226D, 228D, 22B5, 22G5, 22K5, 22KD, 2555, 2557, 2581, 25C1, 26A1, 26B1, 2725, 2755, 2801, 2861, 288D, 28B1, 28KD, 2AA1, 2B25, 2B51, 2BB1, 2C81, 2D6D, 2DA1, 2DDJ, 2DGD, 2G0J, 2GB5, 2GDD, 2GGJ, 2I0L, 2I6L, 2ICL, 2J05, 2JK5, 2K07, 2K2D, 2K55, 2K6D, 2KB5, 2KI7, 2L2D, 2L8D, 2LK7, 302L, 30G3, 320L, 32IL, 332L, 33G3, 33G9, 36AF, 36EF, 382J, 388J, 39EL, 3AAF, 3BBJ, 3CG9, 3E2J, 3E6F, 3E6L, 3EEF, 3FAF, 3G69, 3GI3, 3GI9, 3IG9, 3LEL, 3LG3, 4025, 404H, 4063, 4075, 40AH, 40B5, 40B7, 40GH, 4225, 4363, 4447, 444H, 447H, 4487, 44B7, 44H7, 4525, 4555, 45B5, 4663, 4777, 47GH, 4807, 4B0H, 4BB5, 4BG3, 4EBH, 4G4H, 4GB3, 4HH3, 4I03, 4I63, 4IGH, 5069, 5077, 50KD, 5255, 52IJ, 5455, 5477, 5507, 5527, 556F, 557F, 5587, 56EF, 56FF, 56GD, 56KD, 5771, 57C1, 57EF, 5807, 580J, 589J, 58E7, 58IJ, 58K7, 5905, 5945, 5957, 5969, 5989, 598J, 5999, 59EJ, 59K7, 5AEF, 5B25, 5B55, 5BA5, 5BI7, 5BIJ, 5BK5, 5BK7, 5C21, 5EAF, 5EE7, 5F09, 5F6F, 5F95, 5FKD, 5G6D, 5G89, 5I09, 5I0F, 5I27, 5I6F, 5I89, 5IB7, 5IBJ, 5IFF, 5J77, 5K07, 5K25, 5K27, 5K6F, 5K87, 5KK7, 6013, 60A1, 60AF, 60EL, 6281, 62B1, 62KD, 63G9, 6403, 6643, 66EL, 66G9, 68KD, 69B3, 6A01, 6A0F, 6AAF, 6AFF, 6B21, 6B8D, 6BBD, 6BK1, 6D03, 6D43, 6D69, 6D6F, 6D93, 6D9F, 6E21, 6E81, 6E8D, 6EBD, 6ECL, 6ELL, 6FD9, 6GBD, 6GG9, 6IEF, 6K21, 6K8D, 6KBD, 6KE1, 6L43, 6LB3, 6LDF, 6LEL, 6LG9, 700F, 7027, 7057, 70EH, 70LF, 710H, 71G5, 7207, 7225, 7255, 727L, 7505, 7507, 755F, 75EF, 75F5, 75I7, 766F, 7681, 76CF, 7771, 7781, 77IL, 7861, 7A0H, 7A4H, 7BB5, 7C6F, 7EEH, 7EHL, 7FG5, 7G45, 7HEL, 7I0L, 7I27, 7IE7, 7IGH, 7K07, 7K61, 7K77, 7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, H0D1, H0HJ, H0LL, H1HH, H20D, H22D, H26D, H2I7, H2K7, H30L, H3KJ, H40H, H447, H4G3, H4H3, H4HH, H66L, H6ED, 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5IIIIIIIIIIIIIIIIIIIIIIIIIIF, D555555555555555555555550555, EEAAAAAAAAAAAAAAAAAAAAAAAAAF, HHHHHHHHHHHHHHHHHHHHHHHHEEBH, K66666666666666666666666666F, LLLLLLLLLLLLLLLLLLLLLLLLEEB7, D5555555555555555555555555A55, GGGGGGGGGGGGGGGGGGGGGGGGGGGG3, GIG0000000000000000000000000H, HH00000000000000000000000001H, K0000000000000000000000005KEF, 5BBBBBBBBBBBBBBBBBBBBBBBBBBBBD, HB0000000000000000000000000001, K000000000000000000000000505EF, L7777777777777777777777777772L, 2000000000000000000000000000CB1, C8CCCCCCCCCCCCCCCCCCCCCCCCCCCC9, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKFF, JE0000000000000000000000000000J, K000000000000000000000000000261, A0000000000000000000000000004I4H, HD000000000000000000000000000001, K000000000000000000000000000EC01, K0FFFFFFFFFFFFFFFFFFFFFFFFFFFFCF, D0002222222222222222222222222222D, FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL2L, I700000000000000000000000000000GH, K00000000000000000000000000000E61, 20000000000000000000000000000000JJ, DD5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL0KE7, 44HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, F0000000000000000000000000000000000000000000000000000000L89, J000000000000000000000000000000000000000000000000000000JJIJ, K222222222222222222222222222222222222222222222222222222222DD, KCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCF, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKCF, D55KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, K000000000000000000000000000000000000000000000000000000000000077, 22222222222222222222222222222222222222222222222222222222222222222D, K000000000000000000000000000000000000000000000000000000000000008IJ, DFA00000000000000000000000000000000000000000000000000000000000000005, CC4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJE0J, 777777777777777777777777777777777777777777777777777777777777777777777777EK7, 6000000000000000000000000000000000000000000000000000000000000000000000000000000043, KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKB5, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEEH, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 50000000000000000000000000000000000000000000000000000000000000000000000000000000002C1, K0000000000000000000000000000000000000000000000000000000000000000000000000000000000055EF, H700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000K1, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0I7, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, A400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, E0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000071, 7LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIL, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLK77, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEI7, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIB3, I7G00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 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77EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEK7, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJKJ, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777772L, BKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK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KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK5

===Base 24===

15, 17, 1D, 1H, 1J, 1N, 25, 2B, 2D, 2J, 2N, 31, 37, 3B, 3H, 41, 45, 47, 4B, 4D, 4H, 57, 5B, 5H, 5J, 65, 67, 6D, 6J, 6N, 75, 7B, 7D, 7N, 81, 85, 87, 8J, 97, 9B, 9D, 9H, 9N, A1, AB, AH, AN, B5, B7, BD, BH, BJ, C5, CJ, CN, D1, D5, DJ, E1, EB, ED, EH, EN, F7, FD, FJ, FN, G5, GD, GH, H1, HB, HD, HN, I1, I7, IB, IH, J1, J5, J7, JB, JN, K7, KB, KJ, KN, L5, LH, LJ, MD, MJ, N5, NB, NH, NJ, 101, 10B, 111, 1F1, 1FB, 1GB, 1LB, 201, 221, 22H, 261, 271, 277, 28H, 2A7, 2C7, 2G7, 2H7, 2L1, 2L7, 2MH, 305, 30D, 30J, 33N, 34N, 35D, 35N, 38D, 395, 3A5, 3AJ, 3CD, 3DD, 3DN, 3E5, 3EJ, 3GJ, 3IJ, 3JJ, 3K5, 3KD, 3ND, 43N, 44N, 49J, 4EJ, 4GJ, 4GN, 4NN, 50N, 535, 54N, 551, 55N, 5C1, 5CD, 5E5, 5K1, 5KD, 5LN, 5M5, 5N1, 601, 60B, 61B, 66H, 68B, 691, 6CH, 6FH, 6GB, 6HH, 6MH, 70H, 70J, 711, 761, 771, 77H, 77J, 78H, 7C7, 7CH, 7FH, 7G7, 7H7, 7HH, 7IJ, 7JJ, 7K1, 7M1, 7M7, 80D, 82H, 83N, 88D, 88H, 8AD, 8CD, 8DB, 8DD, 8DH, 8DN, 8GB, 8KD, 8MB, 8MH, 905, 911, 921, 935, 955, 99J, 9AJ, 9G1, 9JJ, 9K5, 9L1, 9M5, A0J, A3J, A95, AA7, AD7, AE5, AG7, AGJ, AI5, AIJ, AJD, AL7, ALD, B01, B0N, B11, B61, B6B, B8N, B91, BIN, BL1, BLN, BNN, C1B, C21, C27, C2H, C3D, C61, C8H, C91, CA7, CB1, CBB, CC7, CCB, CCD, CDD, CFB, CG1, CGB, CK1, CL1, CMB, CMH, D0B, D3D, D3N, D4N, D6B, D6H, D7H, D8B, D8N, DAD, DCD, DCH, DDH, DDN, DG7, DGB, DID, DMN, DND, E05, E4J, EA7, EEJ, EF5, EGJ, EI5, EJJ, EM5, EM7, F01, F21, F51, F8H, F95, FC1, FF1, FFB, FKH, FM5, G0B, G0N, G11, G3N, G6B, G77, G7J, G8B, G8N, G91, GA7, GBB, GC7, GFB, GG1, GGJ, GGN, GK1, GL1, GLN, GMN, GN1, GNN, H0J, H2H, H3J, H4J, H77, HA5, HA7, HE5, HFH, HIJ, HJJ, HKH, HL7, HMH, I0N, I3D, I3J, I3N, I4N, I5D, I95, IA5, IAJ, IE5, IEJ, IF5, IGJ, IJD, IK5, IKD, J0D, J4J, J8D, JAD, JDH, JEJ, JFH, JHH, JKD, JMH, K35, K6H, KCD, KFH, KH5, KLD, KM1, L01, L0B, L0D, L0N, L61, L6B, L8D, LA7, LC7, LDD, LF1, LG7, LGB, LGN, LID, LK1, LKD, LL1, LLB, LLD, LMN, LNN, M0H, M11, M21, M4N, M71, M91, M95, MA5, MA7, MBN, MC7, MF1, MF5, MFB, MFH, MG7, MI5, MIN, ML1, ML7, MLB, MMH, N01, N21, N4N, N71, N8N, NC1, ND7, NE7, NG1, NID, NK1, NL7, NMN, NN7, 11CB, 11MB, 1291, 12G1, 16C1, 16CB, 16K1, 186B, 18CB, 19K1, 1BK1, 1C8B, 1K91, 1KC1, 1KL1, 1L21, 1LC1, 1LM1, 1M61, 1M8B, 1MG1, 206H, 20CH, 20M7, 21C1, 21M1, 2207, 260H, 26KH, 2991, 2C6H, 2CC1, 2CM1, 2F11, 2FHH, 2MC1, 2MK1, 2MM1, 308N, 30GN, 30IN, 30LN, 30MN, 333J, 33JD, 33LD, 343J, 344J, 35I5, 380N, 393J, 394J, 3A3D, 3FI5, 3IMN, 3J3D, 3JID, 3L3D, 3L8N, 3M0N, 3M55, 3NGN, 404J, 408N, 40LN, 434J, 44AJ, 4ILN, 4JAJ, 4L8N, 5091, 5095, 50F1, 50I5, 51L1, 5211, 5291, 52G1, 53ID, 53MN, 5595, 55AD, 56F1, 588N, 58MN, 58ND, 5961, 5991, 5A5D, 5AAD, 5F91, 5GF1, 5GIN, 5I05, 5I55, 5I8D, 5IDD, 5IDN, 5IIN, 5IMN, 5KI5, 5M61, 5M8N, 5N3N, 602H, 6211, 62F1, 62G1, 66C1, 66FB, 66M1, 66MB, 6B21, 6BM1, 6BMB, 6C6B, 6CF1, 6CLB, 6FG1, 6K21, 6K2H, 6KG1, 6KKH, 6L21, 6LCB, 6LM1, 6MB1, 6MBB, 6MG1, 6MK1, 7001, 7027, 7207, 726H, 739J, 793J, 79C1, 7A4J, 7A9J, 7AE7, 7C01, 7CC1, 7FL1, 7G21, 7G9J, 7GAJ, 7GC1, 7HGJ, 7J2H, 7J6H, 7MKH, 800B, 800H, 804N, 806H, 808N, 80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, NK3D, NK8D, 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5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, 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EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEG7, 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===Base 30===

11, 17, 1B, 1D, 1H, 1N, 1T, 21, 27, 2B, 2D, 2J, 2N, 2T, 37, 3B, 3D, 3H, 3J, 3N, 47, 4B, 4H, 4J, 4T, 51, 57, 5D, 5H, 5N, 5T, 61, 6B, 6D, 6H, 6J, 71, 7D, 7H, 7J, 7N, 7T, 81, 8B, 8H, 8N, 8T, 91, 97, 9B, 9D, 9N, A7, AB, AD, AH, B1, B7, BH, BJ, BN, BT, C7, CD, CJ, CN, CT, D7, DB, DJ, DT, E1, EB, ED, EJ, EN, ET, F7, FB, FD, FH, FT, G7, GB, GJ, GN, GT, HB, HD, I1, I7, IH, IN, IT, J1, J7, JH, JN, JT, K1, K7, KD, KH, KJ, L1, LB, LD, LH, LN, LT, M1, MD, MH, MN, N1, NB, NJ, NT, O7, OD, OJ, ON, P1, P7, PB, PJ, PN, Q7, QH, QT, R1, RB, RD, RH, RJ, RT, SD, SH, SJ, SN, T7, TB, TD, TH, 10J, 15J, 1IJ, 1JJ, 1LJ, 1MJ, 1QJ, 22H, 29H, 2EH, 2GH, 30T, 331, 33T, 36T, 39T, 3A1, 3C1, 3G1, 3KT, 3MT, 3OT, 3S1, 3T1, 40D, 40N, 431, 44D, 46N, 48D, 4AN, 4DD, 4DN, 4F1, 4FN, 4GD, 4ID, 4PD, 4S1, 50J, 58J, 59J, 5CB, 5FJ, 5IB, 5IJ, 5JB, 5MB, 5MJ, 5OB, 5SB, 607, 63T, 687, 6E7, 6KT, 6L7, 6M7, 6MT, 6NN, 6QN, 6R7, 6S7, 6ST, 6TT, 70B, 77B, 787, 7KB, 7M7, 7MB, 7SB, 807, 80D, 80J, 84D, 85J, 877, 88J, 89J, 8DD, 8FJ, 8ID, 8IJ, 8JJ, 8M7, 8MJ, 8PD, 8QD, 8R7, 90H, 93T, 95J, 99H, 99J, 9AJ, 9AT, 9EH, 9GH, 9HH, 9HJ, 9JJ, 9MJ, 9OH, 9OT, 9PH, 9ST, 9TT, A01, A0T, A1J, A31, A6N, A6T, AAJ, AAN, AFN, AKN, AKT, ALJ, AMJ, AMT, AO1, AOT, AQ1, AQN, ARN, AT1, B5B, BBD, BCB, BDD, BID, BOB, BPD, BQB, C31, C9H, CC1, CCB, CCH, CF1, CH1, CIB, CKB, CMB, D01, D0H, D2H, D41, D4D, D6N, D8D, D9H, DDH, DDN, DGH, DH1, DHH, DID, DKN, DNN, DO1, DOH, DQN, DS1, EEH, EHH, EM7, EOH, EPH, ER7, F0N, F31, F5J, F8J, F9J, FLJ, FO1, FQ1, FQN, FS1, G01, G4D, G8D, GF1, GGH, GO1, GOH, GQD, GS1, H07, H0J, H0T, H1J, H2H, H31, H4N, H8J, HA1, HAJ, HAT, HC1, HE7, HEH, HFN, HGH, HH7, HIJ, HJJ, HKN, HL7, HNH, HPH, HQN, HS1, HTJ, HTN, I4D, I5B, I5J, I8D, IAJ, IDD, IGD, IIJ, IKB, IMB, IMJ, IOB, IPD, IQD, J8J, J9J, JAJ, JBD, JCB, JFJ, JIJ, JJB, JJD, JLJ, JPD, JQB, K3T, K4N, KAT, KBB, KCB, KMB, KNN, KOB, KOT, KQN, KST, KTT, L0J, L5J, L67, LAJ, LJJ, LQJ, LR7, M0J, M3T, M5B, M7B, M87, M9J, MAT, MFJ, MIJ, MJJ, MKB, MMJ, MOT, MQJ, N07, N0H, N67, N6N, N87, NAN, NDH, NGD, NHH, NKN, NN7, NNH, NPH, NQD, NQN, NR7, O01, O0B, O0H, OAT, OC1, OCH, OEH, OF1, OH1, OKT, OMB, OOT, OPH, OQ1, OQB, OS1, OST, P3T, P6T, P9H, PCH, PDH, PEH, PGD, PHH, PHT, PID, PMT, PPT, PQD, PST, PTT, Q5B, Q6N, Q9J, QAJ, QBB, QBD, QC1, QDN, QFJ, QFN, QGD, QJB, QKN, QLJ, QMB, QND, QNN, QO1, QQ1, QQN, QSB, R4N, R77, R87, RAN, RKN, RM7, RNN, RR7, RS7, S0T, S41, S6T, S87, SBB, SC1, SM7, SMT, SOB, SQ1, SR7, SS1, STT, T0J, T6T, T9T, TAN, TFN, TKN, TKT, TNN, TO1, TOT, TPT, TQ1, TQN, TTN, 18AJ, 19FJ, 1AFJ, 1FAJ, 20CH, 20PH, 2C0H, 2CHH, 2COH, 2H0H, 2HCH, 2HHH, 2POH, 2PPH, 3001, 34Q1, 3F41, 3FF1, 3QF1, 3SPT, 3SST, 3TTT, 40A1, 4441, 44KN, 44QN, 4AC1, 4C01, 4CA1, 4DG1, 4GA1, 4GC1, 4NND, 4NRN, 4OA1, 4Q4N, 4QA1, 4QRN, 4RQN, 550B, 555B, 55KB, 5A5J, 5AJJ, 5BKB, 5JQJ, 5KKB, 5QQB, 5QQJ, 604N, 606T, 60FN, 60KN, 60OT, 60PT, 660N, 660T, 664N, 66AT, 66FN, 66KN, 66TN, 6A0N, 6A9T, 6AAT, 6FKN, 6FRN, 6K0N, 6KFN, 6KKN, 6O6T, 6PAT, 6RFN, 6RRN, 6T6N, 6TRN, 7067, 70R7, 75BB, 77R7, 7C5B, 7CQB, 7E67, 7IQB, 7O5B, 7OIB, 7QOB, 7R07, 7RE7, 8667, 88L7, 88S7, 8E87, 8EE7, 8EL7, 8J8D, 8LL7, 8LS7, 906T, 908J, 90FJ, 90QJ, 90TJ, 92CH, 966T, 99MT, 99PT, 9I0J, 9I8J, 9LFJ, 9LIJ, 9M6T, 9MKT, 9MMT, 9PKT, 9QQJ, 9TLJ, A04N, A0QJ, A3AT, A3ST, A4A1, A4G1, A5QJ, A8QJ, A90J, A9PT, AA3T, AA41, AAAT, AAF1, AAG1, AAPT, AC41, ACS1, AF0J, AFC1, AFIJ, AG41, AGA1, AGG1, AI8J, AIJJ, AIQJ, AJ5J, AJJJ, AQ8J, AQJJ, AQQJ, AS3T, AT4N, AT5J, ATST, B04D, B0QD, BBIB, BBKB, BI0B, BK0B, BKIB, BKSB, BMIB, BMSB, BQ4D, BQQD, BSMB, C00B, C05B, C0A1, C0BB, C0EH, C0GH, C0Q1, C0QB, C2OH, C2PH, C441, C4Q1, C50B, CB0B, CBSB, CEGH, CG2H, CGHH, CHHH, CHOH, COGH, COO1, COOB, COOH, CQ01, CQ0B, CQQB, CQS1, CS0B, D00N, D0PD, D44N, DA0N, DCEH, DCG1, DCQ1, DDC1, DDF1, DDPD, DDQ1, DECH, DF4N, DFA1, DFRN, DG0D, DGD1, DGQ1, DNEH, DPDD, DQA1, DQPD, DRFN, DRRN, E20H, E667, E767, E8L7, E8S7, ECGH, EG2H, EH67, EH77, ES67, F001, F01J, F0MJ, F0QJ, F44N, F4G1, F4RN, F6RN, FA0J, FAC1, FAF1, FAIJ, FC01, FCG1, FFAJ, FFC1, FFG1, FFIJ, FGA1, FJJJ, FJQJ, FKAN, FR6N, FRFN, G09H, G0EH, G0HH, G20H, G341, G3Q1, G4G1, G92H, GA41, GAC1, GC0H, GCEH, GCG1, GCPH, GD31, GDA1, GDG1, GDPD, GE2H, GE9H, GGG1, GGGD, GHCH, GHHH, GP0D, GP2H, GPPD, GQ31, GQA1, GQG1, H00N, H0CH, H3ST, H4O1, H667, H66N, H677, H69T, H99T, H9CH, H9KT, H9LJ, H9PT, H9QJ, HH6N, HHF1, HHFJ, HHH1, HHHT, HHLJ, HHOH, HHQ1, HHST, HK6T, HL9J, HLMJ, HMKT, HMMT, HNM7, HNRN, HOPT, HQ01, HQ5J, HR0N, HR6N, HS77, HSKT, HT01, I0BB, I0JD, I0JJ, I98J, I9LJ, I9QJ, ICBB, IF0J, IFFJ, IFQJ, IIBD, IJBB, IJID, IL8J, ILFJ, IQ8J, IQJJ, IQQJ, IS0B, ISQB, J00D, J05J, J0BB, J0MJ, J0OB, J4QD, J50B, J5QJ, JBKB, JBMB, JDGD, JDQD, JGDD, JI0D, JIIB, JISB, JKIB, JKKB, JM0B, JOIB, JQ4D, JQDD, JQID, JQJJ, JQMJ, JSIB, JSMB, K06N, K0KB, K0KT, K0MT, K0PT, K0SB, K5QB, K60T, K66T, K6AN, K6FN, K6PT, K6RN, K96T, K99T, KA0N, KF6N, KFKN, KI0B, KK0T, KK6N, KK6T, KKFN, KKKB, KKKT, KKQB, KKTN, KMMT, KQ0B, KQQB, KR0N, KS5B, L0M7, L8E7, L98J, LFFJ, LI8J, LIFJ, LL87, LL9J, LLIJ, LLM7, LM77, LML7, LMM7, M09T, M0E7, M0IB, M0L7, M0M7, M0TT, M55J, M5LJ, M60T, M69T, M707, M767, M777, M7E7, M7S7, M8LJ, M96T, M9KT, MC0B, MCOB, ME07, MIBB, MJ0B, MJBB, MJSB, MLLJ, MLS7, MM07, MM0T, MML7, MMOB, MMR7, MOBB, MOCB, MP0T, MP9T, MPKT, MQ0B, MQIB, MQOB, MS07, MSPT, MSSB, MT8J, MTTT, N00D, N04D, N04N, N0FN, N0ND, N0RN, N2OH, N92H, NCGH, ND0D, ND4N, NDPD, NEGH, NEL7, NF4N, NFRN, NGCH, NGEH, NHM7, NME7, NML7, NMM7, NN0D, NN4D, NN4N, NNDN, NNID, NNND, NNRN, NP4D, NSL7, NSS7, O00T, O03T, O0MT, O0PT, O2HH, O2OH, O341, O4A1, O4G1, O4O1, O5KB, O90T, OA41, OBSB, OC5B, OCOB, OG2H, OG31, OH9H, OH9T, OHHH, OHTT, OICB, OISB, OM0T, OM6T, OMPT, OMTT, OO2H, OO9H, OOA1, OOCB, OOGH, OOKB, OOO1, OSKB, OTMT, P00T, P04D, P08D, P09T, P0KT, P90T, PAAT, PKKT, PO0T, PP0D, PPPD, Q00B, Q0AN, Q0D1, Q0F1, Q0IB, Q0JJ, Q0MJ, Q0OB, Q55J, Q5QJ, Q88D, QA0N, QA41, QA4N, QDF1, QDPD, QF41, QFA1, QG31, QIQJ, QJ0J, QJ8D, QJDD, QJJJ, QJMJ, QJQD, QKKB, QKQB, QOKB, QOOB, QP0D, QPDD, QQ8D, QQID, QQJD, QQPD, QQQB, QS01, QSA1, R00N, R067, R06N, R0FN, R0L7, R60N, RE67, RFRN, RRFN, RRRN, S00B, S3AT, S3ST, S50B, S5QB, S99T, SAAT, SC0B, SE67, SGG1, SICB, SIQB, SK5B, SKIB, SKKT, SKPT, SKQB, SM0B, SMMB, SMSB, SOA1, SOG1, SPOT, SQCB, SQQB, SS0B, SS67, ST31, STG1, T00N, T03T, T041, T04N, T0AT, T0C1, T0MT, T0ST, T1AJ, T3F1, T4A1, T4RN, T5JJ, T66N, T8AJ, T9FJ, T9QJ, TA3T, TAA1, TAG1, TAIJ, TCS1, TFA1, TFAJ, TFF1, TFFJ, TFG1, TGA1, TI9J, TIJJ, TIQJ, TL8J, TL9J, TM0T, TM5J, TM8J, TMAJ, TMLJ, TMST, TSF1, TSG1, TT01, TT5J, TT8J, TTA1, TTC1, TTFJ, TTG1, TTLJ, TTMJ, 30441, 30O41, 30OO1, 34O41, 3AATT, 3ASAT, 3O441, 3Q041, 40G41, 40GQ1, 40O41, 40OG1, 40OO1, 44001, 440C1, 440G1, 4444N, 44CG1, 44GQ1, 4AAA1, 4COG1, 4DAA1, 4GQ41, 4KRRN, 4OGG1, 4Q041, 4QGG1, 5005B, 500BB, 500KB, 500QB, 50K0B, 50Q0B, 50QKB, 555AJ, 555LJ, 555QJ, 55J5J, 5B00B, 5B0BB, 5J55J, 5J5JJ, 5JJ5J, 5JJJJ, 5K00B, 5QJ5J, 5QK0B, 6006N, 6009T, 600AT, 6066N, 60R0N, 666N7, 6696T, 66OPT, 67767, 6900T, 696PT, 69P0T, 6F6AN, 6O9PT, 6OP0T, 6OP9T, 6P99T, 755QB, 77767, 7CBBB, 7IBIB, 7IIBB, 7IICB, 7QIIB, 8888D, 88E67, 8L887, 8SSE7, 9099T, 90KKT, 90KMT, 90MPT, 9690T, 96P9T, 990KT, 9990T, 9999T, 99K6T, 9FFFJ, 9HKMT, 9HMPT, 9ILLJ, 9KKMT, 9KKPT, 9M00T, 9T8QJ, 9TFIJ, A008J, A00NN, A055J, A0FJJ, A0I0J, A0I9J, A0J0J, A44C1, A555J, A9FFJ, A9QIJ, A9T8J, AFFA1, AFFFJ, AFFJJ, AFFQJ, AJ00J, AP99T, ASA9T, ASFF1, ASP9T, ASSAT, AT3TT, ATFQJ, ATQIJ, ATT9J, ATTQJ, B088D, B08GD, B0GGD, B0SKB, B8GGD, BG00D, BIISB, BISIB, BS0IB, BS0SB, BSSKB, BSSSB, C0041, C0G41, C0H0H, C0PPH, C0S01, C4001, C4AG1, C4OG1, C5BBB, CG00H, CGE0H, CGGA1, CGQ41, COSSB, CP20H, CPGPH, CPP2H, CPPOH, CPPPH, CS001, CS55B, CSSQB, D00GD, D0A4N, D0FAN, D0GDD, D0N0D, DAAA1, DAAC1, DAFG1, DDAA1, DDGGD, DFFAN, DFGC1, DGDGD, DGG31, DGGA1, DH0AN, DPPPH, DQDD1, DQGG1, E0L87, E88E7, EC00H, EC02H, EE867, EE887, EEL87, ELE87, F00IJ, F0441, F0AG1, F0F0J, F0F41, F0FF1, F0GG1, F4041, F44A1, F44C1, F4A41, F64KN, F6K6N, FAAA1, FAFQJ, FCAA1, FF041, FF0F1, FF64N, FF6KN, FFA41, FFA4N, FFF4N, FFFF1, FFFFN, FFKRN, FFQMJ, FIJ0J, FJ00J, FKKKN, FNNFN, FQQ0J, FQQMJ, G00DD, G00DH, G00PD, G02PH, G0DDD, G0PDD, GAAA1, GC2HH, GDDDD, GDGDD, GDPPH, GG0PD, GGCA1, GGCQ1, GGDD1, GH4Q1, GHHG1, GII0D, GQ441, H009H, H00G1, H00H1, H04Q1, H0ANN, H0H01, H0H9H, H0HG1, H0HO1, H0O41, H0OHH, H0QG1, H40G1, H4G41, H4GG1, H60AN, HAN0N, HF0G1, HFF41, HFFMJ, HFFQJ, HGQ41, HH3TT, HH401, HH441, HH4G1, HH55J, HH66T, HH6OT, HH96T, HHA0N, HHANN, HHC0H, HHCHH, HHH0N, HHHCH, HHHHN, HHHNN, HHKMT, HHO41, HHP9T, HHPKT, HHTMT, HHTT1, HK9MT, HKKMT, HLLFJ, HM66T, HM7R7, HMM67, HMM77, HMMM7, HMTST, HO9HT, HOO41, HOOOH, HOT3T, HOTT1, HPO9T, HSO3T, HTGG1, I000J, I009J, I00BD, I00QJ, I00SB, I08QJ, I0I0D, I0IQB, I0JIB, I0Q0B, I0SCB, I0SIB, I0SSB, IB00D, IBBSB, IBISB, ICSSB, II0CB, II0ID, II0SB, IIIBB, IIQ0B, IIQIB, IISIB, IJ0SB, IJJQJ, ILLLJ, IQ00J, IQC0B, IQIQB, IQQIB, ISSCB, J00MB, J08GD, J0IID, J0QQD, J55BB, J55JJ, J5J5J, J5JJJ, J88GD, JB0SB, JBB0B, JBBSB, JDDDD, JG0ID, JGG0D, JIIID, JJ0QJ, JJ5JJ, JQ08D, JQ0QJ, JQQ0D, JQQ5J, JS55B, JSK0B, JSS5B, K000T, K006T, K00TN, K0FAN, K505B, K6T0N, K9KMT, K9KPT, K9P0T, KIQIB, KK00N, KK05B, KK0AN, KK0KN, KKM9T, KS0IB, KS0QB, KSQIB, KTR6N, KTRRN, L0087, L08S7, LE087, LEE87, LL8LJ, LMEE7, LMSE7, LMSS7, M00CB, M00PT, M066T, M0CQB, M0K6T, M0MQB, M0MSB, M0QQB, M0SMB, M0SST, M666T, M900T, MBBSB, MEE77, MEL77, MELE7, MES77, MESE7, MESS7, MI00B, MIICB, MIISB, MIQCB, MK9PT, ML7L7, MLEL7, MLLE7, MLME7, MM677, MMBIB, MMBSB, MMCQB, MME77, MMEE7, MMICB, MMISB, MMK6T, MMKKT, MMM9T, MMMMT, MMMTT, MMQQB, MMSCB, MMSKT, MMSMB, MMSS7, MMTST, MOIIB, MOOIB, MOSIB, MQCQB, MR007, MR667, MRL07, MS7L7, MSEL7, MSK9T, MSL77, MSSL7, MT00T, MTMMT, N0DDD, N4NNN, N7LE7, N7S77, NE2CH, NE9CH, NEC2H, NEE77, NFFNN, NFNFN, NII0D, NL777, NL7L7, NLES7, NLLL7, NLS77, NOG9H, NRR0N, O6P9T, O9H6T, O9HPT, O9P9T, OCBBB, OG441, OGAG1, OGGA1, OHH6T, OHOOH, OKIIB, OKK5B, OKKIB, OO5BB, OOBIB, OOIIB, OSCSB, OT3TT, OT441, OTG41, OTGG1, P00PH, P0D0D, P0G2H, P0OGH, P0PGH, PA99T, PGP0H, POOOH, PP20H, PP88D, PPP0H, Q0001, Q000N, Q001J, Q00ID, Q00PD, Q044N, Q04G1, Q0I0D, Q0PPD, Q40G1, Q444N, Q44RN, Q4AA1, Q8QQJ, QAAS1, QAFF1, QAFG1, QASG1, QDGG1, QFGG1, QI00D, QIICB, QIIQB, QIQCB, QOIIB, QPP4D, QQ08J, QQ0CB, QQC0B, QQI0J, QQJ5J, QSFG1, R00E7, R0E07, R0NE7, R6F6N, R6FFN, REE07, RELE7, RFF6N, RL0E7, RLE07, RLEE7, RLLE7, RQR0N, RR0QN, RRQ0N, S03O1, S0AF1, S0AG1, S0O31, S0QIB, S0SIB, S30F1, S30O1, S7QIB, SA3PT, SAAA1, SAFG1, SASST, SCSQB, SF0G1, SFFF1, SI0SB, SISSB, SKSSB, SSCQB, SSCSB, SSMIB, SSPAT, SSSKT, SSSMB, SSSSB, STAF1, STF01, T0001, T0031, T0AF1, T0AS1, T0G31, T0R6N, T0T31, T0TF1, T3AAT, T40G1, T4CG1, T4G41, T5LLJ, T8LLJ, TA441, TA98J, TAFJJ, TAQ5J, TASST, TATAT, TC401, TCGG1, TFQIJ, TFQJJ, TFQMJ, TG441, TGC41, TI8LJ, TLLMJ, TMMMT, TMTMT, TQ8QJ, TSS3T, TT3AT, TT9IJ, TTAAT, TTQIJ, TTS31, TTS3T, TTT9J, TTTST, 20000H, 200OOH, 3440O1, 3TAAST, 404CQ1, 4KKKKN, 4KKKRN, 4QQQQD, 505BBB, 50BB0B, 6000AN, 6444RN, 66666N, 666O9T, 66999T, 669P9T, 66N777, 6A444N, 6FF66N, 6FFF6N, 6R666N, 766767, 77S677, 7IBBBB, 8888E7, 8LLLLJ, 8SSSL7, 9000MT, 90K90T, 90KP0T, 90M90T, 99000T, 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===Base 36===

11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 2UJ, 2WN, 2XN, 30Z, 337, 33B, 33H, 33P, 34H, 34P, 37D, 38Z, 39H, 3AB, 3AP, 3AZ, 3D7, 3DH, 3FD, 3FZ, 3HD, 3HH, 3KZ, 3L7, 3LZ, 3MB, 3O7, 3OZ, 3PD, 3PP, 3Q7, 3RH, 3S7, 3SZ, 3UP, 3UZ, 3XB, 40P, 43B, 43H, 46V, 48B, 48V, 49B, 4AP, 4BB, 4E1, 4FP, 4HB, 4HH, 4HP, 4IH, 4MB, 4OP, 4OV, 4PH, 4RH, 4VB, 4VH, 4WH, 4WP, 4WV, 4X1, 4XH, 55T, 565, 58Z, 5C5, 5E7, 5ET, 5EZ, 5IN, 5KT, 5L7, 5MP, 5NN, 5O5, 5O7, 5OP, 5R7, 5RT, 5RZ, 5ST, 5SZ, 5TN, 5TZ, 5W7, 5XN, 5YN, 61T, 625, 62J, 64V, 661, 66J, 691, 6AV, 6EV, 6IJ, 6J1, 6K5, 6O1, 6OT, 6QJ, 6QT, 6RV, 6SJ, 6T1, 6TT, 6U5, 6UV, 6VV, 6W5, 6Y1, 6YT, 72D, 737, 77D, 791, 7A1, 7AZ, 7D7, 7GD, 7I1, 7IN, 7LN, 7M7, 7MN, 7N7, 7NN, 7RN, 7RZ, 7WN, 7X7, 7Z1, 801, 80V, 82D, 83B, 841, 84H, 877, 881, 887, 88B, 88V, 88Z, 89H, 89V, 8A1, 8AB, 8B7, 8BH, 8D1, 8DH, 8EB, 8EV, 8GD, 8GZ, 8HD, 8IV, 8L7, 8LZ, 8M1, 8MB, 8MD, 8O7, 8OB, 8Q7, 8QD, 8RD, 8S7, 8SH, 8SZ, 8UZ, 8W7, 8WV, 8XV, 8Y1, 90H, 91J, 93B, 93H, 94V, 981, 98H, 99J, 9AH, 9BB, 9C5, 9IH, 9IV, 9JV, 9KH, 9M1, 9OB, 9QJ, 9R1, 9RH, 9SH, 9UJ, 9VB, 9VJ, 9W5, 9X1, 9Y1, 9YB, 9YJ, 9YV, A25, A3P, A3Z, A4H, A61, A81, A8B, AAB, AAH, ABB, AC5, AEZ, AHP, AK1, AKB, AKH, ALZ, AMB, AO5, AOZ, APH, AQP, AR1, ARV, ARZ, AUV, AV5, AVB, AVV, AWV, AX1, AXB, AZ1, B2J, B4B, B4V, B6V, B87, B8H, B9B, BAB, BAH, BBH, BBV, BE7, BEB, BGJ, BHB, BIJ, BJB, BJT, BLJ, BOB, BOT, BQ7, BRV, BS7, BTT, BTV, BVB, BVJ, BVT, BWV, BX7, BXH, BYH, BYV, C25, C2J, C2N, C55, C65, C6J, C95, CCN, CDD, CED, CF5, CFJ, CGD, CIN, CJD, CO5, CRN, CUJ, CXJ, CXN, D37, D3P, D41, D55, D5P, D6T, D77, D81, D9H, DA1, DCD, DD1, DD5, DDH, DDP, DE7, DEP, DF5, DFD, DG7, DHH, DI1, DK1, DK5, DKT, DO1, DOP, DP1, DPD, DQ7, DR7, DRH, DS7, DTT, DU1, DWD, DX7, DYH, E0P, E2D, E3Z, E41, E4P, E71, E81, E87, E8B, EAZ, EDP, EED, EEN, EEV, EFZ, EGT, EI1, EKZ, ELB, EMB, EMN, EN7, EO1, EOT, EPP, EPZ, EQ7, ERN, EST, ETN, ETV, EUP, EUZ, EVT, EWD, EWN, EX1, EYB, EYN, EZD, EZP, F2J, F3B, F3Z, F4P, F65, F8B, F8Z, FCJ, FD5, FEZ, FFB, FFD, FG5, FGD, FGZ, FIJ, FJ5, FJJ, FLZ, FPD, FQD, FSZ, FXB, G07, G0D, G0Z, G3D, G3Z, G55, G6T, G7D, G7Z, G87, G8Z, GA5, GCJ, GD7, GE7, GET, GG5, GG7, GGT, GGZ, GI7, GJD, GKZ, GL7, GLT, GQT, GRT, GS7, GST, GU5, GUZ, GW5, GXJ, GZ7, GZZ, H3D, H3H, H4H, H9B, HAH, HED, HGD, HHP, HIH, HKH, HKP, HLN, HNH, HOB, HOP, HQP, HRD, HRH, HRN, HSH, HWD, HWH, HWP, HYN, I41, I6J, I71, I7N, I87, I8H, I9H, I9J, IA1, IAV, IE1, IFJ, IG7, IHH, II1, IIH, IIV, IJV, IK1, IL7, ILJ, ILN, IM1, INN, IOV, IRH, IUV, IX1, IXH, IXV, IYJ, J2D, J4V, J55, J61, J6J, J8B, J8V, J95, J9J, J9V, JA5, JBB, JD5, JDT, JF5, JFJ, JGJ, JGT, JIV, JJ1, JJD, JJV, JK5, JKT, JLB, JLT, JMN, JN5, JNJ, JQJ, JQN, JQT, JRD, JTB, JVJ, JVN, JVV, JWN, JY1, JYJ, K0B, K25, K3P, K65, K81, K95, K9H, KAH, KC5, KEP, KEZ, KG5, KHP, KK1, KKT, KLB, KLP, KLZ, KM1, KMB, KMP, KOT, KP1, KQP, KR1, KRT, KRZ, KTT, KW5, KX1, KZB, L0N, L0P, L8Z, L9B, LAZ, LBJ, LEN, LET, LFB, LFZ, LG7, LGZ, LIJ, LJJ, LKB, LKP, LLB, LLP, LLT, LMB, LMN, LN7, LO7, LOT, LPZ, LQT, LRN, LTN, LTT, LWP, LX7, LYN, M01, M0P, M2D, M2N, M37, M3B, M41, M61, M77, M87, M8D, M91, ME1, ME7, MEB, MFB, MFP, MGD, MI7, MKB, MM7, MMN, MN7, MNN, MO7, MOB, MOP, MPP, MQP, MS7, MW7, MXN, MYN, N5N, N65, N87, N8D, N8H, N95, NCD, NCJ, NDH, NE7, NFJ, NG5, NG7, NGJ, NJ5, NN7, NND, NO5, NQD, NQJ, NRN, NU5, NWH, NWN, NXH, NXN, NYJ, O0Z, O25, O37, O3B, O3P, O3Z, O41, O4B, O5Z, O61, O71, O7Z, O81, OA5, OAP, OBB, OBV, OC5, OEZ, OF5, OG7, OKP, OKZ, OMB, OO5, OOZ, OP5, OPP, OQT, OR1, OR7, OS7, OST, OTB, OTZ, OU5, OVV, OW5, OX7, OXB, OXV, OYV, OZ5, OZ7, P01, P0D, P0N, P3P, P4H, P4P, P5N, P65, P6V, P8V, PD1, PED, PEZ, PHH, PHP, PI1, PIN, PLN, PLP, PLZ, POP, PP1, PPH, PPV, PQD, PQN, PRV, PV5, PVH, PVV, PWH, PWP, PX1, PXV, PYH, PYN, Q07, Q0P, Q2N, Q37, Q3P, Q6J, Q6T, Q7D, Q8D, Q9J, QCD, QCJ, QD7, QED, QFP, QGT, QI7, QIN, QJN, QLJ, QMD, QMN, QMP, QND, QNJ, QOT, QQJ, QTN, QWN, QYJ, QYT, R0V, R0Z, R37, R3H, R4H, R77, R7D, R7N, R7Z, R81, R8V, R91, RA1, RCD, RCN, RD1, RED, REV, REZ, RGT, RGZ, RHD, RIV, RKH, RKZ, RLN, RMD, RNH, RO7, RQN, RRD, RRT, RRZ, RS7, RSH, RT1, RTV, RU1, RUZ, RVN, RVT, RW7, RWH, RX7, RY1, S0J, S6J, S87, S9H, SAZ, SCJ, SET, SFJ, SG7, SGZ, SI7, SM7, SO7, SOT, SQJ, SQT, SRH, SSH, STT, SW7, SX7, SXH, SYH, T0N, T0Z, T1D, T1T, T6V, T8Z, TBB, TBV, TCN, TD1, TEV, TGT, TKT, TLB, TM1, TO1, TOB, TQN, TR1, TRD, TT1, TTB, TTN, TUZ, TVB, TVN, TVT, TWD, TWV, TXB, TXV, TYV, U2J, U3Z, U61, U8V, U95, UA1, UC5, UEP, UEZ, UFJ, UG5, UJ5, ULZ, UO5, UOP, UOZ, UQP, URV, URZ, UUJ, UW5, UWV, UXJ, UXV, UYJ, V0H, V25, V4V, V55, V6V, V9J, V9V, VBB, VBJ, VCJ, VET, VGJ, VHP, VIH, VIN, VJJ, VJN, VLT, VMP, VNJ, VOB, VP5, VQJ, VQT, VRT, VRV, VSH, VSJ, VST, VTB, VTN, VUP, VW5, VWN, VXH, VXN, VYB, W07, W25, W3D, W3H, W4P, W4V, W57, W6V, W7D, W8H, W95, W9H, WAH, WAV, WCN, WD7, WDD, WDH, WE7, WEN, WF5, WGD, WHH, WK5, WKH, WN7, WNN, WPP, WPV, WQP, WR7, WRD, WRH, WRN, WS7, WU5, WUP, WVN, WWH, WWP, WX7, WXH, WXN, WYN, WYV, X0J, X2J, X2N, X4B, X4H, X4V, X6J, X87, X8B, X91, X9B, XAV, XCN, XEB, XFB, XHB, XHH, XHN, XKB, XLJ, XNH, XO1, XQ7, XS7, XSH, XUJ, XWN, XWV, XX7, XXH, XXV, Y0N, Y1J, Y1T, Y2N, Y3H, Y61, Y8V, Y91, YBV, YCN, YEB, YFJ, YHB, YHN, YIH, YJN, YJT, YLJ, YLN, YMB, YMN, YNH, YOB, YOV, YRH, YTB, YTN, YTT, YTV, YVB, YVH, YWH, YWV, YXB, YY1, YYJ, YYT, YYV, Z01, Z3D, Z3Z, Z71, Z8B, ZB7, ZBB, ZD5, ZDD, ZDP, ZG7, ZGD, ZGZ, ZKB, ZLP, ZM1, ZO5, ZP1, ZPD, ZQD, ZU1, ZX1, ZXB, ZZD, 102J, 106T, 10ET, 10GJ, 10LJ, 10TD, 12FJ, 12LJ, 13ED, 13GD, 13MD, 192J, 1D0T, 1DET, 1E3D, 1EFD, 1F0D, 1FWD, 1G0J, 1GGJ, 1GIJ, 1GUJ, 1I0J, 1ISJ, 1JOT, 1JTT, 1M0D, 1M3D, 1MFD, 1MWD, 1OKT, 1OOT, 1Q3D, 1QDT, 1QKT, 1QTD, 1QWD, 1SLJ, 1TDT, 1TFD, 1TOT, 1TTD, 1TTT, 1UGJ, 1ULJ, 1WFD, 1WWD, 20CJ, 20FJ, 20G5, 20IN, 20JD, 20MD, 20RN, 20WD, 20YN, 2295, 25RN, 260J, 26LJ, 288D, 28DD, 28FD, 296J, 29G5, 29K5, 29LJ, 2C05, 2C0J, 2C0N, 2CJJ, 2CK5, 2CLJ, 2CSJ, 2D25, 2DC5, 2DDD, 2DW5, 2F0J, 2FDD, 2FMD, 2FSJ, 2G65, 2G95, 2GK5, 2I0N, 2JC5, 2JCJ, 2JWD, 2K05, 2L6J, 2L9J, 2N0N, 2NIN, 2NN5, 2NSJ, 2NW5, 2NYN, 2Q0N, 2QNN, 2R0N, 2RFD, 2RWD, 2SJJ, 2SXJ, 2SYJ, 2W05, 2WC5, 2WD5, 2WMD, 2XCJ, 2YCJ, 2YJJ, 2YRN, 2YSJ, 300D, 302D, 304B, 3077, 3087, 308D, 30BH, 30DD, 30FP, 30G7, 30GD, 30MD, 30QP, 30YH, 320D, 333D, 333Z, 33WD, 344B, 34FB, 373Z, 37E7, 37EZ, 37G7, 3807, 38BB, 38E7, 38G7, 38HB, 38I7, 38X7, 390B, 394B, 399B, 39EB, 39FB, 39KB, 3ASH, 3AWH, 3B8B, 3BG7, 3BLB, 3BR7, 3BW7, 3C8D, 3D0P, 3D2D, 3DKP, 3E3D, 3E77, 3EBB, 3EFB, 3EKB, 3EQP, 3ERD, 3ERZ, 3EZZ, 3F4B, 3FEP, 3G2D, 3GCD, 3GED, 3GM7, 3GR7, 3GWD, 3GZD, 3H0B, 3HEP, 3HKB, 3IAH, 3ISH, 3K4B, 3KBH, 3KEB, 3KKB, 3L4B, 3L8B, 3M3D, 3MEP, 3MG7, 3MWP, 3OFP, 3OKB, 3OMP, 3OOB, 3OYB, 3P8H, 3PXH, 3QDP, 3QQD, 3QRD, 3QWD, 3R3D, 3R3Z, 3R8D, 3RG7, 3RM7, 3RQD, 3RR7, 3RZ7, 3SIH, 3W0H, 3WED, 3WM7, 3WQD, 3WW7, 3WYH, 3X8H, 3XE7, 3XG7, 3XKH, 3Y0B, 3Y4B, 3Y8H, 3YAH, 3YSH, 3YYB, 3Z4B, 3ZFP, 404V, 4061, 40EB, 40I1, 40IV, 40KH, 40M1, 40R1, 40XB, 40XV, 43QP, 4441, 444H, 444P, 4481, 448H, 4491, 449H, 44EB, 44EV, 44K1, 44KB, 44KP, 44M1, 44R1, 44RV, 44SH, 44U1, 44VP, 44YB, 46I1, 46K1, 46M1, 480H, 488H, 48AH, 48I1, 48K1, 48U1, 4991, 49EV, 49I1, 49O1, 49UV, 4A0H, 4A41, 4A4B, 4A8H, 4A91, 4ALB, 4ASH, 4BXV, 4E4B, 4EEP, 4EFB, 4ELP, 4EUV, 4EXB, 4FOB, 4I01, 4I0V, 4I4V, 4I61, 4I81, 4I91, 4IEV, 4IY1, 4K0H, 4K91, 4KI1, 4KKH, 4KKP, 4KUP, 4L3P, 4L4P, 4LAB, 4LMP, 4MA1, 4MK1, 4MMP, 4MP1, 4MU1, 4O0B, 4OFB, 4OKB, 4OM1, 4OU1, 4OYB, 4P41, 4P81, 4PO1, 4PYV, 4QLP, 4QQP, 4RAV, 4RK1, 4RRV, 4S8H, 4U3P, 4U91, 4U9V, 4UIV, 4UUP, 4VAV, 4VLP, 4VVV, 4X0B, 4X0V, 4XAB, 4XBV, 4XVV, 4XYV, 4Y0V, 4Y41, 4Y4H, 4Y4V, 4Y81, 4YAV, 4YEV, 4YIV, 4YK1, 4YKB, 4YM1, 4YR1, 4YUV, 4YYB, 5025, 503Z, 5095, 50G5, 50PN, 50QN, 50WP, 50Z7, 5205, 52RN, 53FP, 53M7, 53PZ, 53X7, 53ZZ, 54PP, 550N, 553P, 554P, 5557, 555N, 557Z, 5595, 55EN, 55G5, 55GZ, 55LZ, 55OZ, 55QN, 55Z7, 572N, 577Z, 5787, 57CN, 57FZ, 57QN, 57S7, 57ZZ, 58I7, 58X7, 5955, 5A95, 5AEP, 5CMN, 5CWN, 5E2N, 5EAP, 5ECN, 5EKP, 5EQN, 5F95, 5FLP, 5FQP, 5G05, 5GK5, 5GQ7, 5I37, 5II7, 5IM7, 5L3P, 5LCN, 5LFP, 5LGT, 5LKZ, 5LUP, 5LZZ, 5M07, 5N55, 5NQ7, 5NX7, 5OAZ, 5OFZ, 5OLT, 5OOT, 5P25, 5P95, 5PG5, 5PGZ, 5PPN, 5PQP, 5PW5, 5Q0N, 5Q87, 5QCN, 5QKP, 5QN7, 5QPN, 5QPP, 5QQT, 5QTT, 5QWP, 5QX7, 5S37, 5SS7, 5T6T, 5TQT, 5U4P, 5UGZ, 5W0P, 5W2N, 5W5P, 5WLN, 5WP5, 5WQN, 5XM7, 5Y0T, 5YOT, 5Z25, 5Z4P, 5Z57, 5ZG5, 5ZLZ, 5ZQ7, 5ZWP, 6005, 600V, 601J, 6055, 606V, 60A1, 60C5, 60GT, 60JJ, 60LJ, 60M1, 60R1, 60VT, 60YV, 610J, 61FJ, 6401, 64K1, 6505, 665T, 6665, 66A5, 66F5, 66G5, 66GT, 66KT, 66V5, 66XV, 68OV, 68YV, 690J, 69G5, 69O5, 69XJ, 69XV, 6A55, 6AA1, 6AA5, 6AF5, 6C9J, 6CA5, 6CGJ, 6CYJ, 6EK1, 6ER1, 6FLJ, 6FXJ, 6G0J, 6G65, 6GGJ, 6GJ5, 6GLJ, 6I81, 6I9V, 6J65, 6J6V, 6JET, 6JST, 6JWV, 6K01, 6KET, 6KST, 6L0J, 6L6T, 6LFJ, 6LRT, 6LXJ, 6MA1, 6MM1, 6MX1, 6O8V, 6OV5, 6R0T, 6RET, 6RKT, 6S6T, 6T8V, 6U01, 6U9J, 6UE1, 6UGJ, 6UJJ, 6ULJ, 6UU1, 6V0T, 6VA5, 6VF5, 6VG5, 6VKT, 6VXJ, 6WWV, 6WXV, 6X41, 6X6V, 6XIV, 6XJV, 6XOV, 6Y0V, 6YCJ, 6YXV, 7001, 702N, 7041, 707N, 7081, 70GZ, 70I7, 70KZ, 70ND, 70O7, 70OZ, 70R7, 70U1, 70UZ, 70WD, 70YN, 71QD, 71WD, 720N, 72CN, 72YN, 733Z, 73ED, 73ZD, 7401, 7441, 74M1, 74O1, 74R1, 7641, 76K1, 76M1, 76R1, 76U1, 76X1, 773Z, 7741, 7771, 7781, 77FZ, 77G7, 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8KRH, 8KU1, 8KYH, 8KZ1, 8L4B, 8LBB, 8MG7, 8MR7, 8O4V, 8OI1, 8OK1, 8OOV, 8OX1, 8RAV, 8RHH, 8RUV, 8RX1, 8U4V, 8UK1, 8V4B, 8V9B, 8VHH, 8VRH, 8VVV, 8VYV, 8W0H, 8WFD, 8X07, 8XAH, 8XR1, 8XU1, 8XYB, 8Y6V, 8Y8H, 8YBB, 8YFB, 8YHH, 8YKB, 8YLB, 8YUV, 8YVV, 8YYH, 8Z1D, 8Z4B, 8Z8D, 8Z91, 8ZCD, 8ZEZ, 8ZFD, 8ZFZ, 8ZK1, 8ZO1, 8ZX7, 90A5, 90BJ, 90BV, 90EB, 90EV, 90GJ, 90IJ, 90J5, 90JB, 90LB, 90O1, 90OV, 90XB, 90XV, 9225, 92K5, 9441, 94BH, 94EB, 94KB, 94XB, 9505, 9601, 960V, 96A1, 96J5, 96K1, 96V5, 96WV, 96XV, 984B, 98AV, 98BV, 98FB, 994H, 998B, 998V, 999B, 99A1, 99A5, 99E1, 99G5, 99O1, 99O5, 99WH, 9A05, 9A55, 9A95, 9AA5, 9ABV, 9AE1, 9AEV, 9AI1, 9AO1, 9AOV, 9B0J, 9B0V, 9B8V, 9B9H, 9BCJ, 9BFJ, 9BSJ, 9BVV, 9CCJ, 9CLJ, 9CSJ, 9E01, 9E4B, 9EA1, 9EAB, 9EAV, 9EK1, 9F05, 9F25, 9FA5, 9FBJ, 9FFJ, 9FJB, 9FSJ, 9FXJ, 9G25, 9G2J, 9GIJ, 9GSJ, 9HFB, 9HLB, 9IIJ, 9IJ1, 9J01, 9J0B, 9J0J, 9J9B, 9JCJ, 9JE1, 9K05, 9K41, 9K55, 9K61, 9K8B, 9KA5, 9KI1, 9KK5, 9KU5, 9L2J, 9LCJ, 9O05, 9O0V, 9O95, 9OI1, 9OO1, 9OOV, 9OWV, 9RRV, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ

====Additional known quasi-minimal primes (not necessarily the next)====

P&lt;sub&gt;81993&lt;/sub&gt;SZ

==Unsolved families==

Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')) by covering congruence, algebraic factorization, or combine of them.

{|class=wikitable
|base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family
|-
|13||9{5}||(113×13&lt;sup&gt;''n''&lt;/sup&gt;−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|13||A{3}A||(41×13&lt;sup&gt;''n''+1&lt;/sup&gt;+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|16||{3}AF||(16&lt;sup&gt;''n''+2&lt;/sup&gt;+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1{7}||(23×17&lt;sup&gt;''n''&lt;/sup&gt;−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1F{0}7||32×17&lt;sup&gt;''n''+1&lt;/sup&gt;+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||4{7}A||(71×17&lt;sup&gt;''n''+1&lt;/sup&gt;+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||51{0}D||86×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||70F{0}D||2038×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||8{B}9||(139×17&lt;sup&gt;''n''+1&lt;/sup&gt;−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||9{5}9||(149×17&lt;sup&gt;''n''+1&lt;/sup&gt;+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||95{F}||(2543×17&lt;sup&gt;''n''&lt;/sup&gt;−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||A{D}F||(173×17&lt;sup&gt;''n''+1&lt;/sup&gt;+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}B3||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+190||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}DB||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+232||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2BE||(11×17&lt;sup&gt;''n''+3&lt;/sup&gt;−41579)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2E||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;−2411)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}E9||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+773)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}EE||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+853)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||D0G{D}||(60381×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2860381*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||E9{B}||(3963×17&lt;sup&gt;''n''&lt;/sup&gt;−11)/16||29000||[http://factordb.com/index.php?query=%283963*17%5En-11%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||F1{9}||(4105×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16||1000000||[http://factordb.com/index.php?query=%284105*17%5En-9%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||FD0{D}||(72909×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2872909*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||G{7}F||(263×17&lt;sup&gt;''n''+1&lt;/sup&gt;+121)/16||29000||[http://factordb.com/index.php?query=%28263*17%5E%28n%2B1%29%2B121%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||2{7}9D||(47×21&lt;sup&gt;''n''+2&lt;/sup&gt;+953)/20||20000||[http://factordb.com/index.php?query=%2847*21%5E%28n%2B2%29%2B953%29%2F20&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||2F{C}A||(288×21&lt;sup&gt;''n''+1&lt;/sup&gt;−13)/5||20000||[http://factordb.com/index.php?query=%28288*21%5E%28n%2B1%29-13%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||4{3}B||(83×21&lt;sup&gt;''n''+1&lt;/sup&gt;+157)/20||20000||[http://factordb.com/index.php?query=%2883*21%5E%28n%2B1%29%2B157%29%2F20&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||5{0}DJ||5×21&lt;sup&gt;''n''+2&lt;/sup&gt;+292||20000||[http://factordb.com/index.php?query=5*21%5E%28n%2B2%29%2B292&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||7{K}Z||(53×36&lt;sup&gt;''n''+1&lt;/sup&gt;+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||B{0}EUV||11×36&lt;sup&gt;''n''+3&lt;/sup&gt;+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||HM{0}N||634×36&lt;sup&gt;''n''+1&lt;/sup&gt;+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||N{0}YYN||23×36&lt;sup&gt;''n''+3&lt;/sup&gt;+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||O{L}Z||(123×36&lt;sup&gt;''n''+1&lt;/sup&gt;+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||S{0}8H||28×36&lt;sup&gt;''n''+2&lt;/sup&gt;+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|}

For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E

For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family B0{H}6H may or may not be quasi-minimal prime, since another unsolved family is B{H}6H, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D}

=== Base 17 ===

All searched to length 29000

* 1{7}
* 1F{0}7
* 4{7}A
* 51{0}D
* 70F{0}D
* 8{B}9
* 9{5}9
* 95{F}
* A{D}F
* B{0}B3
* B{0}DB
* {B}2BE
* {B}2E
* {B}E9
* {B}EE
* D0G{D}
* E9{B}
* F1{9}
* FD0{D}
* G{7}F

=== Base 21 ===

All searched to length 20000

* 2{7}9D
* 2F{C}A
* 4{3}B
* 5{0}DJ
* {5}FEK
* {7}ID
* 99{0}99H
* {9}0D
* {9}D
* B0{H}6H
* B3{0}EB
* B9{0}E5
* B{D}B
* B{H}6H
* DH{D}
* F{9}D
* {F}35
* G{0}FK
* H{0}7771
* H{D}
* {J}6J

=== Base 36 ===

All searched to length 20000

* 7{K}Z
* B{0}EUV
* HM{0}N
* N{0}YYN
* O{L}Z
* S{0}8H

==Primality certificates for the proven primes &gt; 10&lt;sup&gt;299&lt;/sup&gt;==

See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]]

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime
|-
|9||149||76&lt;sub&gt;329&lt;/sub&gt;2||(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642]
|-
|9||150||27&lt;sub&gt;686&lt;/sub&gt;07||(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486]
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||3×9&lt;sup&gt;1160&lt;/sup&gt;+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423]
|-
|11||1065||A&lt;sub&gt;713&lt;/sub&gt;58||11&lt;sup&gt;715&lt;/sup&gt;−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487]
|-
|11||1066||7&lt;sub&gt;759&lt;/sub&gt;44||(7×11&lt;sup&gt;761&lt;/sup&gt;−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840]
|-
|11||1067||557&lt;sub&gt;1011&lt;/sub&gt;||(607×11&lt;sup&gt;1011&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522]
|-
|13||3165||50&lt;sub&gt;270&lt;/sub&gt;44||5×13&lt;sup&gt;272&lt;/sup&gt;+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005]
|-
|13||3166||9&lt;sub&gt;271&lt;/sub&gt;095||(3×13&lt;sup&gt;274&lt;/sup&gt;−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654]
|-
|13||3167||10&lt;sub&gt;286&lt;/sub&gt;7771||13&lt;sup&gt;290&lt;/sup&gt;+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633]
|-
|13||3168||9&lt;sub&gt;308&lt;/sub&gt;1||(3×13&lt;sup&gt;309&lt;/sup&gt;−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=308&amp;c0=-&amp;EN= 13&lt;sup&gt;308&lt;/sup&gt;−1]
|-
|13||3169||B&lt;sub&gt;341&lt;/sub&gt;C4||(11×13&lt;sup&gt;343&lt;/sup&gt;+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618]
|-
|13||3170||8B&lt;sub&gt;343&lt;/sub&gt;||(107×13&lt;sup&gt;343&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736]
|-
|13||3171||710&lt;sub&gt;371&lt;/sub&gt;111||92×13&lt;sup&gt;374&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609]
|-
|13||3172||75&lt;sub&gt;375&lt;/sub&gt;7||(89×13&lt;sup&gt;376&lt;/sup&gt;+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596]
|-
|13||3173||9B0&lt;sub&gt;391&lt;/sub&gt;9||128×13&lt;sup&gt;392&lt;/sup&gt;+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790]
|-
|13||3174||7B0B&lt;sub&gt;397&lt;/sub&gt;||(15923×13&lt;sup&gt;397&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574]
|-
|13||3175||10&lt;sub&gt;414&lt;/sub&gt;93||13&lt;sup&gt;416&lt;/sup&gt;+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240]
|-
|13||3176||81010&lt;sub&gt;415&lt;/sub&gt;1||17746×13&lt;sup&gt;416&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3177||8110&lt;sub&gt;435&lt;/sub&gt;1||1366×13&lt;sup&gt;436&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3178||B7&lt;sub&gt;486&lt;/sub&gt;||(139×13&lt;sup&gt;486&lt;/sup&gt;−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892]
|-
|13||3179||B&lt;sub&gt;563&lt;/sub&gt;C||(11×13&lt;sup&gt;564&lt;/sup&gt;+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=564&amp;c0=-&amp;EN= 13&lt;sup&gt;564&lt;/sup&gt;−1]
|-
|13||3180||1B&lt;sub&gt;576&lt;/sub&gt;||(23×13&lt;sup&gt;576&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=576&amp;c0=-&amp;EN= 13&lt;sup&gt;576&lt;/sup&gt;−1]
|-
|13||3181||80&lt;sub&gt;693&lt;/sub&gt;87||8×13&lt;sup&gt;695&lt;/sup&gt;+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532]
|-
|13||3182||CC5&lt;sub&gt;713&lt;/sub&gt;||(2021×13&lt;sup&gt;713&lt;/sup&gt;−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353]
|-
|13||3183||B&lt;sub&gt;834&lt;/sub&gt;74||(11×13&lt;sup&gt;836&lt;/sup&gt;−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871]
|-
|13||3184||9&lt;sub&gt;968&lt;/sub&gt;B||(3×13&lt;sup&gt;969&lt;/sup&gt;+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244]
|-
|13||3185||10&lt;sub&gt;1295&lt;/sub&gt;181||13&lt;sup&gt;1298&lt;/sup&gt;+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013]
|-
|13||3186||9&lt;sub&gt;1362&lt;/sub&gt;5||(3×13&lt;sup&gt;1363&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776]
|-
|13||3187||7&lt;sub&gt;1504&lt;/sub&gt;1||(7×13&lt;sup&gt;1505&lt;/sup&gt;−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755]
|-
|13||3188||930&lt;sub&gt;1551&lt;/sub&gt;1||120×13&lt;sup&gt;1552&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3189||720&lt;sub&gt;2297&lt;/sub&gt;2||93×13&lt;sup&gt;2298&lt;/sup&gt;+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910]
|-
|13||3190||1770&lt;sub&gt;2703&lt;/sub&gt;17||267×13&lt;sup&gt;2705&lt;/sup&gt;+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825]
|-
|13||3191||390&lt;sub&gt;6266&lt;/sub&gt;1||48×13&lt;sup&gt;6267&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3192||B0&lt;sub&gt;6540&lt;/sub&gt;BBA||11×13&lt;sup&gt;6543&lt;/sup&gt;+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906]
|-
|13||3193||C&lt;sub&gt;10631&lt;/sub&gt;92||13&lt;sup&gt;10633&lt;/sup&gt;−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750]
|-
|14||649||34D&lt;sub&gt;708&lt;/sub&gt;||47×14&lt;sup&gt;708&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||5×14&lt;sup&gt;19698&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2328||880&lt;sub&gt;246&lt;/sub&gt;7||136×16&lt;sup&gt;247&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is &lt; 10&lt;sup&gt;299&lt;/sup&gt;
|-
|16||2329||D4&lt;sub&gt;263&lt;/sub&gt;D||(199×16&lt;sup&gt;264&lt;/sup&gt;+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238]
|-
|16||2330||E0&lt;sub&gt;261&lt;/sub&gt;4DD||14×16&lt;sup&gt;264&lt;/sup&gt;+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352]
|-
|16||2331||8C0&lt;sub&gt;290&lt;/sub&gt;ED||140×16&lt;sup&gt;292&lt;/sup&gt;+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307]
|-
|16||2332||DA&lt;sub&gt;305&lt;/sub&gt;5||(41×16&lt;sup&gt;306&lt;/sup&gt;−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284]
|-
|16||2333||CE80&lt;sub&gt;422&lt;/sub&gt;D||3304×16&lt;sup&gt;423&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257]
|-
|16||2334||5F&lt;sub&gt;544&lt;/sub&gt;6F||6×16&lt;sup&gt;546&lt;/sup&gt;−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967]
|-
|16||2335||88F&lt;sub&gt;545&lt;/sub&gt;||137×16&lt;sup&gt;545&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2336||BE0&lt;sub&gt;792&lt;/sub&gt;BB||190×16&lt;sup&gt;794&lt;/sup&gt;+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938]
|-
|16||2337||D9&lt;sub&gt;1052&lt;/sub&gt;||(68×16&lt;sup&gt;1052&lt;/sup&gt;−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020]
|-
|16||2338||FAF&lt;sub&gt;1062&lt;/sub&gt;45||251×16&lt;sup&gt;1064&lt;/sup&gt;−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610]
|-
|16||2339||F8&lt;sub&gt;1517&lt;/sub&gt;F||(233×16&lt;sup&gt;1518&lt;/sup&gt;+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824]
|-
|16||2340||20&lt;sub&gt;1713&lt;/sub&gt;321||2×16&lt;sup&gt;1716&lt;/sup&gt;+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735]
|-
|16||2341||300F&lt;sub&gt;1960&lt;/sub&gt;AF||769×16&lt;sup&gt;1962&lt;/sup&gt;−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750]
|-
|16||2342||90&lt;sub&gt;3542&lt;/sub&gt;91||9×16&lt;sup&gt;3544&lt;/sup&gt;+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191]
|-
|16||2343||5BC&lt;sub&gt;3700&lt;/sub&gt;D||(459×16&lt;sup&gt;3701&lt;/sup&gt;+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322]
|-
|16||2344||D0B&lt;sub&gt;17804&lt;/sub&gt;||(3131×16&lt;sup&gt;17804&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511]
|-
|18||547||80&lt;sub&gt;298&lt;/sub&gt;B||8×18&lt;sup&gt;299&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is an algebraic factor of ''N''+1, factordb entry of 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is [http://factordb.com/index.php?id=1100000000900149167]
|-
|18||548||H&lt;sub&gt;766&lt;/sub&gt;FH||18&lt;sup&gt;768&lt;/sup&gt;−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490]
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||12×18&lt;sup&gt;6270&lt;/sup&gt;+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437]
|-
|20||3301||H&lt;sub&gt;247&lt;/sub&gt;A0H||(17×20&lt;sup&gt;250&lt;/sup&gt;−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619]
|-
|20||3302||7&lt;sub&gt;249&lt;/sub&gt;A7||(7×20&lt;sup&gt;251&lt;/sup&gt;+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602]
|-
|20||3303||J7&lt;sub&gt;270&lt;/sub&gt;||(368×20&lt;sup&gt;270&lt;/sup&gt;−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462]
|-
|20||3304||J&lt;sub&gt;330&lt;/sub&gt;CCC7||20&lt;sup&gt;334&lt;/sup&gt;−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572]
|-
|20||3305||40&lt;sub&gt;387&lt;/sub&gt;404B||4×20&lt;sup&gt;391&lt;/sup&gt;+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563]
|-
|20||3306||EC0&lt;sub&gt;429&lt;/sub&gt;7||292×20&lt;sup&gt;430&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702]
|-
|20||3307||G&lt;sub&gt;447&lt;/sub&gt;99||(16×20&lt;sup&gt;449&lt;/sup&gt;−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753]
|-
|20||3308||3A&lt;sub&gt;527&lt;/sub&gt;3||(67×20&lt;sup&gt;528&lt;/sup&gt;−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531]
|-
|20||3309||E&lt;sub&gt;566&lt;/sub&gt;C7||(14×20&lt;sup&gt;568&lt;/sup&gt;−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516]
|-
|20||3310||JCJ&lt;sub&gt;629&lt;/sub&gt;||393×20&lt;sup&gt;629&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|20||3311||J&lt;sub&gt;655&lt;/sub&gt;05J||20&lt;sup&gt;658&lt;/sup&gt;−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052]
|-
|20||3312||50&lt;sub&gt;1163&lt;/sub&gt;AJ||5×20&lt;sup&gt;1165&lt;/sup&gt;+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412]
|-
|20||3313||CD&lt;sub&gt;2449&lt;/sub&gt;||(241×20&lt;sup&gt;2449&lt;/sup&gt;−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915]
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||16×20&lt;sup&gt;6270&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457]
|-
|22||7984||I7G0&lt;sub&gt;254&lt;/sub&gt;H||8882×22&lt;sup&gt;255&lt;/sup&gt;+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788]
|-
|22||7985||D0&lt;sub&gt;255&lt;/sub&gt;5EEF||13×22&lt;sup&gt;259&lt;/sup&gt;+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932]
|-
|22||7986||IK&lt;sub&gt;322&lt;/sub&gt;F||(398×22&lt;sup&gt;323&lt;/sup&gt;−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145]
|-
|22||7987||C0&lt;sub&gt;340&lt;/sub&gt;G9||12×22&lt;sup&gt;342&lt;/sup&gt;+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159]
|-
|22||7988||77E&lt;sub&gt;348&lt;/sub&gt;K7||(485×22&lt;sup&gt;350&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779]
|-
|22||7989||J&lt;sub&gt;379&lt;/sub&gt;KJ||(19×22&lt;sup&gt;381&lt;/sup&gt;+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027]
|-
|22||7990||J&lt;sub&gt;388&lt;/sub&gt;EJ||(19×22&lt;sup&gt;390&lt;/sup&gt;−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729]
|-
|22||7991||DJ&lt;sub&gt;400&lt;/sub&gt;||(292×22&lt;sup&gt;400&lt;/sup&gt;−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110]
|-
|22||7992||E&lt;sub&gt;404&lt;/sub&gt;K7||(2×22&lt;sup&gt;406&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298]
|-
|22||7993||66F&lt;sub&gt;453&lt;/sub&gt;B3||(971×22&lt;sup&gt;455&lt;/sup&gt;−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809]
|-
|22||7994||L0&lt;sub&gt;454&lt;/sub&gt;B63||21×22&lt;sup&gt;457&lt;/sup&gt;+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331]
|-
|22||7995||L&lt;sub&gt;483&lt;/sub&gt;G3||22&lt;sup&gt;485&lt;/sup&gt;−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730]
|-
|22||7996||E60&lt;sub&gt;496&lt;/sub&gt;L||314×22&lt;sup&gt;497&lt;/sup&gt;+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239]
|-
|22||7997||I&lt;sub&gt;626&lt;/sub&gt;AF||(6×22&lt;sup&gt;628&lt;/sup&gt;−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334]
|-
|22||7998||K0&lt;sub&gt;760&lt;/sub&gt;EC1||20×22&lt;sup&gt;763&lt;/sup&gt;+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415]
|-
|22||7999||J0&lt;sub&gt;767&lt;/sub&gt;IGGJ||19×22&lt;sup&gt;771&lt;/sup&gt;+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567]
|-
|22||8000||7&lt;sub&gt;959&lt;/sub&gt;K7||(22&lt;sup&gt;961&lt;/sup&gt;+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817]
|-
|22||8001||L&lt;sub&gt;2385&lt;/sub&gt;KE7||22&lt;sup&gt;2388&lt;/sup&gt;−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774]
|-
|22||8002||7&lt;sub&gt;3815&lt;/sub&gt;2L||(22&lt;sup&gt;3817&lt;/sup&gt;−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839]
|-
|24||3400||I0&lt;sub&gt;241&lt;/sub&gt;I5||18×24&lt;sup&gt;243&lt;/sup&gt;+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037]
|-
|24||3401||D0&lt;sub&gt;259&lt;/sub&gt;KKD||13×24&lt;sup&gt;262&lt;/sup&gt;+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725]
|-
|24||3402||C7&lt;sub&gt;298&lt;/sub&gt;||(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235]
|-
|24||3403||20&lt;sub&gt;313&lt;/sub&gt;7||2×24&lt;sup&gt;314&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241]
|-
|24||3404||BC0&lt;sub&gt;331&lt;/sub&gt;B||276×24&lt;sup&gt;332&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842]
|-
|24||3405||N&lt;sub&gt;2644&lt;/sub&gt;LLN||24&lt;sup&gt;2647&lt;/sup&gt;−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089]
|-
|24||3406||D&lt;sub&gt;2698&lt;/sub&gt;LD||(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876]
|-
|24||3407||A0&lt;sub&gt;2951&lt;/sub&gt;8ID||10×24&lt;sup&gt;2954&lt;/sup&gt;+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654]
|-
|24||3408||88N&lt;sub&gt;5951&lt;/sub&gt;||201×24&lt;sup&gt;5951&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606]
|-
|30||2613||AN&lt;sub&gt;206&lt;/sub&gt;||(313×30&lt;sup&gt;206&lt;/sup&gt;−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073]
|-
|30||2614||M&lt;sub&gt;241&lt;/sub&gt;QB||(22×30&lt;sup&gt;243&lt;/sup&gt;+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295]
|-
|30||2615||M0&lt;sub&gt;547&lt;/sub&gt;SS7||22×30&lt;sup&gt;550&lt;/sup&gt;+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988]
|-
|30||2616||C0&lt;sub&gt;1022&lt;/sub&gt;1||12×30&lt;sup&gt;1023&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|30||2617||5&lt;sub&gt;4882&lt;/sub&gt;J||(5×30&lt;sup&gt;4883&lt;/sup&gt;+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423]
|-
|30||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||25×30&lt;sup&gt;34205&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|}

==Unproven PRPs==

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this PRP
|-
|11||1068||57&lt;sub&gt;62668&lt;/sub&gt;||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000003573679860]
|-
|13||3194||C5&lt;sub&gt;23755&lt;/sub&gt;C||(149×13&lt;sup&gt;23756&lt;/sup&gt;+79)/12||[http://factordb.com/index.php?id=1100000003590647776]
|-
|13||3195||80&lt;sub&gt;32017&lt;/sub&gt;111||8×13&lt;sup&gt;32020&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000000490878060]
|-
|16||2345||DB&lt;sub&gt;32234&lt;/sub&gt;||(206×16&lt;sup&gt;32234&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000002383583629]
|-
|16||2346||4&lt;sub&gt;72785&lt;/sub&gt;DD||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15||[http://factordb.com/index.php?id=1100000003615909841]
|-
|22||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21||[http://factordb.com/index.php?id=1100000003594696838]
|-
|30||2618||I0&lt;sub&gt;24608&lt;/sub&gt;D||18×30&lt;sup&gt;24609&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003593967511]
|}

All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10&lt;sup&gt;16&lt;/sup&gt;. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]])

==Proof==

===Base 2===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.

===Base 3===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (2,1), (2,2)

* Case (1,1):
** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes)
*** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime.

===Base 4===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (2,1), (2,3), (3,1), (3,3)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.

===Base 5===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4)

* Case (1,1):
** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes)
*** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes)
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes)
***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb])
***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes)
**** The smallest prime of the form 1{4}4 is 14444.
**** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes)
*** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes)
**** For the 3{0,3}1 family, we can separate this family to four families:
***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01.
****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime.
***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031.
***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001.
***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31.
****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime.
**** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.
* Case (3,4):
** 34 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes)
**** The smallest prime of the form 4{4}1 is 44441.
**** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime.
* Case (4,2):
** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime.

===Base 6===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes)
*** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes)
**** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 4{0}41 is 40041
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes)
*** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.

===Base 7===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

* Case (1,1):
** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes)
*** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {}
**** If there are &gt;=3 number of 1 and 5 in the {}:
***** If there is 111 in the {}, then we have the prime 11111
***** If there is 115 in the {}, then the prime 115 is a subsequence
***** If there is 151 in the {}, then the prime 115 is a subsequence
***** If there is 155 in the {}, then the prime 155 is a subsequence
***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime
***** If there is 515 in the {}, then the prime 515 is a subsequence
***** If there is 551 in the {}, then the prime 551 is a subsequence
***** If there is 555 in the {}, then the prime 551 is a subsequence
**** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1
***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1
****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1
******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21
******** The smallest prime of the form 11{0}21 is 1100021
******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime
******** The smallest prime of the form 1001{0}21 is 100121
********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families
***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1
****** Since 1051 is prime, we only need to consider the family 15{0}1
******* The smallest prime of the form 15{0}1 is 150001
* Case (1,2):
** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes)
*** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {}
**** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime.
**** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime.
**** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2
***** Since 1022 and 1202 are primes, we only need to consider the number 122
****** 122 is not prime.
**** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime.
* Case (1,3):
** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** 14 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime.
* Case (1,6):
** 16 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime.
* Case (2,2):
** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,6):
** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes)
*** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes)
**** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's)
***** For the 3{0,1}1 family:
****** If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
****** If there are no 1's between (3,1), then the form will be 3{0}1
******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime.
****** If there are no 0's between (3,1), then the form will be 3{1}1
******* The smallest prime of the form 3{1}1 is 31111
****** If there are exactly 1 0's between (3,1), then there must be &lt;3 1's between (3,1), or 31111 will be a subsequence.
******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime.
******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011
******** Neither 3101 nor 3011 is prime.
******* If there are no 1's between (3,1), then the number must be 301
******** 301 is not prime.
***** For the 3{3}3{0,1}1 family:
****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence.
****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1
******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence.
******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime.
******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime.
******* If there are no 1's between (33,1), then the form will be 33{0}1
******** The smallest prime of the form 33{0}1 is 33001
****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1
******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence.
******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301
******** The smallest prime of the form 3{3}301 is 33333301
******* If there are no 0's between (3{3}3,1), then the form is 3{3}31
******** The smallest prime of the form 3{3}31 is 33333333333333331
**** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1
***** Numbers containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1,3}3{0,1,3}1
******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes)
******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes)
********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes)
********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence.
********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime.
********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1
*********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime.
********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime.
***** Numbers not containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1}1
******* If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1
******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime.
******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1
******** If there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111
********* None of 31151, 31511, 35111 are primes.
******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 351
********* 351 is not prime.
******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111
********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes.
********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime.
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1
********* Neither 3051 nor 3501 is prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** If there are &gt;=2 5's in {}, then 553 will be a subsequence.
*** If there are no 5's in {}, then the family will be 3{0,3,6}3
**** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
*** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3
**** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes)
***** The smallest prime of the form 3{0}53 is 300053
* Case (3,4):
** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes)
*** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes)
**** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes)
**** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes)
* Case (3,5):
** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes)
*** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence.
*** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence.
*** If there are at least two 1's in {}, then 115 will be a subsequence.
*** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5
**** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime.
*** If there is no 1's in {}, then the family will be 3{0,4,5}5
**** If there are at least to 4's in {}, then 344 and 445 will be subsequences.
**** If there is no 4's in {}, then the family will be 3{0,5}5
***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5
****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime.
****** The smallest prime of the form 3{5}5 is 35555
**** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5
***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes)
****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime.
* Case (3,6):
** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,2):
** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes)
*** If there is no 5's in {}, then the family will be 4{0,2,4,6}4
**** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4
**** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes)
***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence.
***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4
****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes)
***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4
****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4
******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes)
* Case (4,5):
** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** If there are at least two 5's in {}, then 4555 will be a subsequence.
*** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there is no 5's in {}, then the family will be 4{0}5
**** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime.
* Case (4,6):
** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (5,1):
** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 3's in {}, then 533 will be a subsequence.
*** If there is no 3's in {}, then the family will be 5{0,1}1
**** Since 5011 is prime, we only need to consider the family 5{1}{0}1
***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes)
****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime.
****** The smallest prime of the form 51{0}1 is 5100000001
****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime.
****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime.
*** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1
**** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence.
***** Thus we only need to consider the family 5{0}3{0,1}1
****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime.
****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime.
****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence.
****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1
******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0)
******** The smallest prime of the form 5311{0}1 is 531101
* Case (5,2):
** 52 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 1's in {}, then 113 will be a subsequence.
*** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
*** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime.
* Case (5,4):
** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes)
*** If there are no 5's in {}, then the family will be 5{0,3}4
**** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime.
*** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence.
*** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there are at least two 5's in {}, then 5554 will be a subsequence.
* Case (5,5):
** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes)
*** If there are no 4's in {}, then the family will be 5{0,5}5
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
*** If there are no 5's in {}, then the family will be 5{0,4}5
**** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime.
*** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence.
* Case (5,6):
** 56 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,2):
** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (6,3):
** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,4):
** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are at least three 3's in {}, then 3334 will be a subsequence.
*** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4
**** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence.
**** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence.
**** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence.
**** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence.
**** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence.
**** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence.
**** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence.
**** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence.
**** Thus the number can only be 634
***** 634 is not prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,6):
** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}6
**** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime.
*** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6
**** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime.
*** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence.
*** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence.

===Base 8===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7)

* Case (1,1):
** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes)
**** 171 is not prime.
**** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 &gt; 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 &gt; 1, this factorization is nontrivial), thus cannot be prime.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes)
*** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime)
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes)
*** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes)
*** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes)
**** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime.
**** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes)
***** The smallest prime of the form 33{4}1 is 3344441
***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime.
**** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes)
***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime.
***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes)
****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes.
* Case (3,3):
** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes)
*** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes)
**** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime)
**** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461
***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes)
****** The smallest prime of the form 4{4}41 is 444444441
****** The smallest prime of the form 4{4}641 is 444641
***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes)
****** The smallest prime of the form 4{4}411 is 444444441
****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes)
***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461
****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime)
**** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes)
***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime)
***** The smallest prime of the form 4{4}641 is 444641
* Case (4,3):
** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes)
*** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes)
**** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime.
**** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime.
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes)
*** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes)
**** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb])
**** The smallest prime of the form 4{4}77 is 4444477
**** The smallest prime of the form 4{7}7 is 47777
**** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime)
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes)
*** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes)
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
**** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes)
***** 500525 is not prime.
***** The smallest prime of the form 5{5}25 is 555555555555525
***** The smallest prime of the form 5{5}025 is 55555025
***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes)
***** The smallest prime of the form 5{5}0525 is 5550525
***** The smallest prime of the form 5{5}00525 is 5500525
***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes)
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence)
**** The form is 6{0,1,7}4{0,1,7}1
***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1
****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411
******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41
******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41
********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.
********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime.
********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime.
******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411
******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime)
*** Numbers not containing 4:
**** The form is 6{0,1,7}1
***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1
****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime.
****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1
******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1
******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.
******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime)
* Case (6,3):
** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,7):
** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them
**** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17
***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime.
**** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77
***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime.
**** If there is neither 1 nor 7 between them, then the form is 6{0,4}7
***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes)
****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime.
****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime.
* Case (7,1):
** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes)
*** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes)
**** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes)
***** The smallest prime of the form 7{7}1 is 7777777777771
***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime)
***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime.
***** The smallest prime of the form 7{7}4444441 is 77774444441
****** Since this prime has just 4 7's, we only need to consider the families with &lt;=3 7's
******* The smallest prime of the form 7{4}1 is 744444441
******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime.
******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes)
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.

===Base 10===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,9):
** 19 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 2{0}21 is 20021
**** The smallest prime of the form 22{0}1 is 22000001
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes)
*** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes)
**** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime.
**** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime.
* Case (2,9):
** 29 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,9):
** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** 47 is prime, and thus the only minimal prime in this family.
* Case (4,9):
** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime.
* Case (5,1):
** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes)
**** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes)
***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 5{5}1 is 555555555551
**** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581
***** 581 is not prime.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,7):
** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027
**** The smallest prime of the form 52{0}7 is 5200007
* Case (5,9):
** 59 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,3):
** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,9):
** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49
***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49
****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes)
******* The smallest prime of the form 6{6}49 is 666649
******** Since this prime has just 4 6's, we only need to consider the families with &lt;=3 6's
********* The smallest prime of the form 6{0}49 is 60000049
********* The smallest prime of the form 66{0}49 is 66000049
********* The smallest prime of the form 666{0}49 is 66600049
* Case (7,1):
** 71 is prime, and thus the only minimal prime in this family.
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,9):
** 79 is prime, and thus the only minimal prime in this family.
* Case (8,1):
** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes)
**** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes)
***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime)
***** The smallest prime of the form 80{5}1 is 80555551
* Case (8,3):
** 83 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime.
* Case (8,9):
** 89 is prime, and thus the only minimal prime in this family.
* Case (9,1):
** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes)
**** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021
***** None of 921, 9201, 9021 are primes.
**** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801
***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
* Case (9,3):
** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (9,7):
** 97 is prime, and thus the only minimal prime in this family.
* Case (9,9):
** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9
***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes)
****** The smallest prime of the form 94{6}9 is 946669

===Base 12===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,B):
** 1B is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes)
*** The smallest prime of the form 2{0}1 is 2001
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (2,B):
** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes)
**** None of 20B, 22B, 29B, 209B, 220B are primes.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,B):
** 3B is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes)
**** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 44{A}1 is 44AAA1
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes)
**** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077
* Case (4,B):
** 4B is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes)
*** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime.
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (5,B):
** 5B is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,5):
** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes)
**** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 6{0}A5 is 600A5
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,B):
** 6B is prime, and thus the only minimal prime in this family.
* Case (7,1):
** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes)
*** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1
**** None of 741, 7B1, 74B1 are primes.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,B):
** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes)
**** The smallest prime of the form 7{9}B is 7999B
**** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes)
**** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime)
* Case (8,1):
** 81 is prime, and thus the only minimal prime in this family.
* Case (8,5):
** 85 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** 87 is prime, and thus the only minimal prime in this family.
* Case (8,B):
** 8B is prime, and thus the only minimal prime in this family.
* Case (9,1):
** 91 is prime, and thus the only minimal prime in this family.
* Case (9,5):
** 95 is prime, and thus the only minimal prime in this family.
* Case (9,7):
** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes)
**** None of 947, 977, 997, 9477, 9977 are primes.
* Case (9,B):
** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes)
**** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes)
***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime)
***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime)
**** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes)
***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes)
****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime.
****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime)
****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime.
**** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B
***** None of 99B, 999B, 979B, 9799B, 9979B are primes.
* Case (A,1):
** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes)
**** For the A{A}{0}1 family:
***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form AA{0}1 is AA000001
***** The smallest prime of the form AAA{0}1 is AAA0001
***** The smallest prime of the form AAAA{0}1 is AAAA1
****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc.
**** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime.
* Case (A,5):
** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes)
*** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes)
**** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form A{0}65 is A00065
* Case (A,7):
** A7 is prime, and thus the only minimal prime in this family.
* Case (A,B):
** AB is prime, and thus the only minimal prime in this family.
* Case (B,1):
** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes)
**** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1
***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1
****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime)
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1
****** The smallest prime of the form B{A}01 is BAA01
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
**** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1
***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1
****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1
******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1
******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes.
****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1
******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime)
***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes)
****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B04{A}1 is B04A1
****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime)
**** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1
***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1
****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1
******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime.
******* The smallest prime of the form B{B}AA1 is BBBAA1
****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1
******* The smallest prime of the form B{B}AA1 is BBBAA1
**** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes)
***** The smallest prime of the form B{B}B1 is BBBB1
***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form B{B}BAA1 is BBBAA1
* Case (B,5):
** B5 is prime, and thus the only minimal prime in this family.
* Case (B,7):
** B7 is prime, and thus the only minimal prime in this family.
* Case (B,B):
** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes)
*** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B
**** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B
***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime.
***** For the B{0,B}9B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes)
******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B
******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 &gt; 1, 10^n+5 ≥ 10^1+5 = 15 &gt; 1, this factorization is nontrivial), thus cannot be prime.
***** For the B{0,B}99B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes)
******* The smallest prime of the form B{0}99B is B00099B
******* The smallest prime of the form B{B}99B is BBBBBB99B
***** For the B{0,B}999B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes)
******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes)
******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime)

== Examples of families which can be ruled out as contain no primes &gt; ''b'' ==

It is not known if this problem is solvable:

Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''?

It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime &gt; ''b'' or not. We use two different heuristic strategies to show that such families contain no primes &gt; ''b''.

In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;.

Examples of first strategy: (we can show that the corresponding numbers are &gt; all elements in ''S'', if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial)

* In base 10, all numbers of the form 4{6}9 are divisible by 7
* In base 6, all numbers of the form 4{0}1 are divisible by 5
* In base 15, all numbers of the form 9{6}8 are divisible by 11
* In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5}
* In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3}
* In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5}
* In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13}
* In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17}
* In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13}
* In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13}

Examples of second strategy: (we can show that both factors are &gt; 1, if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial)

* In base 9, all numbers of the form {1} factored as difference of squares
* In base 8, all numbers of the form 1{0}1 factored as sum of cubes
* In base 9, all numbers of the form 3{8} factored as difference of squares
* In base 16, all numbers of the form 8{F} factored as difference of squares
* In base 16, all numbers of the form {F}7 factored as difference of squares
* In base 9, all numbers of the form 3{1} factored as difference of squares
* In base 16, all numbers of the form {4}1 factored as difference of squares
* In base 16, all numbers of the form 1{5} factored as difference of squares
* In base 16, all numbers of the from {C}D factored as ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;
* In base 16, all numbers of the form B{4}1 factored as difference of squares

Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are &gt; all elements in S, and for the part of the second strategy, both factors are &gt; 1, if n makes corresponding numbers &gt; b, thus these factorizations are nontrivial)

* In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd
* In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even

== Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes &gt; ''b'' ==

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-powers factorization

=== 1{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 1{0}3 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== 1{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 1{0}5 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 5: Finite covering set {5}

=== 1{0}6 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== 1{0}7 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 7: Finite covering set {7}

=== 1{0}z ===

(none)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

(none)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {1} ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with ''r''&gt;1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2))

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' == 0 mod 2: Finite covering set {2}

=== 1{2} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 1{3} ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361)
* ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865)

=== 1{4} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization

=== 1{z} ===

(none)

=== 2{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 2{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {2}1 ===

* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 2{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 3{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {3}1 ===

* ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121)

=== 3{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 4{0}1 ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 4{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {4}1 ===

* ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225)

=== 4{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 5{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 5{z} ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}
* ''b'' = 6''m''&lt;sup&gt;2&lt;/sup&gt; with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014)

=== 6{0}1 ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== 6{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 7{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 7{z} ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993)
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Difference-of-cubes factorization

=== 8{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022)
* ''b'' = 467: Finite covering set {3, 5, 7, 19, 37}
* ''b'' = 722: Finite covering set {3, 5, 13, 73, 109}
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Sum-of-cubes factorization
* ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2

=== 8{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 9{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 9{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{0}1 ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{z} ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 307: Finite covering set {5, 11, 29}
* ''b'' = 901: Finite covering set {7, 11, 13, 19}

=== C{0}1 ===

* ''b'' == 1 mod 13: Finite covering set {13}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 296, 901: Finite covering set {7, 11, 13, 19}
* ''b'' = 562, 828, 900: Finite covering set {7, 13, 19}
* ''b'' = 563: Finite covering set {5, 7, 13, 19, 29}
* ''b'' = 597: Finite covering set {5, 13, 29}

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

(none)

=== {y}z ===

(none)

=== z{0}1 ===

(none)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3))
* ''b'' = 4''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt; (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}t ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 7: Finite covering set {7}

=== {z}u ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== {z}v ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 5: Finite covering set {5}

=== {z}w ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== {z}x ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== {z}y ===

* ''b'' == 0 mod 2: Finite covering set {2}

== Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) ==

Format: base (length)

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

(none)

=== 1{0}2 ===

(none)

=== 1{0}3 ===

(none)

=== 1{0}4 ===

53 (13403)

113 (10647)

=== 1{0}z ===

113 (20089)

123 (64371)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

(none)

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

208 (26682)

607 (11032)

828 (19659)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

201 (31276)

222 (52727)

227 (36323)

327 (135983)

425 (11231)

710 (24112)

717 (37508)

719 (13420)

=== {1} ===

152 (270217)

184 (16703)

200 (17807)

311 (36497)

326 (26713)

331 (25033)

371 (15527)

485 (99523)

629 (32233)

649 (43987)

670 (18617)

684 (22573)

691 (62903)

693 (41189)

731 (15427)

752 (32833)

872 (10093)

932 (20431)

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

(none)

=== 1{z} ===

107 (21911)

170 (166429)

278 (43909)

303 (40175)

383 (20957)

515 (58467)

522 (62289)

578 (129469)

590 (15527)

647 (21577)

662 (16591)

698 (127559)

704 (62035)

845 (39407)

938 (40423)

969 (24097)

989 (26869)

=== 2{0}1 ===

101 (192276)

206 (46206)

218 (333926)

236 (161230)

257 (12184)

305 (16808)

467 (126776)

578 (44166)

626 (174204)

695 (94626)

752 (26164)

788 (72918)

869 (49150)

887 (27772)

899 (15732)

932 (13644)

=== 2{z} ===

432 (16003)

=== 3{0}1 ===

912 (132174)

=== 3{z} ===

72 (1119850)

212 (34414)

218 (23050)

270 (89662)

303 (198358)

312 (51566)

422 (21738)

480 (93610)

513 (38032)

527 (46074)

566 (23874)

650 (498102)

686 (16584)

758 (15574)

783 (12508)

800 (33838)

921 (98668)

947 (10056)

=== 4{0}1 ===

107 (32587)

227 (13347)

257 (160423)

355 (10990)

410 (144079)

440 (56087)

452 (14155)

482 (30691)

542 (15983)

579 (67776)

608 (20707)

635 (11723)

650 (96223)

679 (69450)

737 (269303)

740 (58043)

789 (149140)

797 (468703)

920 (103687)

934 (101404)

962 (84235)

=== 4{z} ===

14 (19699)

68 (13575)

254 (15451)

800 (20509)

=== 5{0}1 ===

326 (400786)

350 (20392)

554 (10630)

662 (13390)

926 (40036)

=== 5{z} ===

258 (212135)

272 (148427)

299 (64898)

307 (26263)

354 (25566)

433 (283919)

635 (36163)

678 (40859)

692 (45447)

719 (20552)

768 (70214)

857 (23083)

867 (61411)

972 (36703)

=== 6{0}1 ===

108 (16318)

129 (16797)

409 (369833)

522 (52604)

587 (24120)

643 (164916)

762 (11152)

789 (27297)

986 (21634)

=== 6{z} ===

68 (25396)

332 (15222)

338 (42868)

362 (146342)

488 (33164)

566 (164828)

980 (50878)

986 (12506)

1016 (23336)

=== 7{0}1 ===

398 (17473)

1004 (54849)

=== 7{z} ===

97 (192336)

170 (15423)

194 (38361)

202 (155772)

282 (21413)

283 (164769)

332 (13205)

412 (29792)

560 (19905)

639 (10668)

655 (53009)

811 (31784)

814 (17366)

866 (108591)

908 (61797)

962 (31841)

992 (10605)

997 (15815)

=== 8{0}1 ===

23 (119216)

53 (227184)

158 (123476)

254 (67716)

320 (52004)

410 (279992)

425 (94662)

513 (19076)

518 (11768)

596 (148446)

641 (87702)

684 (23387)

695 (39626)

785 (900326)

788 (11408)

893 (86772)

908 (243440)

920 (107822)

962 (47222)

998 (81240)

1013 (43872)

=== 8{z} ===

138 (35686)

412 (12154)

788 (11326)

990 (23032)

=== 9{0}1 ===

248 (39511)

592 (96870)

=== 9{z} ===

431 (43574)

446 (152028)

458 (126262)

599 (11776)

846 (12781)

=== A{0}1 ===

173 (264235)

198 (47665)

311 (314807)

341 (106009)

449 (18507)

492 (42843)

605 (12395)

708 (17563)

710 (31039)

743 (285479)

744 (137056)

786 (68169)

800 (15105)

802 (149320)

879 (25004)

929 (13065)

977 (125873)

986 (48279)

1004 (10645)

=== A{z} ===

368 (10867)

488 (10231)

534 (80328)

662 (13307)

978 (14066)

=== B{0}1 ===

710 (15272)

740 (33520)

878 (227482)

=== B{z} ===

153 (21660)

186 (112718)

439 (18752)

593 (16064)

602 (36518)

707 (10573)

717 (67707)

=== C{0}1 ===

68 (656922)

219 (29231)

230 (94751)

312 (21163)

334 (83334)

353 (20262)

359 (61295)

457 (10024)

481 (45941)

501 (20140)

593 (42779)

600 (11242)

604 (17371)

641 (26422)

700 (91953)

887 (13961)

919 (45359)

923 (64365)

992 (10300)

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

(none)

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

38 (136212)

83 (21496)

113 (286644)

188 (13508)

401 (103670)

417 (21003)

458 (46900)

494 (21580)

518 (129372)

527 (65822)

602 (17644)

608 (36228)

638 (74528)

663 (47557)

723 (24536)

758 (50564)

833 (12220)

904 (13430)

938 (50008)

950 (16248)

=== z{0}1 ===

202 (46774)

251 (102979)

272 (16681)

297 (14314)

298 (60671)

326 (64757)

347 (69661)

363 (142877)

452 (71941)

543 (10042)

564 (38065)

634 (84823)

788 (13541)

869 (12289)

890 (37377)

953 (60995)

1004 (29685)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

53 (21942)

124 (16426)

175 (31626)

188 (22036)

316 (48538)

365 (25578)

373 (24006)

434 (10090)

530 (11086)

545 (12346)

560 (15072)

596 (12762)

701 (12576)

706 (10656)

821 (13536)

833 (17116)

966 (14820)

983 (11272)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}y ===

317 (13896)

== Bases 2≤''b''≤1024 which have these families as unsolved families ==

Unsolved families are families which are neither primes (&gt;''b'') found nor can be ruled out as contain no primes &gt; ''b''

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608)

1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000)

1{0}3: 646, 718, 998 (length limit: 2000)

1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000)

1{0}5

1{0}6

1{0}7

1{0}8

1{0}9

1{0}A

1{0}B

1{0}C

1{0}D

1{0}E

1{0}F

1{0}G

1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000)

1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000)

10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000)

11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000)

{1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000)

{1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000)

11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z})

{1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000)

{1}z

1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000)

1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995

1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000)

1{z}: 581, 992, 1019 (length limit: ≥100000)

2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000)

2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000)

2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000)

{2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000)

2{z}: 588, 972 (length limit: ≥100000)

3{0}1: 718 (length limit: ≥100000)

3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000)

3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000)

{3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000)

3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000)

4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000)

4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000)

{4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000)

4{z}: 338, 998 (length limit: ≥100000)

5{0}1: 308, 512, 824 (length limit: ≥100000)

5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000)

6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000)

6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000)

7{0}1: (none)

7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000)

8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000)

8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000)

9{0}1: 724, 884 (length limit: ≥100000)

9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000)

A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000)

A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000)

B{0}1: 560, 770, 968 (length limit: ≥100000)

B{z}: 263, 615, 912, 978 (length limit: ≥100000)

C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000)

C{z}

D{0}1

D{z}

E{0}1

E{z}

F{0}1

F{z}

G{0}1

{&amp;#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000)

{&amp;#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143)

&amp;#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000)

${&amp;#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2)

x{z}

y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000)

z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000)

{y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000)

{z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326)

zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z})

{z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000)

{z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1)

{z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000)

{z}k

{z}l

{z}m

{z}n

{z}o

{z}p

{z}q

{z}r

{z}s

{z}t

{z}u

{z}v

{z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000)

{z}x: (none)

{z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000)

== List of lengths for quasi-minimal primes in some simple families ==

[https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024]

NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases &lt;=7, family {z}x in bases &lt;=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime &gt; base) in this base)

RC: this family can be proven to only contain composite numbers (only count numbers &gt; base)

unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers &gt; base)

Background color: red for title (bases or families), green for length &gt; 10000, orange for 2500 &lt; length ≤ 10000, white for length ≤ 2500, cyan for &quot;RC&quot;, pink for &quot;NB&quot;, yellow for &quot;unknown&quot;.

Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families.

== References ==

* [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem]

* [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are &gt; ''b'' for bases ''b''≤16]

* [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary]

* [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia]

* [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime&gt;base is not required) in The Prime Puzzles &amp; Problems Connection]

* [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles &amp; Problems Connection]

* [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16]

* [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime&gt;base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29]

* [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime&gt;base is not required, search limits of lengths: 10000 for all b]

* [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime&gt;base is not required]

* [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime&gt;base is not required]

* [http://www.prothsearch.com/sierp.html the Sierpinski problem]

* [http://www.prothsearch.com/rieselprob.html the Riesel problem]

* [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem]

* [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030]

* [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100]

* [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem]

* [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024]

* [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project]

* [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project]

* [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count]

* [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty]

* [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers]

* [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares]

* [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software]

* [https://sourceforge.net/projects/openpfgw/ pfgw software]

* [http://jpenne.free.fr/index2.html llr software]

* [http://www.ellipsa.eu/public/primo/primo.html PRIMO software]

* [https://primes.utm.edu/prove/index.html website for primality proving]

* [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!]

* [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime&gt;base is not required]

* [https://primes.utm.edu/primes/lists/all.txt top proven primes]

* [http://www.primenumbers.net/prptop/prptop.php top PRPs]

* [http://factordb.com online factor database, including many primes which are minimal primes in a small base]</text>
      <sha1>3pt3gnnboykc1gorluxkf0m4at2bb4h</sha1>
    </revision>
    <revision>
      <id>2412028</id>
      <parentid>2412027</parentid>
      <timestamp>2022-08-05T03:53:26Z</timestamp>
      <contributor>
        <ip>2402:7500:901:2B90:BD69:8142:3D96:F08F</ip>
      </contributor>
      <comment>/* Unsolved families */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1961633" xml:space="preserve">A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime &gt; ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime &gt; 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 &gt; 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 &gt; 10 by changing the order.

(using A−Z to represent digit values 10 to 35)

For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')).

I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E&lt;sub&gt;800873&lt;/sub&gt;, and base 36 has quasi-minimal prime P&lt;sub&gt;81993&lt;/sub&gt;SZ).

Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to:

* Prove that all elements in ''S'', when read as base ''b'' representation, are primes &gt; ''b''.

* Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are &gt; ''b'', are composite.

* Prove that all primes &gt; ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are &gt; ''b'', are composite).

e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to:

* Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes &gt; 10.

* Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are &gt; 10 are composite.

* Prove that all primes &gt; 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers &gt; 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite).

==Condensed table==

{|class=wikitable
|''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form of largest known quasi-minimal prime base ''b''
|-
|2||1||11||2||3
|-
|3||3||111||3||13
|-
|4||5||221||3||41
|-
|5||22||10&lt;sub&gt;93&lt;/sub&gt;13||96||5&lt;sup&gt;95&lt;/sup&gt;+8
|-
|6||11||40041||5||5209
|-
|7||71||3&lt;sub&gt;16&lt;/sub&gt;1||17||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2
|-
|8||75||4&lt;sub&gt;220&lt;/sub&gt;7||221||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||1161||3×9&lt;sup&gt;1160&lt;/sup&gt;+10
|-
|10||77||50&lt;sub&gt;28&lt;/sub&gt;27||31||5×10&lt;sup&gt;30&lt;/sup&gt;+27
|-
|11&lt;sup&gt;*&lt;/sup&gt;||1068||57&lt;sub&gt;62668&lt;/sub&gt;||62669||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10
|-
|12||106||40&lt;sub&gt;39&lt;/sub&gt;77||42||4×12&lt;sup&gt;41&lt;/sup&gt;+91
|-
|13&lt;sup&gt;*&lt;/sup&gt;||3195~3197||80&lt;sub&gt;32017&lt;/sub&gt;111||32021||8×13&lt;sup&gt;32020&lt;/sup&gt;+183
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||19699||5×14&lt;sup&gt;19698&lt;/sup&gt;−1
|-
|15||1284||7&lt;sub&gt;155&lt;/sub&gt;97||157||(15&lt;sup&gt;157&lt;/sup&gt;+59)/2
|-
|16&lt;sup&gt;*&lt;/sup&gt;||2346~2347||4&lt;sub&gt;72785&lt;/sub&gt;DD||72787||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15
|-
|17&lt;sup&gt;*&lt;/sup&gt;||10408~10428||F70&lt;sub&gt;186767&lt;/sub&gt;1||186770||262×17&lt;sup&gt;186768&lt;/sup&gt;+1
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||6271||12×18&lt;sup&gt;6270&lt;/sup&gt;+221
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||6271||16×20&lt;sup&gt;6270&lt;/sup&gt;+13
|-
|21&lt;sup&gt;*&lt;/sup&gt;||13375~13396||CF&lt;sub&gt;479147&lt;/sub&gt;0K||479150||(51×21&lt;sup&gt;479149&lt;/sup&gt;−1243)/4
|-
|22&lt;sup&gt;*&lt;/sup&gt;||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||22003||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||8134||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49
|-
|30&lt;sup&gt;*&lt;/sup&gt;||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||34206||25×30&lt;sup&gt;34205&lt;/sup&gt;−1
|-
|36&lt;sup&gt;*&lt;/sup&gt;||35257~35263||P&lt;sub&gt;81993&lt;/sub&gt;SZ||81995||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7
|}

&lt;sup&gt;*&lt;/sup&gt; Data assumes the primality of the [[w:probable prime|probable prime]]s.

Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes &gt; 10&lt;sup&gt;25000&lt;/sup&gt;), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below.

Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists))

Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists))

==Data for quasi-minimal primes==

===Base 2===

11

===Base 3===

12, 21, 111

===Base 4===

11, 13, 23, 31, 221

===Base 5===

12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013

===Base 6===

11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041

===Base 7===

14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331

===Base 8===

13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447

===Base 9===

12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011

===Base 10===

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027

===Base 11===

12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, 2888882883, 2888888883, 3555555509, 3577777077, 3700000001, 4000000005, 40000005AA, 5377777707, 5555505553, 555555580A, 600000A999, 7000100047, 7000704777, 7007777107, 7057777777, 7070007774, 7077707774, 7077777074, 7100000704, 7470000041, 7701000004, 7707077774, 7770707774, 7777707074, 8888822883, 9555555503, 9900000979, 9999770007, A000144444, A900000066, A999999971, 10000000477, 33333333337, 44444444447, 44444444777, 55A55555552, 60000008883, 68888888306, 68888888883, 70000003999, 70000007717, 70004777777, 70477777777, 77007770004, 77700000477, 77707000704, 77707770074, 77707777774, 77777077774, 77777770004, 83000000006, 97000000999, A0000000001, A0014444444, A4777777771, 100000000057, 305007777777, 305777777777, 370000000007, 377777770007, 377777777107, 700000007474, 707077000074, 707077777774, 707777777717, 770000010004, 771007000007, 777070700004, 777700000704, A95555555555, A99999777777, 1000000003007, 40000000A0041, 58A5555555555, 7004400000004, 7700000000104, 7707000007047, 7707707000004, 7777007000004, 7777700000004, 7777770077704, 7777777710077, 9977777777717, A000000014444, A044444444441, A144444444411, 40000000000401, 45557777777777, 4555AAAAAAAAAA, 59077777777777, 70007777777771, 70077070000074, 70700000004777, 77000007700704, 77700000700047, 77777777770704, 88888888830006, 90900000000799, A0000044444441, 300000000005777, 302000000000002, 55555555A555552, 700000000000174, 770000000000474, 771700000000007, 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7707777777777777777704, 7900000000000000000005, 9777777777777777777707, A999999999999999999999, 10000000000000000000747, 970000000000000000000777, 999900000000000000000007, 3577777777777777777777777, 5555555555555555555555A52, 7000000000000000000777771, 7000000000000000077777771, AAAAAAAAAAAAAAAAAAA000058, 10000000000000000000000044, 77700000000000000000000008, 500777777777777777777777777, 777777777777777777777770774, 855555555555555555555055555, A44444444444444444444444441, 1500000000000000000000000007, 40000000000000000000000000041, 440000000000000000000000000001, 70000000000000000000000000007771, 999999999999999999999999999999991, 95555555555555555555555555555555553, 1900000000000000000000000000000000001, 7777777777777777777777777777777777474, 7777777777777777777777777777777777704, 10000000000000000000000000000000000000307, 50777777777777777777777777777777777777707, 475777777777777777777777777777777777777777, 555555555555555555555555555555555555555A2A, 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===Base 12===

11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077

===Base 13===

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===Base 14===

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===Base 15===

12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797

===Base 16===

11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD

===Base 17===

12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 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====Additional known quasi-minimal primes (not necessarily the next)====

570&lt;sub&gt;51310&lt;/sub&gt;1, 49&lt;sub&gt;111333&lt;/sub&gt;, 970&lt;sub&gt;166047&lt;/sub&gt;1, F70&lt;sub&gt;186767&lt;/sub&gt;1

===Base 18===

11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, 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===Base 20===

13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, GGJ0J, GGJJJ, GJ00J, GJGJJ, GJJJ7, H00EH, H024B, H04B1, H0E4B, H0F41, H0H11, H0HEH, H4E0B, H6I11, HAAA3, HAAG3, HAGA3, HB44B, HBBBB, HBHHH, HGGA3, HGGG3, HH001, HH061, HH3AH, HH6F1, HHG63, HHIA3, HI011, HIII3, I0029, I02I9, I085H, I0A41, I0F29, I2229, I2929, I2I29, I2I99, I2II9, I33A3, I3EHH, I422B, I424B, I4EEB, I4IEB, I80G1, I9IA9, IA099, IA309, IA841, IA939, IA9A9, IAF09, IAF99, IAI99, IAIA9, IB0A1, IB4IB, IB54B, IBB4B, IBBIB, IBE0H, IBH0H, IC929, ICC29, IF841, IFAG1, IGA81, IHB4B, II0A9, II42B, II44B, IIB5B, III4B, IIIA3, IIIIB, J00G1, J00JB, J0601, J06D1, J06I1, J0861, J08B1, J0B01, J0CGJ, J0E0B, J0GA1, J0GG1, J0I2B, J0J2B, J0JIB, J55CJ, J60A1, J60G1, J6D81, JA777, JA7A7, JAJG1, JAJJ1, JAJJ7, JB00B, JB08B, JC00J, JC0GJ, JEECJ, JGGGJ, JGGJJ, JJ0B1, JJ0JB, JJ55J, JJAG1, JJAJ1, JJC0J, JJE0B, JJEEB, JJGG1, JJGGJ, JJGJJ, JJJ01, JJJEB, JJJJB, 17555J, 175EEJ, 1E0007, 1JE55J, 1JJJ5J, 20005H, 2000HH, 222I29, 22E889, 22EE89, 24222B, 2999I9, 29III9, 2A000H, 2B0I0H, 2I2999, 2I9I29, 2III29, 333AF3, 336IA3, 36A3F3, 36I3A3, 388E8H, 3AF333, 3EHHHH, 3HEHHH, 3HHEHH, 40008B, 40054B, 40405B, 40448B, 415E5J, 44440B, 44452B, 44524B, 44E08B, 45444B, 4AAAE7, 4AEEE7, 4E4447, 4GEEE7, 4I544B, 4IEEEB, 500EFJ, 500FCJ, 5055FJ, 50AFFJ, 50EEFJ, 50EF5J, 50F5FJ, 50FFCJ, 52224B, 54EEEB, 5558AH, 555CAJ, 5585AH, 55F55J, 55FFFJ, 5888AH, 588A5H, 5E555J, 5F055J, 5F05CJ, 5F5F5J, 5FFFCJ, 5JEEEB, 60006D, 6000F1, 6000G3, 6001G1, 600A63, 600G1D, 600G6D, 603A33, 606G6D, 608001, 608F89, 608IF9, 60A333, 60A363, 60AAA3, 60CCCD, 60D04D, 60D0DD, 60F8I9, 60FF89, 60GCG3, 636AF3, 63A333, 63A3F3, 63AFF3, 63IIA3, 64AAAD, 660089, 660A33, 660DDD, 660F89, 662289, 666A33, 666D4D, 66A6A3, 66AF33, 66AF63, 66D0DD, 66FF89, 6A33F3, 6AFF33, 6AG6A3, 6AGA63, 6D000D, 6D004D, 6D0D4D, 6F6689, 7000G7, 7060DD, 70AAAD, 70D0CD, 733C33, 7366C3, 73AAA3, 7600DD, 760CDD, 766333, 76664D, 766FC3, 76CCDD, 76DD4D, 770G07, 771JEJ, 7771EJ, 777481, 7777I1, 777F81, 77C7JJ, 77EC77, 77EEC7, 77EJJJ, 77GCG7, 77JECJ, 7A7E77, 7AAAG3, 7AAF63, 7AEEE7, 7C7GG7, 7CCCAD, 7CCCG7, 7CGJJJ, 7D000D, 7J000D, 7JCGGJ, 80005B, 8000G1, 8000I1, 800GA1, 800I01, 80GGA1, 84405B, 84454B, 888EAH, 88E8AH, 8A4441, 8B8EAH, 8E8IF9, 8I0AG1, 902229, 909A09, 909C29, 909FA9, 90IC29, 9222I9, 92I999, 944EE7, 94AEE7, 977GG7, 97AEE7, 97GGG7, 990I29, 999I29, 99I299, 99IIA9, 9A9447, 9A9AF9, 9A9FF9, 9CII29, 9EEEE7, 9FA9A9, 9I9C29, 9I9I29, 9ICI29, 9IIIA9, A1555J, A1E55J, A333C9, A336A3, A33889, A33F63, A36663, A444E7, A4AAE7, A555FJ, A666A3, A6AAA3, A7A7E7, A7AE77, A944E7, A9A9F9, A9AAA9, AA3389, AA9AF9, AAA099, AAA3C9, AAA7E7, AAA939, AAACG9, AAAFF9, AAAI99, AAAIC9, AAE4E7, AAG9A9, AAGAA3, ADE55J, AF6663, AF9FF9, AGA963, AGG6A3, AH4441, AI000H, AI99C9, AI9AA9, AII099, B0004B, B0054B, B0080B, B00H0H, B04IIB, B05B8B, B0A6I1, B0BB8B, B0BIBB, B0E0IH, B0HA0H, B0I0HH, B0I4IB, B0IIBB, B888AH, BB058B, BB05BB, BB080B, BB0I4B, BBB04B, BBB08B, BBIB8B, BE88AH, BEHA0H, BEHHAH, BHEA0H, BHEHAH, BHHA0H, BIE00H, C007A7, C00GG7, C00JA7, C00JAJ, C00JC7, C0C7A7, C0JAC7, C0JJC7, C3G999, CAJJ77, CC0629, CC0929, CCC007, CCCCC7, CCCCG3, CCCGG7, CCCI29, CCI029, CD4GGJ, CE0009, CEFF69, CEFFF9, CEI609, CII029, CIII29, CJACC7, CJAJC7, CJJJA7, D000D7, D000J7, D004CD, D0D007, DCCDC7, DDD04D, DDDAI1, DEJJJJ, DG000D, DG5E5J, DGGG5J, DJ00CJ, DJEEE7, DJJJ5J, E00097, E00A0H, E00AA9, E02289, E028I9, E029I9, E055AJ, E0774J, E0777J, E07JCJ, E0AF5J, E0BI8B, E0E4CJ, E0EA5J, E0ECJJ, E0FFF9, E0H0HH, E0HA0H, E0IF29, E0J5AJ, E0JJAJ, E29299, E444E7, E4EEE7, E4IIIB, E66289, E66629, E66909, E66F69, E69609, E69999, E7774J, E77C77, E7C7C7, E8B8AH, E90029, E90229, E90909, E90FF9, E92299, E94EE7, E97EE7, E990F9, E99I09, E9F669, E9I029, EA8AF9, EAAAI9, EAIAA9, EAJJJJ, EBIBIB, EC7CC7, EE08F9, EE408B, EE8IF9, EEAF5J, EEE84B, EEEC07, EEEC77, EEEE0B, EEF55J, EEFFFJ, EEJ5AJ, EFE009, EFEF09, EFF669, EFFFF9, EGGGG9, EIF669, F004AJ, F00A41, F00JA1, F04481, F055CJ, F0A15J, F0EI09, F0G01J, F0J0GJ, F0J0JJ, F0JJ5J, F14441, F68F89, F68IF9, F800G1, F8E8I9, F99299, FA0009, FA1JJJ, FAAIA9, FAII09, FAII99, FCC929, FCE229, FCE609, FCEF29, FE68F9, FE8FA9, FF0E09, FF68F9, FF9FA9, FFC4AJ, FFC929, FFCE09, FFE009, FFEE89, FFEF09, FFFA99, FFFFFJ, FFFGJJ, FG1JJJ, FI99A9, FIA999, FIAAA9, FIAII9, FIIA09, G00093, G000EJ, G000G9, G00781, G007G3, G00C09, G07GC3, G09033, G0C903, G0CGG3, G0G903, G0G933, G0GCC3, G0GGC9, G30003, G36003, G5000J, G5E0EJ, G60303, G6AGG3, G6GAA3, G7AAA3, G7G363, G90AA9, G90C63, G9AAA9, GAAAA3, GAAGG3, GC0003, GCCC93, GCCCC3, GCG903, GDEEE7, GE0007, GG0CG3, GGA8G1, GGC6G3, GGCCG3, GGCGC9, GGG0I9, GGG363, GGG6G3, GGG75J, GGG8G1, GGG963, GGGAA3, GGGCG3, GGGCG9, H044EB, H0BIBB, H0E00B, H0F011, H0IIEB, H2444B, H400EB, H44IEB, H4EEEB, HBB4IB, HE000B, HEE40B, HEEE4B, HEHHAH, HF0011, HF4441, HH3HEH, HHGAA3, HHGAG3, HHHAG3, HHHEAH, HI0001, HIBIBB, HIEBBB, I0000H, I000A9, I002HH, I00A09, I00A99, I00AG1, I00AI9, I00H3H, I02999, I03HEH, I09AF9, I0A009, I0E0HH, I0I299, I0IIA9, I4445B, I4I45B, I90I29, I99029, I99A09, I9I029, IA3999, IB000H, IB00EH, IB0HEH, IB5B8B, IF2999, II0929, II92I9, II99A9, II9C29, IIAFF9, IIF299, III5BB, III8BB, J0000B, J00081, J000IB, J0010D, J006A1, J00BI1, J00I8B, J00IA1, J05EEJ, J06G81, J0C0JJ, J0E0CJ, J0EEEB, J0J0EB, J5EAEJ, J5EEAJ, J608I1, JE505J, JEEEEB, JJ0001, JJ0JCJ, JJ0JGJ, JJAJC7, JJJAE7, JJJBI1, JJJI8B, 14GGGG1, 1J5555J, 1JGGGG1, 205555H, 20I000H, 222222B, 2929999, 2BI000H, 3333A33, 3333G09, 333A3A3, 333AAA3, 336AAA3, 336AF33, 33II3A3, 363AF33, 3GIIII9, 400445B, 404454B, 404800B, 415555J, 440045B, 440080B, 442222B, 444004B, 444422B, 444444B, 4444G47, 444GGG7, 444I4IB, 44AAA47, 44I44IB, 44I4IIB, 4AAAAA7, 4EEE4E7, 4I4454B, 5000AFJ, 525555H, 555552H, 555555J, 555585H, 58BBBBB, 5BB8BBB, 5E0E55J, 5EBBBBB, 60003A3, 60008I9, 6000DDD, 6006DDD, 60080A1, 60080I1, 600A3A3, 600D8I1, 600F841, 60CCDDD, 60CGGG3, 60DDD0D, 60DDDDD, 60GGG63, 60GGGC3, 66600A3, 66603A3, 66608I9, 6666089, 66666A3, 6666C29, 6666F89, 666AFF3, 66DDDDD, 6800001, 6AGGGG3, 6CCCC29, 6G6AAA3, 6GAGGG3, 6GCCC63, 6GGG633, 700000D, 70000CD, 7000EC7, 70070C7, 700EEC7, 7070EC7, 70EEEC7, 76666CD, 77333A3, 7733A33, 775EE5J, 777A777, 777CGJJ, 777GE5J, 77A7777, 77JCJJJ, 77JJJCJ, 77JJJJJ, 7A77777, 7A777A7, 7AAAA63, 7AAAAAD, 7C00007, 7E77JCJ, 7EE7JCJ, 7EEC007, 7EEEEC7, 7G5E55J, 7GE555J, 7GJJJJJ, 7J6666D, 9000029, 9000299, 9009I29, 900I2I9, 90FAAA9, 90I0299, 90I0I29, 90I9929, 90II029, 90II299, 90II2I9, 94444E7, 97777E7, 9902I99, 99902I9, 9992929, 9992999, 9999929, 99I0C29, 99I9029, 99I92I9, 99III29, 9AAAF09, 9AFFFF9, 9I0AAA9, 9IAAA09, 9II0I29, 9II9929, 9III029, 9III2I9, 9IIIC29, A5F555J, A777E77, A994447, A9AAF99, AA00089, AA6A6A3, AAA0009, AAA3999, AAA9A99, AAA9AA9, AAAAA47, AAAAA77, AAAAA99, AAAAG99, AAAAIA9, AAAD447, AAAGAC9, AAGAAAD, AAGAAC9, AF00009, AGAAA99, AGAGG63, AGGGA63, AI0C999, AJ777A7, B000B8B, B000HEH, B000I0H, B00I00H, B0B805B, B0HHHEH, B0IHE0H, B0III8B, B8B000B, BB8BBBB, BBB8B0B, BEH0HHH, BHHHEHH, C00000J, C000C07, C066629, C0AJJJJ, C0C0A47, C0G000J, C0JAJJJ, C0JJAJJ, C0JJJJJ, C6CCG33, CC004A7, CC00A47, CC6CG33, CC92229, CCCC7G7, CCCGGG3, CE66609, CEF6669, D00007D, D00007J, D00071J, D000C0J, D000I01, D00700D, D00C007, D00C0C7, D00G00D, D0C00C7, D0CDCC7, D444447, D7JJJJJ, DC0000J, DDD7DDD, DDDCCC7, DJ0000J, DJ000EJ, DJJJJ0J, DJJJJJJ, E000009, E00084B, E000889, E000909, E000HAH, E008AF9, E04000B, E0AA8F9, E0CJJJJ, E0HHHHH, E40400B, E6660F9, E666609, E669969, E699669, E7CCC07, E996669, E99I669, E9FAAA9, EB8BBBB, EBBII8B, EBH0HHH, EE092I9, EE44IIB, EE4E4E7, EEBB8BB, EEC0007, EEE0IBB, EEE0IIB, EEEBB8B, EEEE44B, EEJCJJJ, EF00F09, EF0555J, EFAAAA9, EFF0009, EFF00F9, EFF60F9, EI0AAA9, EI660F9, EIIBB8B, EJ0CJJJ, EJ5055J, EJJ0CJJ, F0E00F9, F0E88I9, F0JJJJ1, F299929, F4G555J, F5555CJ, F5555FJ, F6668I9, F8GGGG1, F929999, F999C29, FA84441, FAAA9A9, FCE00F9, FCEF009, FCFEE29, FE00F09, FEE2889, FEF0009, FFAA9A9, FFCEE29, FFFF4AJ, FIII9A9, FIIIAI9, FJJJJ81, G000009, G0000J7, G000303, G000AA9, G000CC3, G000GG3, G003003, G0090C3, G00CG03, G00D007, G00G0G3, G00GG03, G00GG93, G080001, G0G00I9, G0G09C3, G0G0C93, G0G0GG9, G0GG093, G0GGC63, G0GGGG9, G44GG47, G6AAGA3, G999999, GAA9AA9, GCCCGG9, GCGGCC9, GCGGG03, GEE4447, GEEEE47, GG0C007, GG0GC09, GG0GGG3, GG6AAA3, GG6AGA3, GG6GAG3, GGEEE47, GGG00C9, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 21===

12, 18, 1A, 1G, 1K, 21, 25, 2B, 2H, 2J, 34, 38, 3A, 3G, 3K, 45, 4D, 4H, 4J, 52, 54, 58, 61, 65, 6B, 6D, 72, 74, 7A, 7G, 7K, 85, 8B, 8D, 92, 94, 98, 9A, A1, AD, AH, AJ, B2, B8, BA, BK, C5, CB, CH, CJ, D4, D8, DA, DK, ED, EH, EJ, F2, FG, G1, GB, GD, GH, H2, HA, HG, I1, I5, IB, IJ, J2, JA, JK, K1, KB, KD, KJ, 111, 115, 11H, 135, 13H, 13J, 151, 15B, 15H, 164, 175, 17B, 17D, 17J, 191, 19B, 19D, 19H, 1B1, 1B5, 1BB, 1BJ, 1D5, 1DD, 1DJ, 1E4, 1F1, 1F5, 1FD, 1FH, 1HB, 1HD, 1I4, 1JD, 1JH, 1JJ, 22D, 232, 29G, 29K, 2DG, 2F4, 2FK, 31H, 322, 33D, 351, 355, 35B, 35J, 362, 371, 37B, 37D, 37H, 37J, 39B, 39J, 3B5, 3BD, 3BH, 3D1, 3D5, 3DB, 3DD, 3DH, 3E2, 3FJ, 3HD, 3HH, 3HJ, 3J1, 3JB, 3JJ, 414, 481, 49K, 4B4, 4BG, 4C1, 4EB, 4F4, 4F8, 4FA, 4FK, 50G, 51B, 51D, 51H, 531, 535, 53D, 53J, 551, 56A, 56G, 56K, 575, 57J, 595, 59H, 5B1, 5B5, 5BB, 5CA, 5CG, 5CK, 5F1, 5FB, 5FJ, 5GA, 5GG, 5HH, 5IA, 5J5, 5JD, 5JH, 60H, 632, 66H, 66J, 678, 68J, 69G, 6F8, 6FA, 6GJ, 6H8, 6HK, 6IH, 6J4, 6JG, 6KH, 711, 71B, 71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, GGJ, GJ4, GK5, H0K, H15, H1B, H1J, H31, H3D, H3H, H48, H51, H55, H5J, H6K, H75, H84, H88, H91, H95, H9D, H9H, HBD, HC4, HC8, HDJ, HF5, HFB, HFH, HHD, HHJ, HI4, HI8, HJ5, HJB, HKK, I0D, I2D, I6H, I78, I8H, I9K, IFA, IFK, IID, J08, J1J, J31, J35, J3J, J44, J5H, J68, J6G, J71, J7B, J7D, J7H, J8G, J95, J9D, JBD, JBH, JBJ, JCG, JDB, JDH, JE4, JE8, JEG, JF5, JFD, JFJ, JG4, JGG, JH1, JH5, JHB, JHH, JI4, JJ1, JJ5, K0H, K5G, K6H, K9K, KG5, KH4, KKH, 101B, 103D, 1044, 1055, 10DH, 10FB, 10H1, 10H5, 10J1, 110J, 11CD, 11EB, 11ID, 130D, 13C1, 13EB, 141B, 1444, 1471, 14FB, 1505, 15CD, 167H, 16FJ, 16HH, 176H, 17C1, 17IH, 190J, 1BF4, 1BID, 1C04, 1C3D, 1CD1, 1D6H, 1DE1, 1DEB, 1E1B, 1E71, 1E95, 1EFB, 1F0B, 1F4B, 1FB4, 1H01, 1H05, 1H41, 1H6H, 1H6J, 1HC1, 1HE1, 1HF4, 1IDH, 1IHH, 1JE5, 1JF4, 20CD, 20D2, 20ID, 2278, 22F8, 239D, 23FD, 2478, 2708, 27FD, 293D, 29DD, 2A78, 2C0D, 2CF8, 2F0A, 2F9D, 2FAA, 2FDD, 2GFA, 2IF8, 3015, 301D, 301J, 3031, 3035, 305D, 30F1, 30FB, 30FD, 30HB, 30I2, 3105, 31C1, 329D, 330H, 336J, 33E1, 33EB, 33IH, 35CD, 363H, 390H, 39E5, 39ID, 3B0J, 3B6J, 3BC1, 3BEB, 3CC2, 3CF1, 3CFD, 3D32, 3E3B, 3E91, 3EF1, 3EFB, 3F05, 3FC1, 3FE1, 3FEB, 3H01, 3H0B, 3HC1, 3I3H, 3I9D, 404B, 4078, 40E1, 411B, 4131, 4171, 41FB, 4331, 4391, 43BB, 43FB, 43I2, 4432, 444B, 44E1, 477B, 4931, 4971, 499B, 49CG, 4A32, 4A4B, 4A78, 4A9G, 4B1B, 4B31, 4B3B, 4BFB, 4C78, 4E32, 4F1B, 4FB1, 5011, 509D, 50BD, 50DB, 50DD, 50EK, 50FD, 50H1, 50HJ, 50IK, 50JJ, 5101, 5105, 51E1, 530B, 53EB, 53IH, 550D, 550H, 55AB, 55BG, 55DG, 55E5, 55KH, 563H, 56JJ, 5701, 570H, 57EB, 57ID, 57IH, 590D, 59C1, 59CD, 5A3B, 5A7B, 5AGK, 5AIG, 5AKG, 5B0D, 5B9G, 5BCD, 5BGJ, 5BIH, 5C91, 5CD1, 5D0D, 5D0J, 5D6H, 5D6J, 5DC1, 5DG5, 5DIH, 5DJG, 5E0A, 5E11, 5E3B, 5E71, 5EAK, 5EEA, 5EKK, 5F0H, 5F9K, 5G0K, 5G9J, 5GF5, 5GIK, 5H0B, 5H5K, 5H9K, 5HE5, 5HID, 5I3H, 5I7D, 5IKG, 5J6J, 5JE1, 5KAG, 5KEK, 5KFH, 5KH5, 60FK, 63FH, 67JH, 69JH, 6E9K, 6F0K, 6F64, 6FJH, 6H04, 6HE4, 6HFJ, 6IF4, 6J9H, 6J9J, 7015, 7031, 703B, 7055, 705J, 7088, 709B, 709D, 709H, 70DJ, 70I8, 710J, 716J, 730B, 73C1, 73E5, 7501, 769H, 76FJ, 76JJ, 770D, 77E1, 77E5, 77EB, 7888, 78FH, 78H1, 78JH, 790H, 79CD, 79IH, 7B0J, 7B6H, 7C31, 7C91, 7C9D, 7E35, 7E75, 7E91, 7EF1, 7F01, 7F78, 7FCD, 7FH8, 7FIH, 7FJ8, 7H81, 7HC1, 7HF8, 7J6J, 7J78, 7JC1, 7JF8, 7JH8, 7JJ8, 809K, 80H8, 81H1, 81HH, 81J1, 839H, 83FH, 869K, 871H, 8731, 8771, 8788, 879J, 87F1, 87FJ, 87JH, 888H, 8911, 8931, 896K, 8A78, 8A9K, 8CC1, 8E9K, 8F3H, 8F9H, 8FF1, 8FHH, 8FJH, 8G9K, 8GJ8, 8H3J, 8H7H, 8H8K, 8HEK, 8HF1, 8HFJ, 8HH1, 8HJJ, 8HK8, 8J7J, 8J91, 8JC8, 8JFH, 8JJH, 8K78, 8K8H, 8KH8, 901J, 9035, 903B, 9071, 90CG, 90GG, 90JD, 9101, 91E5, 9301, 93E1, 93EB, 95CD, 95DG, 95FK, 967J, 969H, 96HH, 96HJ, 96JJ, 9705, 97C1, 9905, 995K, 999G, 99CD, 99EB, 99K5, 9B5G, 9BE5, 9BID, 9C0G, 9C11, 9C71, 9C9D, 9CEG, 9CGG, 9D05, 9D5G, 9DDG, 9DE5, 9E35, 9E3B, 9E71, 9EB5, 9EEG, 9EF1, 9EFB, 9F0J, 9F6H, 9F9K, 9FC1, 9FIH, 9FKH, 9G35, 9G95, 9GIK, 9GJJ, 9GKG, 9H0H, 9H5K, 9H6H, 9HKH, 9I6K, 9I9H, 9IDD, 9J6J, 9JEB, 9JGJ, 9JID, 9JJG, 9K35, 9K95, 9KEG, A33B, A3F5, A432, A478, A53B, A5IK, A788, A795, A878, A975, A99B, A9CG, A9GG, AAAB, AB44, AB55, ABFB, AC32, AC78, ACFA, AE32, AE5A, AE9K, AEEB, AEFA, AEFK, AF75, AFCA, AFCK, AFEA, AFFB, AG9G, AGFA, AI32, B064, B091, B09H, B0B1, B0D5, B0DH, B0EG, B0FB, B0FH, B0H5, B0HD, B114, B141, B14B, B1B4, B1E1, B411, B431, B444, B4B1, B59G, B5G5, B5JG, B67H, B67J, B69H, B6FH, B6FJ, B6HH, B6JH, B96H, B9DG, B9G5, BB14, BBB4, BC1D, BC31, BCDD, BD05, BD0H, BD6H, BDC1, BDE5, BDIH, BE0G, BE31, BEBB, BEE4, BEIG, BF0B, BF0H, BF6J, BFEB, BFIH, BG0G, BG55, BG95, BGG4, BGGG, BGIG, BGJ5, BH0B, BHEB, BIEG, BIGG, BIHH, BJ01, BJ0J, BJC1, BJF4, BJID, C041, C081, C09G, C0E1, C0FK, C104, C131, C171, C1C4, C1D1, C2F8, C2FA, C39D, C3F1, C478, C6FK, C7FD, C8C1, C90G, C911, C971, C99D, CAFK, CC78, CC81, CCE1, CCF8, CD02, CD31, CDFD, CE01, CE32, CF0A, CF48, CF64, CF68, CF6A, CF91, CFAK, CFDD, CFEA, CFEK, CIF4, CK32, CK9G, CKFA, D01D, D01H, D07B, D0DB, D0F1, D0FD, D0HD, D0HH, D10D, D10H, D16H, D1EB, D1IH, D23D, D3E1, D56H, D63J, D6CG, D6IG, D70B, D70D, D70J, D99G, D9GJ, D9ID, D9JG, DBC1, DBCD, DBE1, DBJG, DC11, DC9D, DCF1, DCFD, DCGG, DCIG, DD0H, DD6J, DDGJ, DDID, DDJG, DE31, DE75, DFC1, DFID, DFIH, DG22, DG6G, DG75, DG9J, DGCG, DGIG, DH6H, DHE5, DI6G, DICG, DIDH, DIFD, DIFH, DIGG, DJCD, DJDG, DJG5, E032, E05A, E0EB, E11B, E2F8, E31B, E331, E375, E395, E3C2, E3F1, E3F5, E3FB, E3I2, E53B, E555, E591, E59B, E5EG, E6F4, E755, E775, E89K, E915, E935, E93B, E96K, E9FB, EA32, EAEB, EAFK, EB44, EB95, ECFA, ECFK, EE32, EEEB, EEFK, EF08, EF35, EF48, EF55, EF64, EF68, EF75, EF95, EFB1, EFCA, EFCK, EFEA, EFIA, EFKK, EGF4, EIF4, EKF4, EKF8, EKFA, F008, F051, F075, F095, F09H, F0CA, F0CK, F0DD, F0FB, F0H1, F0IA, F0JJ, F105, F10B, F10D, F16J, F1B4, F1EB, F1ID, F1J4, F36H, F468, F47B, F491, F4C8, F501, F50B, F648, F66A, F69H, F705, F778, F864, F8F1, F90D, F99K, F9C1, F9HK, FA6A, FA75, FA7B, FA95, FAEA, FAFB, FB0D, FB14, FBCD, FBE1, FBJ4, FC1D, FC68, FC9D, FCAA, FCCK, FCD1, FCEA, FD6H, FDIH, FE0K, FE1B, FE64, FE95, 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====Additional known quasi-minimal primes (not necessarily the next)====

40&lt;sub&gt;47333&lt;/sub&gt;9G, CF&lt;sub&gt;479147&lt;/sub&gt;0K

===Base 22===

11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, IHH, IIJ, IJ7, IL7, J07, J55, J69, J8J, J99, J9J, JA5, JAH, JE7, JEH, JHH, JI9, JJ5, JJ9, JJH, JK7, K2J, K51, K5D, K75, K81, KA1, KB1, KB7, KBJ, KD1, KEJ, KG5, KIF, KJ5, KKD, KKJ, L0D, L47, L7F, L97, LAF, LD3, LD9, LDD, LEF, LGD, LI7, LJ7, LJJ, LLD, 104H, 10D5, 1205, 12B5, 140H, 1433, 144H, 14AH, 14B3, 16ED, 1AIH, 1B43, 1DD5, 1DDD, 1E6D, 1EGD, 1G05, 1GDD, 1GED, 1GGD, 1HB3, 1HHH, 1IAH, 200L, 2015, 2051, 20A1, 20DJ, 20GD, 20IL, 21B5, 21DD, 220D, 226D, 228D, 22B5, 22G5, 22K5, 22KD, 2555, 2557, 2581, 25C1, 26A1, 26B1, 2725, 2755, 2801, 2861, 288D, 28B1, 28KD, 2AA1, 2B25, 2B51, 2BB1, 2C81, 2D6D, 2DA1, 2DDJ, 2DGD, 2G0J, 2GB5, 2GDD, 2GGJ, 2I0L, 2I6L, 2ICL, 2J05, 2JK5, 2K07, 2K2D, 2K55, 2K6D, 2KB5, 2KI7, 2L2D, 2L8D, 2LK7, 302L, 30G3, 320L, 32IL, 332L, 33G3, 33G9, 36AF, 36EF, 382J, 388J, 39EL, 3AAF, 3BBJ, 3CG9, 3E2J, 3E6F, 3E6L, 3EEF, 3FAF, 3G69, 3GI3, 3GI9, 3IG9, 3LEL, 3LG3, 4025, 404H, 4063, 4075, 40AH, 40B5, 40B7, 40GH, 4225, 4363, 4447, 444H, 447H, 4487, 44B7, 44H7, 4525, 4555, 45B5, 4663, 4777, 47GH, 4807, 4B0H, 4BB5, 4BG3, 4EBH, 4G4H, 4GB3, 4HH3, 4I03, 4I63, 4IGH, 5069, 5077, 50KD, 5255, 52IJ, 5455, 5477, 5507, 5527, 556F, 557F, 5587, 56EF, 56FF, 56GD, 56KD, 5771, 57C1, 57EF, 5807, 580J, 589J, 58E7, 58IJ, 58K7, 5905, 5945, 5957, 5969, 5989, 598J, 5999, 59EJ, 59K7, 5AEF, 5B25, 5B55, 5BA5, 5BI7, 5BIJ, 5BK5, 5BK7, 5C21, 5EAF, 5EE7, 5F09, 5F6F, 5F95, 5FKD, 5G6D, 5G89, 5I09, 5I0F, 5I27, 5I6F, 5I89, 5IB7, 5IBJ, 5IFF, 5J77, 5K07, 5K25, 5K27, 5K6F, 5K87, 5KK7, 6013, 60A1, 60AF, 60EL, 6281, 62B1, 62KD, 63G9, 6403, 6643, 66EL, 66G9, 68KD, 69B3, 6A01, 6A0F, 6AAF, 6AFF, 6B21, 6B8D, 6BBD, 6BK1, 6D03, 6D43, 6D69, 6D6F, 6D93, 6D9F, 6E21, 6E81, 6E8D, 6EBD, 6ECL, 6ELL, 6FD9, 6GBD, 6GG9, 6IEF, 6K21, 6K8D, 6KBD, 6KE1, 6L43, 6LB3, 6LDF, 6LEL, 6LG9, 700F, 7027, 7057, 70EH, 70LF, 710H, 71G5, 7207, 7225, 7255, 727L, 7505, 7507, 755F, 75EF, 75F5, 75I7, 766F, 7681, 76CF, 7771, 7781, 77IL, 7861, 7A0H, 7A4H, 7BB5, 7C6F, 7EEH, 7EHL, 7FG5, 7G45, 7HEL, 7I0L, 7I27, 7IE7, 7IGH, 7K07, 7K61, 7K77, 7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, H0D1, H0HJ, H0LL, H1HH, H20D, H22D, H26D, H2I7, H2K7, H30L, H3KJ, H40H, H447, H4G3, H4H3, H4HH, H66L, H6ED, 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5IIIIIIIIIIIIIIIIIIIIIIIIIIF, D555555555555555555555550555, EEAAAAAAAAAAAAAAAAAAAAAAAAAF, HHHHHHHHHHHHHHHHHHHHHHHHEEBH, K66666666666666666666666666F, LLLLLLLLLLLLLLLLLLLLLLLLEEB7, D5555555555555555555555555A55, GGGGGGGGGGGGGGGGGGGGGGGGGGGG3, GIG0000000000000000000000000H, HH00000000000000000000000001H, K0000000000000000000000005KEF, 5BBBBBBBBBBBBBBBBBBBBBBBBBBBBD, HB0000000000000000000000000001, K000000000000000000000000505EF, L7777777777777777777777777772L, 2000000000000000000000000000CB1, C8CCCCCCCCCCCCCCCCCCCCCCCCCCCC9, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKFF, JE0000000000000000000000000000J, K000000000000000000000000000261, A0000000000000000000000000004I4H, HD000000000000000000000000000001, K000000000000000000000000000EC01, K0FFFFFFFFFFFFFFFFFFFFFFFFFFFFCF, D0002222222222222222222222222222D, FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL2L, I700000000000000000000000000000GH, K00000000000000000000000000000E61, 20000000000000000000000000000000JJ, DD5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL0KE7, 44HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, F0000000000000000000000000000000000000000000000000000000L89, J000000000000000000000000000000000000000000000000000000JJIJ, K222222222222222222222222222222222222222222222222222222222DD, KCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCF, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKCF, D55KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, K000000000000000000000000000000000000000000000000000000000000077, 22222222222222222222222222222222222222222222222222222222222222222D, K000000000000000000000000000000000000000000000000000000000000008IJ, DFA00000000000000000000000000000000000000000000000000000000000000005, CC4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJE0J, 777777777777777777777777777777777777777777777777777777777777777777777777EK7, 6000000000000000000000000000000000000000000000000000000000000000000000000000000043, KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKB5, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEEH, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 50000000000000000000000000000000000000000000000000000000000000000000000000000000002C1, K0000000000000000000000000000000000000000000000000000000000000000000000000000000000055EF, H700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000K1, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0I7, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, A400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, E0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000071, 7LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIL, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLK77, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEI7, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIB3, I7G00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 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77EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEK7, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJKJ, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777772L, BKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK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KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK5

===Base 24===

15, 17, 1D, 1H, 1J, 1N, 25, 2B, 2D, 2J, 2N, 31, 37, 3B, 3H, 41, 45, 47, 4B, 4D, 4H, 57, 5B, 5H, 5J, 65, 67, 6D, 6J, 6N, 75, 7B, 7D, 7N, 81, 85, 87, 8J, 97, 9B, 9D, 9H, 9N, A1, AB, AH, AN, B5, B7, BD, BH, BJ, C5, CJ, CN, D1, D5, DJ, E1, EB, ED, EH, EN, F7, FD, FJ, FN, G5, GD, GH, H1, HB, HD, HN, I1, I7, IB, IH, J1, J5, J7, JB, JN, K7, KB, KJ, KN, L5, LH, LJ, MD, MJ, N5, NB, NH, NJ, 101, 10B, 111, 1F1, 1FB, 1GB, 1LB, 201, 221, 22H, 261, 271, 277, 28H, 2A7, 2C7, 2G7, 2H7, 2L1, 2L7, 2MH, 305, 30D, 30J, 33N, 34N, 35D, 35N, 38D, 395, 3A5, 3AJ, 3CD, 3DD, 3DN, 3E5, 3EJ, 3GJ, 3IJ, 3JJ, 3K5, 3KD, 3ND, 43N, 44N, 49J, 4EJ, 4GJ, 4GN, 4NN, 50N, 535, 54N, 551, 55N, 5C1, 5CD, 5E5, 5K1, 5KD, 5LN, 5M5, 5N1, 601, 60B, 61B, 66H, 68B, 691, 6CH, 6FH, 6GB, 6HH, 6MH, 70H, 70J, 711, 761, 771, 77H, 77J, 78H, 7C7, 7CH, 7FH, 7G7, 7H7, 7HH, 7IJ, 7JJ, 7K1, 7M1, 7M7, 80D, 82H, 83N, 88D, 88H, 8AD, 8CD, 8DB, 8DD, 8DH, 8DN, 8GB, 8KD, 8MB, 8MH, 905, 911, 921, 935, 955, 99J, 9AJ, 9G1, 9JJ, 9K5, 9L1, 9M5, A0J, A3J, A95, AA7, AD7, AE5, AG7, AGJ, AI5, AIJ, AJD, AL7, ALD, B01, B0N, B11, B61, B6B, B8N, B91, BIN, BL1, BLN, BNN, C1B, C21, C27, C2H, C3D, C61, C8H, C91, CA7, CB1, CBB, CC7, CCB, CCD, CDD, CFB, CG1, CGB, CK1, CL1, CMB, CMH, D0B, D3D, D3N, D4N, D6B, D6H, D7H, D8B, D8N, DAD, DCD, DCH, DDH, DDN, DG7, DGB, DID, DMN, DND, E05, E4J, EA7, EEJ, EF5, EGJ, EI5, EJJ, EM5, EM7, F01, F21, F51, F8H, F95, FC1, FF1, FFB, FKH, FM5, G0B, G0N, G11, G3N, G6B, G77, G7J, G8B, G8N, G91, GA7, GBB, GC7, GFB, GG1, GGJ, GGN, GK1, GL1, GLN, GMN, GN1, GNN, H0J, H2H, H3J, H4J, H77, HA5, HA7, HE5, HFH, HIJ, HJJ, HKH, HL7, HMH, I0N, I3D, I3J, I3N, I4N, I5D, I95, IA5, IAJ, IE5, IEJ, IF5, IGJ, IJD, IK5, IKD, J0D, J4J, J8D, JAD, JDH, JEJ, JFH, JHH, JKD, JMH, K35, K6H, KCD, KFH, KH5, KLD, KM1, L01, L0B, L0D, L0N, L61, L6B, L8D, LA7, LC7, LDD, LF1, LG7, LGB, LGN, LID, LK1, LKD, LL1, LLB, LLD, LMN, LNN, M0H, M11, M21, M4N, M71, M91, M95, MA5, MA7, MBN, MC7, MF1, MF5, MFB, MFH, MG7, MI5, MIN, ML1, ML7, MLB, MMH, N01, N21, N4N, N71, N8N, NC1, ND7, NE7, NG1, NID, NK1, NL7, NMN, NN7, 11CB, 11MB, 1291, 12G1, 16C1, 16CB, 16K1, 186B, 18CB, 19K1, 1BK1, 1C8B, 1K91, 1KC1, 1KL1, 1L21, 1LC1, 1LM1, 1M61, 1M8B, 1MG1, 206H, 20CH, 20M7, 21C1, 21M1, 2207, 260H, 26KH, 2991, 2C6H, 2CC1, 2CM1, 2F11, 2FHH, 2MC1, 2MK1, 2MM1, 308N, 30GN, 30IN, 30LN, 30MN, 333J, 33JD, 33LD, 343J, 344J, 35I5, 380N, 393J, 394J, 3A3D, 3FI5, 3IMN, 3J3D, 3JID, 3L3D, 3L8N, 3M0N, 3M55, 3NGN, 404J, 408N, 40LN, 434J, 44AJ, 4ILN, 4JAJ, 4L8N, 5091, 5095, 50F1, 50I5, 51L1, 5211, 5291, 52G1, 53ID, 53MN, 5595, 55AD, 56F1, 588N, 58MN, 58ND, 5961, 5991, 5A5D, 5AAD, 5F91, 5GF1, 5GIN, 5I05, 5I55, 5I8D, 5IDD, 5IDN, 5IIN, 5IMN, 5KI5, 5M61, 5M8N, 5N3N, 602H, 6211, 62F1, 62G1, 66C1, 66FB, 66M1, 66MB, 6B21, 6BM1, 6BMB, 6C6B, 6CF1, 6CLB, 6FG1, 6K21, 6K2H, 6KG1, 6KKH, 6L21, 6LCB, 6LM1, 6MB1, 6MBB, 6MG1, 6MK1, 7001, 7027, 7207, 726H, 739J, 793J, 79C1, 7A4J, 7A9J, 7AE7, 7C01, 7CC1, 7FL1, 7G21, 7G9J, 7GAJ, 7GC1, 7HGJ, 7J2H, 7J6H, 7MKH, 800B, 800H, 804N, 806H, 808N, 80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, NK3D, NK8D, 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5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, 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EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEG7, 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===Base 30===

11, 17, 1B, 1D, 1H, 1N, 1T, 21, 27, 2B, 2D, 2J, 2N, 2T, 37, 3B, 3D, 3H, 3J, 3N, 47, 4B, 4H, 4J, 4T, 51, 57, 5D, 5H, 5N, 5T, 61, 6B, 6D, 6H, 6J, 71, 7D, 7H, 7J, 7N, 7T, 81, 8B, 8H, 8N, 8T, 91, 97, 9B, 9D, 9N, A7, AB, AD, AH, B1, B7, BH, BJ, BN, BT, C7, CD, CJ, CN, CT, D7, DB, DJ, DT, E1, EB, ED, EJ, EN, ET, F7, FB, FD, FH, FT, G7, GB, GJ, GN, GT, HB, HD, I1, I7, IH, IN, IT, J1, J7, JH, JN, JT, K1, K7, KD, KH, KJ, L1, LB, LD, LH, LN, LT, M1, MD, MH, MN, N1, NB, NJ, NT, O7, OD, OJ, ON, P1, P7, PB, PJ, PN, Q7, QH, QT, R1, RB, RD, RH, RJ, RT, SD, SH, SJ, SN, T7, TB, TD, TH, 10J, 15J, 1IJ, 1JJ, 1LJ, 1MJ, 1QJ, 22H, 29H, 2EH, 2GH, 30T, 331, 33T, 36T, 39T, 3A1, 3C1, 3G1, 3KT, 3MT, 3OT, 3S1, 3T1, 40D, 40N, 431, 44D, 46N, 48D, 4AN, 4DD, 4DN, 4F1, 4FN, 4GD, 4ID, 4PD, 4S1, 50J, 58J, 59J, 5CB, 5FJ, 5IB, 5IJ, 5JB, 5MB, 5MJ, 5OB, 5SB, 607, 63T, 687, 6E7, 6KT, 6L7, 6M7, 6MT, 6NN, 6QN, 6R7, 6S7, 6ST, 6TT, 70B, 77B, 787, 7KB, 7M7, 7MB, 7SB, 807, 80D, 80J, 84D, 85J, 877, 88J, 89J, 8DD, 8FJ, 8ID, 8IJ, 8JJ, 8M7, 8MJ, 8PD, 8QD, 8R7, 90H, 93T, 95J, 99H, 99J, 9AJ, 9AT, 9EH, 9GH, 9HH, 9HJ, 9JJ, 9MJ, 9OH, 9OT, 9PH, 9ST, 9TT, A01, A0T, A1J, A31, A6N, A6T, AAJ, AAN, AFN, AKN, AKT, ALJ, AMJ, AMT, AO1, AOT, AQ1, AQN, ARN, AT1, B5B, BBD, BCB, BDD, BID, BOB, BPD, BQB, C31, C9H, CC1, CCB, CCH, CF1, CH1, CIB, CKB, CMB, D01, D0H, D2H, D41, D4D, D6N, D8D, D9H, DDH, DDN, DGH, DH1, DHH, DID, DKN, DNN, DO1, DOH, DQN, DS1, EEH, EHH, EM7, EOH, EPH, ER7, F0N, F31, F5J, F8J, F9J, FLJ, FO1, FQ1, FQN, FS1, G01, G4D, G8D, GF1, GGH, GO1, GOH, GQD, GS1, H07, H0J, H0T, H1J, H2H, H31, H4N, H8J, HA1, HAJ, HAT, HC1, HE7, HEH, HFN, HGH, HH7, HIJ, HJJ, HKN, HL7, HNH, HPH, HQN, HS1, HTJ, HTN, I4D, I5B, I5J, I8D, IAJ, IDD, IGD, IIJ, IKB, IMB, IMJ, IOB, IPD, IQD, J8J, J9J, JAJ, JBD, JCB, JFJ, JIJ, JJB, JJD, JLJ, JPD, JQB, K3T, K4N, KAT, KBB, KCB, KMB, KNN, KOB, KOT, KQN, KST, KTT, L0J, L5J, L67, LAJ, LJJ, LQJ, LR7, M0J, M3T, M5B, M7B, M87, M9J, MAT, MFJ, MIJ, MJJ, MKB, MMJ, MOT, MQJ, N07, N0H, N67, N6N, N87, NAN, NDH, NGD, NHH, NKN, NN7, NNH, NPH, NQD, NQN, NR7, O01, O0B, O0H, OAT, OC1, OCH, OEH, OF1, OH1, OKT, OMB, OOT, OPH, OQ1, OQB, OS1, OST, P3T, P6T, P9H, PCH, PDH, PEH, PGD, PHH, PHT, PID, PMT, PPT, PQD, PST, PTT, Q5B, Q6N, Q9J, QAJ, QBB, QBD, QC1, QDN, QFJ, QFN, QGD, QJB, QKN, QLJ, QMB, QND, QNN, QO1, QQ1, QQN, QSB, R4N, R77, R87, RAN, RKN, RM7, RNN, RR7, RS7, S0T, S41, S6T, S87, SBB, SC1, SM7, SMT, SOB, SQ1, SR7, SS1, STT, T0J, T6T, T9T, TAN, TFN, TKN, TKT, TNN, TO1, TOT, TPT, TQ1, TQN, TTN, 18AJ, 19FJ, 1AFJ, 1FAJ, 20CH, 20PH, 2C0H, 2CHH, 2COH, 2H0H, 2HCH, 2HHH, 2POH, 2PPH, 3001, 34Q1, 3F41, 3FF1, 3QF1, 3SPT, 3SST, 3TTT, 40A1, 4441, 44KN, 44QN, 4AC1, 4C01, 4CA1, 4DG1, 4GA1, 4GC1, 4NND, 4NRN, 4OA1, 4Q4N, 4QA1, 4QRN, 4RQN, 550B, 555B, 55KB, 5A5J, 5AJJ, 5BKB, 5JQJ, 5KKB, 5QQB, 5QQJ, 604N, 606T, 60FN, 60KN, 60OT, 60PT, 660N, 660T, 664N, 66AT, 66FN, 66KN, 66TN, 6A0N, 6A9T, 6AAT, 6FKN, 6FRN, 6K0N, 6KFN, 6KKN, 6O6T, 6PAT, 6RFN, 6RRN, 6T6N, 6TRN, 7067, 70R7, 75BB, 77R7, 7C5B, 7CQB, 7E67, 7IQB, 7O5B, 7OIB, 7QOB, 7R07, 7RE7, 8667, 88L7, 88S7, 8E87, 8EE7, 8EL7, 8J8D, 8LL7, 8LS7, 906T, 908J, 90FJ, 90QJ, 90TJ, 92CH, 966T, 99MT, 99PT, 9I0J, 9I8J, 9LFJ, 9LIJ, 9M6T, 9MKT, 9MMT, 9PKT, 9QQJ, 9TLJ, A04N, A0QJ, A3AT, A3ST, A4A1, A4G1, A5QJ, A8QJ, A90J, A9PT, AA3T, AA41, AAAT, AAF1, AAG1, AAPT, AC41, ACS1, AF0J, AFC1, AFIJ, AG41, AGA1, AGG1, AI8J, AIJJ, AIQJ, AJ5J, AJJJ, AQ8J, AQJJ, AQQJ, AS3T, AT4N, AT5J, ATST, B04D, B0QD, BBIB, BBKB, BI0B, BK0B, BKIB, BKSB, BMIB, BMSB, BQ4D, BQQD, BSMB, C00B, C05B, C0A1, C0BB, C0EH, C0GH, C0Q1, C0QB, C2OH, C2PH, C441, C4Q1, C50B, CB0B, CBSB, CEGH, CG2H, CGHH, CHHH, CHOH, COGH, COO1, COOB, COOH, CQ01, CQ0B, CQQB, CQS1, CS0B, D00N, D0PD, D44N, DA0N, DCEH, DCG1, DCQ1, DDC1, DDF1, DDPD, DDQ1, DECH, DF4N, DFA1, DFRN, DG0D, DGD1, DGQ1, DNEH, DPDD, DQA1, DQPD, DRFN, DRRN, E20H, E667, E767, E8L7, E8S7, ECGH, EG2H, EH67, EH77, ES67, F001, F01J, F0MJ, F0QJ, F44N, F4G1, F4RN, F6RN, FA0J, FAC1, FAF1, FAIJ, FC01, FCG1, FFAJ, FFC1, FFG1, FFIJ, FGA1, FJJJ, FJQJ, FKAN, FR6N, FRFN, G09H, G0EH, G0HH, G20H, G341, G3Q1, G4G1, G92H, GA41, GAC1, GC0H, GCEH, GCG1, GCPH, GD31, GDA1, GDG1, GDPD, GE2H, GE9H, GGG1, GGGD, GHCH, GHHH, GP0D, GP2H, GPPD, GQ31, GQA1, GQG1, H00N, H0CH, H3ST, H4O1, H667, H66N, H677, H69T, H99T, H9CH, H9KT, H9LJ, H9PT, H9QJ, HH6N, HHF1, HHFJ, HHH1, HHHT, HHLJ, HHOH, HHQ1, HHST, HK6T, HL9J, HLMJ, HMKT, HMMT, HNM7, HNRN, HOPT, HQ01, HQ5J, HR0N, HR6N, HS77, HSKT, HT01, I0BB, I0JD, I0JJ, I98J, I9LJ, I9QJ, ICBB, IF0J, IFFJ, IFQJ, IIBD, IJBB, IJID, IL8J, ILFJ, IQ8J, IQJJ, IQQJ, IS0B, ISQB, J00D, J05J, J0BB, J0MJ, J0OB, J4QD, J50B, J5QJ, JBKB, JBMB, JDGD, JDQD, JGDD, JI0D, JIIB, JISB, JKIB, JKKB, JM0B, JOIB, JQ4D, JQDD, JQID, JQJJ, JQMJ, JSIB, JSMB, K06N, K0KB, K0KT, K0MT, K0PT, K0SB, K5QB, K60T, K66T, K6AN, K6FN, K6PT, K6RN, K96T, K99T, KA0N, KF6N, KFKN, KI0B, KK0T, KK6N, KK6T, KKFN, KKKB, KKKT, KKQB, KKTN, KMMT, KQ0B, KQQB, KR0N, KS5B, L0M7, L8E7, L98J, LFFJ, LI8J, LIFJ, LL87, LL9J, LLIJ, LLM7, LM77, LML7, LMM7, M09T, M0E7, M0IB, M0L7, M0M7, M0TT, M55J, M5LJ, M60T, M69T, M707, M767, M777, M7E7, M7S7, M8LJ, M96T, M9KT, MC0B, MCOB, ME07, MIBB, MJ0B, MJBB, MJSB, MLLJ, MLS7, MM07, MM0T, MML7, MMOB, MMR7, MOBB, MOCB, MP0T, MP9T, MPKT, MQ0B, MQIB, MQOB, MS07, MSPT, MSSB, MT8J, MTTT, N00D, N04D, N04N, N0FN, N0ND, N0RN, N2OH, N92H, NCGH, ND0D, ND4N, NDPD, NEGH, NEL7, NF4N, NFRN, NGCH, NGEH, NHM7, NME7, NML7, NMM7, NN0D, NN4D, NN4N, NNDN, NNID, NNND, NNRN, NP4D, NSL7, NSS7, O00T, O03T, O0MT, O0PT, O2HH, O2OH, O341, O4A1, O4G1, O4O1, O5KB, O90T, OA41, OBSB, OC5B, OCOB, OG2H, OG31, OH9H, OH9T, OHHH, OHTT, OICB, OISB, OM0T, OM6T, OMPT, OMTT, OO2H, OO9H, OOA1, OOCB, OOGH, OOKB, OOO1, OSKB, OTMT, P00T, P04D, P08D, P09T, P0KT, P90T, PAAT, PKKT, PO0T, PP0D, PPPD, Q00B, Q0AN, Q0D1, Q0F1, Q0IB, Q0JJ, Q0MJ, Q0OB, Q55J, Q5QJ, Q88D, QA0N, QA41, QA4N, QDF1, QDPD, QF41, QFA1, QG31, QIQJ, QJ0J, QJ8D, QJDD, QJJJ, QJMJ, QJQD, QKKB, QKQB, QOKB, QOOB, QP0D, QPDD, QQ8D, QQID, QQJD, QQPD, QQQB, QS01, QSA1, R00N, R067, R06N, R0FN, R0L7, R60N, RE67, RFRN, RRFN, RRRN, S00B, S3AT, S3ST, S50B, S5QB, S99T, SAAT, SC0B, SE67, SGG1, SICB, SIQB, SK5B, SKIB, SKKT, SKPT, SKQB, SM0B, SMMB, SMSB, SOA1, SOG1, SPOT, SQCB, SQQB, SS0B, SS67, ST31, STG1, T00N, T03T, T041, T04N, T0AT, T0C1, T0MT, T0ST, T1AJ, T3F1, T4A1, T4RN, T5JJ, T66N, T8AJ, T9FJ, T9QJ, TA3T, TAA1, TAG1, TAIJ, TCS1, TFA1, TFAJ, TFF1, TFFJ, TFG1, TGA1, TI9J, TIJJ, TIQJ, TL8J, TL9J, TM0T, TM5J, TM8J, TMAJ, TMLJ, TMST, TSF1, TSG1, TT01, TT5J, TT8J, TTA1, TTC1, TTFJ, TTG1, TTLJ, TTMJ, 30441, 30O41, 30OO1, 34O41, 3AATT, 3ASAT, 3O441, 3Q041, 40G41, 40GQ1, 40O41, 40OG1, 40OO1, 44001, 440C1, 440G1, 4444N, 44CG1, 44GQ1, 4AAA1, 4COG1, 4DAA1, 4GQ41, 4KRRN, 4OGG1, 4Q041, 4QGG1, 5005B, 500BB, 500KB, 500QB, 50K0B, 50Q0B, 50QKB, 555AJ, 555LJ, 555QJ, 55J5J, 5B00B, 5B0BB, 5J55J, 5J5JJ, 5JJ5J, 5JJJJ, 5K00B, 5QJ5J, 5QK0B, 6006N, 6009T, 600AT, 6066N, 60R0N, 666N7, 6696T, 66OPT, 67767, 6900T, 696PT, 69P0T, 6F6AN, 6O9PT, 6OP0T, 6OP9T, 6P99T, 755QB, 77767, 7CBBB, 7IBIB, 7IIBB, 7IICB, 7QIIB, 8888D, 88E67, 8L887, 8SSE7, 9099T, 90KKT, 90KMT, 90MPT, 9690T, 96P9T, 990KT, 9990T, 9999T, 99K6T, 9FFFJ, 9HKMT, 9HMPT, 9ILLJ, 9KKMT, 9KKPT, 9M00T, 9T8QJ, 9TFIJ, A008J, A00NN, A055J, A0FJJ, A0I0J, A0I9J, A0J0J, A44C1, A555J, A9FFJ, A9QIJ, A9T8J, AFFA1, AFFFJ, AFFJJ, AFFQJ, AJ00J, AP99T, ASA9T, ASFF1, ASP9T, ASSAT, AT3TT, ATFQJ, ATQIJ, ATT9J, ATTQJ, B088D, B08GD, B0GGD, B0SKB, B8GGD, BG00D, BIISB, BISIB, BS0IB, BS0SB, BSSKB, BSSSB, C0041, C0G41, C0H0H, C0PPH, C0S01, C4001, C4AG1, C4OG1, C5BBB, CG00H, CGE0H, CGGA1, CGQ41, COSSB, CP20H, CPGPH, CPP2H, CPPOH, CPPPH, CS001, CS55B, CSSQB, D00GD, D0A4N, D0FAN, D0GDD, D0N0D, DAAA1, DAAC1, DAFG1, DDAA1, DDGGD, DFFAN, DFGC1, DGDGD, DGG31, DGGA1, DH0AN, DPPPH, DQDD1, DQGG1, E0L87, E88E7, EC00H, EC02H, EE867, EE887, EEL87, ELE87, F00IJ, F0441, F0AG1, F0F0J, F0F41, F0FF1, F0GG1, F4041, F44A1, F44C1, F4A41, F64KN, F6K6N, FAAA1, FAFQJ, FCAA1, FF041, FF0F1, FF64N, FF6KN, FFA41, FFA4N, FFF4N, FFFF1, FFFFN, FFKRN, FFQMJ, FIJ0J, FJ00J, FKKKN, FNNFN, FQQ0J, FQQMJ, G00DD, G00DH, G00PD, G02PH, G0DDD, G0PDD, GAAA1, GC2HH, GDDDD, GDGDD, GDPPH, GG0PD, GGCA1, GGCQ1, GGDD1, GH4Q1, GHHG1, GII0D, GQ441, H009H, H00G1, H00H1, H04Q1, H0ANN, H0H01, H0H9H, H0HG1, H0HO1, H0O41, H0OHH, H0QG1, H40G1, H4G41, H4GG1, H60AN, HAN0N, HF0G1, HFF41, HFFMJ, HFFQJ, HGQ41, HH3TT, HH401, HH441, HH4G1, HH55J, HH66T, HH6OT, HH96T, HHA0N, HHANN, HHC0H, HHCHH, HHH0N, HHHCH, HHHHN, HHHNN, HHKMT, HHO41, HHP9T, HHPKT, HHTMT, HHTT1, HK9MT, HKKMT, HLLFJ, HM66T, HM7R7, HMM67, HMM77, HMMM7, HMTST, HO9HT, HOO41, HOOOH, HOT3T, HOTT1, HPO9T, HSO3T, HTGG1, I000J, I009J, I00BD, I00QJ, I00SB, I08QJ, I0I0D, I0IQB, I0JIB, I0Q0B, I0SCB, I0SIB, I0SSB, IB00D, IBBSB, IBISB, ICSSB, II0CB, II0ID, II0SB, IIIBB, IIQ0B, IIQIB, IISIB, IJ0SB, IJJQJ, ILLLJ, IQ00J, IQC0B, IQIQB, IQQIB, ISSCB, J00MB, J08GD, J0IID, J0QQD, J55BB, J55JJ, J5J5J, J5JJJ, J88GD, JB0SB, JBB0B, JBBSB, JDDDD, JG0ID, JGG0D, JIIID, JJ0QJ, JJ5JJ, JQ08D, JQ0QJ, JQQ0D, JQQ5J, JS55B, JSK0B, JSS5B, K000T, K006T, K00TN, K0FAN, K505B, K6T0N, K9KMT, K9KPT, K9P0T, KIQIB, KK00N, KK05B, KK0AN, KK0KN, KKM9T, KS0IB, KS0QB, KSQIB, KTR6N, KTRRN, L0087, L08S7, LE087, LEE87, LL8LJ, LMEE7, LMSE7, LMSS7, M00CB, M00PT, M066T, M0CQB, M0K6T, M0MQB, M0MSB, M0QQB, M0SMB, M0SST, M666T, M900T, MBBSB, MEE77, MEL77, MELE7, MES77, MESE7, MESS7, MI00B, MIICB, MIISB, MIQCB, MK9PT, ML7L7, MLEL7, MLLE7, MLME7, MM677, MMBIB, MMBSB, MMCQB, MME77, MMEE7, MMICB, MMISB, MMK6T, MMKKT, MMM9T, MMMMT, MMMTT, MMQQB, MMSCB, MMSKT, MMSMB, MMSS7, MMTST, MOIIB, MOOIB, MOSIB, MQCQB, MR007, MR667, MRL07, MS7L7, MSEL7, MSK9T, MSL77, MSSL7, MT00T, MTMMT, N0DDD, N4NNN, N7LE7, N7S77, NE2CH, NE9CH, NEC2H, NEE77, NFFNN, NFNFN, NII0D, NL777, NL7L7, NLES7, NLLL7, NLS77, NOG9H, NRR0N, O6P9T, O9H6T, O9HPT, O9P9T, OCBBB, OG441, OGAG1, OGGA1, OHH6T, OHOOH, OKIIB, OKK5B, OKKIB, OO5BB, OOBIB, OOIIB, OSCSB, OT3TT, OT441, OTG41, OTGG1, P00PH, P0D0D, P0G2H, P0OGH, P0PGH, PA99T, PGP0H, POOOH, PP20H, PP88D, PPP0H, Q0001, Q000N, Q001J, Q00ID, Q00PD, Q044N, Q04G1, Q0I0D, Q0PPD, Q40G1, Q444N, Q44RN, Q4AA1, Q8QQJ, QAAS1, QAFF1, QAFG1, QASG1, QDGG1, QFGG1, QI00D, QIICB, QIIQB, QIQCB, QOIIB, QPP4D, QQ08J, QQ0CB, QQC0B, QQI0J, QQJ5J, QSFG1, R00E7, R0E07, R0NE7, R6F6N, R6FFN, REE07, RELE7, RFF6N, RL0E7, RLE07, RLEE7, RLLE7, RQR0N, RR0QN, RRQ0N, S03O1, S0AF1, S0AG1, S0O31, S0QIB, S0SIB, S30F1, S30O1, S7QIB, SA3PT, SAAA1, SAFG1, SASST, SCSQB, SF0G1, SFFF1, SI0SB, SISSB, SKSSB, SSCQB, SSCSB, SSMIB, SSPAT, SSSKT, SSSMB, SSSSB, STAF1, STF01, T0001, T0031, T0AF1, T0AS1, T0G31, T0R6N, T0T31, T0TF1, T3AAT, T40G1, T4CG1, T4G41, T5LLJ, T8LLJ, TA441, TA98J, TAFJJ, TAQ5J, TASST, TATAT, TC401, TCGG1, TFQIJ, TFQJJ, TFQMJ, TG441, TGC41, TI8LJ, TLLMJ, TMMMT, TMTMT, TQ8QJ, TSS3T, TT3AT, TT9IJ, TTAAT, TTQIJ, TTS31, TTS3T, TTT9J, TTTST, 20000H, 200OOH, 3440O1, 3TAAST, 404CQ1, 4KKKKN, 4KKKRN, 4QQQQD, 505BBB, 50BB0B, 6000AN, 6444RN, 66666N, 666O9T, 66999T, 669P9T, 66N777, 6A444N, 6FF66N, 6FFF6N, 6R666N, 766767, 77S677, 7IBBBB, 8888E7, 8LLLLJ, 8SSSL7, 9000MT, 90K90T, 90KP0T, 90M90T, 99000T, 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===Base 36===

11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 2UJ, 2WN, 2XN, 30Z, 337, 33B, 33H, 33P, 34H, 34P, 37D, 38Z, 39H, 3AB, 3AP, 3AZ, 3D7, 3DH, 3FD, 3FZ, 3HD, 3HH, 3KZ, 3L7, 3LZ, 3MB, 3O7, 3OZ, 3PD, 3PP, 3Q7, 3RH, 3S7, 3SZ, 3UP, 3UZ, 3XB, 40P, 43B, 43H, 46V, 48B, 48V, 49B, 4AP, 4BB, 4E1, 4FP, 4HB, 4HH, 4HP, 4IH, 4MB, 4OP, 4OV, 4PH, 4RH, 4VB, 4VH, 4WH, 4WP, 4WV, 4X1, 4XH, 55T, 565, 58Z, 5C5, 5E7, 5ET, 5EZ, 5IN, 5KT, 5L7, 5MP, 5NN, 5O5, 5O7, 5OP, 5R7, 5RT, 5RZ, 5ST, 5SZ, 5TN, 5TZ, 5W7, 5XN, 5YN, 61T, 625, 62J, 64V, 661, 66J, 691, 6AV, 6EV, 6IJ, 6J1, 6K5, 6O1, 6OT, 6QJ, 6QT, 6RV, 6SJ, 6T1, 6TT, 6U5, 6UV, 6VV, 6W5, 6Y1, 6YT, 72D, 737, 77D, 791, 7A1, 7AZ, 7D7, 7GD, 7I1, 7IN, 7LN, 7M7, 7MN, 7N7, 7NN, 7RN, 7RZ, 7WN, 7X7, 7Z1, 801, 80V, 82D, 83B, 841, 84H, 877, 881, 887, 88B, 88V, 88Z, 89H, 89V, 8A1, 8AB, 8B7, 8BH, 8D1, 8DH, 8EB, 8EV, 8GD, 8GZ, 8HD, 8IV, 8L7, 8LZ, 8M1, 8MB, 8MD, 8O7, 8OB, 8Q7, 8QD, 8RD, 8S7, 8SH, 8SZ, 8UZ, 8W7, 8WV, 8XV, 8Y1, 90H, 91J, 93B, 93H, 94V, 981, 98H, 99J, 9AH, 9BB, 9C5, 9IH, 9IV, 9JV, 9KH, 9M1, 9OB, 9QJ, 9R1, 9RH, 9SH, 9UJ, 9VB, 9VJ, 9W5, 9X1, 9Y1, 9YB, 9YJ, 9YV, A25, A3P, A3Z, A4H, A61, A81, A8B, AAB, AAH, ABB, AC5, AEZ, AHP, AK1, AKB, AKH, ALZ, AMB, AO5, AOZ, APH, AQP, AR1, ARV, ARZ, AUV, AV5, AVB, AVV, AWV, AX1, AXB, AZ1, B2J, B4B, B4V, B6V, B87, B8H, B9B, BAB, BAH, BBH, BBV, BE7, BEB, BGJ, BHB, BIJ, BJB, BJT, BLJ, BOB, BOT, BQ7, BRV, BS7, BTT, BTV, BVB, BVJ, BVT, BWV, BX7, BXH, BYH, BYV, C25, C2J, C2N, C55, C65, C6J, C95, CCN, CDD, CED, CF5, CFJ, CGD, CIN, CJD, CO5, CRN, CUJ, CXJ, CXN, D37, D3P, D41, D55, D5P, D6T, D77, D81, D9H, DA1, DCD, DD1, DD5, DDH, DDP, DE7, DEP, DF5, DFD, DG7, DHH, DI1, DK1, DK5, DKT, DO1, DOP, DP1, DPD, DQ7, DR7, DRH, DS7, DTT, DU1, DWD, DX7, DYH, E0P, E2D, E3Z, E41, E4P, E71, E81, E87, E8B, EAZ, EDP, EED, EEN, EEV, EFZ, EGT, EI1, EKZ, ELB, EMB, EMN, EN7, EO1, EOT, EPP, EPZ, EQ7, ERN, EST, ETN, ETV, EUP, EUZ, EVT, EWD, EWN, EX1, EYB, EYN, EZD, EZP, F2J, F3B, F3Z, F4P, F65, F8B, F8Z, FCJ, FD5, FEZ, FFB, FFD, FG5, FGD, FGZ, FIJ, FJ5, FJJ, FLZ, FPD, FQD, FSZ, FXB, G07, G0D, G0Z, G3D, G3Z, G55, G6T, G7D, G7Z, G87, G8Z, GA5, GCJ, GD7, GE7, GET, GG5, GG7, GGT, GGZ, GI7, GJD, GKZ, GL7, GLT, GQT, GRT, GS7, GST, GU5, GUZ, GW5, GXJ, GZ7, GZZ, H3D, H3H, H4H, H9B, HAH, HED, HGD, HHP, HIH, HKH, HKP, HLN, HNH, HOB, HOP, HQP, HRD, HRH, HRN, HSH, HWD, HWH, HWP, HYN, I41, I6J, I71, I7N, I87, I8H, I9H, I9J, IA1, IAV, IE1, IFJ, IG7, IHH, II1, IIH, IIV, IJV, IK1, IL7, ILJ, ILN, IM1, INN, IOV, IRH, IUV, IX1, IXH, IXV, IYJ, J2D, J4V, J55, J61, J6J, J8B, J8V, J95, J9J, J9V, JA5, JBB, JD5, JDT, JF5, JFJ, JGJ, JGT, JIV, JJ1, JJD, JJV, JK5, JKT, JLB, JLT, JMN, JN5, JNJ, JQJ, JQN, JQT, JRD, JTB, JVJ, JVN, JVV, JWN, JY1, JYJ, K0B, K25, K3P, K65, K81, K95, K9H, KAH, KC5, KEP, KEZ, KG5, KHP, KK1, KKT, KLB, KLP, KLZ, KM1, KMB, KMP, KOT, KP1, KQP, KR1, KRT, KRZ, KTT, KW5, KX1, KZB, L0N, L0P, L8Z, L9B, LAZ, LBJ, LEN, LET, LFB, LFZ, LG7, LGZ, LIJ, LJJ, LKB, LKP, LLB, LLP, LLT, LMB, LMN, LN7, LO7, LOT, LPZ, LQT, LRN, LTN, LTT, LWP, LX7, LYN, M01, M0P, M2D, M2N, M37, M3B, M41, M61, M77, M87, M8D, M91, ME1, ME7, MEB, MFB, MFP, MGD, MI7, MKB, MM7, MMN, MN7, MNN, MO7, MOB, MOP, MPP, MQP, MS7, MW7, MXN, MYN, N5N, N65, N87, N8D, N8H, N95, NCD, NCJ, NDH, NE7, NFJ, NG5, NG7, NGJ, NJ5, NN7, NND, NO5, NQD, NQJ, NRN, NU5, NWH, NWN, NXH, NXN, NYJ, O0Z, O25, O37, O3B, O3P, O3Z, O41, O4B, O5Z, O61, O71, O7Z, O81, OA5, OAP, OBB, OBV, OC5, OEZ, OF5, OG7, OKP, OKZ, OMB, OO5, OOZ, OP5, OPP, OQT, OR1, OR7, OS7, OST, OTB, OTZ, OU5, OVV, OW5, OX7, OXB, OXV, OYV, OZ5, OZ7, P01, P0D, P0N, P3P, P4H, P4P, P5N, P65, P6V, P8V, PD1, PED, PEZ, PHH, PHP, PI1, PIN, PLN, PLP, PLZ, POP, PP1, PPH, PPV, PQD, PQN, PRV, PV5, PVH, PVV, PWH, PWP, PX1, PXV, PYH, PYN, Q07, Q0P, Q2N, Q37, Q3P, Q6J, Q6T, Q7D, Q8D, Q9J, QCD, QCJ, QD7, QED, QFP, QGT, QI7, QIN, QJN, QLJ, QMD, QMN, QMP, QND, QNJ, QOT, QQJ, QTN, QWN, QYJ, QYT, R0V, R0Z, R37, R3H, R4H, R77, R7D, R7N, R7Z, R81, R8V, R91, RA1, RCD, RCN, RD1, RED, REV, REZ, RGT, RGZ, RHD, RIV, RKH, RKZ, RLN, RMD, RNH, RO7, RQN, RRD, RRT, RRZ, RS7, RSH, RT1, RTV, RU1, RUZ, RVN, RVT, RW7, RWH, RX7, RY1, S0J, S6J, S87, S9H, SAZ, SCJ, SET, SFJ, SG7, SGZ, SI7, SM7, SO7, SOT, SQJ, SQT, SRH, SSH, STT, SW7, SX7, SXH, SYH, T0N, T0Z, T1D, T1T, T6V, T8Z, TBB, TBV, TCN, TD1, TEV, TGT, TKT, TLB, TM1, TO1, TOB, TQN, TR1, TRD, TT1, TTB, TTN, TUZ, TVB, TVN, TVT, TWD, TWV, TXB, TXV, TYV, U2J, U3Z, U61, U8V, U95, UA1, UC5, UEP, UEZ, UFJ, UG5, UJ5, ULZ, UO5, UOP, UOZ, UQP, URV, URZ, UUJ, UW5, UWV, UXJ, UXV, UYJ, V0H, V25, V4V, V55, V6V, V9J, V9V, VBB, VBJ, VCJ, VET, VGJ, VHP, VIH, VIN, VJJ, VJN, VLT, VMP, VNJ, VOB, VP5, VQJ, VQT, VRT, VRV, VSH, VSJ, VST, VTB, VTN, VUP, VW5, VWN, VXH, VXN, VYB, W07, W25, W3D, W3H, W4P, W4V, W57, W6V, W7D, W8H, W95, W9H, WAH, WAV, WCN, WD7, WDD, WDH, WE7, WEN, WF5, WGD, WHH, WK5, WKH, WN7, WNN, WPP, WPV, WQP, WR7, WRD, WRH, WRN, WS7, WU5, WUP, WVN, WWH, WWP, WX7, WXH, WXN, WYN, WYV, X0J, X2J, X2N, X4B, X4H, X4V, X6J, X87, X8B, X91, X9B, XAV, XCN, XEB, XFB, XHB, XHH, XHN, XKB, XLJ, XNH, XO1, XQ7, XS7, XSH, XUJ, XWN, XWV, XX7, XXH, XXV, Y0N, Y1J, Y1T, Y2N, Y3H, Y61, Y8V, Y91, YBV, YCN, YEB, YFJ, YHB, YHN, YIH, YJN, YJT, YLJ, YLN, YMB, YMN, YNH, YOB, YOV, YRH, YTB, YTN, YTT, YTV, YVB, YVH, YWH, YWV, YXB, YY1, YYJ, YYT, YYV, Z01, Z3D, Z3Z, Z71, Z8B, ZB7, ZBB, ZD5, ZDD, ZDP, ZG7, ZGD, ZGZ, ZKB, ZLP, ZM1, ZO5, ZP1, ZPD, ZQD, ZU1, ZX1, ZXB, ZZD, 102J, 106T, 10ET, 10GJ, 10LJ, 10TD, 12FJ, 12LJ, 13ED, 13GD, 13MD, 192J, 1D0T, 1DET, 1E3D, 1EFD, 1F0D, 1FWD, 1G0J, 1GGJ, 1GIJ, 1GUJ, 1I0J, 1ISJ, 1JOT, 1JTT, 1M0D, 1M3D, 1MFD, 1MWD, 1OKT, 1OOT, 1Q3D, 1QDT, 1QKT, 1QTD, 1QWD, 1SLJ, 1TDT, 1TFD, 1TOT, 1TTD, 1TTT, 1UGJ, 1ULJ, 1WFD, 1WWD, 20CJ, 20FJ, 20G5, 20IN, 20JD, 20MD, 20RN, 20WD, 20YN, 2295, 25RN, 260J, 26LJ, 288D, 28DD, 28FD, 296J, 29G5, 29K5, 29LJ, 2C05, 2C0J, 2C0N, 2CJJ, 2CK5, 2CLJ, 2CSJ, 2D25, 2DC5, 2DDD, 2DW5, 2F0J, 2FDD, 2FMD, 2FSJ, 2G65, 2G95, 2GK5, 2I0N, 2JC5, 2JCJ, 2JWD, 2K05, 2L6J, 2L9J, 2N0N, 2NIN, 2NN5, 2NSJ, 2NW5, 2NYN, 2Q0N, 2QNN, 2R0N, 2RFD, 2RWD, 2SJJ, 2SXJ, 2SYJ, 2W05, 2WC5, 2WD5, 2WMD, 2XCJ, 2YCJ, 2YJJ, 2YRN, 2YSJ, 300D, 302D, 304B, 3077, 3087, 308D, 30BH, 30DD, 30FP, 30G7, 30GD, 30MD, 30QP, 30YH, 320D, 333D, 333Z, 33WD, 344B, 34FB, 373Z, 37E7, 37EZ, 37G7, 3807, 38BB, 38E7, 38G7, 38HB, 38I7, 38X7, 390B, 394B, 399B, 39EB, 39FB, 39KB, 3ASH, 3AWH, 3B8B, 3BG7, 3BLB, 3BR7, 3BW7, 3C8D, 3D0P, 3D2D, 3DKP, 3E3D, 3E77, 3EBB, 3EFB, 3EKB, 3EQP, 3ERD, 3ERZ, 3EZZ, 3F4B, 3FEP, 3G2D, 3GCD, 3GED, 3GM7, 3GR7, 3GWD, 3GZD, 3H0B, 3HEP, 3HKB, 3IAH, 3ISH, 3K4B, 3KBH, 3KEB, 3KKB, 3L4B, 3L8B, 3M3D, 3MEP, 3MG7, 3MWP, 3OFP, 3OKB, 3OMP, 3OOB, 3OYB, 3P8H, 3PXH, 3QDP, 3QQD, 3QRD, 3QWD, 3R3D, 3R3Z, 3R8D, 3RG7, 3RM7, 3RQD, 3RR7, 3RZ7, 3SIH, 3W0H, 3WED, 3WM7, 3WQD, 3WW7, 3WYH, 3X8H, 3XE7, 3XG7, 3XKH, 3Y0B, 3Y4B, 3Y8H, 3YAH, 3YSH, 3YYB, 3Z4B, 3ZFP, 404V, 4061, 40EB, 40I1, 40IV, 40KH, 40M1, 40R1, 40XB, 40XV, 43QP, 4441, 444H, 444P, 4481, 448H, 4491, 449H, 44EB, 44EV, 44K1, 44KB, 44KP, 44M1, 44R1, 44RV, 44SH, 44U1, 44VP, 44YB, 46I1, 46K1, 46M1, 480H, 488H, 48AH, 48I1, 48K1, 48U1, 4991, 49EV, 49I1, 49O1, 49UV, 4A0H, 4A41, 4A4B, 4A8H, 4A91, 4ALB, 4ASH, 4BXV, 4E4B, 4EEP, 4EFB, 4ELP, 4EUV, 4EXB, 4FOB, 4I01, 4I0V, 4I4V, 4I61, 4I81, 4I91, 4IEV, 4IY1, 4K0H, 4K91, 4KI1, 4KKH, 4KKP, 4KUP, 4L3P, 4L4P, 4LAB, 4LMP, 4MA1, 4MK1, 4MMP, 4MP1, 4MU1, 4O0B, 4OFB, 4OKB, 4OM1, 4OU1, 4OYB, 4P41, 4P81, 4PO1, 4PYV, 4QLP, 4QQP, 4RAV, 4RK1, 4RRV, 4S8H, 4U3P, 4U91, 4U9V, 4UIV, 4UUP, 4VAV, 4VLP, 4VVV, 4X0B, 4X0V, 4XAB, 4XBV, 4XVV, 4XYV, 4Y0V, 4Y41, 4Y4H, 4Y4V, 4Y81, 4YAV, 4YEV, 4YIV, 4YK1, 4YKB, 4YM1, 4YR1, 4YUV, 4YYB, 5025, 503Z, 5095, 50G5, 50PN, 50QN, 50WP, 50Z7, 5205, 52RN, 53FP, 53M7, 53PZ, 53X7, 53ZZ, 54PP, 550N, 553P, 554P, 5557, 555N, 557Z, 5595, 55EN, 55G5, 55GZ, 55LZ, 55OZ, 55QN, 55Z7, 572N, 577Z, 5787, 57CN, 57FZ, 57QN, 57S7, 57ZZ, 58I7, 58X7, 5955, 5A95, 5AEP, 5CMN, 5CWN, 5E2N, 5EAP, 5ECN, 5EKP, 5EQN, 5F95, 5FLP, 5FQP, 5G05, 5GK5, 5GQ7, 5I37, 5II7, 5IM7, 5L3P, 5LCN, 5LFP, 5LGT, 5LKZ, 5LUP, 5LZZ, 5M07, 5N55, 5NQ7, 5NX7, 5OAZ, 5OFZ, 5OLT, 5OOT, 5P25, 5P95, 5PG5, 5PGZ, 5PPN, 5PQP, 5PW5, 5Q0N, 5Q87, 5QCN, 5QKP, 5QN7, 5QPN, 5QPP, 5QQT, 5QTT, 5QWP, 5QX7, 5S37, 5SS7, 5T6T, 5TQT, 5U4P, 5UGZ, 5W0P, 5W2N, 5W5P, 5WLN, 5WP5, 5WQN, 5XM7, 5Y0T, 5YOT, 5Z25, 5Z4P, 5Z57, 5ZG5, 5ZLZ, 5ZQ7, 5ZWP, 6005, 600V, 601J, 6055, 606V, 60A1, 60C5, 60GT, 60JJ, 60LJ, 60M1, 60R1, 60VT, 60YV, 610J, 61FJ, 6401, 64K1, 6505, 665T, 6665, 66A5, 66F5, 66G5, 66GT, 66KT, 66V5, 66XV, 68OV, 68YV, 690J, 69G5, 69O5, 69XJ, 69XV, 6A55, 6AA1, 6AA5, 6AF5, 6C9J, 6CA5, 6CGJ, 6CYJ, 6EK1, 6ER1, 6FLJ, 6FXJ, 6G0J, 6G65, 6GGJ, 6GJ5, 6GLJ, 6I81, 6I9V, 6J65, 6J6V, 6JET, 6JST, 6JWV, 6K01, 6KET, 6KST, 6L0J, 6L6T, 6LFJ, 6LRT, 6LXJ, 6MA1, 6MM1, 6MX1, 6O8V, 6OV5, 6R0T, 6RET, 6RKT, 6S6T, 6T8V, 6U01, 6U9J, 6UE1, 6UGJ, 6UJJ, 6ULJ, 6UU1, 6V0T, 6VA5, 6VF5, 6VG5, 6VKT, 6VXJ, 6WWV, 6WXV, 6X41, 6X6V, 6XIV, 6XJV, 6XOV, 6Y0V, 6YCJ, 6YXV, 7001, 702N, 7041, 707N, 7081, 70GZ, 70I7, 70KZ, 70ND, 70O7, 70OZ, 70R7, 70U1, 70UZ, 70WD, 70YN, 71QD, 71WD, 720N, 72CN, 72YN, 733Z, 73ED, 73ZD, 7401, 7441, 74M1, 74O1, 74R1, 7641, 76K1, 76M1, 76R1, 76U1, 76X1, 773Z, 7741, 7771, 7781, 77FZ, 77G7, 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8KRH, 8KU1, 8KYH, 8KZ1, 8L4B, 8LBB, 8MG7, 8MR7, 8O4V, 8OI1, 8OK1, 8OOV, 8OX1, 8RAV, 8RHH, 8RUV, 8RX1, 8U4V, 8UK1, 8V4B, 8V9B, 8VHH, 8VRH, 8VVV, 8VYV, 8W0H, 8WFD, 8X07, 8XAH, 8XR1, 8XU1, 8XYB, 8Y6V, 8Y8H, 8YBB, 8YFB, 8YHH, 8YKB, 8YLB, 8YUV, 8YVV, 8YYH, 8Z1D, 8Z4B, 8Z8D, 8Z91, 8ZCD, 8ZEZ, 8ZFD, 8ZFZ, 8ZK1, 8ZO1, 8ZX7, 90A5, 90BJ, 90BV, 90EB, 90EV, 90GJ, 90IJ, 90J5, 90JB, 90LB, 90O1, 90OV, 90XB, 90XV, 9225, 92K5, 9441, 94BH, 94EB, 94KB, 94XB, 9505, 9601, 960V, 96A1, 96J5, 96K1, 96V5, 96WV, 96XV, 984B, 98AV, 98BV, 98FB, 994H, 998B, 998V, 999B, 99A1, 99A5, 99E1, 99G5, 99O1, 99O5, 99WH, 9A05, 9A55, 9A95, 9AA5, 9ABV, 9AE1, 9AEV, 9AI1, 9AO1, 9AOV, 9B0J, 9B0V, 9B8V, 9B9H, 9BCJ, 9BFJ, 9BSJ, 9BVV, 9CCJ, 9CLJ, 9CSJ, 9E01, 9E4B, 9EA1, 9EAB, 9EAV, 9EK1, 9F05, 9F25, 9FA5, 9FBJ, 9FFJ, 9FJB, 9FSJ, 9FXJ, 9G25, 9G2J, 9GIJ, 9GSJ, 9HFB, 9HLB, 9IIJ, 9IJ1, 9J01, 9J0B, 9J0J, 9J9B, 9JCJ, 9JE1, 9K05, 9K41, 9K55, 9K61, 9K8B, 9KA5, 9KI1, 9KK5, 9KU5, 9L2J, 9LCJ, 9O05, 9O0V, 9O95, 9OI1, 9OO1, 9OOV, 9OWV, 9RRV, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ

====Additional known quasi-minimal primes (not necessarily the next)====

P&lt;sub&gt;81993&lt;/sub&gt;SZ

==Unsolved families==

Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')) by covering congruence, algebraic factorization, or combine of them.

{|class=wikitable
|base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family
|-
|13||9{5}||(113×13&lt;sup&gt;''n''&lt;/sup&gt;−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|13||A{3}A||(41×13&lt;sup&gt;''n''+1&lt;/sup&gt;+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|16||{3}AF||(16&lt;sup&gt;''n''+2&lt;/sup&gt;+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1{7}||(23×17&lt;sup&gt;''n''&lt;/sup&gt;−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1F{0}7||32×17&lt;sup&gt;''n''+1&lt;/sup&gt;+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||4{7}A||(71×17&lt;sup&gt;''n''+1&lt;/sup&gt;+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||51{0}D||86×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||70F{0}D||2038×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||8{B}9||(139×17&lt;sup&gt;''n''+1&lt;/sup&gt;−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||9{5}9||(149×17&lt;sup&gt;''n''+1&lt;/sup&gt;+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||95{F}||(2543×17&lt;sup&gt;''n''&lt;/sup&gt;−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||A{D}F||(173×17&lt;sup&gt;''n''+1&lt;/sup&gt;+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}B3||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+190||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}DB||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+232||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2BE||(11×17&lt;sup&gt;''n''+3&lt;/sup&gt;−41579)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2E||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;−2411)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}E9||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+773)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}EE||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+853)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||D0G{D}||(60381×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2860381*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||E9{B}||(3963×17&lt;sup&gt;''n''&lt;/sup&gt;−11)/16||29000||[http://factordb.com/index.php?query=%283963*17%5En-11%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||F1{9}||(4105×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16||1000000||[http://factordb.com/index.php?query=%284105*17%5En-9%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||FD0{D}||(72909×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2872909*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||G{7}F||(263×17&lt;sup&gt;''n''+1&lt;/sup&gt;+121)/16||29000||[http://factordb.com/index.php?query=%28263*17%5E%28n%2B1%29%2B121%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||2{7}9D||(47×21&lt;sup&gt;''n''+2&lt;/sup&gt;+953)/20||20000||[http://factordb.com/index.php?query=%2847*21%5E%28n%2B2%29%2B953%29%2F20&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||2F{C}A||(288×21&lt;sup&gt;''n''+1&lt;/sup&gt;−13)/5||20000||[http://factordb.com/index.php?query=%28288*21%5E%28n%2B1%29-13%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||4{3}B||(83×21&lt;sup&gt;''n''+1&lt;/sup&gt;+157)/20||20000||[http://factordb.com/index.php?query=%2883*21%5E%28n%2B1%29%2B157%29%2F20&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||5{0}DJ||5×21&lt;sup&gt;''n''+2&lt;/sup&gt;+292||20000||[http://factordb.com/index.php?query=5*21%5E%28n%2B2%29%2B292&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||{5}FEK||(21&lt;sup&gt;''n''+3&lt;/sup&gt;+18455)/4||20000||[http://factordb.com/index.php?query=%2821%5E%28n%2B3%29%2B18455%29%2F4&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||{7}ID||(7×21&lt;sup&gt;''n''+2&lt;/sup&gt;+4733)/20||20000||[http://factordb.com/index.php?query=%287*21%5E%28n%2B2%29%2B4733%29%2F20&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||7{K}Z||(53×36&lt;sup&gt;''n''+1&lt;/sup&gt;+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||B{0}EUV||11×36&lt;sup&gt;''n''+3&lt;/sup&gt;+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||HM{0}N||634×36&lt;sup&gt;''n''+1&lt;/sup&gt;+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||N{0}YYN||23×36&lt;sup&gt;''n''+3&lt;/sup&gt;+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||O{L}Z||(123×36&lt;sup&gt;''n''+1&lt;/sup&gt;+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||S{0}8H||28×36&lt;sup&gt;''n''+2&lt;/sup&gt;+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|}

For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E

For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family B0{H}6H may or may not be quasi-minimal prime, since another unsolved family is B{H}6H, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D}

=== Base 17 ===

All searched to length 29000

* 1{7}
* 1F{0}7
* 4{7}A
* 51{0}D
* 70F{0}D
* 8{B}9
* 9{5}9
* 95{F}
* A{D}F
* B{0}B3
* B{0}DB
* {B}2BE
* {B}2E
* {B}E9
* {B}EE
* D0G{D}
* E9{B}
* F1{9}
* FD0{D}
* G{7}F

=== Base 21 ===

All searched to length 20000

* 2{7}9D
* 2F{C}A
* 4{3}B
* 5{0}DJ
* {5}FEK
* {7}ID
* 99{0}99H
* {9}0D
* {9}D
* B0{H}6H
* B3{0}EB
* B9{0}E5
* B{D}B
* B{H}6H
* DH{D}
* F{9}D
* {F}35
* G{0}FK
* H{0}7771
* H{D}
* {J}6J

=== Base 36 ===

All searched to length 20000

* 7{K}Z
* B{0}EUV
* HM{0}N
* N{0}YYN
* O{L}Z
* S{0}8H

==Primality certificates for the proven primes &gt; 10&lt;sup&gt;299&lt;/sup&gt;==

See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]]

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime
|-
|9||149||76&lt;sub&gt;329&lt;/sub&gt;2||(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642]
|-
|9||150||27&lt;sub&gt;686&lt;/sub&gt;07||(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486]
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||3×9&lt;sup&gt;1160&lt;/sup&gt;+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423]
|-
|11||1065||A&lt;sub&gt;713&lt;/sub&gt;58||11&lt;sup&gt;715&lt;/sup&gt;−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487]
|-
|11||1066||7&lt;sub&gt;759&lt;/sub&gt;44||(7×11&lt;sup&gt;761&lt;/sup&gt;−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840]
|-
|11||1067||557&lt;sub&gt;1011&lt;/sub&gt;||(607×11&lt;sup&gt;1011&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522]
|-
|13||3165||50&lt;sub&gt;270&lt;/sub&gt;44||5×13&lt;sup&gt;272&lt;/sup&gt;+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005]
|-
|13||3166||9&lt;sub&gt;271&lt;/sub&gt;095||(3×13&lt;sup&gt;274&lt;/sup&gt;−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654]
|-
|13||3167||10&lt;sub&gt;286&lt;/sub&gt;7771||13&lt;sup&gt;290&lt;/sup&gt;+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633]
|-
|13||3168||9&lt;sub&gt;308&lt;/sub&gt;1||(3×13&lt;sup&gt;309&lt;/sup&gt;−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=308&amp;c0=-&amp;EN= 13&lt;sup&gt;308&lt;/sup&gt;−1]
|-
|13||3169||B&lt;sub&gt;341&lt;/sub&gt;C4||(11×13&lt;sup&gt;343&lt;/sup&gt;+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618]
|-
|13||3170||8B&lt;sub&gt;343&lt;/sub&gt;||(107×13&lt;sup&gt;343&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736]
|-
|13||3171||710&lt;sub&gt;371&lt;/sub&gt;111||92×13&lt;sup&gt;374&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609]
|-
|13||3172||75&lt;sub&gt;375&lt;/sub&gt;7||(89×13&lt;sup&gt;376&lt;/sup&gt;+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596]
|-
|13||3173||9B0&lt;sub&gt;391&lt;/sub&gt;9||128×13&lt;sup&gt;392&lt;/sup&gt;+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790]
|-
|13||3174||7B0B&lt;sub&gt;397&lt;/sub&gt;||(15923×13&lt;sup&gt;397&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574]
|-
|13||3175||10&lt;sub&gt;414&lt;/sub&gt;93||13&lt;sup&gt;416&lt;/sup&gt;+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240]
|-
|13||3176||81010&lt;sub&gt;415&lt;/sub&gt;1||17746×13&lt;sup&gt;416&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3177||8110&lt;sub&gt;435&lt;/sub&gt;1||1366×13&lt;sup&gt;436&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3178||B7&lt;sub&gt;486&lt;/sub&gt;||(139×13&lt;sup&gt;486&lt;/sup&gt;−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892]
|-
|13||3179||B&lt;sub&gt;563&lt;/sub&gt;C||(11×13&lt;sup&gt;564&lt;/sup&gt;+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=564&amp;c0=-&amp;EN= 13&lt;sup&gt;564&lt;/sup&gt;−1]
|-
|13||3180||1B&lt;sub&gt;576&lt;/sub&gt;||(23×13&lt;sup&gt;576&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=576&amp;c0=-&amp;EN= 13&lt;sup&gt;576&lt;/sup&gt;−1]
|-
|13||3181||80&lt;sub&gt;693&lt;/sub&gt;87||8×13&lt;sup&gt;695&lt;/sup&gt;+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532]
|-
|13||3182||CC5&lt;sub&gt;713&lt;/sub&gt;||(2021×13&lt;sup&gt;713&lt;/sup&gt;−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353]
|-
|13||3183||B&lt;sub&gt;834&lt;/sub&gt;74||(11×13&lt;sup&gt;836&lt;/sup&gt;−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871]
|-
|13||3184||9&lt;sub&gt;968&lt;/sub&gt;B||(3×13&lt;sup&gt;969&lt;/sup&gt;+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244]
|-
|13||3185||10&lt;sub&gt;1295&lt;/sub&gt;181||13&lt;sup&gt;1298&lt;/sup&gt;+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013]
|-
|13||3186||9&lt;sub&gt;1362&lt;/sub&gt;5||(3×13&lt;sup&gt;1363&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776]
|-
|13||3187||7&lt;sub&gt;1504&lt;/sub&gt;1||(7×13&lt;sup&gt;1505&lt;/sup&gt;−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755]
|-
|13||3188||930&lt;sub&gt;1551&lt;/sub&gt;1||120×13&lt;sup&gt;1552&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3189||720&lt;sub&gt;2297&lt;/sub&gt;2||93×13&lt;sup&gt;2298&lt;/sup&gt;+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910]
|-
|13||3190||1770&lt;sub&gt;2703&lt;/sub&gt;17||267×13&lt;sup&gt;2705&lt;/sup&gt;+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825]
|-
|13||3191||390&lt;sub&gt;6266&lt;/sub&gt;1||48×13&lt;sup&gt;6267&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3192||B0&lt;sub&gt;6540&lt;/sub&gt;BBA||11×13&lt;sup&gt;6543&lt;/sup&gt;+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906]
|-
|13||3193||C&lt;sub&gt;10631&lt;/sub&gt;92||13&lt;sup&gt;10633&lt;/sup&gt;−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750]
|-
|14||649||34D&lt;sub&gt;708&lt;/sub&gt;||47×14&lt;sup&gt;708&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||5×14&lt;sup&gt;19698&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2328||880&lt;sub&gt;246&lt;/sub&gt;7||136×16&lt;sup&gt;247&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is &lt; 10&lt;sup&gt;299&lt;/sup&gt;
|-
|16||2329||D4&lt;sub&gt;263&lt;/sub&gt;D||(199×16&lt;sup&gt;264&lt;/sup&gt;+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238]
|-
|16||2330||E0&lt;sub&gt;261&lt;/sub&gt;4DD||14×16&lt;sup&gt;264&lt;/sup&gt;+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352]
|-
|16||2331||8C0&lt;sub&gt;290&lt;/sub&gt;ED||140×16&lt;sup&gt;292&lt;/sup&gt;+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307]
|-
|16||2332||DA&lt;sub&gt;305&lt;/sub&gt;5||(41×16&lt;sup&gt;306&lt;/sup&gt;−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284]
|-
|16||2333||CE80&lt;sub&gt;422&lt;/sub&gt;D||3304×16&lt;sup&gt;423&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257]
|-
|16||2334||5F&lt;sub&gt;544&lt;/sub&gt;6F||6×16&lt;sup&gt;546&lt;/sup&gt;−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967]
|-
|16||2335||88F&lt;sub&gt;545&lt;/sub&gt;||137×16&lt;sup&gt;545&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2336||BE0&lt;sub&gt;792&lt;/sub&gt;BB||190×16&lt;sup&gt;794&lt;/sup&gt;+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938]
|-
|16||2337||D9&lt;sub&gt;1052&lt;/sub&gt;||(68×16&lt;sup&gt;1052&lt;/sup&gt;−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020]
|-
|16||2338||FAF&lt;sub&gt;1062&lt;/sub&gt;45||251×16&lt;sup&gt;1064&lt;/sup&gt;−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610]
|-
|16||2339||F8&lt;sub&gt;1517&lt;/sub&gt;F||(233×16&lt;sup&gt;1518&lt;/sup&gt;+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824]
|-
|16||2340||20&lt;sub&gt;1713&lt;/sub&gt;321||2×16&lt;sup&gt;1716&lt;/sup&gt;+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735]
|-
|16||2341||300F&lt;sub&gt;1960&lt;/sub&gt;AF||769×16&lt;sup&gt;1962&lt;/sup&gt;−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750]
|-
|16||2342||90&lt;sub&gt;3542&lt;/sub&gt;91||9×16&lt;sup&gt;3544&lt;/sup&gt;+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191]
|-
|16||2343||5BC&lt;sub&gt;3700&lt;/sub&gt;D||(459×16&lt;sup&gt;3701&lt;/sup&gt;+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322]
|-
|16||2344||D0B&lt;sub&gt;17804&lt;/sub&gt;||(3131×16&lt;sup&gt;17804&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511]
|-
|18||547||80&lt;sub&gt;298&lt;/sub&gt;B||8×18&lt;sup&gt;299&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is an algebraic factor of ''N''+1, factordb entry of 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is [http://factordb.com/index.php?id=1100000000900149167]
|-
|18||548||H&lt;sub&gt;766&lt;/sub&gt;FH||18&lt;sup&gt;768&lt;/sup&gt;−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490]
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||12×18&lt;sup&gt;6270&lt;/sup&gt;+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437]
|-
|20||3301||H&lt;sub&gt;247&lt;/sub&gt;A0H||(17×20&lt;sup&gt;250&lt;/sup&gt;−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619]
|-
|20||3302||7&lt;sub&gt;249&lt;/sub&gt;A7||(7×20&lt;sup&gt;251&lt;/sup&gt;+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602]
|-
|20||3303||J7&lt;sub&gt;270&lt;/sub&gt;||(368×20&lt;sup&gt;270&lt;/sup&gt;−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462]
|-
|20||3304||J&lt;sub&gt;330&lt;/sub&gt;CCC7||20&lt;sup&gt;334&lt;/sup&gt;−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572]
|-
|20||3305||40&lt;sub&gt;387&lt;/sub&gt;404B||4×20&lt;sup&gt;391&lt;/sup&gt;+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563]
|-
|20||3306||EC0&lt;sub&gt;429&lt;/sub&gt;7||292×20&lt;sup&gt;430&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702]
|-
|20||3307||G&lt;sub&gt;447&lt;/sub&gt;99||(16×20&lt;sup&gt;449&lt;/sup&gt;−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753]
|-
|20||3308||3A&lt;sub&gt;527&lt;/sub&gt;3||(67×20&lt;sup&gt;528&lt;/sup&gt;−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531]
|-
|20||3309||E&lt;sub&gt;566&lt;/sub&gt;C7||(14×20&lt;sup&gt;568&lt;/sup&gt;−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516]
|-
|20||3310||JCJ&lt;sub&gt;629&lt;/sub&gt;||393×20&lt;sup&gt;629&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|20||3311||J&lt;sub&gt;655&lt;/sub&gt;05J||20&lt;sup&gt;658&lt;/sup&gt;−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052]
|-
|20||3312||50&lt;sub&gt;1163&lt;/sub&gt;AJ||5×20&lt;sup&gt;1165&lt;/sup&gt;+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412]
|-
|20||3313||CD&lt;sub&gt;2449&lt;/sub&gt;||(241×20&lt;sup&gt;2449&lt;/sup&gt;−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915]
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||16×20&lt;sup&gt;6270&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457]
|-
|22||7984||I7G0&lt;sub&gt;254&lt;/sub&gt;H||8882×22&lt;sup&gt;255&lt;/sup&gt;+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788]
|-
|22||7985||D0&lt;sub&gt;255&lt;/sub&gt;5EEF||13×22&lt;sup&gt;259&lt;/sup&gt;+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932]
|-
|22||7986||IK&lt;sub&gt;322&lt;/sub&gt;F||(398×22&lt;sup&gt;323&lt;/sup&gt;−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145]
|-
|22||7987||C0&lt;sub&gt;340&lt;/sub&gt;G9||12×22&lt;sup&gt;342&lt;/sup&gt;+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159]
|-
|22||7988||77E&lt;sub&gt;348&lt;/sub&gt;K7||(485×22&lt;sup&gt;350&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779]
|-
|22||7989||J&lt;sub&gt;379&lt;/sub&gt;KJ||(19×22&lt;sup&gt;381&lt;/sup&gt;+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027]
|-
|22||7990||J&lt;sub&gt;388&lt;/sub&gt;EJ||(19×22&lt;sup&gt;390&lt;/sup&gt;−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729]
|-
|22||7991||DJ&lt;sub&gt;400&lt;/sub&gt;||(292×22&lt;sup&gt;400&lt;/sup&gt;−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110]
|-
|22||7992||E&lt;sub&gt;404&lt;/sub&gt;K7||(2×22&lt;sup&gt;406&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298]
|-
|22||7993||66F&lt;sub&gt;453&lt;/sub&gt;B3||(971×22&lt;sup&gt;455&lt;/sup&gt;−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809]
|-
|22||7994||L0&lt;sub&gt;454&lt;/sub&gt;B63||21×22&lt;sup&gt;457&lt;/sup&gt;+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331]
|-
|22||7995||L&lt;sub&gt;483&lt;/sub&gt;G3||22&lt;sup&gt;485&lt;/sup&gt;−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730]
|-
|22||7996||E60&lt;sub&gt;496&lt;/sub&gt;L||314×22&lt;sup&gt;497&lt;/sup&gt;+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239]
|-
|22||7997||I&lt;sub&gt;626&lt;/sub&gt;AF||(6×22&lt;sup&gt;628&lt;/sup&gt;−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334]
|-
|22||7998||K0&lt;sub&gt;760&lt;/sub&gt;EC1||20×22&lt;sup&gt;763&lt;/sup&gt;+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415]
|-
|22||7999||J0&lt;sub&gt;767&lt;/sub&gt;IGGJ||19×22&lt;sup&gt;771&lt;/sup&gt;+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567]
|-
|22||8000||7&lt;sub&gt;959&lt;/sub&gt;K7||(22&lt;sup&gt;961&lt;/sup&gt;+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817]
|-
|22||8001||L&lt;sub&gt;2385&lt;/sub&gt;KE7||22&lt;sup&gt;2388&lt;/sup&gt;−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774]
|-
|22||8002||7&lt;sub&gt;3815&lt;/sub&gt;2L||(22&lt;sup&gt;3817&lt;/sup&gt;−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839]
|-
|24||3400||I0&lt;sub&gt;241&lt;/sub&gt;I5||18×24&lt;sup&gt;243&lt;/sup&gt;+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037]
|-
|24||3401||D0&lt;sub&gt;259&lt;/sub&gt;KKD||13×24&lt;sup&gt;262&lt;/sup&gt;+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725]
|-
|24||3402||C7&lt;sub&gt;298&lt;/sub&gt;||(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235]
|-
|24||3403||20&lt;sub&gt;313&lt;/sub&gt;7||2×24&lt;sup&gt;314&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241]
|-
|24||3404||BC0&lt;sub&gt;331&lt;/sub&gt;B||276×24&lt;sup&gt;332&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842]
|-
|24||3405||N&lt;sub&gt;2644&lt;/sub&gt;LLN||24&lt;sup&gt;2647&lt;/sup&gt;−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089]
|-
|24||3406||D&lt;sub&gt;2698&lt;/sub&gt;LD||(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876]
|-
|24||3407||A0&lt;sub&gt;2951&lt;/sub&gt;8ID||10×24&lt;sup&gt;2954&lt;/sup&gt;+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654]
|-
|24||3408||88N&lt;sub&gt;5951&lt;/sub&gt;||201×24&lt;sup&gt;5951&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606]
|-
|30||2613||AN&lt;sub&gt;206&lt;/sub&gt;||(313×30&lt;sup&gt;206&lt;/sup&gt;−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073]
|-
|30||2614||M&lt;sub&gt;241&lt;/sub&gt;QB||(22×30&lt;sup&gt;243&lt;/sup&gt;+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295]
|-
|30||2615||M0&lt;sub&gt;547&lt;/sub&gt;SS7||22×30&lt;sup&gt;550&lt;/sup&gt;+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988]
|-
|30||2616||C0&lt;sub&gt;1022&lt;/sub&gt;1||12×30&lt;sup&gt;1023&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|30||2617||5&lt;sub&gt;4882&lt;/sub&gt;J||(5×30&lt;sup&gt;4883&lt;/sup&gt;+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423]
|-
|30||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||25×30&lt;sup&gt;34205&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|}

==Unproven PRPs==

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this PRP
|-
|11||1068||57&lt;sub&gt;62668&lt;/sub&gt;||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000003573679860]
|-
|13||3194||C5&lt;sub&gt;23755&lt;/sub&gt;C||(149×13&lt;sup&gt;23756&lt;/sup&gt;+79)/12||[http://factordb.com/index.php?id=1100000003590647776]
|-
|13||3195||80&lt;sub&gt;32017&lt;/sub&gt;111||8×13&lt;sup&gt;32020&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000000490878060]
|-
|16||2345||DB&lt;sub&gt;32234&lt;/sub&gt;||(206×16&lt;sup&gt;32234&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000002383583629]
|-
|16||2346||4&lt;sub&gt;72785&lt;/sub&gt;DD||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15||[http://factordb.com/index.php?id=1100000003615909841]
|-
|22||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21||[http://factordb.com/index.php?id=1100000003594696838]
|-
|30||2618||I0&lt;sub&gt;24608&lt;/sub&gt;D||18×30&lt;sup&gt;24609&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003593967511]
|}

All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10&lt;sup&gt;16&lt;/sup&gt;. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]])

==Proof==

===Base 2===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.

===Base 3===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (2,1), (2,2)

* Case (1,1):
** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes)
*** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime.

===Base 4===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (2,1), (2,3), (3,1), (3,3)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.

===Base 5===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4)

* Case (1,1):
** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes)
*** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes)
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes)
***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb])
***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes)
**** The smallest prime of the form 1{4}4 is 14444.
**** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes)
*** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes)
**** For the 3{0,3}1 family, we can separate this family to four families:
***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01.
****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime.
***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031.
***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001.
***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31.
****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime.
**** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.
* Case (3,4):
** 34 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes)
**** The smallest prime of the form 4{4}1 is 44441.
**** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime.
* Case (4,2):
** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime.

===Base 6===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes)
*** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes)
**** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 4{0}41 is 40041
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes)
*** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.

===Base 7===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

* Case (1,1):
** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes)
*** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {}
**** If there are &gt;=3 number of 1 and 5 in the {}:
***** If there is 111 in the {}, then we have the prime 11111
***** If there is 115 in the {}, then the prime 115 is a subsequence
***** If there is 151 in the {}, then the prime 115 is a subsequence
***** If there is 155 in the {}, then the prime 155 is a subsequence
***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime
***** If there is 515 in the {}, then the prime 515 is a subsequence
***** If there is 551 in the {}, then the prime 551 is a subsequence
***** If there is 555 in the {}, then the prime 551 is a subsequence
**** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1
***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1
****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1
******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21
******** The smallest prime of the form 11{0}21 is 1100021
******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime
******** The smallest prime of the form 1001{0}21 is 100121
********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families
***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1
****** Since 1051 is prime, we only need to consider the family 15{0}1
******* The smallest prime of the form 15{0}1 is 150001
* Case (1,2):
** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes)
*** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {}
**** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime.
**** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime.
**** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2
***** Since 1022 and 1202 are primes, we only need to consider the number 122
****** 122 is not prime.
**** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime.
* Case (1,3):
** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** 14 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime.
* Case (1,6):
** 16 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime.
* Case (2,2):
** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,6):
** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes)
*** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes)
**** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's)
***** For the 3{0,1}1 family:
****** If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
****** If there are no 1's between (3,1), then the form will be 3{0}1
******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime.
****** If there are no 0's between (3,1), then the form will be 3{1}1
******* The smallest prime of the form 3{1}1 is 31111
****** If there are exactly 1 0's between (3,1), then there must be &lt;3 1's between (3,1), or 31111 will be a subsequence.
******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime.
******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011
******** Neither 3101 nor 3011 is prime.
******* If there are no 1's between (3,1), then the number must be 301
******** 301 is not prime.
***** For the 3{3}3{0,1}1 family:
****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence.
****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1
******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence.
******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime.
******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime.
******* If there are no 1's between (33,1), then the form will be 33{0}1
******** The smallest prime of the form 33{0}1 is 33001
****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1
******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence.
******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301
******** The smallest prime of the form 3{3}301 is 33333301
******* If there are no 0's between (3{3}3,1), then the form is 3{3}31
******** The smallest prime of the form 3{3}31 is 33333333333333331
**** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1
***** Numbers containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1,3}3{0,1,3}1
******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes)
******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes)
********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes)
********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence.
********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime.
********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1
*********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime.
********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime.
***** Numbers not containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1}1
******* If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1
******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime.
******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1
******** If there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111
********* None of 31151, 31511, 35111 are primes.
******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 351
********* 351 is not prime.
******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111
********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes.
********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime.
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1
********* Neither 3051 nor 3501 is prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** If there are &gt;=2 5's in {}, then 553 will be a subsequence.
*** If there are no 5's in {}, then the family will be 3{0,3,6}3
**** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
*** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3
**** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes)
***** The smallest prime of the form 3{0}53 is 300053
* Case (3,4):
** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes)
*** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes)
**** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes)
**** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes)
* Case (3,5):
** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes)
*** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence.
*** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence.
*** If there are at least two 1's in {}, then 115 will be a subsequence.
*** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5
**** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime.
*** If there is no 1's in {}, then the family will be 3{0,4,5}5
**** If there are at least to 4's in {}, then 344 and 445 will be subsequences.
**** If there is no 4's in {}, then the family will be 3{0,5}5
***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5
****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime.
****** The smallest prime of the form 3{5}5 is 35555
**** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5
***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes)
****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime.
* Case (3,6):
** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,2):
** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes)
*** If there is no 5's in {}, then the family will be 4{0,2,4,6}4
**** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4
**** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes)
***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence.
***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4
****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes)
***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4
****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4
******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes)
* Case (4,5):
** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** If there are at least two 5's in {}, then 4555 will be a subsequence.
*** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there is no 5's in {}, then the family will be 4{0}5
**** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime.
* Case (4,6):
** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (5,1):
** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 3's in {}, then 533 will be a subsequence.
*** If there is no 3's in {}, then the family will be 5{0,1}1
**** Since 5011 is prime, we only need to consider the family 5{1}{0}1
***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes)
****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime.
****** The smallest prime of the form 51{0}1 is 5100000001
****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime.
****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime.
*** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1
**** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence.
***** Thus we only need to consider the family 5{0}3{0,1}1
****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime.
****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime.
****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence.
****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1
******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0)
******** The smallest prime of the form 5311{0}1 is 531101
* Case (5,2):
** 52 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 1's in {}, then 113 will be a subsequence.
*** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
*** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime.
* Case (5,4):
** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes)
*** If there are no 5's in {}, then the family will be 5{0,3}4
**** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime.
*** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence.
*** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there are at least two 5's in {}, then 5554 will be a subsequence.
* Case (5,5):
** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes)
*** If there are no 4's in {}, then the family will be 5{0,5}5
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
*** If there are no 5's in {}, then the family will be 5{0,4}5
**** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime.
*** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence.
* Case (5,6):
** 56 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,2):
** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (6,3):
** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,4):
** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are at least three 3's in {}, then 3334 will be a subsequence.
*** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4
**** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence.
**** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence.
**** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence.
**** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence.
**** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence.
**** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence.
**** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence.
**** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence.
**** Thus the number can only be 634
***** 634 is not prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,6):
** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}6
**** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime.
*** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6
**** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime.
*** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence.
*** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence.

===Base 8===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7)

* Case (1,1):
** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes)
**** 171 is not prime.
**** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 &gt; 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 &gt; 1, this factorization is nontrivial), thus cannot be prime.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes)
*** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime)
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes)
*** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes)
*** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes)
**** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime.
**** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes)
***** The smallest prime of the form 33{4}1 is 3344441
***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime.
**** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes)
***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime.
***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes)
****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes.
* Case (3,3):
** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes)
*** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes)
**** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime)
**** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461
***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes)
****** The smallest prime of the form 4{4}41 is 444444441
****** The smallest prime of the form 4{4}641 is 444641
***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes)
****** The smallest prime of the form 4{4}411 is 444444441
****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes)
***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461
****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime)
**** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes)
***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime)
***** The smallest prime of the form 4{4}641 is 444641
* Case (4,3):
** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes)
*** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes)
**** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime.
**** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime.
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes)
*** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes)
**** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb])
**** The smallest prime of the form 4{4}77 is 4444477
**** The smallest prime of the form 4{7}7 is 47777
**** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime)
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes)
*** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes)
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
**** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes)
***** 500525 is not prime.
***** The smallest prime of the form 5{5}25 is 555555555555525
***** The smallest prime of the form 5{5}025 is 55555025
***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes)
***** The smallest prime of the form 5{5}0525 is 5550525
***** The smallest prime of the form 5{5}00525 is 5500525
***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes)
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence)
**** The form is 6{0,1,7}4{0,1,7}1
***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1
****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411
******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41
******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41
********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.
********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime.
********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime.
******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411
******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime)
*** Numbers not containing 4:
**** The form is 6{0,1,7}1
***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1
****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime.
****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1
******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1
******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.
******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime)
* Case (6,3):
** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,7):
** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them
**** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17
***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime.
**** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77
***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime.
**** If there is neither 1 nor 7 between them, then the form is 6{0,4}7
***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes)
****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime.
****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime.
* Case (7,1):
** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes)
*** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes)
**** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes)
***** The smallest prime of the form 7{7}1 is 7777777777771
***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime)
***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime.
***** The smallest prime of the form 7{7}4444441 is 77774444441
****** Since this prime has just 4 7's, we only need to consider the families with &lt;=3 7's
******* The smallest prime of the form 7{4}1 is 744444441
******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime.
******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes)
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.

===Base 10===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,9):
** 19 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 2{0}21 is 20021
**** The smallest prime of the form 22{0}1 is 22000001
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes)
*** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes)
**** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime.
**** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime.
* Case (2,9):
** 29 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,9):
** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** 47 is prime, and thus the only minimal prime in this family.
* Case (4,9):
** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime.
* Case (5,1):
** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes)
**** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes)
***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 5{5}1 is 555555555551
**** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581
***** 581 is not prime.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,7):
** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027
**** The smallest prime of the form 52{0}7 is 5200007
* Case (5,9):
** 59 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,3):
** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,9):
** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49
***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49
****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes)
******* The smallest prime of the form 6{6}49 is 666649
******** Since this prime has just 4 6's, we only need to consider the families with &lt;=3 6's
********* The smallest prime of the form 6{0}49 is 60000049
********* The smallest prime of the form 66{0}49 is 66000049
********* The smallest prime of the form 666{0}49 is 66600049
* Case (7,1):
** 71 is prime, and thus the only minimal prime in this family.
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,9):
** 79 is prime, and thus the only minimal prime in this family.
* Case (8,1):
** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes)
**** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes)
***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime)
***** The smallest prime of the form 80{5}1 is 80555551
* Case (8,3):
** 83 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime.
* Case (8,9):
** 89 is prime, and thus the only minimal prime in this family.
* Case (9,1):
** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes)
**** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021
***** None of 921, 9201, 9021 are primes.
**** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801
***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
* Case (9,3):
** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (9,7):
** 97 is prime, and thus the only minimal prime in this family.
* Case (9,9):
** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9
***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes)
****** The smallest prime of the form 94{6}9 is 946669

===Base 12===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,B):
** 1B is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes)
*** The smallest prime of the form 2{0}1 is 2001
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (2,B):
** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes)
**** None of 20B, 22B, 29B, 209B, 220B are primes.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,B):
** 3B is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes)
**** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 44{A}1 is 44AAA1
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes)
**** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077
* Case (4,B):
** 4B is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes)
*** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime.
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (5,B):
** 5B is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,5):
** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes)
**** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 6{0}A5 is 600A5
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,B):
** 6B is prime, and thus the only minimal prime in this family.
* Case (7,1):
** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes)
*** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1
**** None of 741, 7B1, 74B1 are primes.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,B):
** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes)
**** The smallest prime of the form 7{9}B is 7999B
**** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes)
**** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime)
* Case (8,1):
** 81 is prime, and thus the only minimal prime in this family.
* Case (8,5):
** 85 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** 87 is prime, and thus the only minimal prime in this family.
* Case (8,B):
** 8B is prime, and thus the only minimal prime in this family.
* Case (9,1):
** 91 is prime, and thus the only minimal prime in this family.
* Case (9,5):
** 95 is prime, and thus the only minimal prime in this family.
* Case (9,7):
** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes)
**** None of 947, 977, 997, 9477, 9977 are primes.
* Case (9,B):
** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes)
**** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes)
***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime)
***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime)
**** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes)
***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes)
****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime.
****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime)
****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime.
**** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B
***** None of 99B, 999B, 979B, 9799B, 9979B are primes.
* Case (A,1):
** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes)
**** For the A{A}{0}1 family:
***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form AA{0}1 is AA000001
***** The smallest prime of the form AAA{0}1 is AAA0001
***** The smallest prime of the form AAAA{0}1 is AAAA1
****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc.
**** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime.
* Case (A,5):
** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes)
*** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes)
**** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form A{0}65 is A00065
* Case (A,7):
** A7 is prime, and thus the only minimal prime in this family.
* Case (A,B):
** AB is prime, and thus the only minimal prime in this family.
* Case (B,1):
** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes)
**** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1
***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1
****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime)
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1
****** The smallest prime of the form B{A}01 is BAA01
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
**** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1
***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1
****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1
******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1
******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes.
****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1
******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime)
***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes)
****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B04{A}1 is B04A1
****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime)
**** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1
***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1
****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1
******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime.
******* The smallest prime of the form B{B}AA1 is BBBAA1
****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1
******* The smallest prime of the form B{B}AA1 is BBBAA1
**** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes)
***** The smallest prime of the form B{B}B1 is BBBB1
***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form B{B}BAA1 is BBBAA1
* Case (B,5):
** B5 is prime, and thus the only minimal prime in this family.
* Case (B,7):
** B7 is prime, and thus the only minimal prime in this family.
* Case (B,B):
** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes)
*** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B
**** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B
***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime.
***** For the B{0,B}9B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes)
******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B
******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 &gt; 1, 10^n+5 ≥ 10^1+5 = 15 &gt; 1, this factorization is nontrivial), thus cannot be prime.
***** For the B{0,B}99B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes)
******* The smallest prime of the form B{0}99B is B00099B
******* The smallest prime of the form B{B}99B is BBBBBB99B
***** For the B{0,B}999B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes)
******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes)
******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime)

== Examples of families which can be ruled out as contain no primes &gt; ''b'' ==

It is not known if this problem is solvable:

Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''?

It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime &gt; ''b'' or not. We use two different heuristic strategies to show that such families contain no primes &gt; ''b''.

In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;.

Examples of first strategy: (we can show that the corresponding numbers are &gt; all elements in ''S'', if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial)

* In base 10, all numbers of the form 4{6}9 are divisible by 7
* In base 6, all numbers of the form 4{0}1 are divisible by 5
* In base 15, all numbers of the form 9{6}8 are divisible by 11
* In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5}
* In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3}
* In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5}
* In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13}
* In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17}
* In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13}
* In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13}

Examples of second strategy: (we can show that both factors are &gt; 1, if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial)

* In base 9, all numbers of the form {1} factored as difference of squares
* In base 8, all numbers of the form 1{0}1 factored as sum of cubes
* In base 9, all numbers of the form 3{8} factored as difference of squares
* In base 16, all numbers of the form 8{F} factored as difference of squares
* In base 16, all numbers of the form {F}7 factored as difference of squares
* In base 9, all numbers of the form 3{1} factored as difference of squares
* In base 16, all numbers of the form {4}1 factored as difference of squares
* In base 16, all numbers of the form 1{5} factored as difference of squares
* In base 16, all numbers of the from {C}D factored as ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;
* In base 16, all numbers of the form B{4}1 factored as difference of squares

Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are &gt; all elements in S, and for the part of the second strategy, both factors are &gt; 1, if n makes corresponding numbers &gt; b, thus these factorizations are nontrivial)

* In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd
* In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even

== Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes &gt; ''b'' ==

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-powers factorization

=== 1{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 1{0}3 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== 1{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 1{0}5 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 5: Finite covering set {5}

=== 1{0}6 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== 1{0}7 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 7: Finite covering set {7}

=== 1{0}z ===

(none)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

(none)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {1} ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with ''r''&gt;1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2))

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' == 0 mod 2: Finite covering set {2}

=== 1{2} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 1{3} ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361)
* ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865)

=== 1{4} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization

=== 1{z} ===

(none)

=== 2{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 2{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {2}1 ===

* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 2{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 3{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {3}1 ===

* ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121)

=== 3{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 4{0}1 ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 4{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {4}1 ===

* ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225)

=== 4{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 5{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 5{z} ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}
* ''b'' = 6''m''&lt;sup&gt;2&lt;/sup&gt; with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014)

=== 6{0}1 ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== 6{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 7{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 7{z} ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993)
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Difference-of-cubes factorization

=== 8{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022)
* ''b'' = 467: Finite covering set {3, 5, 7, 19, 37}
* ''b'' = 722: Finite covering set {3, 5, 13, 73, 109}
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Sum-of-cubes factorization
* ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2

=== 8{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 9{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 9{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{0}1 ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{z} ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 307: Finite covering set {5, 11, 29}
* ''b'' = 901: Finite covering set {7, 11, 13, 19}

=== C{0}1 ===

* ''b'' == 1 mod 13: Finite covering set {13}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 296, 901: Finite covering set {7, 11, 13, 19}
* ''b'' = 562, 828, 900: Finite covering set {7, 13, 19}
* ''b'' = 563: Finite covering set {5, 7, 13, 19, 29}
* ''b'' = 597: Finite covering set {5, 13, 29}

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

(none)

=== {y}z ===

(none)

=== z{0}1 ===

(none)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3))
* ''b'' = 4''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt; (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}t ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 7: Finite covering set {7}

=== {z}u ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== {z}v ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 5: Finite covering set {5}

=== {z}w ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== {z}x ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== {z}y ===

* ''b'' == 0 mod 2: Finite covering set {2}

== Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) ==

Format: base (length)

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

(none)

=== 1{0}2 ===

(none)

=== 1{0}3 ===

(none)

=== 1{0}4 ===

53 (13403)

113 (10647)

=== 1{0}z ===

113 (20089)

123 (64371)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

(none)

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

208 (26682)

607 (11032)

828 (19659)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

201 (31276)

222 (52727)

227 (36323)

327 (135983)

425 (11231)

710 (24112)

717 (37508)

719 (13420)

=== {1} ===

152 (270217)

184 (16703)

200 (17807)

311 (36497)

326 (26713)

331 (25033)

371 (15527)

485 (99523)

629 (32233)

649 (43987)

670 (18617)

684 (22573)

691 (62903)

693 (41189)

731 (15427)

752 (32833)

872 (10093)

932 (20431)

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

(none)

=== 1{z} ===

107 (21911)

170 (166429)

278 (43909)

303 (40175)

383 (20957)

515 (58467)

522 (62289)

578 (129469)

590 (15527)

647 (21577)

662 (16591)

698 (127559)

704 (62035)

845 (39407)

938 (40423)

969 (24097)

989 (26869)

=== 2{0}1 ===

101 (192276)

206 (46206)

218 (333926)

236 (161230)

257 (12184)

305 (16808)

467 (126776)

578 (44166)

626 (174204)

695 (94626)

752 (26164)

788 (72918)

869 (49150)

887 (27772)

899 (15732)

932 (13644)

=== 2{z} ===

432 (16003)

=== 3{0}1 ===

912 (132174)

=== 3{z} ===

72 (1119850)

212 (34414)

218 (23050)

270 (89662)

303 (198358)

312 (51566)

422 (21738)

480 (93610)

513 (38032)

527 (46074)

566 (23874)

650 (498102)

686 (16584)

758 (15574)

783 (12508)

800 (33838)

921 (98668)

947 (10056)

=== 4{0}1 ===

107 (32587)

227 (13347)

257 (160423)

355 (10990)

410 (144079)

440 (56087)

452 (14155)

482 (30691)

542 (15983)

579 (67776)

608 (20707)

635 (11723)

650 (96223)

679 (69450)

737 (269303)

740 (58043)

789 (149140)

797 (468703)

920 (103687)

934 (101404)

962 (84235)

=== 4{z} ===

14 (19699)

68 (13575)

254 (15451)

800 (20509)

=== 5{0}1 ===

326 (400786)

350 (20392)

554 (10630)

662 (13390)

926 (40036)

=== 5{z} ===

258 (212135)

272 (148427)

299 (64898)

307 (26263)

354 (25566)

433 (283919)

635 (36163)

678 (40859)

692 (45447)

719 (20552)

768 (70214)

857 (23083)

867 (61411)

972 (36703)

=== 6{0}1 ===

108 (16318)

129 (16797)

409 (369833)

522 (52604)

587 (24120)

643 (164916)

762 (11152)

789 (27297)

986 (21634)

=== 6{z} ===

68 (25396)

332 (15222)

338 (42868)

362 (146342)

488 (33164)

566 (164828)

980 (50878)

986 (12506)

1016 (23336)

=== 7{0}1 ===

398 (17473)

1004 (54849)

=== 7{z} ===

97 (192336)

170 (15423)

194 (38361)

202 (155772)

282 (21413)

283 (164769)

332 (13205)

412 (29792)

560 (19905)

639 (10668)

655 (53009)

811 (31784)

814 (17366)

866 (108591)

908 (61797)

962 (31841)

992 (10605)

997 (15815)

=== 8{0}1 ===

23 (119216)

53 (227184)

158 (123476)

254 (67716)

320 (52004)

410 (279992)

425 (94662)

513 (19076)

518 (11768)

596 (148446)

641 (87702)

684 (23387)

695 (39626)

785 (900326)

788 (11408)

893 (86772)

908 (243440)

920 (107822)

962 (47222)

998 (81240)

1013 (43872)

=== 8{z} ===

138 (35686)

412 (12154)

788 (11326)

990 (23032)

=== 9{0}1 ===

248 (39511)

592 (96870)

=== 9{z} ===

431 (43574)

446 (152028)

458 (126262)

599 (11776)

846 (12781)

=== A{0}1 ===

173 (264235)

198 (47665)

311 (314807)

341 (106009)

449 (18507)

492 (42843)

605 (12395)

708 (17563)

710 (31039)

743 (285479)

744 (137056)

786 (68169)

800 (15105)

802 (149320)

879 (25004)

929 (13065)

977 (125873)

986 (48279)

1004 (10645)

=== A{z} ===

368 (10867)

488 (10231)

534 (80328)

662 (13307)

978 (14066)

=== B{0}1 ===

710 (15272)

740 (33520)

878 (227482)

=== B{z} ===

153 (21660)

186 (112718)

439 (18752)

593 (16064)

602 (36518)

707 (10573)

717 (67707)

=== C{0}1 ===

68 (656922)

219 (29231)

230 (94751)

312 (21163)

334 (83334)

353 (20262)

359 (61295)

457 (10024)

481 (45941)

501 (20140)

593 (42779)

600 (11242)

604 (17371)

641 (26422)

700 (91953)

887 (13961)

919 (45359)

923 (64365)

992 (10300)

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

(none)

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

38 (136212)

83 (21496)

113 (286644)

188 (13508)

401 (103670)

417 (21003)

458 (46900)

494 (21580)

518 (129372)

527 (65822)

602 (17644)

608 (36228)

638 (74528)

663 (47557)

723 (24536)

758 (50564)

833 (12220)

904 (13430)

938 (50008)

950 (16248)

=== z{0}1 ===

202 (46774)

251 (102979)

272 (16681)

297 (14314)

298 (60671)

326 (64757)

347 (69661)

363 (142877)

452 (71941)

543 (10042)

564 (38065)

634 (84823)

788 (13541)

869 (12289)

890 (37377)

953 (60995)

1004 (29685)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

53 (21942)

124 (16426)

175 (31626)

188 (22036)

316 (48538)

365 (25578)

373 (24006)

434 (10090)

530 (11086)

545 (12346)

560 (15072)

596 (12762)

701 (12576)

706 (10656)

821 (13536)

833 (17116)

966 (14820)

983 (11272)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}y ===

317 (13896)

== Bases 2≤''b''≤1024 which have these families as unsolved families ==

Unsolved families are families which are neither primes (&gt;''b'') found nor can be ruled out as contain no primes &gt; ''b''

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608)

1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000)

1{0}3: 646, 718, 998 (length limit: 2000)

1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000)

1{0}5

1{0}6

1{0}7

1{0}8

1{0}9

1{0}A

1{0}B

1{0}C

1{0}D

1{0}E

1{0}F

1{0}G

1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000)

1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000)

10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000)

11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000)

{1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000)

{1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000)

11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z})

{1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000)

{1}z

1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000)

1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995

1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000)

1{z}: 581, 992, 1019 (length limit: ≥100000)

2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000)

2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000)

2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000)

{2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000)

2{z}: 588, 972 (length limit: ≥100000)

3{0}1: 718 (length limit: ≥100000)

3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000)

3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000)

{3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000)

3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000)

4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000)

4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000)

{4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000)

4{z}: 338, 998 (length limit: ≥100000)

5{0}1: 308, 512, 824 (length limit: ≥100000)

5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000)

6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000)

6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000)

7{0}1: (none)

7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000)

8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000)

8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000)

9{0}1: 724, 884 (length limit: ≥100000)

9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000)

A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000)

A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000)

B{0}1: 560, 770, 968 (length limit: ≥100000)

B{z}: 263, 615, 912, 978 (length limit: ≥100000)

C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000)

C{z}

D{0}1

D{z}

E{0}1

E{z}

F{0}1

F{z}

G{0}1

{&amp;#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000)

{&amp;#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143)

&amp;#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000)

${&amp;#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2)

x{z}

y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000)

z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000)

{y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000)

{z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326)

zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z})

{z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000)

{z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1)

{z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000)

{z}k

{z}l

{z}m

{z}n

{z}o

{z}p

{z}q

{z}r

{z}s

{z}t

{z}u

{z}v

{z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000)

{z}x: (none)

{z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000)

== List of lengths for quasi-minimal primes in some simple families ==

[https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024]

NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases &lt;=7, family {z}x in bases &lt;=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime &gt; base) in this base)

RC: this family can be proven to only contain composite numbers (only count numbers &gt; base)

unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers &gt; base)

Background color: red for title (bases or families), green for length &gt; 10000, orange for 2500 &lt; length ≤ 10000, white for length ≤ 2500, cyan for &quot;RC&quot;, pink for &quot;NB&quot;, yellow for &quot;unknown&quot;.

Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families.

== References ==

* [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem]

* [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are &gt; ''b'' for bases ''b''≤16]

* [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary]

* [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia]

* [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime&gt;base is not required) in The Prime Puzzles &amp; Problems Connection]

* [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles &amp; Problems Connection]

* [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16]

* [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime&gt;base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29]

* [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime&gt;base is not required, search limits of lengths: 10000 for all b]

* [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime&gt;base is not required]

* [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime&gt;base is not required]

* [http://www.prothsearch.com/sierp.html the Sierpinski problem]

* [http://www.prothsearch.com/rieselprob.html the Riesel problem]

* [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem]

* [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030]

* [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100]

* [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem]

* [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024]

* [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project]

* [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project]

* [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count]

* [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty]

* [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers]

* [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares]

* [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software]

* [https://sourceforge.net/projects/openpfgw/ pfgw software]

* [http://jpenne.free.fr/index2.html llr software]

* [http://www.ellipsa.eu/public/primo/primo.html PRIMO software]

* [https://primes.utm.edu/prove/index.html website for primality proving]

* [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!]

* [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime&gt;base is not required]

* [https://primes.utm.edu/primes/lists/all.txt top proven primes]

* [http://www.primenumbers.net/prptop/prptop.php top PRPs]

* [http://factordb.com online factor database, including many primes which are minimal primes in a small base]</text>
      <sha1>1xpcbx74lyj6qmjavc52w9vnflw9mus</sha1>
    </revision>
    <revision>
      <id>2412070</id>
      <parentid>2412028</parentid>
      <timestamp>2022-08-05T07:58:27Z</timestamp>
      <contributor>
        <ip>2402:7500:901:2B90:A16F:2B1F:1CD3:7BC3</ip>
      </contributor>
      <comment>/* Unsolved families */ B0(H^n)6H = 4637 * (1^n) and cannot be prime</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1961623" xml:space="preserve">A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime &gt; ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime &gt; 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 &gt; 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 &gt; 10 by changing the order.

(using A−Z to represent digit values 10 to 35)

For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')).

I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E&lt;sub&gt;800873&lt;/sub&gt;, and base 36 has quasi-minimal prime P&lt;sub&gt;81993&lt;/sub&gt;SZ).

Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to:

* Prove that all elements in ''S'', when read as base ''b'' representation, are primes &gt; ''b''.

* Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are &gt; ''b'', are composite.

* Prove that all primes &gt; ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are &gt; ''b'', are composite).

e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to:

* Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes &gt; 10.

* Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are &gt; 10 are composite.

* Prove that all primes &gt; 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers &gt; 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite).

==Condensed table==

{|class=wikitable
|''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form of largest known quasi-minimal prime base ''b''
|-
|2||1||11||2||3
|-
|3||3||111||3||13
|-
|4||5||221||3||41
|-
|5||22||10&lt;sub&gt;93&lt;/sub&gt;13||96||5&lt;sup&gt;95&lt;/sup&gt;+8
|-
|6||11||40041||5||5209
|-
|7||71||3&lt;sub&gt;16&lt;/sub&gt;1||17||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2
|-
|8||75||4&lt;sub&gt;220&lt;/sub&gt;7||221||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||1161||3×9&lt;sup&gt;1160&lt;/sup&gt;+10
|-
|10||77||50&lt;sub&gt;28&lt;/sub&gt;27||31||5×10&lt;sup&gt;30&lt;/sup&gt;+27
|-
|11&lt;sup&gt;*&lt;/sup&gt;||1068||57&lt;sub&gt;62668&lt;/sub&gt;||62669||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10
|-
|12||106||40&lt;sub&gt;39&lt;/sub&gt;77||42||4×12&lt;sup&gt;41&lt;/sup&gt;+91
|-
|13&lt;sup&gt;*&lt;/sup&gt;||3195~3197||80&lt;sub&gt;32017&lt;/sub&gt;111||32021||8×13&lt;sup&gt;32020&lt;/sup&gt;+183
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||19699||5×14&lt;sup&gt;19698&lt;/sup&gt;−1
|-
|15||1284||7&lt;sub&gt;155&lt;/sub&gt;97||157||(15&lt;sup&gt;157&lt;/sup&gt;+59)/2
|-
|16&lt;sup&gt;*&lt;/sup&gt;||2346~2347||4&lt;sub&gt;72785&lt;/sub&gt;DD||72787||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15
|-
|17&lt;sup&gt;*&lt;/sup&gt;||10408~10428||F70&lt;sub&gt;186767&lt;/sub&gt;1||186770||262×17&lt;sup&gt;186768&lt;/sup&gt;+1
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||6271||12×18&lt;sup&gt;6270&lt;/sup&gt;+221
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||6271||16×20&lt;sup&gt;6270&lt;/sup&gt;+13
|-
|21&lt;sup&gt;*&lt;/sup&gt;||13375~13396||CF&lt;sub&gt;479147&lt;/sub&gt;0K||479150||(51×21&lt;sup&gt;479149&lt;/sup&gt;−1243)/4
|-
|22&lt;sup&gt;*&lt;/sup&gt;||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||22003||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||8134||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49
|-
|30&lt;sup&gt;*&lt;/sup&gt;||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||34206||25×30&lt;sup&gt;34205&lt;/sup&gt;−1
|-
|36&lt;sup&gt;*&lt;/sup&gt;||35257~35263||P&lt;sub&gt;81993&lt;/sub&gt;SZ||81995||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7
|}

&lt;sup&gt;*&lt;/sup&gt; Data assumes the primality of the [[w:probable prime|probable prime]]s.

Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes &gt; 10&lt;sup&gt;25000&lt;/sup&gt;), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below.

Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists))

Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists))

==Data for quasi-minimal primes==

===Base 2===

11

===Base 3===

12, 21, 111

===Base 4===

11, 13, 23, 31, 221

===Base 5===

12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013

===Base 6===

11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041

===Base 7===

14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331

===Base 8===

13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447

===Base 9===

12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011

===Base 10===

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027

===Base 11===

12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, 2888882883, 2888888883, 3555555509, 3577777077, 3700000001, 4000000005, 40000005AA, 5377777707, 5555505553, 555555580A, 600000A999, 7000100047, 7000704777, 7007777107, 7057777777, 7070007774, 7077707774, 7077777074, 7100000704, 7470000041, 7701000004, 7707077774, 7770707774, 7777707074, 8888822883, 9555555503, 9900000979, 9999770007, A000144444, A900000066, A999999971, 10000000477, 33333333337, 44444444447, 44444444777, 55A55555552, 60000008883, 68888888306, 68888888883, 70000003999, 70000007717, 70004777777, 70477777777, 77007770004, 77700000477, 77707000704, 77707770074, 77707777774, 77777077774, 77777770004, 83000000006, 97000000999, A0000000001, A0014444444, A4777777771, 100000000057, 305007777777, 305777777777, 370000000007, 377777770007, 377777777107, 700000007474, 707077000074, 707077777774, 707777777717, 770000010004, 771007000007, 777070700004, 777700000704, A95555555555, A99999777777, 1000000003007, 40000000A0041, 58A5555555555, 7004400000004, 7700000000104, 7707000007047, 7707707000004, 7777007000004, 7777700000004, 7777770077704, 7777777710077, 9977777777717, A000000014444, A044444444441, A144444444411, 40000000000401, 45557777777777, 4555AAAAAAAAAA, 59077777777777, 70007777777771, 70077070000074, 70700000004777, 77000007700704, 77700000700047, 77777777770704, 88888888830006, 90900000000799, A0000044444441, 300000000005777, 302000000000002, 55555555A555552, 700000000000174, 770000000000474, 771700000000007, 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7707777777777777777704, 7900000000000000000005, 9777777777777777777707, A999999999999999999999, 10000000000000000000747, 970000000000000000000777, 999900000000000000000007, 3577777777777777777777777, 5555555555555555555555A52, 7000000000000000000777771, 7000000000000000077777771, AAAAAAAAAAAAAAAAAAA000058, 10000000000000000000000044, 77700000000000000000000008, 500777777777777777777777777, 777777777777777777777770774, 855555555555555555555055555, A44444444444444444444444441, 1500000000000000000000000007, 40000000000000000000000000041, 440000000000000000000000000001, 70000000000000000000000000007771, 999999999999999999999999999999991, 95555555555555555555555555555555553, 1900000000000000000000000000000000001, 7777777777777777777777777777777777474, 7777777777777777777777777777777777704, 10000000000000000000000000000000000000307, 50777777777777777777777777777777777777707, 475777777777777777777777777777777777777777, 555555555555555555555555555555555555555A2A, 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===Base 12===

11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077

===Base 13===

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===Base 14===

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===Base 15===

12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797

===Base 16===

11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD

===Base 17===

12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 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====Additional known quasi-minimal primes (not necessarily the next)====

570&lt;sub&gt;51310&lt;/sub&gt;1, 49&lt;sub&gt;111333&lt;/sub&gt;, 970&lt;sub&gt;166047&lt;/sub&gt;1, F70&lt;sub&gt;186767&lt;/sub&gt;1

===Base 18===

11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, 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===Base 20===

13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, GGJ0J, GGJJJ, GJ00J, GJGJJ, GJJJ7, H00EH, H024B, H04B1, H0E4B, H0F41, H0H11, H0HEH, H4E0B, H6I11, HAAA3, HAAG3, HAGA3, HB44B, HBBBB, HBHHH, HGGA3, HGGG3, HH001, HH061, HH3AH, HH6F1, HHG63, HHIA3, HI011, HIII3, I0029, I02I9, I085H, I0A41, I0F29, I2229, I2929, I2I29, I2I99, I2II9, I33A3, I3EHH, I422B, I424B, I4EEB, I4IEB, I80G1, I9IA9, IA099, IA309, IA841, IA939, IA9A9, IAF09, IAF99, IAI99, IAIA9, IB0A1, IB4IB, IB54B, IBB4B, IBBIB, IBE0H, IBH0H, IC929, ICC29, IF841, IFAG1, IGA81, IHB4B, II0A9, II42B, II44B, IIB5B, III4B, IIIA3, IIIIB, J00G1, J00JB, J0601, J06D1, J06I1, J0861, J08B1, J0B01, J0CGJ, J0E0B, J0GA1, J0GG1, J0I2B, J0J2B, J0JIB, J55CJ, J60A1, J60G1, J6D81, JA777, JA7A7, JAJG1, JAJJ1, JAJJ7, JB00B, JB08B, JC00J, JC0GJ, JEECJ, JGGGJ, JGGJJ, JJ0B1, JJ0JB, JJ55J, JJAG1, JJAJ1, JJC0J, JJE0B, JJEEB, JJGG1, JJGGJ, JJGJJ, JJJ01, JJJEB, JJJJB, 17555J, 175EEJ, 1E0007, 1JE55J, 1JJJ5J, 20005H, 2000HH, 222I29, 22E889, 22EE89, 24222B, 2999I9, 29III9, 2A000H, 2B0I0H, 2I2999, 2I9I29, 2III29, 333AF3, 336IA3, 36A3F3, 36I3A3, 388E8H, 3AF333, 3EHHHH, 3HEHHH, 3HHEHH, 40008B, 40054B, 40405B, 40448B, 415E5J, 44440B, 44452B, 44524B, 44E08B, 45444B, 4AAAE7, 4AEEE7, 4E4447, 4GEEE7, 4I544B, 4IEEEB, 500EFJ, 500FCJ, 5055FJ, 50AFFJ, 50EEFJ, 50EF5J, 50F5FJ, 50FFCJ, 52224B, 54EEEB, 5558AH, 555CAJ, 5585AH, 55F55J, 55FFFJ, 5888AH, 588A5H, 5E555J, 5F055J, 5F05CJ, 5F5F5J, 5FFFCJ, 5JEEEB, 60006D, 6000F1, 6000G3, 6001G1, 600A63, 600G1D, 600G6D, 603A33, 606G6D, 608001, 608F89, 608IF9, 60A333, 60A363, 60AAA3, 60CCCD, 60D04D, 60D0DD, 60F8I9, 60FF89, 60GCG3, 636AF3, 63A333, 63A3F3, 63AFF3, 63IIA3, 64AAAD, 660089, 660A33, 660DDD, 660F89, 662289, 666A33, 666D4D, 66A6A3, 66AF33, 66AF63, 66D0DD, 66FF89, 6A33F3, 6AFF33, 6AG6A3, 6AGA63, 6D000D, 6D004D, 6D0D4D, 6F6689, 7000G7, 7060DD, 70AAAD, 70D0CD, 733C33, 7366C3, 73AAA3, 7600DD, 760CDD, 766333, 76664D, 766FC3, 76CCDD, 76DD4D, 770G07, 771JEJ, 7771EJ, 777481, 7777I1, 777F81, 77C7JJ, 77EC77, 77EEC7, 77EJJJ, 77GCG7, 77JECJ, 7A7E77, 7AAAG3, 7AAF63, 7AEEE7, 7C7GG7, 7CCCAD, 7CCCG7, 7CGJJJ, 7D000D, 7J000D, 7JCGGJ, 80005B, 8000G1, 8000I1, 800GA1, 800I01, 80GGA1, 84405B, 84454B, 888EAH, 88E8AH, 8A4441, 8B8EAH, 8E8IF9, 8I0AG1, 902229, 909A09, 909C29, 909FA9, 90IC29, 9222I9, 92I999, 944EE7, 94AEE7, 977GG7, 97AEE7, 97GGG7, 990I29, 999I29, 99I299, 99IIA9, 9A9447, 9A9AF9, 9A9FF9, 9CII29, 9EEEE7, 9FA9A9, 9I9C29, 9I9I29, 9ICI29, 9IIIA9, A1555J, A1E55J, A333C9, A336A3, A33889, A33F63, A36663, A444E7, A4AAE7, A555FJ, A666A3, A6AAA3, A7A7E7, A7AE77, A944E7, A9A9F9, A9AAA9, AA3389, AA9AF9, AAA099, AAA3C9, AAA7E7, AAA939, AAACG9, AAAFF9, AAAI99, AAAIC9, AAE4E7, AAG9A9, AAGAA3, ADE55J, AF6663, AF9FF9, AGA963, AGG6A3, AH4441, AI000H, AI99C9, AI9AA9, AII099, B0004B, B0054B, B0080B, B00H0H, B04IIB, B05B8B, B0A6I1, B0BB8B, B0BIBB, B0E0IH, B0HA0H, B0I0HH, B0I4IB, B0IIBB, B888AH, BB058B, BB05BB, BB080B, BB0I4B, BBB04B, BBB08B, BBIB8B, BE88AH, BEHA0H, BEHHAH, BHEA0H, BHEHAH, BHHA0H, BIE00H, C007A7, C00GG7, C00JA7, C00JAJ, C00JC7, C0C7A7, C0JAC7, C0JJC7, C3G999, CAJJ77, CC0629, CC0929, CCC007, CCCCC7, CCCCG3, CCCGG7, CCCI29, CCI029, CD4GGJ, CE0009, CEFF69, CEFFF9, CEI609, CII029, CIII29, CJACC7, CJAJC7, CJJJA7, D000D7, D000J7, D004CD, D0D007, DCCDC7, DDD04D, DDDAI1, DEJJJJ, DG000D, DG5E5J, DGGG5J, DJ00CJ, DJEEE7, DJJJ5J, E00097, E00A0H, E00AA9, E02289, E028I9, E029I9, E055AJ, E0774J, E0777J, E07JCJ, E0AF5J, E0BI8B, E0E4CJ, E0EA5J, E0ECJJ, E0FFF9, E0H0HH, E0HA0H, E0IF29, E0J5AJ, E0JJAJ, E29299, E444E7, E4EEE7, E4IIIB, E66289, E66629, E66909, E66F69, E69609, E69999, E7774J, E77C77, E7C7C7, E8B8AH, E90029, E90229, E90909, E90FF9, E92299, E94EE7, E97EE7, E990F9, E99I09, E9F669, E9I029, EA8AF9, EAAAI9, EAIAA9, EAJJJJ, EBIBIB, EC7CC7, EE08F9, EE408B, EE8IF9, EEAF5J, EEE84B, EEEC07, EEEC77, EEEE0B, EEF55J, EEFFFJ, EEJ5AJ, EFE009, EFEF09, EFF669, EFFFF9, EGGGG9, EIF669, F004AJ, F00A41, F00JA1, F04481, F055CJ, F0A15J, F0EI09, F0G01J, F0J0GJ, F0J0JJ, F0JJ5J, F14441, F68F89, F68IF9, F800G1, F8E8I9, F99299, FA0009, FA1JJJ, FAAIA9, FAII09, FAII99, FCC929, FCE229, FCE609, FCEF29, FE68F9, FE8FA9, FF0E09, FF68F9, FF9FA9, FFC4AJ, FFC929, FFCE09, FFE009, FFEE89, FFEF09, FFFA99, FFFFFJ, FFFGJJ, FG1JJJ, FI99A9, FIA999, FIAAA9, FIAII9, FIIA09, G00093, G000EJ, G000G9, G00781, G007G3, G00C09, G07GC3, G09033, G0C903, G0CGG3, G0G903, G0G933, G0GCC3, G0GGC9, G30003, G36003, G5000J, G5E0EJ, G60303, G6AGG3, G6GAA3, G7AAA3, G7G363, G90AA9, G90C63, G9AAA9, GAAAA3, GAAGG3, GC0003, GCCC93, GCCCC3, GCG903, GDEEE7, GE0007, GG0CG3, GGA8G1, GGC6G3, GGCCG3, GGCGC9, GGG0I9, GGG363, GGG6G3, GGG75J, GGG8G1, GGG963, GGGAA3, GGGCG3, GGGCG9, H044EB, H0BIBB, H0E00B, H0F011, H0IIEB, H2444B, H400EB, H44IEB, H4EEEB, HBB4IB, HE000B, HEE40B, HEEE4B, HEHHAH, HF0011, HF4441, HH3HEH, HHGAA3, HHGAG3, HHHAG3, HHHEAH, HI0001, HIBIBB, HIEBBB, I0000H, I000A9, I002HH, I00A09, I00A99, I00AG1, I00AI9, I00H3H, I02999, I03HEH, I09AF9, I0A009, I0E0HH, I0I299, I0IIA9, I4445B, I4I45B, I90I29, I99029, I99A09, I9I029, IA3999, IB000H, IB00EH, IB0HEH, IB5B8B, IF2999, II0929, II92I9, II99A9, II9C29, IIAFF9, IIF299, III5BB, III8BB, J0000B, J00081, J000IB, J0010D, J006A1, J00BI1, J00I8B, J00IA1, J05EEJ, J06G81, J0C0JJ, J0E0CJ, J0EEEB, J0J0EB, J5EAEJ, J5EEAJ, J608I1, JE505J, JEEEEB, JJ0001, JJ0JCJ, JJ0JGJ, JJAJC7, JJJAE7, JJJBI1, JJJI8B, 14GGGG1, 1J5555J, 1JGGGG1, 205555H, 20I000H, 222222B, 2929999, 2BI000H, 3333A33, 3333G09, 333A3A3, 333AAA3, 336AAA3, 336AF33, 33II3A3, 363AF33, 3GIIII9, 400445B, 404454B, 404800B, 415555J, 440045B, 440080B, 442222B, 444004B, 444422B, 444444B, 4444G47, 444GGG7, 444I4IB, 44AAA47, 44I44IB, 44I4IIB, 4AAAAA7, 4EEE4E7, 4I4454B, 5000AFJ, 525555H, 555552H, 555555J, 555585H, 58BBBBB, 5BB8BBB, 5E0E55J, 5EBBBBB, 60003A3, 60008I9, 6000DDD, 6006DDD, 60080A1, 60080I1, 600A3A3, 600D8I1, 600F841, 60CCDDD, 60CGGG3, 60DDD0D, 60DDDDD, 60GGG63, 60GGGC3, 66600A3, 66603A3, 66608I9, 6666089, 66666A3, 6666C29, 6666F89, 666AFF3, 66DDDDD, 6800001, 6AGGGG3, 6CCCC29, 6G6AAA3, 6GAGGG3, 6GCCC63, 6GGG633, 700000D, 70000CD, 7000EC7, 70070C7, 700EEC7, 7070EC7, 70EEEC7, 76666CD, 77333A3, 7733A33, 775EE5J, 777A777, 777CGJJ, 777GE5J, 77A7777, 77JCJJJ, 77JJJCJ, 77JJJJJ, 7A77777, 7A777A7, 7AAAA63, 7AAAAAD, 7C00007, 7E77JCJ, 7EE7JCJ, 7EEC007, 7EEEEC7, 7G5E55J, 7GE555J, 7GJJJJJ, 7J6666D, 9000029, 9000299, 9009I29, 900I2I9, 90FAAA9, 90I0299, 90I0I29, 90I9929, 90II029, 90II299, 90II2I9, 94444E7, 97777E7, 9902I99, 99902I9, 9992929, 9992999, 9999929, 99I0C29, 99I9029, 99I92I9, 99III29, 9AAAF09, 9AFFFF9, 9I0AAA9, 9IAAA09, 9II0I29, 9II9929, 9III029, 9III2I9, 9IIIC29, A5F555J, A777E77, A994447, A9AAF99, AA00089, AA6A6A3, AAA0009, AAA3999, AAA9A99, AAA9AA9, AAAAA47, AAAAA77, AAAAA99, AAAAG99, AAAAIA9, AAAD447, AAAGAC9, AAGAAAD, AAGAAC9, AF00009, AGAAA99, AGAGG63, AGGGA63, AI0C999, AJ777A7, B000B8B, B000HEH, B000I0H, B00I00H, B0B805B, B0HHHEH, B0IHE0H, B0III8B, B8B000B, BB8BBBB, BBB8B0B, BEH0HHH, BHHHEHH, C00000J, C000C07, C066629, C0AJJJJ, C0C0A47, C0G000J, C0JAJJJ, C0JJAJJ, C0JJJJJ, C6CCG33, CC004A7, CC00A47, CC6CG33, CC92229, CCCC7G7, CCCGGG3, CE66609, CEF6669, D00007D, D00007J, D00071J, D000C0J, D000I01, D00700D, D00C007, D00C0C7, D00G00D, D0C00C7, D0CDCC7, D444447, D7JJJJJ, DC0000J, DDD7DDD, DDDCCC7, DJ0000J, DJ000EJ, DJJJJ0J, DJJJJJJ, E000009, E00084B, E000889, E000909, E000HAH, E008AF9, E04000B, E0AA8F9, E0CJJJJ, E0HHHHH, E40400B, E6660F9, E666609, E669969, E699669, E7CCC07, E996669, E99I669, E9FAAA9, EB8BBBB, EBBII8B, EBH0HHH, EE092I9, EE44IIB, EE4E4E7, EEBB8BB, EEC0007, EEE0IBB, EEE0IIB, EEEBB8B, EEEE44B, EEJCJJJ, EF00F09, EF0555J, EFAAAA9, EFF0009, EFF00F9, EFF60F9, EI0AAA9, EI660F9, EIIBB8B, EJ0CJJJ, EJ5055J, EJJ0CJJ, F0E00F9, F0E88I9, F0JJJJ1, F299929, F4G555J, F5555CJ, F5555FJ, F6668I9, F8GGGG1, F929999, F999C29, FA84441, FAAA9A9, FCE00F9, FCEF009, FCFEE29, FE00F09, FEE2889, FEF0009, FFAA9A9, FFCEE29, FFFF4AJ, FIII9A9, FIIIAI9, FJJJJ81, G000009, G0000J7, G000303, G000AA9, G000CC3, G000GG3, G003003, G0090C3, G00CG03, G00D007, G00G0G3, G00GG03, G00GG93, G080001, G0G00I9, G0G09C3, G0G0C93, G0G0GG9, G0GG093, G0GGC63, G0GGGG9, G44GG47, G6AAGA3, G999999, GAA9AA9, GCCCGG9, GCGGCC9, GCGGG03, GEE4447, GEEEE47, GG0C007, GG0GC09, GG0GGG3, GG6AAA3, GG6AGA3, GG6GAG3, GGEEE47, GGG00C9, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 21===

12, 18, 1A, 1G, 1K, 21, 25, 2B, 2H, 2J, 34, 38, 3A, 3G, 3K, 45, 4D, 4H, 4J, 52, 54, 58, 61, 65, 6B, 6D, 72, 74, 7A, 7G, 7K, 85, 8B, 8D, 92, 94, 98, 9A, A1, AD, AH, AJ, B2, B8, BA, BK, C5, CB, CH, CJ, D4, D8, DA, DK, ED, EH, EJ, F2, FG, G1, GB, GD, GH, H2, HA, HG, I1, I5, IB, IJ, J2, JA, JK, K1, KB, KD, KJ, 111, 115, 11H, 135, 13H, 13J, 151, 15B, 15H, 164, 175, 17B, 17D, 17J, 191, 19B, 19D, 19H, 1B1, 1B5, 1BB, 1BJ, 1D5, 1DD, 1DJ, 1E4, 1F1, 1F5, 1FD, 1FH, 1HB, 1HD, 1I4, 1JD, 1JH, 1JJ, 22D, 232, 29G, 29K, 2DG, 2F4, 2FK, 31H, 322, 33D, 351, 355, 35B, 35J, 362, 371, 37B, 37D, 37H, 37J, 39B, 39J, 3B5, 3BD, 3BH, 3D1, 3D5, 3DB, 3DD, 3DH, 3E2, 3FJ, 3HD, 3HH, 3HJ, 3J1, 3JB, 3JJ, 414, 481, 49K, 4B4, 4BG, 4C1, 4EB, 4F4, 4F8, 4FA, 4FK, 50G, 51B, 51D, 51H, 531, 535, 53D, 53J, 551, 56A, 56G, 56K, 575, 57J, 595, 59H, 5B1, 5B5, 5BB, 5CA, 5CG, 5CK, 5F1, 5FB, 5FJ, 5GA, 5GG, 5HH, 5IA, 5J5, 5JD, 5JH, 60H, 632, 66H, 66J, 678, 68J, 69G, 6F8, 6FA, 6GJ, 6H8, 6HK, 6IH, 6J4, 6JG, 6KH, 711, 71B, 71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, GGJ, GJ4, GK5, H0K, H15, H1B, H1J, H31, H3D, H3H, H48, H51, H55, H5J, H6K, H75, H84, H88, H91, H95, H9D, H9H, HBD, HC4, HC8, HDJ, HF5, HFB, HFH, HHD, HHJ, HI4, HI8, HJ5, HJB, HKK, I0D, I2D, I6H, I78, I8H, I9K, IFA, IFK, IID, J08, J1J, J31, J35, J3J, J44, J5H, J68, J6G, J71, J7B, J7D, J7H, J8G, J95, J9D, JBD, JBH, JBJ, JCG, JDB, JDH, JE4, JE8, JEG, JF5, JFD, JFJ, JG4, JGG, JH1, JH5, JHB, JHH, JI4, JJ1, JJ5, K0H, K5G, K6H, K9K, KG5, KH4, KKH, 101B, 103D, 1044, 1055, 10DH, 10FB, 10H1, 10H5, 10J1, 110J, 11CD, 11EB, 11ID, 130D, 13C1, 13EB, 141B, 1444, 1471, 14FB, 1505, 15CD, 167H, 16FJ, 16HH, 176H, 17C1, 17IH, 190J, 1BF4, 1BID, 1C04, 1C3D, 1CD1, 1D6H, 1DE1, 1DEB, 1E1B, 1E71, 1E95, 1EFB, 1F0B, 1F4B, 1FB4, 1H01, 1H05, 1H41, 1H6H, 1H6J, 1HC1, 1HE1, 1HF4, 1IDH, 1IHH, 1JE5, 1JF4, 20CD, 20D2, 20ID, 2278, 22F8, 239D, 23FD, 2478, 2708, 27FD, 293D, 29DD, 2A78, 2C0D, 2CF8, 2F0A, 2F9D, 2FAA, 2FDD, 2GFA, 2IF8, 3015, 301D, 301J, 3031, 3035, 305D, 30F1, 30FB, 30FD, 30HB, 30I2, 3105, 31C1, 329D, 330H, 336J, 33E1, 33EB, 33IH, 35CD, 363H, 390H, 39E5, 39ID, 3B0J, 3B6J, 3BC1, 3BEB, 3CC2, 3CF1, 3CFD, 3D32, 3E3B, 3E91, 3EF1, 3EFB, 3F05, 3FC1, 3FE1, 3FEB, 3H01, 3H0B, 3HC1, 3I3H, 3I9D, 404B, 4078, 40E1, 411B, 4131, 4171, 41FB, 4331, 4391, 43BB, 43FB, 43I2, 4432, 444B, 44E1, 477B, 4931, 4971, 499B, 49CG, 4A32, 4A4B, 4A78, 4A9G, 4B1B, 4B31, 4B3B, 4BFB, 4C78, 4E32, 4F1B, 4FB1, 5011, 509D, 50BD, 50DB, 50DD, 50EK, 50FD, 50H1, 50HJ, 50IK, 50JJ, 5101, 5105, 51E1, 530B, 53EB, 53IH, 550D, 550H, 55AB, 55BG, 55DG, 55E5, 55KH, 563H, 56JJ, 5701, 570H, 57EB, 57ID, 57IH, 590D, 59C1, 59CD, 5A3B, 5A7B, 5AGK, 5AIG, 5AKG, 5B0D, 5B9G, 5BCD, 5BGJ, 5BIH, 5C91, 5CD1, 5D0D, 5D0J, 5D6H, 5D6J, 5DC1, 5DG5, 5DIH, 5DJG, 5E0A, 5E11, 5E3B, 5E71, 5EAK, 5EEA, 5EKK, 5F0H, 5F9K, 5G0K, 5G9J, 5GF5, 5GIK, 5H0B, 5H5K, 5H9K, 5HE5, 5HID, 5I3H, 5I7D, 5IKG, 5J6J, 5JE1, 5KAG, 5KEK, 5KFH, 5KH5, 60FK, 63FH, 67JH, 69JH, 6E9K, 6F0K, 6F64, 6FJH, 6H04, 6HE4, 6HFJ, 6IF4, 6J9H, 6J9J, 7015, 7031, 703B, 7055, 705J, 7088, 709B, 709D, 709H, 70DJ, 70I8, 710J, 716J, 730B, 73C1, 73E5, 7501, 769H, 76FJ, 76JJ, 770D, 77E1, 77E5, 77EB, 7888, 78FH, 78H1, 78JH, 790H, 79CD, 79IH, 7B0J, 7B6H, 7C31, 7C91, 7C9D, 7E35, 7E75, 7E91, 7EF1, 7F01, 7F78, 7FCD, 7FH8, 7FIH, 7FJ8, 7H81, 7HC1, 7HF8, 7J6J, 7J78, 7JC1, 7JF8, 7JH8, 7JJ8, 809K, 80H8, 81H1, 81HH, 81J1, 839H, 83FH, 869K, 871H, 8731, 8771, 8788, 879J, 87F1, 87FJ, 87JH, 888H, 8911, 8931, 896K, 8A78, 8A9K, 8CC1, 8E9K, 8F3H, 8F9H, 8FF1, 8FHH, 8FJH, 8G9K, 8GJ8, 8H3J, 8H7H, 8H8K, 8HEK, 8HF1, 8HFJ, 8HH1, 8HJJ, 8HK8, 8J7J, 8J91, 8JC8, 8JFH, 8JJH, 8K78, 8K8H, 8KH8, 901J, 9035, 903B, 9071, 90CG, 90GG, 90JD, 9101, 91E5, 9301, 93E1, 93EB, 95CD, 95DG, 95FK, 967J, 969H, 96HH, 96HJ, 96JJ, 9705, 97C1, 9905, 995K, 999G, 99CD, 99EB, 99K5, 9B5G, 9BE5, 9BID, 9C0G, 9C11, 9C71, 9C9D, 9CEG, 9CGG, 9D05, 9D5G, 9DDG, 9DE5, 9E35, 9E3B, 9E71, 9EB5, 9EEG, 9EF1, 9EFB, 9F0J, 9F6H, 9F9K, 9FC1, 9FIH, 9FKH, 9G35, 9G95, 9GIK, 9GJJ, 9GKG, 9H0H, 9H5K, 9H6H, 9HKH, 9I6K, 9I9H, 9IDD, 9J6J, 9JEB, 9JGJ, 9JID, 9JJG, 9K35, 9K95, 9KEG, A33B, A3F5, A432, A478, A53B, A5IK, A788, A795, A878, A975, A99B, A9CG, A9GG, AAAB, AB44, AB55, ABFB, AC32, AC78, ACFA, AE32, AE5A, AE9K, AEEB, AEFA, AEFK, AF75, AFCA, AFCK, AFEA, AFFB, AG9G, AGFA, AI32, B064, B091, B09H, B0B1, B0D5, B0DH, B0EG, B0FB, B0FH, B0H5, B0HD, B114, B141, B14B, B1B4, B1E1, B411, B431, B444, B4B1, B59G, B5G5, B5JG, B67H, B67J, B69H, B6FH, B6FJ, B6HH, B6JH, B96H, B9DG, B9G5, BB14, BBB4, BC1D, BC31, BCDD, BD05, BD0H, BD6H, BDC1, BDE5, BDIH, BE0G, BE31, BEBB, BEE4, BEIG, BF0B, BF0H, BF6J, BFEB, BFIH, BG0G, BG55, BG95, BGG4, BGGG, BGIG, BGJ5, BH0B, BHEB, BIEG, BIGG, BIHH, BJ01, BJ0J, BJC1, BJF4, BJID, C041, C081, C09G, C0E1, C0FK, C104, C131, C171, C1C4, C1D1, C2F8, C2FA, C39D, C3F1, C478, C6FK, C7FD, C8C1, C90G, C911, C971, C99D, CAFK, CC78, CC81, CCE1, CCF8, CD02, CD31, CDFD, CE01, CE32, CF0A, CF48, CF64, CF68, CF6A, CF91, CFAK, CFDD, CFEA, CFEK, CIF4, CK32, CK9G, CKFA, D01D, D01H, D07B, D0DB, D0F1, D0FD, D0HD, D0HH, D10D, D10H, D16H, D1EB, D1IH, D23D, D3E1, D56H, D63J, D6CG, D6IG, D70B, D70D, D70J, D99G, D9GJ, D9ID, D9JG, DBC1, DBCD, DBE1, DBJG, DC11, DC9D, DCF1, DCFD, DCGG, DCIG, DD0H, DD6J, DDGJ, DDID, DDJG, DE31, DE75, DFC1, DFID, DFIH, DG22, DG6G, DG75, DG9J, DGCG, DGIG, DH6H, DHE5, DI6G, DICG, DIDH, DIFD, DIFH, DIGG, DJCD, DJDG, DJG5, E032, E05A, E0EB, E11B, E2F8, E31B, E331, E375, E395, E3C2, E3F1, E3F5, E3FB, E3I2, E53B, E555, E591, E59B, E5EG, E6F4, E755, E775, E89K, E915, E935, E93B, E96K, E9FB, EA32, EAEB, EAFK, EB44, EB95, ECFA, ECFK, EE32, EEEB, EEFK, EF08, EF35, EF48, EF55, EF64, EF68, EF75, EF95, EFB1, EFCA, EFCK, EFEA, EFIA, EFKK, EGF4, EIF4, EKF4, EKF8, EKFA, F008, F051, F075, F095, F09H, F0CA, F0CK, F0DD, F0FB, F0H1, F0IA, F0JJ, F105, F10B, F10D, F16J, F1B4, F1EB, F1ID, F1J4, F36H, F468, F47B, F491, F4C8, F501, F50B, F648, F66A, F69H, F705, F778, F864, F8F1, F90D, F99K, F9C1, F9HK, FA6A, FA75, FA7B, FA95, FAEA, FAFB, FB0D, FB14, FBCD, FBE1, FBJ4, FC1D, FC68, FC9D, FCAA, FCCK, FCD1, FCEA, FD6H, FDIH, FE0K, FE1B, FE64, FE95, 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====Additional known quasi-minimal primes (not necessarily the next)====

40&lt;sub&gt;47333&lt;/sub&gt;9G, CF&lt;sub&gt;479147&lt;/sub&gt;0K

===Base 22===

11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, IHH, IIJ, IJ7, IL7, J07, J55, J69, J8J, J99, J9J, JA5, JAH, JE7, JEH, JHH, JI9, JJ5, JJ9, JJH, JK7, K2J, K51, K5D, K75, K81, KA1, KB1, KB7, KBJ, KD1, KEJ, KG5, KIF, KJ5, KKD, KKJ, L0D, L47, L7F, L97, LAF, LD3, LD9, LDD, LEF, LGD, LI7, LJ7, LJJ, LLD, 104H, 10D5, 1205, 12B5, 140H, 1433, 144H, 14AH, 14B3, 16ED, 1AIH, 1B43, 1DD5, 1DDD, 1E6D, 1EGD, 1G05, 1GDD, 1GED, 1GGD, 1HB3, 1HHH, 1IAH, 200L, 2015, 2051, 20A1, 20DJ, 20GD, 20IL, 21B5, 21DD, 220D, 226D, 228D, 22B5, 22G5, 22K5, 22KD, 2555, 2557, 2581, 25C1, 26A1, 26B1, 2725, 2755, 2801, 2861, 288D, 28B1, 28KD, 2AA1, 2B25, 2B51, 2BB1, 2C81, 2D6D, 2DA1, 2DDJ, 2DGD, 2G0J, 2GB5, 2GDD, 2GGJ, 2I0L, 2I6L, 2ICL, 2J05, 2JK5, 2K07, 2K2D, 2K55, 2K6D, 2KB5, 2KI7, 2L2D, 2L8D, 2LK7, 302L, 30G3, 320L, 32IL, 332L, 33G3, 33G9, 36AF, 36EF, 382J, 388J, 39EL, 3AAF, 3BBJ, 3CG9, 3E2J, 3E6F, 3E6L, 3EEF, 3FAF, 3G69, 3GI3, 3GI9, 3IG9, 3LEL, 3LG3, 4025, 404H, 4063, 4075, 40AH, 40B5, 40B7, 40GH, 4225, 4363, 4447, 444H, 447H, 4487, 44B7, 44H7, 4525, 4555, 45B5, 4663, 4777, 47GH, 4807, 4B0H, 4BB5, 4BG3, 4EBH, 4G4H, 4GB3, 4HH3, 4I03, 4I63, 4IGH, 5069, 5077, 50KD, 5255, 52IJ, 5455, 5477, 5507, 5527, 556F, 557F, 5587, 56EF, 56FF, 56GD, 56KD, 5771, 57C1, 57EF, 5807, 580J, 589J, 58E7, 58IJ, 58K7, 5905, 5945, 5957, 5969, 5989, 598J, 5999, 59EJ, 59K7, 5AEF, 5B25, 5B55, 5BA5, 5BI7, 5BIJ, 5BK5, 5BK7, 5C21, 5EAF, 5EE7, 5F09, 5F6F, 5F95, 5FKD, 5G6D, 5G89, 5I09, 5I0F, 5I27, 5I6F, 5I89, 5IB7, 5IBJ, 5IFF, 5J77, 5K07, 5K25, 5K27, 5K6F, 5K87, 5KK7, 6013, 60A1, 60AF, 60EL, 6281, 62B1, 62KD, 63G9, 6403, 6643, 66EL, 66G9, 68KD, 69B3, 6A01, 6A0F, 6AAF, 6AFF, 6B21, 6B8D, 6BBD, 6BK1, 6D03, 6D43, 6D69, 6D6F, 6D93, 6D9F, 6E21, 6E81, 6E8D, 6EBD, 6ECL, 6ELL, 6FD9, 6GBD, 6GG9, 6IEF, 6K21, 6K8D, 6KBD, 6KE1, 6L43, 6LB3, 6LDF, 6LEL, 6LG9, 700F, 7027, 7057, 70EH, 70LF, 710H, 71G5, 7207, 7225, 7255, 727L, 7505, 7507, 755F, 75EF, 75F5, 75I7, 766F, 7681, 76CF, 7771, 7781, 77IL, 7861, 7A0H, 7A4H, 7BB5, 7C6F, 7EEH, 7EHL, 7FG5, 7G45, 7HEL, 7I0L, 7I27, 7IE7, 7IGH, 7K07, 7K61, 7K77, 7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, H0D1, H0HJ, H0LL, H1HH, H20D, H22D, H26D, H2I7, H2K7, H30L, H3KJ, H40H, H447, H4G3, H4H3, H4HH, H66L, H6ED, 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5IIIIIIIIIIIIIIIIIIIIIIIIIIF, D555555555555555555555550555, EEAAAAAAAAAAAAAAAAAAAAAAAAAF, HHHHHHHHHHHHHHHHHHHHHHHHEEBH, K66666666666666666666666666F, LLLLLLLLLLLLLLLLLLLLLLLLEEB7, D5555555555555555555555555A55, GGGGGGGGGGGGGGGGGGGGGGGGGGGG3, GIG0000000000000000000000000H, HH00000000000000000000000001H, K0000000000000000000000005KEF, 5BBBBBBBBBBBBBBBBBBBBBBBBBBBBD, HB0000000000000000000000000001, K000000000000000000000000505EF, L7777777777777777777777777772L, 2000000000000000000000000000CB1, C8CCCCCCCCCCCCCCCCCCCCCCCCCCCC9, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKFF, JE0000000000000000000000000000J, K000000000000000000000000000261, A0000000000000000000000000004I4H, HD000000000000000000000000000001, K000000000000000000000000000EC01, K0FFFFFFFFFFFFFFFFFFFFFFFFFFFFCF, D0002222222222222222222222222222D, FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL2L, I700000000000000000000000000000GH, K00000000000000000000000000000E61, 20000000000000000000000000000000JJ, DD5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL0KE7, 44HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, F0000000000000000000000000000000000000000000000000000000L89, J000000000000000000000000000000000000000000000000000000JJIJ, K222222222222222222222222222222222222222222222222222222222DD, KCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCF, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKCF, D55KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, K000000000000000000000000000000000000000000000000000000000000077, 22222222222222222222222222222222222222222222222222222222222222222D, K000000000000000000000000000000000000000000000000000000000000008IJ, DFA00000000000000000000000000000000000000000000000000000000000000005, CC4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJE0J, 777777777777777777777777777777777777777777777777777777777777777777777777EK7, 6000000000000000000000000000000000000000000000000000000000000000000000000000000043, KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKB5, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEEH, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 50000000000000000000000000000000000000000000000000000000000000000000000000000000002C1, K0000000000000000000000000000000000000000000000000000000000000000000000000000000000055EF, H700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000K1, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0I7, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, A400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, E0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000071, 7LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIL, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLK77, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEI7, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIB3, I7G00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 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77EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEK7, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJKJ, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777772L, BKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK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KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK5

===Base 24===

15, 17, 1D, 1H, 1J, 1N, 25, 2B, 2D, 2J, 2N, 31, 37, 3B, 3H, 41, 45, 47, 4B, 4D, 4H, 57, 5B, 5H, 5J, 65, 67, 6D, 6J, 6N, 75, 7B, 7D, 7N, 81, 85, 87, 8J, 97, 9B, 9D, 9H, 9N, A1, AB, AH, AN, B5, B7, BD, BH, BJ, C5, CJ, CN, D1, D5, DJ, E1, EB, ED, EH, EN, F7, FD, FJ, FN, G5, GD, GH, H1, HB, HD, HN, I1, I7, IB, IH, J1, J5, J7, JB, JN, K7, KB, KJ, KN, L5, LH, LJ, MD, MJ, N5, NB, NH, NJ, 101, 10B, 111, 1F1, 1FB, 1GB, 1LB, 201, 221, 22H, 261, 271, 277, 28H, 2A7, 2C7, 2G7, 2H7, 2L1, 2L7, 2MH, 305, 30D, 30J, 33N, 34N, 35D, 35N, 38D, 395, 3A5, 3AJ, 3CD, 3DD, 3DN, 3E5, 3EJ, 3GJ, 3IJ, 3JJ, 3K5, 3KD, 3ND, 43N, 44N, 49J, 4EJ, 4GJ, 4GN, 4NN, 50N, 535, 54N, 551, 55N, 5C1, 5CD, 5E5, 5K1, 5KD, 5LN, 5M5, 5N1, 601, 60B, 61B, 66H, 68B, 691, 6CH, 6FH, 6GB, 6HH, 6MH, 70H, 70J, 711, 761, 771, 77H, 77J, 78H, 7C7, 7CH, 7FH, 7G7, 7H7, 7HH, 7IJ, 7JJ, 7K1, 7M1, 7M7, 80D, 82H, 83N, 88D, 88H, 8AD, 8CD, 8DB, 8DD, 8DH, 8DN, 8GB, 8KD, 8MB, 8MH, 905, 911, 921, 935, 955, 99J, 9AJ, 9G1, 9JJ, 9K5, 9L1, 9M5, A0J, A3J, A95, AA7, AD7, AE5, AG7, AGJ, AI5, AIJ, AJD, AL7, ALD, B01, B0N, B11, B61, B6B, B8N, B91, BIN, BL1, BLN, BNN, C1B, C21, C27, C2H, C3D, C61, C8H, C91, CA7, CB1, CBB, CC7, CCB, CCD, CDD, CFB, CG1, CGB, CK1, CL1, CMB, CMH, D0B, D3D, D3N, D4N, D6B, D6H, D7H, D8B, D8N, DAD, DCD, DCH, DDH, DDN, DG7, DGB, DID, DMN, DND, E05, E4J, EA7, EEJ, EF5, EGJ, EI5, EJJ, EM5, EM7, F01, F21, F51, F8H, F95, FC1, FF1, FFB, FKH, FM5, G0B, G0N, G11, G3N, G6B, G77, G7J, G8B, G8N, G91, GA7, GBB, GC7, GFB, GG1, GGJ, GGN, GK1, GL1, GLN, GMN, GN1, GNN, H0J, H2H, H3J, H4J, H77, HA5, HA7, HE5, HFH, HIJ, HJJ, HKH, HL7, HMH, I0N, I3D, I3J, I3N, I4N, I5D, I95, IA5, IAJ, IE5, IEJ, IF5, IGJ, IJD, IK5, IKD, J0D, J4J, J8D, JAD, JDH, JEJ, JFH, JHH, JKD, JMH, K35, K6H, KCD, KFH, KH5, KLD, KM1, L01, L0B, L0D, L0N, L61, L6B, L8D, LA7, LC7, LDD, LF1, LG7, LGB, LGN, LID, LK1, LKD, LL1, LLB, LLD, LMN, LNN, M0H, M11, M21, M4N, M71, M91, M95, MA5, MA7, MBN, MC7, MF1, MF5, MFB, MFH, MG7, MI5, MIN, ML1, ML7, MLB, MMH, N01, N21, N4N, N71, N8N, NC1, ND7, NE7, NG1, NID, NK1, NL7, NMN, NN7, 11CB, 11MB, 1291, 12G1, 16C1, 16CB, 16K1, 186B, 18CB, 19K1, 1BK1, 1C8B, 1K91, 1KC1, 1KL1, 1L21, 1LC1, 1LM1, 1M61, 1M8B, 1MG1, 206H, 20CH, 20M7, 21C1, 21M1, 2207, 260H, 26KH, 2991, 2C6H, 2CC1, 2CM1, 2F11, 2FHH, 2MC1, 2MK1, 2MM1, 308N, 30GN, 30IN, 30LN, 30MN, 333J, 33JD, 33LD, 343J, 344J, 35I5, 380N, 393J, 394J, 3A3D, 3FI5, 3IMN, 3J3D, 3JID, 3L3D, 3L8N, 3M0N, 3M55, 3NGN, 404J, 408N, 40LN, 434J, 44AJ, 4ILN, 4JAJ, 4L8N, 5091, 5095, 50F1, 50I5, 51L1, 5211, 5291, 52G1, 53ID, 53MN, 5595, 55AD, 56F1, 588N, 58MN, 58ND, 5961, 5991, 5A5D, 5AAD, 5F91, 5GF1, 5GIN, 5I05, 5I55, 5I8D, 5IDD, 5IDN, 5IIN, 5IMN, 5KI5, 5M61, 5M8N, 5N3N, 602H, 6211, 62F1, 62G1, 66C1, 66FB, 66M1, 66MB, 6B21, 6BM1, 6BMB, 6C6B, 6CF1, 6CLB, 6FG1, 6K21, 6K2H, 6KG1, 6KKH, 6L21, 6LCB, 6LM1, 6MB1, 6MBB, 6MG1, 6MK1, 7001, 7027, 7207, 726H, 739J, 793J, 79C1, 7A4J, 7A9J, 7AE7, 7C01, 7CC1, 7FL1, 7G21, 7G9J, 7GAJ, 7GC1, 7HGJ, 7J2H, 7J6H, 7MKH, 800B, 800H, 804N, 806H, 808N, 80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, NK3D, NK8D, 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5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, 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EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEG7, 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===Base 30===

11, 17, 1B, 1D, 1H, 1N, 1T, 21, 27, 2B, 2D, 2J, 2N, 2T, 37, 3B, 3D, 3H, 3J, 3N, 47, 4B, 4H, 4J, 4T, 51, 57, 5D, 5H, 5N, 5T, 61, 6B, 6D, 6H, 6J, 71, 7D, 7H, 7J, 7N, 7T, 81, 8B, 8H, 8N, 8T, 91, 97, 9B, 9D, 9N, A7, AB, AD, AH, B1, B7, BH, BJ, BN, BT, C7, CD, CJ, CN, CT, D7, DB, DJ, DT, E1, EB, ED, EJ, EN, ET, F7, FB, FD, FH, FT, G7, GB, GJ, GN, GT, HB, HD, I1, I7, IH, IN, IT, J1, J7, JH, JN, JT, K1, K7, KD, KH, KJ, L1, LB, LD, LH, LN, LT, M1, MD, MH, MN, N1, NB, NJ, NT, O7, OD, OJ, ON, P1, P7, PB, PJ, PN, Q7, QH, QT, R1, RB, RD, RH, RJ, RT, SD, SH, SJ, SN, T7, TB, TD, TH, 10J, 15J, 1IJ, 1JJ, 1LJ, 1MJ, 1QJ, 22H, 29H, 2EH, 2GH, 30T, 331, 33T, 36T, 39T, 3A1, 3C1, 3G1, 3KT, 3MT, 3OT, 3S1, 3T1, 40D, 40N, 431, 44D, 46N, 48D, 4AN, 4DD, 4DN, 4F1, 4FN, 4GD, 4ID, 4PD, 4S1, 50J, 58J, 59J, 5CB, 5FJ, 5IB, 5IJ, 5JB, 5MB, 5MJ, 5OB, 5SB, 607, 63T, 687, 6E7, 6KT, 6L7, 6M7, 6MT, 6NN, 6QN, 6R7, 6S7, 6ST, 6TT, 70B, 77B, 787, 7KB, 7M7, 7MB, 7SB, 807, 80D, 80J, 84D, 85J, 877, 88J, 89J, 8DD, 8FJ, 8ID, 8IJ, 8JJ, 8M7, 8MJ, 8PD, 8QD, 8R7, 90H, 93T, 95J, 99H, 99J, 9AJ, 9AT, 9EH, 9GH, 9HH, 9HJ, 9JJ, 9MJ, 9OH, 9OT, 9PH, 9ST, 9TT, A01, A0T, A1J, A31, A6N, A6T, AAJ, AAN, AFN, AKN, AKT, ALJ, AMJ, AMT, AO1, AOT, AQ1, AQN, ARN, AT1, B5B, BBD, BCB, BDD, BID, BOB, BPD, BQB, C31, C9H, CC1, CCB, CCH, CF1, CH1, CIB, CKB, CMB, D01, D0H, D2H, D41, D4D, D6N, D8D, D9H, DDH, DDN, DGH, DH1, DHH, DID, DKN, DNN, DO1, DOH, DQN, DS1, EEH, EHH, EM7, EOH, EPH, ER7, F0N, F31, F5J, F8J, F9J, FLJ, FO1, FQ1, FQN, FS1, G01, G4D, G8D, GF1, GGH, GO1, GOH, GQD, GS1, H07, H0J, H0T, H1J, H2H, H31, H4N, H8J, HA1, HAJ, HAT, HC1, HE7, HEH, HFN, HGH, HH7, HIJ, HJJ, HKN, HL7, HNH, HPH, HQN, HS1, HTJ, HTN, I4D, I5B, I5J, I8D, IAJ, IDD, IGD, IIJ, IKB, IMB, IMJ, IOB, IPD, IQD, J8J, J9J, JAJ, JBD, JCB, JFJ, JIJ, JJB, JJD, JLJ, JPD, JQB, K3T, K4N, KAT, KBB, KCB, KMB, KNN, KOB, KOT, KQN, KST, KTT, L0J, L5J, L67, LAJ, LJJ, LQJ, LR7, M0J, M3T, M5B, M7B, M87, M9J, MAT, MFJ, MIJ, MJJ, MKB, MMJ, MOT, MQJ, N07, N0H, N67, N6N, N87, NAN, NDH, NGD, NHH, NKN, NN7, NNH, NPH, NQD, NQN, NR7, O01, O0B, O0H, OAT, OC1, OCH, OEH, OF1, OH1, OKT, OMB, OOT, OPH, OQ1, OQB, OS1, OST, P3T, P6T, P9H, PCH, PDH, PEH, PGD, PHH, PHT, PID, PMT, PPT, PQD, PST, PTT, Q5B, Q6N, Q9J, QAJ, QBB, QBD, QC1, QDN, QFJ, QFN, QGD, QJB, QKN, QLJ, QMB, QND, QNN, QO1, QQ1, QQN, QSB, R4N, R77, R87, RAN, RKN, RM7, RNN, RR7, RS7, S0T, S41, S6T, S87, SBB, SC1, SM7, SMT, SOB, SQ1, SR7, SS1, STT, T0J, T6T, T9T, TAN, TFN, TKN, TKT, TNN, TO1, TOT, TPT, TQ1, TQN, TTN, 18AJ, 19FJ, 1AFJ, 1FAJ, 20CH, 20PH, 2C0H, 2CHH, 2COH, 2H0H, 2HCH, 2HHH, 2POH, 2PPH, 3001, 34Q1, 3F41, 3FF1, 3QF1, 3SPT, 3SST, 3TTT, 40A1, 4441, 44KN, 44QN, 4AC1, 4C01, 4CA1, 4DG1, 4GA1, 4GC1, 4NND, 4NRN, 4OA1, 4Q4N, 4QA1, 4QRN, 4RQN, 550B, 555B, 55KB, 5A5J, 5AJJ, 5BKB, 5JQJ, 5KKB, 5QQB, 5QQJ, 604N, 606T, 60FN, 60KN, 60OT, 60PT, 660N, 660T, 664N, 66AT, 66FN, 66KN, 66TN, 6A0N, 6A9T, 6AAT, 6FKN, 6FRN, 6K0N, 6KFN, 6KKN, 6O6T, 6PAT, 6RFN, 6RRN, 6T6N, 6TRN, 7067, 70R7, 75BB, 77R7, 7C5B, 7CQB, 7E67, 7IQB, 7O5B, 7OIB, 7QOB, 7R07, 7RE7, 8667, 88L7, 88S7, 8E87, 8EE7, 8EL7, 8J8D, 8LL7, 8LS7, 906T, 908J, 90FJ, 90QJ, 90TJ, 92CH, 966T, 99MT, 99PT, 9I0J, 9I8J, 9LFJ, 9LIJ, 9M6T, 9MKT, 9MMT, 9PKT, 9QQJ, 9TLJ, A04N, A0QJ, A3AT, A3ST, A4A1, A4G1, A5QJ, A8QJ, A90J, A9PT, AA3T, AA41, AAAT, AAF1, AAG1, AAPT, AC41, ACS1, AF0J, AFC1, AFIJ, AG41, AGA1, AGG1, AI8J, AIJJ, AIQJ, AJ5J, AJJJ, AQ8J, AQJJ, AQQJ, AS3T, AT4N, AT5J, ATST, B04D, B0QD, BBIB, BBKB, BI0B, BK0B, BKIB, BKSB, BMIB, BMSB, BQ4D, BQQD, BSMB, C00B, C05B, C0A1, C0BB, C0EH, C0GH, C0Q1, C0QB, C2OH, C2PH, C441, C4Q1, C50B, CB0B, CBSB, CEGH, CG2H, CGHH, CHHH, CHOH, COGH, COO1, COOB, COOH, CQ01, CQ0B, CQQB, CQS1, CS0B, D00N, D0PD, D44N, DA0N, DCEH, DCG1, DCQ1, DDC1, DDF1, DDPD, DDQ1, DECH, DF4N, DFA1, DFRN, DG0D, DGD1, DGQ1, DNEH, DPDD, DQA1, DQPD, DRFN, DRRN, E20H, E667, E767, E8L7, E8S7, ECGH, EG2H, EH67, EH77, ES67, F001, F01J, F0MJ, F0QJ, F44N, F4G1, F4RN, F6RN, FA0J, FAC1, FAF1, FAIJ, FC01, FCG1, FFAJ, FFC1, FFG1, FFIJ, FGA1, FJJJ, FJQJ, FKAN, FR6N, FRFN, G09H, G0EH, G0HH, G20H, G341, G3Q1, G4G1, G92H, GA41, GAC1, GC0H, GCEH, GCG1, GCPH, GD31, GDA1, GDG1, GDPD, GE2H, GE9H, GGG1, GGGD, GHCH, GHHH, GP0D, GP2H, GPPD, GQ31, GQA1, GQG1, H00N, H0CH, H3ST, H4O1, H667, H66N, H677, H69T, H99T, H9CH, H9KT, H9LJ, H9PT, H9QJ, HH6N, HHF1, HHFJ, HHH1, HHHT, HHLJ, HHOH, HHQ1, HHST, HK6T, HL9J, HLMJ, HMKT, HMMT, HNM7, HNRN, HOPT, HQ01, HQ5J, HR0N, HR6N, HS77, HSKT, HT01, I0BB, I0JD, I0JJ, I98J, I9LJ, I9QJ, ICBB, IF0J, IFFJ, IFQJ, IIBD, IJBB, IJID, IL8J, ILFJ, IQ8J, IQJJ, IQQJ, IS0B, ISQB, J00D, J05J, J0BB, J0MJ, J0OB, J4QD, J50B, J5QJ, JBKB, JBMB, JDGD, JDQD, JGDD, JI0D, JIIB, JISB, JKIB, JKKB, JM0B, JOIB, JQ4D, JQDD, JQID, JQJJ, JQMJ, JSIB, JSMB, K06N, K0KB, K0KT, K0MT, K0PT, K0SB, K5QB, K60T, K66T, K6AN, K6FN, K6PT, K6RN, K96T, K99T, KA0N, KF6N, KFKN, KI0B, KK0T, KK6N, KK6T, KKFN, KKKB, KKKT, KKQB, KKTN, KMMT, KQ0B, KQQB, KR0N, KS5B, L0M7, L8E7, L98J, LFFJ, LI8J, LIFJ, LL87, LL9J, LLIJ, LLM7, LM77, LML7, LMM7, M09T, M0E7, M0IB, M0L7, M0M7, M0TT, M55J, M5LJ, M60T, M69T, M707, M767, M777, M7E7, M7S7, M8LJ, M96T, M9KT, MC0B, MCOB, ME07, MIBB, MJ0B, MJBB, MJSB, MLLJ, MLS7, MM07, MM0T, MML7, MMOB, MMR7, MOBB, MOCB, MP0T, MP9T, MPKT, MQ0B, MQIB, MQOB, MS07, MSPT, MSSB, MT8J, MTTT, N00D, N04D, N04N, N0FN, N0ND, N0RN, N2OH, N92H, NCGH, ND0D, ND4N, NDPD, NEGH, NEL7, NF4N, NFRN, NGCH, NGEH, NHM7, NME7, NML7, NMM7, NN0D, NN4D, NN4N, NNDN, NNID, NNND, NNRN, NP4D, NSL7, NSS7, O00T, O03T, O0MT, O0PT, O2HH, O2OH, O341, O4A1, O4G1, O4O1, O5KB, O90T, OA41, OBSB, OC5B, OCOB, OG2H, OG31, OH9H, OH9T, OHHH, OHTT, OICB, OISB, OM0T, OM6T, OMPT, OMTT, OO2H, OO9H, OOA1, OOCB, OOGH, OOKB, OOO1, OSKB, OTMT, P00T, P04D, P08D, P09T, P0KT, P90T, PAAT, PKKT, PO0T, PP0D, PPPD, Q00B, Q0AN, Q0D1, Q0F1, Q0IB, Q0JJ, Q0MJ, Q0OB, Q55J, Q5QJ, Q88D, QA0N, QA41, QA4N, QDF1, QDPD, QF41, QFA1, QG31, QIQJ, QJ0J, QJ8D, QJDD, QJJJ, QJMJ, QJQD, QKKB, QKQB, QOKB, QOOB, QP0D, QPDD, QQ8D, QQID, QQJD, QQPD, QQQB, QS01, QSA1, R00N, R067, R06N, R0FN, R0L7, R60N, RE67, RFRN, RRFN, RRRN, S00B, S3AT, S3ST, S50B, S5QB, S99T, SAAT, SC0B, SE67, SGG1, SICB, SIQB, SK5B, SKIB, SKKT, SKPT, SKQB, SM0B, SMMB, SMSB, SOA1, SOG1, SPOT, SQCB, SQQB, SS0B, SS67, ST31, STG1, T00N, T03T, T041, T04N, T0AT, T0C1, T0MT, T0ST, T1AJ, T3F1, T4A1, T4RN, T5JJ, T66N, T8AJ, T9FJ, T9QJ, TA3T, TAA1, TAG1, TAIJ, TCS1, TFA1, TFAJ, TFF1, TFFJ, TFG1, TGA1, TI9J, TIJJ, TIQJ, TL8J, TL9J, TM0T, TM5J, TM8J, TMAJ, TMLJ, TMST, TSF1, TSG1, TT01, TT5J, TT8J, TTA1, TTC1, TTFJ, TTG1, TTLJ, TTMJ, 30441, 30O41, 30OO1, 34O41, 3AATT, 3ASAT, 3O441, 3Q041, 40G41, 40GQ1, 40O41, 40OG1, 40OO1, 44001, 440C1, 440G1, 4444N, 44CG1, 44GQ1, 4AAA1, 4COG1, 4DAA1, 4GQ41, 4KRRN, 4OGG1, 4Q041, 4QGG1, 5005B, 500BB, 500KB, 500QB, 50K0B, 50Q0B, 50QKB, 555AJ, 555LJ, 555QJ, 55J5J, 5B00B, 5B0BB, 5J55J, 5J5JJ, 5JJ5J, 5JJJJ, 5K00B, 5QJ5J, 5QK0B, 6006N, 6009T, 600AT, 6066N, 60R0N, 666N7, 6696T, 66OPT, 67767, 6900T, 696PT, 69P0T, 6F6AN, 6O9PT, 6OP0T, 6OP9T, 6P99T, 755QB, 77767, 7CBBB, 7IBIB, 7IIBB, 7IICB, 7QIIB, 8888D, 88E67, 8L887, 8SSE7, 9099T, 90KKT, 90KMT, 90MPT, 9690T, 96P9T, 990KT, 9990T, 9999T, 99K6T, 9FFFJ, 9HKMT, 9HMPT, 9ILLJ, 9KKMT, 9KKPT, 9M00T, 9T8QJ, 9TFIJ, A008J, A00NN, A055J, A0FJJ, A0I0J, A0I9J, A0J0J, A44C1, A555J, A9FFJ, A9QIJ, A9T8J, AFFA1, AFFFJ, AFFJJ, AFFQJ, AJ00J, AP99T, ASA9T, ASFF1, ASP9T, ASSAT, AT3TT, ATFQJ, ATQIJ, ATT9J, ATTQJ, B088D, B08GD, B0GGD, B0SKB, B8GGD, BG00D, BIISB, BISIB, BS0IB, BS0SB, BSSKB, BSSSB, C0041, C0G41, C0H0H, C0PPH, C0S01, C4001, C4AG1, C4OG1, C5BBB, CG00H, CGE0H, CGGA1, CGQ41, COSSB, CP20H, CPGPH, CPP2H, CPPOH, CPPPH, CS001, CS55B, CSSQB, D00GD, D0A4N, D0FAN, D0GDD, D0N0D, DAAA1, DAAC1, DAFG1, DDAA1, DDGGD, DFFAN, DFGC1, DGDGD, DGG31, DGGA1, DH0AN, DPPPH, DQDD1, DQGG1, E0L87, E88E7, EC00H, EC02H, EE867, EE887, EEL87, ELE87, F00IJ, F0441, F0AG1, F0F0J, F0F41, F0FF1, F0GG1, F4041, F44A1, F44C1, F4A41, F64KN, F6K6N, FAAA1, FAFQJ, FCAA1, FF041, FF0F1, FF64N, FF6KN, FFA41, FFA4N, FFF4N, FFFF1, FFFFN, FFKRN, FFQMJ, FIJ0J, FJ00J, FKKKN, FNNFN, FQQ0J, FQQMJ, G00DD, G00DH, G00PD, G02PH, G0DDD, G0PDD, GAAA1, GC2HH, GDDDD, GDGDD, GDPPH, GG0PD, GGCA1, GGCQ1, GGDD1, GH4Q1, GHHG1, GII0D, GQ441, H009H, H00G1, H00H1, H04Q1, H0ANN, H0H01, H0H9H, H0HG1, H0HO1, H0O41, H0OHH, H0QG1, H40G1, H4G41, H4GG1, H60AN, HAN0N, HF0G1, HFF41, HFFMJ, HFFQJ, HGQ41, HH3TT, HH401, HH441, HH4G1, HH55J, HH66T, HH6OT, HH96T, HHA0N, HHANN, HHC0H, HHCHH, HHH0N, HHHCH, HHHHN, HHHNN, HHKMT, HHO41, HHP9T, HHPKT, HHTMT, HHTT1, HK9MT, HKKMT, HLLFJ, HM66T, HM7R7, HMM67, HMM77, HMMM7, HMTST, HO9HT, HOO41, HOOOH, HOT3T, HOTT1, HPO9T, HSO3T, HTGG1, I000J, I009J, I00BD, I00QJ, I00SB, I08QJ, I0I0D, I0IQB, I0JIB, I0Q0B, I0SCB, I0SIB, I0SSB, IB00D, IBBSB, IBISB, ICSSB, II0CB, II0ID, II0SB, IIIBB, IIQ0B, IIQIB, IISIB, IJ0SB, IJJQJ, ILLLJ, IQ00J, IQC0B, IQIQB, IQQIB, ISSCB, J00MB, J08GD, J0IID, J0QQD, J55BB, J55JJ, J5J5J, J5JJJ, J88GD, JB0SB, JBB0B, JBBSB, JDDDD, JG0ID, JGG0D, JIIID, JJ0QJ, JJ5JJ, JQ08D, JQ0QJ, JQQ0D, JQQ5J, JS55B, JSK0B, JSS5B, K000T, K006T, K00TN, K0FAN, K505B, K6T0N, K9KMT, K9KPT, K9P0T, KIQIB, KK00N, KK05B, KK0AN, KK0KN, KKM9T, KS0IB, KS0QB, KSQIB, KTR6N, KTRRN, L0087, L08S7, LE087, LEE87, LL8LJ, LMEE7, LMSE7, LMSS7, M00CB, M00PT, M066T, M0CQB, M0K6T, M0MQB, M0MSB, M0QQB, M0SMB, M0SST, M666T, M900T, MBBSB, MEE77, MEL77, MELE7, MES77, MESE7, MESS7, MI00B, MIICB, MIISB, MIQCB, MK9PT, ML7L7, MLEL7, MLLE7, MLME7, MM677, MMBIB, MMBSB, MMCQB, MME77, MMEE7, MMICB, MMISB, MMK6T, MMKKT, MMM9T, MMMMT, MMMTT, MMQQB, MMSCB, MMSKT, MMSMB, MMSS7, MMTST, MOIIB, MOOIB, MOSIB, MQCQB, MR007, MR667, MRL07, MS7L7, MSEL7, MSK9T, MSL77, MSSL7, MT00T, MTMMT, N0DDD, N4NNN, N7LE7, N7S77, NE2CH, NE9CH, NEC2H, NEE77, NFFNN, NFNFN, NII0D, NL777, NL7L7, NLES7, NLLL7, NLS77, NOG9H, NRR0N, O6P9T, O9H6T, O9HPT, O9P9T, OCBBB, OG441, OGAG1, OGGA1, OHH6T, OHOOH, OKIIB, OKK5B, OKKIB, OO5BB, OOBIB, OOIIB, OSCSB, OT3TT, OT441, OTG41, OTGG1, P00PH, P0D0D, P0G2H, P0OGH, P0PGH, PA99T, PGP0H, POOOH, PP20H, PP88D, PPP0H, Q0001, Q000N, Q001J, Q00ID, Q00PD, Q044N, Q04G1, Q0I0D, Q0PPD, Q40G1, Q444N, Q44RN, Q4AA1, Q8QQJ, QAAS1, QAFF1, QAFG1, QASG1, QDGG1, QFGG1, QI00D, QIICB, QIIQB, QIQCB, QOIIB, QPP4D, QQ08J, QQ0CB, QQC0B, QQI0J, QQJ5J, QSFG1, R00E7, R0E07, R0NE7, R6F6N, R6FFN, REE07, RELE7, RFF6N, RL0E7, RLE07, RLEE7, RLLE7, RQR0N, RR0QN, RRQ0N, S03O1, S0AF1, S0AG1, S0O31, S0QIB, S0SIB, S30F1, S30O1, S7QIB, SA3PT, SAAA1, SAFG1, SASST, SCSQB, SF0G1, SFFF1, SI0SB, SISSB, SKSSB, SSCQB, SSCSB, SSMIB, SSPAT, SSSKT, SSSMB, SSSSB, STAF1, STF01, T0001, T0031, T0AF1, T0AS1, T0G31, T0R6N, T0T31, T0TF1, T3AAT, T40G1, T4CG1, T4G41, T5LLJ, T8LLJ, TA441, TA98J, TAFJJ, TAQ5J, TASST, TATAT, TC401, TCGG1, TFQIJ, TFQJJ, TFQMJ, TG441, TGC41, TI8LJ, TLLMJ, TMMMT, TMTMT, TQ8QJ, TSS3T, TT3AT, TT9IJ, TTAAT, TTQIJ, TTS31, TTS3T, TTT9J, TTTST, 20000H, 200OOH, 3440O1, 3TAAST, 404CQ1, 4KKKKN, 4KKKRN, 4QQQQD, 505BBB, 50BB0B, 6000AN, 6444RN, 66666N, 666O9T, 66999T, 669P9T, 66N777, 6A444N, 6FF66N, 6FFF6N, 6R666N, 766767, 77S677, 7IBBBB, 8888E7, 8LLLLJ, 8SSSL7, 9000MT, 90K90T, 90KP0T, 90M90T, 99000T, 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===Base 36===

11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 2UJ, 2WN, 2XN, 30Z, 337, 33B, 33H, 33P, 34H, 34P, 37D, 38Z, 39H, 3AB, 3AP, 3AZ, 3D7, 3DH, 3FD, 3FZ, 3HD, 3HH, 3KZ, 3L7, 3LZ, 3MB, 3O7, 3OZ, 3PD, 3PP, 3Q7, 3RH, 3S7, 3SZ, 3UP, 3UZ, 3XB, 40P, 43B, 43H, 46V, 48B, 48V, 49B, 4AP, 4BB, 4E1, 4FP, 4HB, 4HH, 4HP, 4IH, 4MB, 4OP, 4OV, 4PH, 4RH, 4VB, 4VH, 4WH, 4WP, 4WV, 4X1, 4XH, 55T, 565, 58Z, 5C5, 5E7, 5ET, 5EZ, 5IN, 5KT, 5L7, 5MP, 5NN, 5O5, 5O7, 5OP, 5R7, 5RT, 5RZ, 5ST, 5SZ, 5TN, 5TZ, 5W7, 5XN, 5YN, 61T, 625, 62J, 64V, 661, 66J, 691, 6AV, 6EV, 6IJ, 6J1, 6K5, 6O1, 6OT, 6QJ, 6QT, 6RV, 6SJ, 6T1, 6TT, 6U5, 6UV, 6VV, 6W5, 6Y1, 6YT, 72D, 737, 77D, 791, 7A1, 7AZ, 7D7, 7GD, 7I1, 7IN, 7LN, 7M7, 7MN, 7N7, 7NN, 7RN, 7RZ, 7WN, 7X7, 7Z1, 801, 80V, 82D, 83B, 841, 84H, 877, 881, 887, 88B, 88V, 88Z, 89H, 89V, 8A1, 8AB, 8B7, 8BH, 8D1, 8DH, 8EB, 8EV, 8GD, 8GZ, 8HD, 8IV, 8L7, 8LZ, 8M1, 8MB, 8MD, 8O7, 8OB, 8Q7, 8QD, 8RD, 8S7, 8SH, 8SZ, 8UZ, 8W7, 8WV, 8XV, 8Y1, 90H, 91J, 93B, 93H, 94V, 981, 98H, 99J, 9AH, 9BB, 9C5, 9IH, 9IV, 9JV, 9KH, 9M1, 9OB, 9QJ, 9R1, 9RH, 9SH, 9UJ, 9VB, 9VJ, 9W5, 9X1, 9Y1, 9YB, 9YJ, 9YV, A25, A3P, A3Z, A4H, A61, A81, A8B, AAB, AAH, ABB, AC5, AEZ, AHP, AK1, AKB, AKH, ALZ, AMB, AO5, AOZ, APH, AQP, AR1, ARV, ARZ, AUV, AV5, AVB, AVV, AWV, AX1, AXB, AZ1, B2J, B4B, B4V, B6V, B87, B8H, B9B, BAB, BAH, BBH, BBV, BE7, BEB, BGJ, BHB, BIJ, BJB, BJT, BLJ, BOB, BOT, BQ7, BRV, BS7, BTT, BTV, BVB, BVJ, BVT, BWV, BX7, BXH, BYH, BYV, C25, C2J, C2N, C55, C65, C6J, C95, CCN, CDD, CED, CF5, CFJ, CGD, CIN, CJD, CO5, CRN, CUJ, CXJ, CXN, D37, D3P, D41, D55, D5P, D6T, D77, D81, D9H, DA1, DCD, DD1, DD5, DDH, DDP, DE7, DEP, DF5, DFD, DG7, DHH, DI1, DK1, DK5, DKT, DO1, DOP, DP1, DPD, DQ7, DR7, DRH, DS7, DTT, DU1, DWD, DX7, DYH, E0P, E2D, E3Z, E41, E4P, E71, E81, E87, E8B, EAZ, EDP, EED, EEN, EEV, EFZ, EGT, EI1, EKZ, ELB, EMB, EMN, EN7, EO1, EOT, EPP, EPZ, EQ7, ERN, EST, ETN, ETV, EUP, EUZ, EVT, EWD, EWN, EX1, EYB, EYN, EZD, EZP, F2J, F3B, F3Z, F4P, F65, F8B, F8Z, FCJ, FD5, FEZ, FFB, FFD, FG5, FGD, FGZ, FIJ, FJ5, FJJ, FLZ, FPD, FQD, FSZ, FXB, G07, G0D, G0Z, G3D, G3Z, G55, G6T, G7D, G7Z, G87, G8Z, GA5, GCJ, GD7, GE7, GET, GG5, GG7, GGT, GGZ, GI7, GJD, GKZ, GL7, GLT, GQT, GRT, GS7, GST, GU5, GUZ, GW5, GXJ, GZ7, GZZ, H3D, H3H, H4H, H9B, HAH, HED, HGD, HHP, HIH, HKH, HKP, HLN, HNH, HOB, HOP, HQP, HRD, HRH, HRN, HSH, HWD, HWH, HWP, HYN, I41, I6J, I71, I7N, I87, I8H, I9H, I9J, IA1, IAV, IE1, IFJ, IG7, IHH, II1, IIH, IIV, IJV, IK1, IL7, ILJ, ILN, IM1, INN, IOV, IRH, IUV, IX1, IXH, IXV, IYJ, J2D, J4V, J55, J61, J6J, J8B, J8V, J95, J9J, J9V, JA5, JBB, JD5, JDT, JF5, JFJ, JGJ, JGT, JIV, JJ1, JJD, JJV, JK5, JKT, JLB, JLT, JMN, JN5, JNJ, JQJ, JQN, JQT, JRD, JTB, JVJ, JVN, JVV, JWN, JY1, JYJ, K0B, K25, K3P, K65, K81, K95, K9H, KAH, KC5, KEP, KEZ, KG5, KHP, KK1, KKT, KLB, KLP, KLZ, KM1, KMB, KMP, KOT, KP1, KQP, KR1, KRT, KRZ, KTT, KW5, KX1, KZB, L0N, L0P, L8Z, L9B, LAZ, LBJ, LEN, LET, LFB, LFZ, LG7, LGZ, LIJ, LJJ, LKB, LKP, LLB, LLP, LLT, LMB, LMN, LN7, LO7, LOT, LPZ, LQT, LRN, LTN, LTT, LWP, LX7, LYN, M01, M0P, M2D, M2N, M37, M3B, M41, M61, M77, M87, M8D, M91, ME1, ME7, MEB, MFB, MFP, MGD, MI7, MKB, MM7, MMN, MN7, MNN, MO7, MOB, MOP, MPP, MQP, MS7, MW7, MXN, MYN, N5N, N65, N87, N8D, N8H, N95, NCD, NCJ, NDH, NE7, NFJ, NG5, NG7, NGJ, NJ5, NN7, NND, NO5, NQD, NQJ, NRN, NU5, NWH, NWN, NXH, NXN, NYJ, O0Z, O25, O37, O3B, O3P, O3Z, O41, O4B, O5Z, O61, O71, O7Z, O81, OA5, OAP, OBB, OBV, OC5, OEZ, OF5, OG7, OKP, OKZ, OMB, OO5, OOZ, OP5, OPP, OQT, OR1, OR7, OS7, OST, OTB, OTZ, OU5, OVV, OW5, OX7, OXB, OXV, OYV, OZ5, OZ7, P01, P0D, P0N, P3P, P4H, P4P, P5N, P65, P6V, P8V, PD1, PED, PEZ, PHH, PHP, PI1, PIN, PLN, PLP, PLZ, POP, PP1, PPH, PPV, PQD, PQN, PRV, PV5, PVH, PVV, PWH, PWP, PX1, PXV, PYH, PYN, Q07, Q0P, Q2N, Q37, Q3P, Q6J, Q6T, Q7D, Q8D, Q9J, QCD, QCJ, QD7, QED, QFP, QGT, QI7, QIN, QJN, QLJ, QMD, QMN, QMP, QND, QNJ, QOT, QQJ, QTN, QWN, QYJ, QYT, R0V, R0Z, R37, R3H, R4H, R77, R7D, R7N, R7Z, R81, R8V, R91, RA1, RCD, RCN, RD1, RED, REV, REZ, RGT, RGZ, RHD, RIV, RKH, RKZ, RLN, RMD, RNH, RO7, RQN, RRD, RRT, RRZ, RS7, RSH, RT1, RTV, RU1, RUZ, RVN, RVT, RW7, RWH, RX7, RY1, S0J, S6J, S87, S9H, SAZ, SCJ, SET, SFJ, SG7, SGZ, SI7, SM7, SO7, SOT, SQJ, SQT, SRH, SSH, STT, SW7, SX7, SXH, SYH, T0N, T0Z, T1D, T1T, T6V, T8Z, TBB, TBV, TCN, TD1, TEV, TGT, TKT, TLB, TM1, TO1, TOB, TQN, TR1, TRD, TT1, TTB, TTN, TUZ, TVB, TVN, TVT, TWD, TWV, TXB, TXV, TYV, U2J, U3Z, U61, U8V, U95, UA1, UC5, UEP, UEZ, UFJ, UG5, UJ5, ULZ, UO5, UOP, UOZ, UQP, URV, URZ, UUJ, UW5, UWV, UXJ, UXV, UYJ, V0H, V25, V4V, V55, V6V, V9J, V9V, VBB, VBJ, VCJ, VET, VGJ, VHP, VIH, VIN, VJJ, VJN, VLT, VMP, VNJ, VOB, VP5, VQJ, VQT, VRT, VRV, VSH, VSJ, VST, VTB, VTN, VUP, VW5, VWN, VXH, VXN, VYB, W07, W25, W3D, W3H, W4P, W4V, W57, W6V, W7D, W8H, W95, W9H, WAH, WAV, WCN, WD7, WDD, WDH, WE7, WEN, WF5, WGD, WHH, WK5, WKH, WN7, WNN, WPP, WPV, WQP, WR7, WRD, WRH, WRN, WS7, WU5, WUP, WVN, WWH, WWP, WX7, WXH, WXN, WYN, WYV, X0J, X2J, X2N, X4B, X4H, X4V, X6J, X87, X8B, X91, X9B, XAV, XCN, XEB, XFB, XHB, XHH, XHN, XKB, XLJ, XNH, XO1, XQ7, XS7, XSH, XUJ, XWN, XWV, XX7, XXH, XXV, Y0N, Y1J, Y1T, Y2N, Y3H, Y61, Y8V, Y91, YBV, YCN, YEB, YFJ, YHB, YHN, YIH, YJN, YJT, YLJ, YLN, YMB, YMN, YNH, YOB, YOV, YRH, YTB, YTN, YTT, YTV, YVB, YVH, YWH, YWV, YXB, YY1, YYJ, YYT, YYV, Z01, Z3D, Z3Z, Z71, Z8B, ZB7, ZBB, ZD5, ZDD, ZDP, ZG7, ZGD, ZGZ, ZKB, ZLP, ZM1, ZO5, ZP1, ZPD, ZQD, ZU1, ZX1, ZXB, ZZD, 102J, 106T, 10ET, 10GJ, 10LJ, 10TD, 12FJ, 12LJ, 13ED, 13GD, 13MD, 192J, 1D0T, 1DET, 1E3D, 1EFD, 1F0D, 1FWD, 1G0J, 1GGJ, 1GIJ, 1GUJ, 1I0J, 1ISJ, 1JOT, 1JTT, 1M0D, 1M3D, 1MFD, 1MWD, 1OKT, 1OOT, 1Q3D, 1QDT, 1QKT, 1QTD, 1QWD, 1SLJ, 1TDT, 1TFD, 1TOT, 1TTD, 1TTT, 1UGJ, 1ULJ, 1WFD, 1WWD, 20CJ, 20FJ, 20G5, 20IN, 20JD, 20MD, 20RN, 20WD, 20YN, 2295, 25RN, 260J, 26LJ, 288D, 28DD, 28FD, 296J, 29G5, 29K5, 29LJ, 2C05, 2C0J, 2C0N, 2CJJ, 2CK5, 2CLJ, 2CSJ, 2D25, 2DC5, 2DDD, 2DW5, 2F0J, 2FDD, 2FMD, 2FSJ, 2G65, 2G95, 2GK5, 2I0N, 2JC5, 2JCJ, 2JWD, 2K05, 2L6J, 2L9J, 2N0N, 2NIN, 2NN5, 2NSJ, 2NW5, 2NYN, 2Q0N, 2QNN, 2R0N, 2RFD, 2RWD, 2SJJ, 2SXJ, 2SYJ, 2W05, 2WC5, 2WD5, 2WMD, 2XCJ, 2YCJ, 2YJJ, 2YRN, 2YSJ, 300D, 302D, 304B, 3077, 3087, 308D, 30BH, 30DD, 30FP, 30G7, 30GD, 30MD, 30QP, 30YH, 320D, 333D, 333Z, 33WD, 344B, 34FB, 373Z, 37E7, 37EZ, 37G7, 3807, 38BB, 38E7, 38G7, 38HB, 38I7, 38X7, 390B, 394B, 399B, 39EB, 39FB, 39KB, 3ASH, 3AWH, 3B8B, 3BG7, 3BLB, 3BR7, 3BW7, 3C8D, 3D0P, 3D2D, 3DKP, 3E3D, 3E77, 3EBB, 3EFB, 3EKB, 3EQP, 3ERD, 3ERZ, 3EZZ, 3F4B, 3FEP, 3G2D, 3GCD, 3GED, 3GM7, 3GR7, 3GWD, 3GZD, 3H0B, 3HEP, 3HKB, 3IAH, 3ISH, 3K4B, 3KBH, 3KEB, 3KKB, 3L4B, 3L8B, 3M3D, 3MEP, 3MG7, 3MWP, 3OFP, 3OKB, 3OMP, 3OOB, 3OYB, 3P8H, 3PXH, 3QDP, 3QQD, 3QRD, 3QWD, 3R3D, 3R3Z, 3R8D, 3RG7, 3RM7, 3RQD, 3RR7, 3RZ7, 3SIH, 3W0H, 3WED, 3WM7, 3WQD, 3WW7, 3WYH, 3X8H, 3XE7, 3XG7, 3XKH, 3Y0B, 3Y4B, 3Y8H, 3YAH, 3YSH, 3YYB, 3Z4B, 3ZFP, 404V, 4061, 40EB, 40I1, 40IV, 40KH, 40M1, 40R1, 40XB, 40XV, 43QP, 4441, 444H, 444P, 4481, 448H, 4491, 449H, 44EB, 44EV, 44K1, 44KB, 44KP, 44M1, 44R1, 44RV, 44SH, 44U1, 44VP, 44YB, 46I1, 46K1, 46M1, 480H, 488H, 48AH, 48I1, 48K1, 48U1, 4991, 49EV, 49I1, 49O1, 49UV, 4A0H, 4A41, 4A4B, 4A8H, 4A91, 4ALB, 4ASH, 4BXV, 4E4B, 4EEP, 4EFB, 4ELP, 4EUV, 4EXB, 4FOB, 4I01, 4I0V, 4I4V, 4I61, 4I81, 4I91, 4IEV, 4IY1, 4K0H, 4K91, 4KI1, 4KKH, 4KKP, 4KUP, 4L3P, 4L4P, 4LAB, 4LMP, 4MA1, 4MK1, 4MMP, 4MP1, 4MU1, 4O0B, 4OFB, 4OKB, 4OM1, 4OU1, 4OYB, 4P41, 4P81, 4PO1, 4PYV, 4QLP, 4QQP, 4RAV, 4RK1, 4RRV, 4S8H, 4U3P, 4U91, 4U9V, 4UIV, 4UUP, 4VAV, 4VLP, 4VVV, 4X0B, 4X0V, 4XAB, 4XBV, 4XVV, 4XYV, 4Y0V, 4Y41, 4Y4H, 4Y4V, 4Y81, 4YAV, 4YEV, 4YIV, 4YK1, 4YKB, 4YM1, 4YR1, 4YUV, 4YYB, 5025, 503Z, 5095, 50G5, 50PN, 50QN, 50WP, 50Z7, 5205, 52RN, 53FP, 53M7, 53PZ, 53X7, 53ZZ, 54PP, 550N, 553P, 554P, 5557, 555N, 557Z, 5595, 55EN, 55G5, 55GZ, 55LZ, 55OZ, 55QN, 55Z7, 572N, 577Z, 5787, 57CN, 57FZ, 57QN, 57S7, 57ZZ, 58I7, 58X7, 5955, 5A95, 5AEP, 5CMN, 5CWN, 5E2N, 5EAP, 5ECN, 5EKP, 5EQN, 5F95, 5FLP, 5FQP, 5G05, 5GK5, 5GQ7, 5I37, 5II7, 5IM7, 5L3P, 5LCN, 5LFP, 5LGT, 5LKZ, 5LUP, 5LZZ, 5M07, 5N55, 5NQ7, 5NX7, 5OAZ, 5OFZ, 5OLT, 5OOT, 5P25, 5P95, 5PG5, 5PGZ, 5PPN, 5PQP, 5PW5, 5Q0N, 5Q87, 5QCN, 5QKP, 5QN7, 5QPN, 5QPP, 5QQT, 5QTT, 5QWP, 5QX7, 5S37, 5SS7, 5T6T, 5TQT, 5U4P, 5UGZ, 5W0P, 5W2N, 5W5P, 5WLN, 5WP5, 5WQN, 5XM7, 5Y0T, 5YOT, 5Z25, 5Z4P, 5Z57, 5ZG5, 5ZLZ, 5ZQ7, 5ZWP, 6005, 600V, 601J, 6055, 606V, 60A1, 60C5, 60GT, 60JJ, 60LJ, 60M1, 60R1, 60VT, 60YV, 610J, 61FJ, 6401, 64K1, 6505, 665T, 6665, 66A5, 66F5, 66G5, 66GT, 66KT, 66V5, 66XV, 68OV, 68YV, 690J, 69G5, 69O5, 69XJ, 69XV, 6A55, 6AA1, 6AA5, 6AF5, 6C9J, 6CA5, 6CGJ, 6CYJ, 6EK1, 6ER1, 6FLJ, 6FXJ, 6G0J, 6G65, 6GGJ, 6GJ5, 6GLJ, 6I81, 6I9V, 6J65, 6J6V, 6JET, 6JST, 6JWV, 6K01, 6KET, 6KST, 6L0J, 6L6T, 6LFJ, 6LRT, 6LXJ, 6MA1, 6MM1, 6MX1, 6O8V, 6OV5, 6R0T, 6RET, 6RKT, 6S6T, 6T8V, 6U01, 6U9J, 6UE1, 6UGJ, 6UJJ, 6ULJ, 6UU1, 6V0T, 6VA5, 6VF5, 6VG5, 6VKT, 6VXJ, 6WWV, 6WXV, 6X41, 6X6V, 6XIV, 6XJV, 6XOV, 6Y0V, 6YCJ, 6YXV, 7001, 702N, 7041, 707N, 7081, 70GZ, 70I7, 70KZ, 70ND, 70O7, 70OZ, 70R7, 70U1, 70UZ, 70WD, 70YN, 71QD, 71WD, 720N, 72CN, 72YN, 733Z, 73ED, 73ZD, 7401, 7441, 74M1, 74O1, 74R1, 7641, 76K1, 76M1, 76R1, 76U1, 76X1, 773Z, 7741, 7771, 7781, 77FZ, 77G7, 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8KRH, 8KU1, 8KYH, 8KZ1, 8L4B, 8LBB, 8MG7, 8MR7, 8O4V, 8OI1, 8OK1, 8OOV, 8OX1, 8RAV, 8RHH, 8RUV, 8RX1, 8U4V, 8UK1, 8V4B, 8V9B, 8VHH, 8VRH, 8VVV, 8VYV, 8W0H, 8WFD, 8X07, 8XAH, 8XR1, 8XU1, 8XYB, 8Y6V, 8Y8H, 8YBB, 8YFB, 8YHH, 8YKB, 8YLB, 8YUV, 8YVV, 8YYH, 8Z1D, 8Z4B, 8Z8D, 8Z91, 8ZCD, 8ZEZ, 8ZFD, 8ZFZ, 8ZK1, 8ZO1, 8ZX7, 90A5, 90BJ, 90BV, 90EB, 90EV, 90GJ, 90IJ, 90J5, 90JB, 90LB, 90O1, 90OV, 90XB, 90XV, 9225, 92K5, 9441, 94BH, 94EB, 94KB, 94XB, 9505, 9601, 960V, 96A1, 96J5, 96K1, 96V5, 96WV, 96XV, 984B, 98AV, 98BV, 98FB, 994H, 998B, 998V, 999B, 99A1, 99A5, 99E1, 99G5, 99O1, 99O5, 99WH, 9A05, 9A55, 9A95, 9AA5, 9ABV, 9AE1, 9AEV, 9AI1, 9AO1, 9AOV, 9B0J, 9B0V, 9B8V, 9B9H, 9BCJ, 9BFJ, 9BSJ, 9BVV, 9CCJ, 9CLJ, 9CSJ, 9E01, 9E4B, 9EA1, 9EAB, 9EAV, 9EK1, 9F05, 9F25, 9FA5, 9FBJ, 9FFJ, 9FJB, 9FSJ, 9FXJ, 9G25, 9G2J, 9GIJ, 9GSJ, 9HFB, 9HLB, 9IIJ, 9IJ1, 9J01, 9J0B, 9J0J, 9J9B, 9JCJ, 9JE1, 9K05, 9K41, 9K55, 9K61, 9K8B, 9KA5, 9KI1, 9KK5, 9KU5, 9L2J, 9LCJ, 9O05, 9O0V, 9O95, 9OI1, 9OO1, 9OOV, 9OWV, 9RRV, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ

====Additional known quasi-minimal primes (not necessarily the next)====

P&lt;sub&gt;81993&lt;/sub&gt;SZ

==Unsolved families==

Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')) by covering congruence, algebraic factorization, or combine of them.

{|class=wikitable
|base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family
|-
|13||9{5}||(113×13&lt;sup&gt;''n''&lt;/sup&gt;−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|13||A{3}A||(41×13&lt;sup&gt;''n''+1&lt;/sup&gt;+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|16||{3}AF||(16&lt;sup&gt;''n''+2&lt;/sup&gt;+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1{7}||(23×17&lt;sup&gt;''n''&lt;/sup&gt;−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1F{0}7||32×17&lt;sup&gt;''n''+1&lt;/sup&gt;+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||4{7}A||(71×17&lt;sup&gt;''n''+1&lt;/sup&gt;+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||51{0}D||86×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||70F{0}D||2038×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||8{B}9||(139×17&lt;sup&gt;''n''+1&lt;/sup&gt;−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||9{5}9||(149×17&lt;sup&gt;''n''+1&lt;/sup&gt;+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||95{F}||(2543×17&lt;sup&gt;''n''&lt;/sup&gt;−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||A{D}F||(173×17&lt;sup&gt;''n''+1&lt;/sup&gt;+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}B3||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+190||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}DB||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+232||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2BE||(11×17&lt;sup&gt;''n''+3&lt;/sup&gt;−41579)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2E||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;−2411)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}E9||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+773)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}EE||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+853)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||D0G{D}||(60381×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2860381*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||E9{B}||(3963×17&lt;sup&gt;''n''&lt;/sup&gt;−11)/16||29000||[http://factordb.com/index.php?query=%283963*17%5En-11%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||F1{9}||(4105×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16||1000000||[http://factordb.com/index.php?query=%284105*17%5En-9%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||FD0{D}||(72909×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2872909*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||G{7}F||(263×17&lt;sup&gt;''n''+1&lt;/sup&gt;+121)/16||29000||[http://factordb.com/index.php?query=%28263*17%5E%28n%2B1%29%2B121%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||2{7}9D||(47×21&lt;sup&gt;''n''+2&lt;/sup&gt;+953)/20||20000||[http://factordb.com/index.php?query=%2847*21%5E%28n%2B2%29%2B953%29%2F20&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||2F{C}A||(288×21&lt;sup&gt;''n''+1&lt;/sup&gt;−13)/5||20000||[http://factordb.com/index.php?query=%28288*21%5E%28n%2B1%29-13%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||4{3}B||(83×21&lt;sup&gt;''n''+1&lt;/sup&gt;+157)/20||20000||[http://factordb.com/index.php?query=%2883*21%5E%28n%2B1%29%2B157%29%2F20&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||5{0}DJ||5×21&lt;sup&gt;''n''+2&lt;/sup&gt;+292||20000||[http://factordb.com/index.php?query=5*21%5E%28n%2B2%29%2B292&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||{5}FEK||(21&lt;sup&gt;''n''+3&lt;/sup&gt;+18455)/4||20000||[http://factordb.com/index.php?query=%2821%5E%28n%2B3%29%2B18455%29%2F4&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||{7}ID||(7×21&lt;sup&gt;''n''+2&lt;/sup&gt;+4733)/20||20000||[http://factordb.com/index.php?query=%287*21%5E%28n%2B2%29%2B4733%29%2F20&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||7{K}Z||(53×36&lt;sup&gt;''n''+1&lt;/sup&gt;+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||B{0}EUV||11×36&lt;sup&gt;''n''+3&lt;/sup&gt;+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||HM{0}N||634×36&lt;sup&gt;''n''+1&lt;/sup&gt;+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||N{0}YYN||23×36&lt;sup&gt;''n''+3&lt;/sup&gt;+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||O{L}Z||(123×36&lt;sup&gt;''n''+1&lt;/sup&gt;+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||S{0}8H||28×36&lt;sup&gt;''n''+2&lt;/sup&gt;+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|}

For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E

For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family B0{H}6H may or may not be quasi-minimal prime, since another unsolved family is B{H}6H, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D}

=== Base 17 ===

All searched to length 29000

* 1{7}
* 1F{0}7
* 4{7}A
* 51{0}D
* 70F{0}D
* 8{B}9
* 9{5}9
* 95{F}
* A{D}F
* B{0}B3
* B{0}DB
* {B}2BE
* {B}2E
* {B}E9
* {B}EE
* D0G{D}
* E9{B}
* F1{9}
* FD0{D}
* G{7}F

=== Base 21 ===

All searched to length 20000

* 2{7}9D
* 2F{C}A
* 4{3}B
* 5{0}DJ
* {5}FEK
* {7}ID
* 99{0}99H
* {9}0D
* {9}D
* B3{0}EB
* B9{0}E5
* B{D}B
* B{H}6H
* DH{D}
* F{9}D
* {F}35
* G{0}FK
* H{0}7771
* H{D}
* {J}6J

=== Base 36 ===

All searched to length 20000

* 7{K}Z
* B{0}EUV
* HM{0}N
* N{0}YYN
* O{L}Z
* S{0}8H

==Primality certificates for the proven primes &gt; 10&lt;sup&gt;299&lt;/sup&gt;==

See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]]

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime
|-
|9||149||76&lt;sub&gt;329&lt;/sub&gt;2||(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642]
|-
|9||150||27&lt;sub&gt;686&lt;/sub&gt;07||(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486]
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||3×9&lt;sup&gt;1160&lt;/sup&gt;+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423]
|-
|11||1065||A&lt;sub&gt;713&lt;/sub&gt;58||11&lt;sup&gt;715&lt;/sup&gt;−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487]
|-
|11||1066||7&lt;sub&gt;759&lt;/sub&gt;44||(7×11&lt;sup&gt;761&lt;/sup&gt;−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840]
|-
|11||1067||557&lt;sub&gt;1011&lt;/sub&gt;||(607×11&lt;sup&gt;1011&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522]
|-
|13||3165||50&lt;sub&gt;270&lt;/sub&gt;44||5×13&lt;sup&gt;272&lt;/sup&gt;+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005]
|-
|13||3166||9&lt;sub&gt;271&lt;/sub&gt;095||(3×13&lt;sup&gt;274&lt;/sup&gt;−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654]
|-
|13||3167||10&lt;sub&gt;286&lt;/sub&gt;7771||13&lt;sup&gt;290&lt;/sup&gt;+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633]
|-
|13||3168||9&lt;sub&gt;308&lt;/sub&gt;1||(3×13&lt;sup&gt;309&lt;/sup&gt;−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=308&amp;c0=-&amp;EN= 13&lt;sup&gt;308&lt;/sup&gt;−1]
|-
|13||3169||B&lt;sub&gt;341&lt;/sub&gt;C4||(11×13&lt;sup&gt;343&lt;/sup&gt;+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618]
|-
|13||3170||8B&lt;sub&gt;343&lt;/sub&gt;||(107×13&lt;sup&gt;343&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736]
|-
|13||3171||710&lt;sub&gt;371&lt;/sub&gt;111||92×13&lt;sup&gt;374&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609]
|-
|13||3172||75&lt;sub&gt;375&lt;/sub&gt;7||(89×13&lt;sup&gt;376&lt;/sup&gt;+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596]
|-
|13||3173||9B0&lt;sub&gt;391&lt;/sub&gt;9||128×13&lt;sup&gt;392&lt;/sup&gt;+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790]
|-
|13||3174||7B0B&lt;sub&gt;397&lt;/sub&gt;||(15923×13&lt;sup&gt;397&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574]
|-
|13||3175||10&lt;sub&gt;414&lt;/sub&gt;93||13&lt;sup&gt;416&lt;/sup&gt;+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240]
|-
|13||3176||81010&lt;sub&gt;415&lt;/sub&gt;1||17746×13&lt;sup&gt;416&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3177||8110&lt;sub&gt;435&lt;/sub&gt;1||1366×13&lt;sup&gt;436&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3178||B7&lt;sub&gt;486&lt;/sub&gt;||(139×13&lt;sup&gt;486&lt;/sup&gt;−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892]
|-
|13||3179||B&lt;sub&gt;563&lt;/sub&gt;C||(11×13&lt;sup&gt;564&lt;/sup&gt;+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=564&amp;c0=-&amp;EN= 13&lt;sup&gt;564&lt;/sup&gt;−1]
|-
|13||3180||1B&lt;sub&gt;576&lt;/sub&gt;||(23×13&lt;sup&gt;576&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=576&amp;c0=-&amp;EN= 13&lt;sup&gt;576&lt;/sup&gt;−1]
|-
|13||3181||80&lt;sub&gt;693&lt;/sub&gt;87||8×13&lt;sup&gt;695&lt;/sup&gt;+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532]
|-
|13||3182||CC5&lt;sub&gt;713&lt;/sub&gt;||(2021×13&lt;sup&gt;713&lt;/sup&gt;−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353]
|-
|13||3183||B&lt;sub&gt;834&lt;/sub&gt;74||(11×13&lt;sup&gt;836&lt;/sup&gt;−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871]
|-
|13||3184||9&lt;sub&gt;968&lt;/sub&gt;B||(3×13&lt;sup&gt;969&lt;/sup&gt;+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244]
|-
|13||3185||10&lt;sub&gt;1295&lt;/sub&gt;181||13&lt;sup&gt;1298&lt;/sup&gt;+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013]
|-
|13||3186||9&lt;sub&gt;1362&lt;/sub&gt;5||(3×13&lt;sup&gt;1363&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776]
|-
|13||3187||7&lt;sub&gt;1504&lt;/sub&gt;1||(7×13&lt;sup&gt;1505&lt;/sup&gt;−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755]
|-
|13||3188||930&lt;sub&gt;1551&lt;/sub&gt;1||120×13&lt;sup&gt;1552&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3189||720&lt;sub&gt;2297&lt;/sub&gt;2||93×13&lt;sup&gt;2298&lt;/sup&gt;+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910]
|-
|13||3190||1770&lt;sub&gt;2703&lt;/sub&gt;17||267×13&lt;sup&gt;2705&lt;/sup&gt;+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825]
|-
|13||3191||390&lt;sub&gt;6266&lt;/sub&gt;1||48×13&lt;sup&gt;6267&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3192||B0&lt;sub&gt;6540&lt;/sub&gt;BBA||11×13&lt;sup&gt;6543&lt;/sup&gt;+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906]
|-
|13||3193||C&lt;sub&gt;10631&lt;/sub&gt;92||13&lt;sup&gt;10633&lt;/sup&gt;−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750]
|-
|14||649||34D&lt;sub&gt;708&lt;/sub&gt;||47×14&lt;sup&gt;708&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||5×14&lt;sup&gt;19698&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2328||880&lt;sub&gt;246&lt;/sub&gt;7||136×16&lt;sup&gt;247&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is &lt; 10&lt;sup&gt;299&lt;/sup&gt;
|-
|16||2329||D4&lt;sub&gt;263&lt;/sub&gt;D||(199×16&lt;sup&gt;264&lt;/sup&gt;+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238]
|-
|16||2330||E0&lt;sub&gt;261&lt;/sub&gt;4DD||14×16&lt;sup&gt;264&lt;/sup&gt;+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352]
|-
|16||2331||8C0&lt;sub&gt;290&lt;/sub&gt;ED||140×16&lt;sup&gt;292&lt;/sup&gt;+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307]
|-
|16||2332||DA&lt;sub&gt;305&lt;/sub&gt;5||(41×16&lt;sup&gt;306&lt;/sup&gt;−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284]
|-
|16||2333||CE80&lt;sub&gt;422&lt;/sub&gt;D||3304×16&lt;sup&gt;423&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257]
|-
|16||2334||5F&lt;sub&gt;544&lt;/sub&gt;6F||6×16&lt;sup&gt;546&lt;/sup&gt;−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967]
|-
|16||2335||88F&lt;sub&gt;545&lt;/sub&gt;||137×16&lt;sup&gt;545&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2336||BE0&lt;sub&gt;792&lt;/sub&gt;BB||190×16&lt;sup&gt;794&lt;/sup&gt;+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938]
|-
|16||2337||D9&lt;sub&gt;1052&lt;/sub&gt;||(68×16&lt;sup&gt;1052&lt;/sup&gt;−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020]
|-
|16||2338||FAF&lt;sub&gt;1062&lt;/sub&gt;45||251×16&lt;sup&gt;1064&lt;/sup&gt;−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610]
|-
|16||2339||F8&lt;sub&gt;1517&lt;/sub&gt;F||(233×16&lt;sup&gt;1518&lt;/sup&gt;+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824]
|-
|16||2340||20&lt;sub&gt;1713&lt;/sub&gt;321||2×16&lt;sup&gt;1716&lt;/sup&gt;+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735]
|-
|16||2341||300F&lt;sub&gt;1960&lt;/sub&gt;AF||769×16&lt;sup&gt;1962&lt;/sup&gt;−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750]
|-
|16||2342||90&lt;sub&gt;3542&lt;/sub&gt;91||9×16&lt;sup&gt;3544&lt;/sup&gt;+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191]
|-
|16||2343||5BC&lt;sub&gt;3700&lt;/sub&gt;D||(459×16&lt;sup&gt;3701&lt;/sup&gt;+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322]
|-
|16||2344||D0B&lt;sub&gt;17804&lt;/sub&gt;||(3131×16&lt;sup&gt;17804&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511]
|-
|18||547||80&lt;sub&gt;298&lt;/sub&gt;B||8×18&lt;sup&gt;299&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is an algebraic factor of ''N''+1, factordb entry of 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is [http://factordb.com/index.php?id=1100000000900149167]
|-
|18||548||H&lt;sub&gt;766&lt;/sub&gt;FH||18&lt;sup&gt;768&lt;/sup&gt;−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490]
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||12×18&lt;sup&gt;6270&lt;/sup&gt;+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437]
|-
|20||3301||H&lt;sub&gt;247&lt;/sub&gt;A0H||(17×20&lt;sup&gt;250&lt;/sup&gt;−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619]
|-
|20||3302||7&lt;sub&gt;249&lt;/sub&gt;A7||(7×20&lt;sup&gt;251&lt;/sup&gt;+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602]
|-
|20||3303||J7&lt;sub&gt;270&lt;/sub&gt;||(368×20&lt;sup&gt;270&lt;/sup&gt;−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462]
|-
|20||3304||J&lt;sub&gt;330&lt;/sub&gt;CCC7||20&lt;sup&gt;334&lt;/sup&gt;−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572]
|-
|20||3305||40&lt;sub&gt;387&lt;/sub&gt;404B||4×20&lt;sup&gt;391&lt;/sup&gt;+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563]
|-
|20||3306||EC0&lt;sub&gt;429&lt;/sub&gt;7||292×20&lt;sup&gt;430&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702]
|-
|20||3307||G&lt;sub&gt;447&lt;/sub&gt;99||(16×20&lt;sup&gt;449&lt;/sup&gt;−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753]
|-
|20||3308||3A&lt;sub&gt;527&lt;/sub&gt;3||(67×20&lt;sup&gt;528&lt;/sup&gt;−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531]
|-
|20||3309||E&lt;sub&gt;566&lt;/sub&gt;C7||(14×20&lt;sup&gt;568&lt;/sup&gt;−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516]
|-
|20||3310||JCJ&lt;sub&gt;629&lt;/sub&gt;||393×20&lt;sup&gt;629&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|20||3311||J&lt;sub&gt;655&lt;/sub&gt;05J||20&lt;sup&gt;658&lt;/sup&gt;−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052]
|-
|20||3312||50&lt;sub&gt;1163&lt;/sub&gt;AJ||5×20&lt;sup&gt;1165&lt;/sup&gt;+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412]
|-
|20||3313||CD&lt;sub&gt;2449&lt;/sub&gt;||(241×20&lt;sup&gt;2449&lt;/sup&gt;−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915]
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||16×20&lt;sup&gt;6270&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457]
|-
|22||7984||I7G0&lt;sub&gt;254&lt;/sub&gt;H||8882×22&lt;sup&gt;255&lt;/sup&gt;+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788]
|-
|22||7985||D0&lt;sub&gt;255&lt;/sub&gt;5EEF||13×22&lt;sup&gt;259&lt;/sup&gt;+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932]
|-
|22||7986||IK&lt;sub&gt;322&lt;/sub&gt;F||(398×22&lt;sup&gt;323&lt;/sup&gt;−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145]
|-
|22||7987||C0&lt;sub&gt;340&lt;/sub&gt;G9||12×22&lt;sup&gt;342&lt;/sup&gt;+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159]
|-
|22||7988||77E&lt;sub&gt;348&lt;/sub&gt;K7||(485×22&lt;sup&gt;350&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779]
|-
|22||7989||J&lt;sub&gt;379&lt;/sub&gt;KJ||(19×22&lt;sup&gt;381&lt;/sup&gt;+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027]
|-
|22||7990||J&lt;sub&gt;388&lt;/sub&gt;EJ||(19×22&lt;sup&gt;390&lt;/sup&gt;−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729]
|-
|22||7991||DJ&lt;sub&gt;400&lt;/sub&gt;||(292×22&lt;sup&gt;400&lt;/sup&gt;−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110]
|-
|22||7992||E&lt;sub&gt;404&lt;/sub&gt;K7||(2×22&lt;sup&gt;406&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298]
|-
|22||7993||66F&lt;sub&gt;453&lt;/sub&gt;B3||(971×22&lt;sup&gt;455&lt;/sup&gt;−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809]
|-
|22||7994||L0&lt;sub&gt;454&lt;/sub&gt;B63||21×22&lt;sup&gt;457&lt;/sup&gt;+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331]
|-
|22||7995||L&lt;sub&gt;483&lt;/sub&gt;G3||22&lt;sup&gt;485&lt;/sup&gt;−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730]
|-
|22||7996||E60&lt;sub&gt;496&lt;/sub&gt;L||314×22&lt;sup&gt;497&lt;/sup&gt;+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239]
|-
|22||7997||I&lt;sub&gt;626&lt;/sub&gt;AF||(6×22&lt;sup&gt;628&lt;/sup&gt;−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334]
|-
|22||7998||K0&lt;sub&gt;760&lt;/sub&gt;EC1||20×22&lt;sup&gt;763&lt;/sup&gt;+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415]
|-
|22||7999||J0&lt;sub&gt;767&lt;/sub&gt;IGGJ||19×22&lt;sup&gt;771&lt;/sup&gt;+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567]
|-
|22||8000||7&lt;sub&gt;959&lt;/sub&gt;K7||(22&lt;sup&gt;961&lt;/sup&gt;+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817]
|-
|22||8001||L&lt;sub&gt;2385&lt;/sub&gt;KE7||22&lt;sup&gt;2388&lt;/sup&gt;−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774]
|-
|22||8002||7&lt;sub&gt;3815&lt;/sub&gt;2L||(22&lt;sup&gt;3817&lt;/sup&gt;−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839]
|-
|24||3400||I0&lt;sub&gt;241&lt;/sub&gt;I5||18×24&lt;sup&gt;243&lt;/sup&gt;+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037]
|-
|24||3401||D0&lt;sub&gt;259&lt;/sub&gt;KKD||13×24&lt;sup&gt;262&lt;/sup&gt;+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725]
|-
|24||3402||C7&lt;sub&gt;298&lt;/sub&gt;||(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235]
|-
|24||3403||20&lt;sub&gt;313&lt;/sub&gt;7||2×24&lt;sup&gt;314&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241]
|-
|24||3404||BC0&lt;sub&gt;331&lt;/sub&gt;B||276×24&lt;sup&gt;332&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842]
|-
|24||3405||N&lt;sub&gt;2644&lt;/sub&gt;LLN||24&lt;sup&gt;2647&lt;/sup&gt;−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089]
|-
|24||3406||D&lt;sub&gt;2698&lt;/sub&gt;LD||(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876]
|-
|24||3407||A0&lt;sub&gt;2951&lt;/sub&gt;8ID||10×24&lt;sup&gt;2954&lt;/sup&gt;+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654]
|-
|24||3408||88N&lt;sub&gt;5951&lt;/sub&gt;||201×24&lt;sup&gt;5951&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606]
|-
|30||2613||AN&lt;sub&gt;206&lt;/sub&gt;||(313×30&lt;sup&gt;206&lt;/sup&gt;−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073]
|-
|30||2614||M&lt;sub&gt;241&lt;/sub&gt;QB||(22×30&lt;sup&gt;243&lt;/sup&gt;+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295]
|-
|30||2615||M0&lt;sub&gt;547&lt;/sub&gt;SS7||22×30&lt;sup&gt;550&lt;/sup&gt;+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988]
|-
|30||2616||C0&lt;sub&gt;1022&lt;/sub&gt;1||12×30&lt;sup&gt;1023&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|30||2617||5&lt;sub&gt;4882&lt;/sub&gt;J||(5×30&lt;sup&gt;4883&lt;/sup&gt;+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423]
|-
|30||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||25×30&lt;sup&gt;34205&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|}

==Unproven PRPs==

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this PRP
|-
|11||1068||57&lt;sub&gt;62668&lt;/sub&gt;||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000003573679860]
|-
|13||3194||C5&lt;sub&gt;23755&lt;/sub&gt;C||(149×13&lt;sup&gt;23756&lt;/sup&gt;+79)/12||[http://factordb.com/index.php?id=1100000003590647776]
|-
|13||3195||80&lt;sub&gt;32017&lt;/sub&gt;111||8×13&lt;sup&gt;32020&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000000490878060]
|-
|16||2345||DB&lt;sub&gt;32234&lt;/sub&gt;||(206×16&lt;sup&gt;32234&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000002383583629]
|-
|16||2346||4&lt;sub&gt;72785&lt;/sub&gt;DD||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15||[http://factordb.com/index.php?id=1100000003615909841]
|-
|22||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21||[http://factordb.com/index.php?id=1100000003594696838]
|-
|30||2618||I0&lt;sub&gt;24608&lt;/sub&gt;D||18×30&lt;sup&gt;24609&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003593967511]
|}

All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10&lt;sup&gt;16&lt;/sup&gt;. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]])

==Proof==

===Base 2===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.

===Base 3===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (2,1), (2,2)

* Case (1,1):
** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes)
*** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime.

===Base 4===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (2,1), (2,3), (3,1), (3,3)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.

===Base 5===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4)

* Case (1,1):
** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes)
*** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes)
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes)
***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb])
***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes)
**** The smallest prime of the form 1{4}4 is 14444.
**** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes)
*** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes)
**** For the 3{0,3}1 family, we can separate this family to four families:
***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01.
****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime.
***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031.
***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001.
***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31.
****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime.
**** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.
* Case (3,4):
** 34 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes)
**** The smallest prime of the form 4{4}1 is 44441.
**** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime.
* Case (4,2):
** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime.

===Base 6===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes)
*** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes)
**** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 4{0}41 is 40041
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes)
*** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.

===Base 7===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

* Case (1,1):
** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes)
*** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {}
**** If there are &gt;=3 number of 1 and 5 in the {}:
***** If there is 111 in the {}, then we have the prime 11111
***** If there is 115 in the {}, then the prime 115 is a subsequence
***** If there is 151 in the {}, then the prime 115 is a subsequence
***** If there is 155 in the {}, then the prime 155 is a subsequence
***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime
***** If there is 515 in the {}, then the prime 515 is a subsequence
***** If there is 551 in the {}, then the prime 551 is a subsequence
***** If there is 555 in the {}, then the prime 551 is a subsequence
**** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1
***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1
****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1
******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21
******** The smallest prime of the form 11{0}21 is 1100021
******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime
******** The smallest prime of the form 1001{0}21 is 100121
********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families
***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1
****** Since 1051 is prime, we only need to consider the family 15{0}1
******* The smallest prime of the form 15{0}1 is 150001
* Case (1,2):
** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes)
*** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {}
**** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime.
**** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime.
**** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2
***** Since 1022 and 1202 are primes, we only need to consider the number 122
****** 122 is not prime.
**** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime.
* Case (1,3):
** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** 14 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime.
* Case (1,6):
** 16 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime.
* Case (2,2):
** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,6):
** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes)
*** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes)
**** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's)
***** For the 3{0,1}1 family:
****** If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
****** If there are no 1's between (3,1), then the form will be 3{0}1
******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime.
****** If there are no 0's between (3,1), then the form will be 3{1}1
******* The smallest prime of the form 3{1}1 is 31111
****** If there are exactly 1 0's between (3,1), then there must be &lt;3 1's between (3,1), or 31111 will be a subsequence.
******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime.
******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011
******** Neither 3101 nor 3011 is prime.
******* If there are no 1's between (3,1), then the number must be 301
******** 301 is not prime.
***** For the 3{3}3{0,1}1 family:
****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence.
****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1
******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence.
******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime.
******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime.
******* If there are no 1's between (33,1), then the form will be 33{0}1
******** The smallest prime of the form 33{0}1 is 33001
****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1
******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence.
******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301
******** The smallest prime of the form 3{3}301 is 33333301
******* If there are no 0's between (3{3}3,1), then the form is 3{3}31
******** The smallest prime of the form 3{3}31 is 33333333333333331
**** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1
***** Numbers containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1,3}3{0,1,3}1
******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes)
******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes)
********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes)
********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence.
********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime.
********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1
*********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime.
********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime.
***** Numbers not containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1}1
******* If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1
******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime.
******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1
******** If there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111
********* None of 31151, 31511, 35111 are primes.
******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 351
********* 351 is not prime.
******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111
********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes.
********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime.
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1
********* Neither 3051 nor 3501 is prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** If there are &gt;=2 5's in {}, then 553 will be a subsequence.
*** If there are no 5's in {}, then the family will be 3{0,3,6}3
**** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
*** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3
**** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes)
***** The smallest prime of the form 3{0}53 is 300053
* Case (3,4):
** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes)
*** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes)
**** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes)
**** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes)
* Case (3,5):
** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes)
*** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence.
*** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence.
*** If there are at least two 1's in {}, then 115 will be a subsequence.
*** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5
**** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime.
*** If there is no 1's in {}, then the family will be 3{0,4,5}5
**** If there are at least to 4's in {}, then 344 and 445 will be subsequences.
**** If there is no 4's in {}, then the family will be 3{0,5}5
***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5
****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime.
****** The smallest prime of the form 3{5}5 is 35555
**** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5
***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes)
****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime.
* Case (3,6):
** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,2):
** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes)
*** If there is no 5's in {}, then the family will be 4{0,2,4,6}4
**** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4
**** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes)
***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence.
***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4
****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes)
***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4
****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4
******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes)
* Case (4,5):
** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** If there are at least two 5's in {}, then 4555 will be a subsequence.
*** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there is no 5's in {}, then the family will be 4{0}5
**** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime.
* Case (4,6):
** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (5,1):
** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 3's in {}, then 533 will be a subsequence.
*** If there is no 3's in {}, then the family will be 5{0,1}1
**** Since 5011 is prime, we only need to consider the family 5{1}{0}1
***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes)
****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime.
****** The smallest prime of the form 51{0}1 is 5100000001
****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime.
****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime.
*** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1
**** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence.
***** Thus we only need to consider the family 5{0}3{0,1}1
****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime.
****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime.
****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence.
****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1
******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0)
******** The smallest prime of the form 5311{0}1 is 531101
* Case (5,2):
** 52 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 1's in {}, then 113 will be a subsequence.
*** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
*** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime.
* Case (5,4):
** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes)
*** If there are no 5's in {}, then the family will be 5{0,3}4
**** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime.
*** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence.
*** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there are at least two 5's in {}, then 5554 will be a subsequence.
* Case (5,5):
** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes)
*** If there are no 4's in {}, then the family will be 5{0,5}5
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
*** If there are no 5's in {}, then the family will be 5{0,4}5
**** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime.
*** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence.
* Case (5,6):
** 56 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,2):
** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (6,3):
** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,4):
** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are at least three 3's in {}, then 3334 will be a subsequence.
*** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4
**** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence.
**** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence.
**** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence.
**** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence.
**** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence.
**** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence.
**** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence.
**** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence.
**** Thus the number can only be 634
***** 634 is not prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,6):
** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}6
**** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime.
*** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6
**** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime.
*** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence.
*** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence.

===Base 8===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7)

* Case (1,1):
** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes)
**** 171 is not prime.
**** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 &gt; 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 &gt; 1, this factorization is nontrivial), thus cannot be prime.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes)
*** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime)
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes)
*** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes)
*** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes)
**** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime.
**** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes)
***** The smallest prime of the form 33{4}1 is 3344441
***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime.
**** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes)
***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime.
***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes)
****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes.
* Case (3,3):
** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes)
*** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes)
**** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime)
**** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461
***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes)
****** The smallest prime of the form 4{4}41 is 444444441
****** The smallest prime of the form 4{4}641 is 444641
***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes)
****** The smallest prime of the form 4{4}411 is 444444441
****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes)
***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461
****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime)
**** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes)
***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime)
***** The smallest prime of the form 4{4}641 is 444641
* Case (4,3):
** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes)
*** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes)
**** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime.
**** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime.
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes)
*** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes)
**** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb])
**** The smallest prime of the form 4{4}77 is 4444477
**** The smallest prime of the form 4{7}7 is 47777
**** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime)
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes)
*** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes)
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
**** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes)
***** 500525 is not prime.
***** The smallest prime of the form 5{5}25 is 555555555555525
***** The smallest prime of the form 5{5}025 is 55555025
***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes)
***** The smallest prime of the form 5{5}0525 is 5550525
***** The smallest prime of the form 5{5}00525 is 5500525
***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes)
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence)
**** The form is 6{0,1,7}4{0,1,7}1
***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1
****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411
******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41
******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41
********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.
********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime.
********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime.
******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411
******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime)
*** Numbers not containing 4:
**** The form is 6{0,1,7}1
***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1
****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime.
****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1
******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1
******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.
******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime)
* Case (6,3):
** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,7):
** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them
**** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17
***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime.
**** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77
***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime.
**** If there is neither 1 nor 7 between them, then the form is 6{0,4}7
***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes)
****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime.
****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime.
* Case (7,1):
** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes)
*** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes)
**** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes)
***** The smallest prime of the form 7{7}1 is 7777777777771
***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime)
***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime.
***** The smallest prime of the form 7{7}4444441 is 77774444441
****** Since this prime has just 4 7's, we only need to consider the families with &lt;=3 7's
******* The smallest prime of the form 7{4}1 is 744444441
******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime.
******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes)
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.

===Base 10===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,9):
** 19 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 2{0}21 is 20021
**** The smallest prime of the form 22{0}1 is 22000001
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes)
*** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes)
**** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime.
**** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime.
* Case (2,9):
** 29 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,9):
** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** 47 is prime, and thus the only minimal prime in this family.
* Case (4,9):
** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime.
* Case (5,1):
** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes)
**** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes)
***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 5{5}1 is 555555555551
**** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581
***** 581 is not prime.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,7):
** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027
**** The smallest prime of the form 52{0}7 is 5200007
* Case (5,9):
** 59 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,3):
** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,9):
** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49
***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49
****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes)
******* The smallest prime of the form 6{6}49 is 666649
******** Since this prime has just 4 6's, we only need to consider the families with &lt;=3 6's
********* The smallest prime of the form 6{0}49 is 60000049
********* The smallest prime of the form 66{0}49 is 66000049
********* The smallest prime of the form 666{0}49 is 66600049
* Case (7,1):
** 71 is prime, and thus the only minimal prime in this family.
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,9):
** 79 is prime, and thus the only minimal prime in this family.
* Case (8,1):
** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes)
**** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes)
***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime)
***** The smallest prime of the form 80{5}1 is 80555551
* Case (8,3):
** 83 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime.
* Case (8,9):
** 89 is prime, and thus the only minimal prime in this family.
* Case (9,1):
** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes)
**** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021
***** None of 921, 9201, 9021 are primes.
**** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801
***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
* Case (9,3):
** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (9,7):
** 97 is prime, and thus the only minimal prime in this family.
* Case (9,9):
** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9
***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes)
****** The smallest prime of the form 94{6}9 is 946669

===Base 12===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,B):
** 1B is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes)
*** The smallest prime of the form 2{0}1 is 2001
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (2,B):
** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes)
**** None of 20B, 22B, 29B, 209B, 220B are primes.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,B):
** 3B is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes)
**** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 44{A}1 is 44AAA1
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes)
**** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077
* Case (4,B):
** 4B is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes)
*** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime.
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (5,B):
** 5B is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,5):
** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes)
**** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 6{0}A5 is 600A5
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,B):
** 6B is prime, and thus the only minimal prime in this family.
* Case (7,1):
** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes)
*** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1
**** None of 741, 7B1, 74B1 are primes.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,B):
** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes)
**** The smallest prime of the form 7{9}B is 7999B
**** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes)
**** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime)
* Case (8,1):
** 81 is prime, and thus the only minimal prime in this family.
* Case (8,5):
** 85 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** 87 is prime, and thus the only minimal prime in this family.
* Case (8,B):
** 8B is prime, and thus the only minimal prime in this family.
* Case (9,1):
** 91 is prime, and thus the only minimal prime in this family.
* Case (9,5):
** 95 is prime, and thus the only minimal prime in this family.
* Case (9,7):
** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes)
**** None of 947, 977, 997, 9477, 9977 are primes.
* Case (9,B):
** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes)
**** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes)
***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime)
***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime)
**** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes)
***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes)
****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime.
****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime)
****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime.
**** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B
***** None of 99B, 999B, 979B, 9799B, 9979B are primes.
* Case (A,1):
** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes)
**** For the A{A}{0}1 family:
***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form AA{0}1 is AA000001
***** The smallest prime of the form AAA{0}1 is AAA0001
***** The smallest prime of the form AAAA{0}1 is AAAA1
****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc.
**** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime.
* Case (A,5):
** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes)
*** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes)
**** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form A{0}65 is A00065
* Case (A,7):
** A7 is prime, and thus the only minimal prime in this family.
* Case (A,B):
** AB is prime, and thus the only minimal prime in this family.
* Case (B,1):
** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes)
**** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1
***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1
****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime)
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1
****** The smallest prime of the form B{A}01 is BAA01
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
**** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1
***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1
****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1
******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1
******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes.
****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1
******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime)
***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes)
****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B04{A}1 is B04A1
****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime)
**** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1
***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1
****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1
******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime.
******* The smallest prime of the form B{B}AA1 is BBBAA1
****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1
******* The smallest prime of the form B{B}AA1 is BBBAA1
**** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes)
***** The smallest prime of the form B{B}B1 is BBBB1
***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form B{B}BAA1 is BBBAA1
* Case (B,5):
** B5 is prime, and thus the only minimal prime in this family.
* Case (B,7):
** B7 is prime, and thus the only minimal prime in this family.
* Case (B,B):
** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes)
*** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B
**** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B
***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime.
***** For the B{0,B}9B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes)
******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B
******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 &gt; 1, 10^n+5 ≥ 10^1+5 = 15 &gt; 1, this factorization is nontrivial), thus cannot be prime.
***** For the B{0,B}99B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes)
******* The smallest prime of the form B{0}99B is B00099B
******* The smallest prime of the form B{B}99B is BBBBBB99B
***** For the B{0,B}999B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes)
******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes)
******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime)

== Examples of families which can be ruled out as contain no primes &gt; ''b'' ==

It is not known if this problem is solvable:

Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''?

It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime &gt; ''b'' or not. We use two different heuristic strategies to show that such families contain no primes &gt; ''b''.

In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;.

Examples of first strategy: (we can show that the corresponding numbers are &gt; all elements in ''S'', if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial)

* In base 10, all numbers of the form 4{6}9 are divisible by 7
* In base 6, all numbers of the form 4{0}1 are divisible by 5
* In base 15, all numbers of the form 9{6}8 are divisible by 11
* In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5}
* In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3}
* In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5}
* In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13}
* In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17}
* In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13}
* In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13}

Examples of second strategy: (we can show that both factors are &gt; 1, if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial)

* In base 9, all numbers of the form {1} factored as difference of squares
* In base 8, all numbers of the form 1{0}1 factored as sum of cubes
* In base 9, all numbers of the form 3{8} factored as difference of squares
* In base 16, all numbers of the form 8{F} factored as difference of squares
* In base 16, all numbers of the form {F}7 factored as difference of squares
* In base 9, all numbers of the form 3{1} factored as difference of squares
* In base 16, all numbers of the form {4}1 factored as difference of squares
* In base 16, all numbers of the form 1{5} factored as difference of squares
* In base 16, all numbers of the from {C}D factored as ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;
* In base 16, all numbers of the form B{4}1 factored as difference of squares

Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are &gt; all elements in S, and for the part of the second strategy, both factors are &gt; 1, if n makes corresponding numbers &gt; b, thus these factorizations are nontrivial)

* In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd
* In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even

== Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes &gt; ''b'' ==

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-powers factorization

=== 1{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 1{0}3 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== 1{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 1{0}5 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 5: Finite covering set {5}

=== 1{0}6 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== 1{0}7 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 7: Finite covering set {7}

=== 1{0}z ===

(none)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

(none)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {1} ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with ''r''&gt;1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2))

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' == 0 mod 2: Finite covering set {2}

=== 1{2} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 1{3} ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361)
* ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865)

=== 1{4} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization

=== 1{z} ===

(none)

=== 2{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 2{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {2}1 ===

* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 2{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 3{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {3}1 ===

* ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121)

=== 3{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 4{0}1 ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 4{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {4}1 ===

* ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225)

=== 4{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 5{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 5{z} ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}
* ''b'' = 6''m''&lt;sup&gt;2&lt;/sup&gt; with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014)

=== 6{0}1 ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== 6{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 7{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 7{z} ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993)
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Difference-of-cubes factorization

=== 8{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022)
* ''b'' = 467: Finite covering set {3, 5, 7, 19, 37}
* ''b'' = 722: Finite covering set {3, 5, 13, 73, 109}
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Sum-of-cubes factorization
* ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2

=== 8{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 9{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 9{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{0}1 ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{z} ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 307: Finite covering set {5, 11, 29}
* ''b'' = 901: Finite covering set {7, 11, 13, 19}

=== C{0}1 ===

* ''b'' == 1 mod 13: Finite covering set {13}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 296, 901: Finite covering set {7, 11, 13, 19}
* ''b'' = 562, 828, 900: Finite covering set {7, 13, 19}
* ''b'' = 563: Finite covering set {5, 7, 13, 19, 29}
* ''b'' = 597: Finite covering set {5, 13, 29}

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

(none)

=== {y}z ===

(none)

=== z{0}1 ===

(none)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3))
* ''b'' = 4''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt; (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}t ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 7: Finite covering set {7}

=== {z}u ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== {z}v ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 5: Finite covering set {5}

=== {z}w ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== {z}x ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== {z}y ===

* ''b'' == 0 mod 2: Finite covering set {2}

== Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) ==

Format: base (length)

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

(none)

=== 1{0}2 ===

(none)

=== 1{0}3 ===

(none)

=== 1{0}4 ===

53 (13403)

113 (10647)

=== 1{0}z ===

113 (20089)

123 (64371)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

(none)

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

208 (26682)

607 (11032)

828 (19659)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

201 (31276)

222 (52727)

227 (36323)

327 (135983)

425 (11231)

710 (24112)

717 (37508)

719 (13420)

=== {1} ===

152 (270217)

184 (16703)

200 (17807)

311 (36497)

326 (26713)

331 (25033)

371 (15527)

485 (99523)

629 (32233)

649 (43987)

670 (18617)

684 (22573)

691 (62903)

693 (41189)

731 (15427)

752 (32833)

872 (10093)

932 (20431)

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

(none)

=== 1{z} ===

107 (21911)

170 (166429)

278 (43909)

303 (40175)

383 (20957)

515 (58467)

522 (62289)

578 (129469)

590 (15527)

647 (21577)

662 (16591)

698 (127559)

704 (62035)

845 (39407)

938 (40423)

969 (24097)

989 (26869)

=== 2{0}1 ===

101 (192276)

206 (46206)

218 (333926)

236 (161230)

257 (12184)

305 (16808)

467 (126776)

578 (44166)

626 (174204)

695 (94626)

752 (26164)

788 (72918)

869 (49150)

887 (27772)

899 (15732)

932 (13644)

=== 2{z} ===

432 (16003)

=== 3{0}1 ===

912 (132174)

=== 3{z} ===

72 (1119850)

212 (34414)

218 (23050)

270 (89662)

303 (198358)

312 (51566)

422 (21738)

480 (93610)

513 (38032)

527 (46074)

566 (23874)

650 (498102)

686 (16584)

758 (15574)

783 (12508)

800 (33838)

921 (98668)

947 (10056)

=== 4{0}1 ===

107 (32587)

227 (13347)

257 (160423)

355 (10990)

410 (144079)

440 (56087)

452 (14155)

482 (30691)

542 (15983)

579 (67776)

608 (20707)

635 (11723)

650 (96223)

679 (69450)

737 (269303)

740 (58043)

789 (149140)

797 (468703)

920 (103687)

934 (101404)

962 (84235)

=== 4{z} ===

14 (19699)

68 (13575)

254 (15451)

800 (20509)

=== 5{0}1 ===

326 (400786)

350 (20392)

554 (10630)

662 (13390)

926 (40036)

=== 5{z} ===

258 (212135)

272 (148427)

299 (64898)

307 (26263)

354 (25566)

433 (283919)

635 (36163)

678 (40859)

692 (45447)

719 (20552)

768 (70214)

857 (23083)

867 (61411)

972 (36703)

=== 6{0}1 ===

108 (16318)

129 (16797)

409 (369833)

522 (52604)

587 (24120)

643 (164916)

762 (11152)

789 (27297)

986 (21634)

=== 6{z} ===

68 (25396)

332 (15222)

338 (42868)

362 (146342)

488 (33164)

566 (164828)

980 (50878)

986 (12506)

1016 (23336)

=== 7{0}1 ===

398 (17473)

1004 (54849)

=== 7{z} ===

97 (192336)

170 (15423)

194 (38361)

202 (155772)

282 (21413)

283 (164769)

332 (13205)

412 (29792)

560 (19905)

639 (10668)

655 (53009)

811 (31784)

814 (17366)

866 (108591)

908 (61797)

962 (31841)

992 (10605)

997 (15815)

=== 8{0}1 ===

23 (119216)

53 (227184)

158 (123476)

254 (67716)

320 (52004)

410 (279992)

425 (94662)

513 (19076)

518 (11768)

596 (148446)

641 (87702)

684 (23387)

695 (39626)

785 (900326)

788 (11408)

893 (86772)

908 (243440)

920 (107822)

962 (47222)

998 (81240)

1013 (43872)

=== 8{z} ===

138 (35686)

412 (12154)

788 (11326)

990 (23032)

=== 9{0}1 ===

248 (39511)

592 (96870)

=== 9{z} ===

431 (43574)

446 (152028)

458 (126262)

599 (11776)

846 (12781)

=== A{0}1 ===

173 (264235)

198 (47665)

311 (314807)

341 (106009)

449 (18507)

492 (42843)

605 (12395)

708 (17563)

710 (31039)

743 (285479)

744 (137056)

786 (68169)

800 (15105)

802 (149320)

879 (25004)

929 (13065)

977 (125873)

986 (48279)

1004 (10645)

=== A{z} ===

368 (10867)

488 (10231)

534 (80328)

662 (13307)

978 (14066)

=== B{0}1 ===

710 (15272)

740 (33520)

878 (227482)

=== B{z} ===

153 (21660)

186 (112718)

439 (18752)

593 (16064)

602 (36518)

707 (10573)

717 (67707)

=== C{0}1 ===

68 (656922)

219 (29231)

230 (94751)

312 (21163)

334 (83334)

353 (20262)

359 (61295)

457 (10024)

481 (45941)

501 (20140)

593 (42779)

600 (11242)

604 (17371)

641 (26422)

700 (91953)

887 (13961)

919 (45359)

923 (64365)

992 (10300)

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

(none)

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

38 (136212)

83 (21496)

113 (286644)

188 (13508)

401 (103670)

417 (21003)

458 (46900)

494 (21580)

518 (129372)

527 (65822)

602 (17644)

608 (36228)

638 (74528)

663 (47557)

723 (24536)

758 (50564)

833 (12220)

904 (13430)

938 (50008)

950 (16248)

=== z{0}1 ===

202 (46774)

251 (102979)

272 (16681)

297 (14314)

298 (60671)

326 (64757)

347 (69661)

363 (142877)

452 (71941)

543 (10042)

564 (38065)

634 (84823)

788 (13541)

869 (12289)

890 (37377)

953 (60995)

1004 (29685)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

53 (21942)

124 (16426)

175 (31626)

188 (22036)

316 (48538)

365 (25578)

373 (24006)

434 (10090)

530 (11086)

545 (12346)

560 (15072)

596 (12762)

701 (12576)

706 (10656)

821 (13536)

833 (17116)

966 (14820)

983 (11272)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}y ===

317 (13896)

== Bases 2≤''b''≤1024 which have these families as unsolved families ==

Unsolved families are families which are neither primes (&gt;''b'') found nor can be ruled out as contain no primes &gt; ''b''

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608)

1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000)

1{0}3: 646, 718, 998 (length limit: 2000)

1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000)

1{0}5

1{0}6

1{0}7

1{0}8

1{0}9

1{0}A

1{0}B

1{0}C

1{0}D

1{0}E

1{0}F

1{0}G

1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000)

1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000)

10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000)

11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000)

{1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000)

{1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000)

11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z})

{1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000)

{1}z

1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000)

1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995

1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000)

1{z}: 581, 992, 1019 (length limit: ≥100000)

2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000)

2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000)

2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000)

{2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000)

2{z}: 588, 972 (length limit: ≥100000)

3{0}1: 718 (length limit: ≥100000)

3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000)

3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000)

{3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000)

3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000)

4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000)

4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000)

{4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000)

4{z}: 338, 998 (length limit: ≥100000)

5{0}1: 308, 512, 824 (length limit: ≥100000)

5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000)

6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000)

6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000)

7{0}1: (none)

7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000)

8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000)

8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000)

9{0}1: 724, 884 (length limit: ≥100000)

9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000)

A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000)

A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000)

B{0}1: 560, 770, 968 (length limit: ≥100000)

B{z}: 263, 615, 912, 978 (length limit: ≥100000)

C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000)

C{z}

D{0}1

D{z}

E{0}1

E{z}

F{0}1

F{z}

G{0}1

{&amp;#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000)

{&amp;#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143)

&amp;#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000)

${&amp;#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2)

x{z}

y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000)

z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000)

{y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000)

{z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326)

zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z})

{z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000)

{z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1)

{z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000)

{z}k

{z}l

{z}m

{z}n

{z}o

{z}p

{z}q

{z}r

{z}s

{z}t

{z}u

{z}v

{z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000)

{z}x: (none)

{z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000)

== List of lengths for quasi-minimal primes in some simple families ==

[https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024]

NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases &lt;=7, family {z}x in bases &lt;=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime &gt; base) in this base)

RC: this family can be proven to only contain composite numbers (only count numbers &gt; base)

unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers &gt; base)

Background color: red for title (bases or families), green for length &gt; 10000, orange for 2500 &lt; length ≤ 10000, white for length ≤ 2500, cyan for &quot;RC&quot;, pink for &quot;NB&quot;, yellow for &quot;unknown&quot;.

Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families.

== References ==

* [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem]

* [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are &gt; ''b'' for bases ''b''≤16]

* [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary]

* [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia]

* [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime&gt;base is not required) in The Prime Puzzles &amp; Problems Connection]

* [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles &amp; Problems Connection]

* [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16]

* [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime&gt;base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29]

* [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime&gt;base is not required, search limits of lengths: 10000 for all b]

* [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime&gt;base is not required]

* [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime&gt;base is not required]

* [http://www.prothsearch.com/sierp.html the Sierpinski problem]

* [http://www.prothsearch.com/rieselprob.html the Riesel problem]

* [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem]

* [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030]

* [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100]

* [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem]

* [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024]

* [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project]

* [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project]

* [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count]

* [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty]

* [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers]

* [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares]

* [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software]

* [https://sourceforge.net/projects/openpfgw/ pfgw software]

* [http://jpenne.free.fr/index2.html llr software]

* [http://www.ellipsa.eu/public/primo/primo.html PRIMO software]

* [https://primes.utm.edu/prove/index.html website for primality proving]

* [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!]

* [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime&gt;base is not required]

* [https://primes.utm.edu/primes/lists/all.txt top proven primes]

* [http://www.primenumbers.net/prptop/prptop.php top PRPs]

* [http://factordb.com online factor database, including many primes which are minimal primes in a small base]</text>
      <sha1>6osyra7ml3no4bnoyxbr04c20bb8u7u</sha1>
    </revision>
    <revision>
      <id>2412071</id>
      <parentid>2412070</parentid>
      <timestamp>2022-08-05T07:58:52Z</timestamp>
      <contributor>
        <ip>2402:7500:901:2B90:A16F:2B1F:1CD3:7BC3</ip>
      </contributor>
      <comment>/* Condensed table */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1961623" xml:space="preserve">A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime &gt; ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime &gt; 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 &gt; 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 &gt; 10 by changing the order.

(using A−Z to represent digit values 10 to 35)

For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')).

I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E&lt;sub&gt;800873&lt;/sub&gt;, and base 36 has quasi-minimal prime P&lt;sub&gt;81993&lt;/sub&gt;SZ).

Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to:

* Prove that all elements in ''S'', when read as base ''b'' representation, are primes &gt; ''b''.

* Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are &gt; ''b'', are composite.

* Prove that all primes &gt; ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are &gt; ''b'', are composite).

e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to:

* Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes &gt; 10.

* Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are &gt; 10 are composite.

* Prove that all primes &gt; 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers &gt; 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite).

==Condensed table==

{|class=wikitable
|''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form of largest known quasi-minimal prime base ''b''
|-
|2||1||11||2||3
|-
|3||3||111||3||13
|-
|4||5||221||3||41
|-
|5||22||10&lt;sub&gt;93&lt;/sub&gt;13||96||5&lt;sup&gt;95&lt;/sup&gt;+8
|-
|6||11||40041||5||5209
|-
|7||71||3&lt;sub&gt;16&lt;/sub&gt;1||17||(7&lt;sup&gt;17&lt;/sup&gt;−5)/2
|-
|8||75||4&lt;sub&gt;220&lt;/sub&gt;7||221||(4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||1161||3×9&lt;sup&gt;1160&lt;/sup&gt;+10
|-
|10||77||50&lt;sub&gt;28&lt;/sub&gt;27||31||5×10&lt;sup&gt;30&lt;/sup&gt;+27
|-
|11&lt;sup&gt;*&lt;/sup&gt;||1068||57&lt;sub&gt;62668&lt;/sub&gt;||62669||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10
|-
|12||106||40&lt;sub&gt;39&lt;/sub&gt;77||42||4×12&lt;sup&gt;41&lt;/sup&gt;+91
|-
|13&lt;sup&gt;*&lt;/sup&gt;||3195~3197||80&lt;sub&gt;32017&lt;/sub&gt;111||32021||8×13&lt;sup&gt;32020&lt;/sup&gt;+183
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||19699||5×14&lt;sup&gt;19698&lt;/sup&gt;−1
|-
|15||1284||7&lt;sub&gt;155&lt;/sub&gt;97||157||(15&lt;sup&gt;157&lt;/sup&gt;+59)/2
|-
|16&lt;sup&gt;*&lt;/sup&gt;||2346~2347||4&lt;sub&gt;72785&lt;/sub&gt;DD||72787||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15
|-
|17&lt;sup&gt;*&lt;/sup&gt;||10408~10428||F70&lt;sub&gt;186767&lt;/sub&gt;1||186770||262×17&lt;sup&gt;186768&lt;/sup&gt;+1
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||6271||12×18&lt;sup&gt;6270&lt;/sup&gt;+221
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||6271||16×20&lt;sup&gt;6270&lt;/sup&gt;+13
|-
|21&lt;sup&gt;*&lt;/sup&gt;||13375~13395||CF&lt;sub&gt;479147&lt;/sub&gt;0K||479150||(51×21&lt;sup&gt;479149&lt;/sup&gt;−1243)/4
|-
|22&lt;sup&gt;*&lt;/sup&gt;||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||22003||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||8134||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49
|-
|30&lt;sup&gt;*&lt;/sup&gt;||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||34206||25×30&lt;sup&gt;34205&lt;/sup&gt;−1
|-
|36&lt;sup&gt;*&lt;/sup&gt;||35257~35263||P&lt;sub&gt;81993&lt;/sub&gt;SZ||81995||(5×36&lt;sup&gt;81995&lt;/sup&gt;+821)/7
|}

&lt;sup&gt;*&lt;/sup&gt; Data assumes the primality of the [[w:probable prime|probable prime]]s.

Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes &gt; 10&lt;sup&gt;25000&lt;/sup&gt;), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below.

Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists))

Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists))

==Data for quasi-minimal primes==

===Base 2===

11

===Base 3===

12, 21, 111

===Base 4===

11, 13, 23, 31, 221

===Base 5===

12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013

===Base 6===

11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041

===Base 7===

14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331

===Base 8===

13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447

===Base 9===

12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011

===Base 10===

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027

===Base 11===

12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, 2888882883, 2888888883, 3555555509, 3577777077, 3700000001, 4000000005, 40000005AA, 5377777707, 5555505553, 555555580A, 600000A999, 7000100047, 7000704777, 7007777107, 7057777777, 7070007774, 7077707774, 7077777074, 7100000704, 7470000041, 7701000004, 7707077774, 7770707774, 7777707074, 8888822883, 9555555503, 9900000979, 9999770007, A000144444, A900000066, A999999971, 10000000477, 33333333337, 44444444447, 44444444777, 55A55555552, 60000008883, 68888888306, 68888888883, 70000003999, 70000007717, 70004777777, 70477777777, 77007770004, 77700000477, 77707000704, 77707770074, 77707777774, 77777077774, 77777770004, 83000000006, 97000000999, A0000000001, A0014444444, A4777777771, 100000000057, 305007777777, 305777777777, 370000000007, 377777770007, 377777777107, 700000007474, 707077000074, 707077777774, 707777777717, 770000010004, 771007000007, 777070700004, 777700000704, A95555555555, A99999777777, 1000000003007, 40000000A0041, 58A5555555555, 7004400000004, 7700000000104, 7707000007047, 7707707000004, 7777007000004, 7777700000004, 7777770077704, 7777777710077, 9977777777717, A000000014444, A044444444441, A144444444411, 40000000000401, 45557777777777, 4555AAAAAAAAAA, 59077777777777, 70007777777771, 70077070000074, 70700000004777, 77000007700704, 77700000700047, 77777777770704, 88888888830006, 90900000000799, A0000044444441, 300000000005777, 302000000000002, 55555555A555552, 700000000000174, 770000000000474, 771700000000007, 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7707777777777777777704, 7900000000000000000005, 9777777777777777777707, A999999999999999999999, 10000000000000000000747, 970000000000000000000777, 999900000000000000000007, 3577777777777777777777777, 5555555555555555555555A52, 7000000000000000000777771, 7000000000000000077777771, AAAAAAAAAAAAAAAAAAA000058, 10000000000000000000000044, 77700000000000000000000008, 500777777777777777777777777, 777777777777777777777770774, 855555555555555555555055555, A44444444444444444444444441, 1500000000000000000000000007, 40000000000000000000000000041, 440000000000000000000000000001, 70000000000000000000000000007771, 999999999999999999999999999999991, 95555555555555555555555555555555553, 1900000000000000000000000000000000001, 7777777777777777777777777777777777474, 7777777777777777777777777777777777704, 10000000000000000000000000000000000000307, 50777777777777777777777777777777777777707, 475777777777777777777777777777777777777777, 555555555555555555555555555555555555555A2A, 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===Base 12===

11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077

===Base 13===

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===Base 14===

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===Base 15===

12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797

===Base 16===

11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD

===Base 17===

12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 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====Additional known quasi-minimal primes (not necessarily the next)====

570&lt;sub&gt;51310&lt;/sub&gt;1, 49&lt;sub&gt;111333&lt;/sub&gt;, 970&lt;sub&gt;166047&lt;/sub&gt;1, F70&lt;sub&gt;186767&lt;/sub&gt;1

===Base 18===

11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, 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===Base 20===

13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, GGJ0J, GGJJJ, GJ00J, GJGJJ, GJJJ7, H00EH, H024B, H04B1, H0E4B, H0F41, H0H11, H0HEH, H4E0B, H6I11, HAAA3, HAAG3, HAGA3, HB44B, HBBBB, HBHHH, HGGA3, HGGG3, HH001, HH061, HH3AH, HH6F1, HHG63, HHIA3, HI011, HIII3, I0029, I02I9, I085H, I0A41, I0F29, I2229, I2929, I2I29, I2I99, I2II9, I33A3, I3EHH, I422B, I424B, I4EEB, I4IEB, I80G1, I9IA9, IA099, IA309, IA841, IA939, IA9A9, IAF09, IAF99, IAI99, IAIA9, IB0A1, IB4IB, IB54B, IBB4B, IBBIB, IBE0H, IBH0H, IC929, ICC29, IF841, IFAG1, IGA81, IHB4B, II0A9, II42B, II44B, IIB5B, III4B, IIIA3, IIIIB, J00G1, J00JB, J0601, J06D1, J06I1, J0861, J08B1, J0B01, J0CGJ, J0E0B, J0GA1, J0GG1, J0I2B, J0J2B, J0JIB, J55CJ, J60A1, J60G1, J6D81, JA777, JA7A7, JAJG1, JAJJ1, JAJJ7, JB00B, JB08B, JC00J, JC0GJ, JEECJ, JGGGJ, JGGJJ, JJ0B1, JJ0JB, JJ55J, JJAG1, JJAJ1, JJC0J, JJE0B, JJEEB, JJGG1, JJGGJ, JJGJJ, JJJ01, JJJEB, JJJJB, 17555J, 175EEJ, 1E0007, 1JE55J, 1JJJ5J, 20005H, 2000HH, 222I29, 22E889, 22EE89, 24222B, 2999I9, 29III9, 2A000H, 2B0I0H, 2I2999, 2I9I29, 2III29, 333AF3, 336IA3, 36A3F3, 36I3A3, 388E8H, 3AF333, 3EHHHH, 3HEHHH, 3HHEHH, 40008B, 40054B, 40405B, 40448B, 415E5J, 44440B, 44452B, 44524B, 44E08B, 45444B, 4AAAE7, 4AEEE7, 4E4447, 4GEEE7, 4I544B, 4IEEEB, 500EFJ, 500FCJ, 5055FJ, 50AFFJ, 50EEFJ, 50EF5J, 50F5FJ, 50FFCJ, 52224B, 54EEEB, 5558AH, 555CAJ, 5585AH, 55F55J, 55FFFJ, 5888AH, 588A5H, 5E555J, 5F055J, 5F05CJ, 5F5F5J, 5FFFCJ, 5JEEEB, 60006D, 6000F1, 6000G3, 6001G1, 600A63, 600G1D, 600G6D, 603A33, 606G6D, 608001, 608F89, 608IF9, 60A333, 60A363, 60AAA3, 60CCCD, 60D04D, 60D0DD, 60F8I9, 60FF89, 60GCG3, 636AF3, 63A333, 63A3F3, 63AFF3, 63IIA3, 64AAAD, 660089, 660A33, 660DDD, 660F89, 662289, 666A33, 666D4D, 66A6A3, 66AF33, 66AF63, 66D0DD, 66FF89, 6A33F3, 6AFF33, 6AG6A3, 6AGA63, 6D000D, 6D004D, 6D0D4D, 6F6689, 7000G7, 7060DD, 70AAAD, 70D0CD, 733C33, 7366C3, 73AAA3, 7600DD, 760CDD, 766333, 76664D, 766FC3, 76CCDD, 76DD4D, 770G07, 771JEJ, 7771EJ, 777481, 7777I1, 777F81, 77C7JJ, 77EC77, 77EEC7, 77EJJJ, 77GCG7, 77JECJ, 7A7E77, 7AAAG3, 7AAF63, 7AEEE7, 7C7GG7, 7CCCAD, 7CCCG7, 7CGJJJ, 7D000D, 7J000D, 7JCGGJ, 80005B, 8000G1, 8000I1, 800GA1, 800I01, 80GGA1, 84405B, 84454B, 888EAH, 88E8AH, 8A4441, 8B8EAH, 8E8IF9, 8I0AG1, 902229, 909A09, 909C29, 909FA9, 90IC29, 9222I9, 92I999, 944EE7, 94AEE7, 977GG7, 97AEE7, 97GGG7, 990I29, 999I29, 99I299, 99IIA9, 9A9447, 9A9AF9, 9A9FF9, 9CII29, 9EEEE7, 9FA9A9, 9I9C29, 9I9I29, 9ICI29, 9IIIA9, A1555J, A1E55J, A333C9, A336A3, A33889, A33F63, A36663, A444E7, A4AAE7, A555FJ, A666A3, A6AAA3, A7A7E7, A7AE77, A944E7, A9A9F9, A9AAA9, AA3389, AA9AF9, AAA099, AAA3C9, AAA7E7, AAA939, AAACG9, AAAFF9, AAAI99, AAAIC9, AAE4E7, AAG9A9, AAGAA3, ADE55J, AF6663, AF9FF9, AGA963, AGG6A3, AH4441, AI000H, AI99C9, AI9AA9, AII099, B0004B, B0054B, B0080B, B00H0H, B04IIB, B05B8B, B0A6I1, B0BB8B, B0BIBB, B0E0IH, B0HA0H, B0I0HH, B0I4IB, B0IIBB, B888AH, BB058B, BB05BB, BB080B, BB0I4B, BBB04B, BBB08B, BBIB8B, BE88AH, BEHA0H, BEHHAH, BHEA0H, BHEHAH, BHHA0H, BIE00H, C007A7, C00GG7, C00JA7, C00JAJ, C00JC7, C0C7A7, C0JAC7, C0JJC7, C3G999, CAJJ77, CC0629, CC0929, CCC007, CCCCC7, CCCCG3, CCCGG7, CCCI29, CCI029, CD4GGJ, CE0009, CEFF69, CEFFF9, CEI609, CII029, CIII29, CJACC7, CJAJC7, CJJJA7, D000D7, D000J7, D004CD, D0D007, DCCDC7, DDD04D, DDDAI1, DEJJJJ, DG000D, DG5E5J, DGGG5J, DJ00CJ, DJEEE7, DJJJ5J, E00097, E00A0H, E00AA9, E02289, E028I9, E029I9, E055AJ, E0774J, E0777J, E07JCJ, E0AF5J, E0BI8B, E0E4CJ, E0EA5J, E0ECJJ, E0FFF9, E0H0HH, E0HA0H, E0IF29, E0J5AJ, E0JJAJ, E29299, E444E7, E4EEE7, E4IIIB, E66289, E66629, E66909, E66F69, E69609, E69999, E7774J, E77C77, E7C7C7, E8B8AH, E90029, E90229, E90909, E90FF9, E92299, E94EE7, E97EE7, E990F9, E99I09, E9F669, E9I029, EA8AF9, EAAAI9, EAIAA9, EAJJJJ, EBIBIB, EC7CC7, EE08F9, EE408B, EE8IF9, EEAF5J, EEE84B, EEEC07, EEEC77, EEEE0B, EEF55J, EEFFFJ, EEJ5AJ, EFE009, EFEF09, EFF669, EFFFF9, EGGGG9, EIF669, F004AJ, F00A41, F00JA1, F04481, F055CJ, F0A15J, F0EI09, F0G01J, F0J0GJ, F0J0JJ, F0JJ5J, F14441, F68F89, F68IF9, F800G1, F8E8I9, F99299, FA0009, FA1JJJ, FAAIA9, FAII09, FAII99, FCC929, FCE229, FCE609, FCEF29, FE68F9, FE8FA9, FF0E09, FF68F9, FF9FA9, FFC4AJ, FFC929, FFCE09, FFE009, FFEE89, FFEF09, FFFA99, FFFFFJ, FFFGJJ, FG1JJJ, FI99A9, FIA999, FIAAA9, FIAII9, FIIA09, G00093, G000EJ, G000G9, G00781, G007G3, G00C09, G07GC3, G09033, G0C903, G0CGG3, G0G903, G0G933, G0GCC3, G0GGC9, G30003, G36003, G5000J, G5E0EJ, G60303, G6AGG3, G6GAA3, G7AAA3, G7G363, G90AA9, G90C63, G9AAA9, GAAAA3, GAAGG3, GC0003, GCCC93, GCCCC3, GCG903, GDEEE7, GE0007, GG0CG3, GGA8G1, GGC6G3, GGCCG3, GGCGC9, GGG0I9, GGG363, GGG6G3, GGG75J, GGG8G1, GGG963, GGGAA3, GGGCG3, GGGCG9, H044EB, H0BIBB, H0E00B, H0F011, H0IIEB, H2444B, H400EB, H44IEB, H4EEEB, HBB4IB, HE000B, HEE40B, HEEE4B, HEHHAH, HF0011, HF4441, HH3HEH, HHGAA3, HHGAG3, HHHAG3, HHHEAH, HI0001, HIBIBB, HIEBBB, I0000H, I000A9, I002HH, I00A09, I00A99, I00AG1, I00AI9, I00H3H, I02999, I03HEH, I09AF9, I0A009, I0E0HH, I0I299, I0IIA9, I4445B, I4I45B, I90I29, I99029, I99A09, I9I029, IA3999, IB000H, IB00EH, IB0HEH, IB5B8B, IF2999, II0929, II92I9, II99A9, II9C29, IIAFF9, IIF299, III5BB, III8BB, J0000B, J00081, J000IB, J0010D, J006A1, J00BI1, J00I8B, J00IA1, J05EEJ, J06G81, J0C0JJ, J0E0CJ, J0EEEB, J0J0EB, J5EAEJ, J5EEAJ, J608I1, JE505J, JEEEEB, JJ0001, JJ0JCJ, JJ0JGJ, JJAJC7, JJJAE7, JJJBI1, JJJI8B, 14GGGG1, 1J5555J, 1JGGGG1, 205555H, 20I000H, 222222B, 2929999, 2BI000H, 3333A33, 3333G09, 333A3A3, 333AAA3, 336AAA3, 336AF33, 33II3A3, 363AF33, 3GIIII9, 400445B, 404454B, 404800B, 415555J, 440045B, 440080B, 442222B, 444004B, 444422B, 444444B, 4444G47, 444GGG7, 444I4IB, 44AAA47, 44I44IB, 44I4IIB, 4AAAAA7, 4EEE4E7, 4I4454B, 5000AFJ, 525555H, 555552H, 555555J, 555585H, 58BBBBB, 5BB8BBB, 5E0E55J, 5EBBBBB, 60003A3, 60008I9, 6000DDD, 6006DDD, 60080A1, 60080I1, 600A3A3, 600D8I1, 600F841, 60CCDDD, 60CGGG3, 60DDD0D, 60DDDDD, 60GGG63, 60GGGC3, 66600A3, 66603A3, 66608I9, 6666089, 66666A3, 6666C29, 6666F89, 666AFF3, 66DDDDD, 6800001, 6AGGGG3, 6CCCC29, 6G6AAA3, 6GAGGG3, 6GCCC63, 6GGG633, 700000D, 70000CD, 7000EC7, 70070C7, 700EEC7, 7070EC7, 70EEEC7, 76666CD, 77333A3, 7733A33, 775EE5J, 777A777, 777CGJJ, 777GE5J, 77A7777, 77JCJJJ, 77JJJCJ, 77JJJJJ, 7A77777, 7A777A7, 7AAAA63, 7AAAAAD, 7C00007, 7E77JCJ, 7EE7JCJ, 7EEC007, 7EEEEC7, 7G5E55J, 7GE555J, 7GJJJJJ, 7J6666D, 9000029, 9000299, 9009I29, 900I2I9, 90FAAA9, 90I0299, 90I0I29, 90I9929, 90II029, 90II299, 90II2I9, 94444E7, 97777E7, 9902I99, 99902I9, 9992929, 9992999, 9999929, 99I0C29, 99I9029, 99I92I9, 99III29, 9AAAF09, 9AFFFF9, 9I0AAA9, 9IAAA09, 9II0I29, 9II9929, 9III029, 9III2I9, 9IIIC29, A5F555J, A777E77, A994447, A9AAF99, AA00089, AA6A6A3, AAA0009, AAA3999, AAA9A99, AAA9AA9, AAAAA47, AAAAA77, AAAAA99, AAAAG99, AAAAIA9, AAAD447, AAAGAC9, AAGAAAD, AAGAAC9, AF00009, AGAAA99, AGAGG63, AGGGA63, AI0C999, AJ777A7, B000B8B, B000HEH, B000I0H, B00I00H, B0B805B, B0HHHEH, B0IHE0H, B0III8B, B8B000B, BB8BBBB, BBB8B0B, BEH0HHH, BHHHEHH, C00000J, C000C07, C066629, C0AJJJJ, C0C0A47, C0G000J, C0JAJJJ, C0JJAJJ, C0JJJJJ, C6CCG33, CC004A7, CC00A47, CC6CG33, CC92229, CCCC7G7, CCCGGG3, CE66609, CEF6669, D00007D, D00007J, D00071J, D000C0J, D000I01, D00700D, D00C007, D00C0C7, D00G00D, D0C00C7, D0CDCC7, D444447, D7JJJJJ, DC0000J, DDD7DDD, DDDCCC7, DJ0000J, DJ000EJ, DJJJJ0J, DJJJJJJ, E000009, E00084B, E000889, E000909, E000HAH, E008AF9, E04000B, E0AA8F9, E0CJJJJ, E0HHHHH, E40400B, E6660F9, E666609, E669969, E699669, E7CCC07, E996669, E99I669, E9FAAA9, EB8BBBB, EBBII8B, EBH0HHH, EE092I9, EE44IIB, EE4E4E7, EEBB8BB, EEC0007, EEE0IBB, EEE0IIB, EEEBB8B, EEEE44B, EEJCJJJ, EF00F09, EF0555J, EFAAAA9, EFF0009, EFF00F9, EFF60F9, EI0AAA9, EI660F9, EIIBB8B, EJ0CJJJ, EJ5055J, EJJ0CJJ, F0E00F9, F0E88I9, F0JJJJ1, F299929, F4G555J, F5555CJ, F5555FJ, F6668I9, F8GGGG1, F929999, F999C29, FA84441, FAAA9A9, FCE00F9, FCEF009, FCFEE29, FE00F09, FEE2889, FEF0009, FFAA9A9, FFCEE29, FFFF4AJ, FIII9A9, FIIIAI9, FJJJJ81, G000009, G0000J7, G000303, G000AA9, G000CC3, G000GG3, G003003, G0090C3, G00CG03, G00D007, G00G0G3, G00GG03, G00GG93, G080001, G0G00I9, G0G09C3, G0G0C93, G0G0GG9, G0GG093, G0GGC63, G0GGGG9, G44GG47, G6AAGA3, G999999, GAA9AA9, GCCCGG9, GCGGCC9, GCGGG03, GEE4447, GEEEE47, GG0C007, GG0GC09, GG0GGG3, GG6AAA3, GG6AGA3, GG6GAG3, GGEEE47, GGG00C9, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 21===

12, 18, 1A, 1G, 1K, 21, 25, 2B, 2H, 2J, 34, 38, 3A, 3G, 3K, 45, 4D, 4H, 4J, 52, 54, 58, 61, 65, 6B, 6D, 72, 74, 7A, 7G, 7K, 85, 8B, 8D, 92, 94, 98, 9A, A1, AD, AH, AJ, B2, B8, BA, BK, C5, CB, CH, CJ, D4, D8, DA, DK, ED, EH, EJ, F2, FG, G1, GB, GD, GH, H2, HA, HG, I1, I5, IB, IJ, J2, JA, JK, K1, KB, KD, KJ, 111, 115, 11H, 135, 13H, 13J, 151, 15B, 15H, 164, 175, 17B, 17D, 17J, 191, 19B, 19D, 19H, 1B1, 1B5, 1BB, 1BJ, 1D5, 1DD, 1DJ, 1E4, 1F1, 1F5, 1FD, 1FH, 1HB, 1HD, 1I4, 1JD, 1JH, 1JJ, 22D, 232, 29G, 29K, 2DG, 2F4, 2FK, 31H, 322, 33D, 351, 355, 35B, 35J, 362, 371, 37B, 37D, 37H, 37J, 39B, 39J, 3B5, 3BD, 3BH, 3D1, 3D5, 3DB, 3DD, 3DH, 3E2, 3FJ, 3HD, 3HH, 3HJ, 3J1, 3JB, 3JJ, 414, 481, 49K, 4B4, 4BG, 4C1, 4EB, 4F4, 4F8, 4FA, 4FK, 50G, 51B, 51D, 51H, 531, 535, 53D, 53J, 551, 56A, 56G, 56K, 575, 57J, 595, 59H, 5B1, 5B5, 5BB, 5CA, 5CG, 5CK, 5F1, 5FB, 5FJ, 5GA, 5GG, 5HH, 5IA, 5J5, 5JD, 5JH, 60H, 632, 66H, 66J, 678, 68J, 69G, 6F8, 6FA, 6GJ, 6H8, 6HK, 6IH, 6J4, 6JG, 6KH, 711, 71B, 71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, GGJ, GJ4, GK5, H0K, H15, H1B, H1J, H31, H3D, H3H, H48, H51, H55, H5J, H6K, H75, H84, H88, H91, H95, H9D, H9H, HBD, HC4, HC8, HDJ, HF5, HFB, HFH, HHD, HHJ, HI4, HI8, HJ5, HJB, HKK, I0D, I2D, I6H, I78, I8H, I9K, IFA, IFK, IID, J08, J1J, J31, J35, J3J, J44, J5H, J68, J6G, J71, J7B, J7D, J7H, J8G, J95, J9D, JBD, JBH, JBJ, JCG, JDB, JDH, JE4, JE8, JEG, JF5, JFD, JFJ, JG4, JGG, JH1, JH5, JHB, JHH, JI4, JJ1, JJ5, K0H, K5G, K6H, K9K, KG5, KH4, KKH, 101B, 103D, 1044, 1055, 10DH, 10FB, 10H1, 10H5, 10J1, 110J, 11CD, 11EB, 11ID, 130D, 13C1, 13EB, 141B, 1444, 1471, 14FB, 1505, 15CD, 167H, 16FJ, 16HH, 176H, 17C1, 17IH, 190J, 1BF4, 1BID, 1C04, 1C3D, 1CD1, 1D6H, 1DE1, 1DEB, 1E1B, 1E71, 1E95, 1EFB, 1F0B, 1F4B, 1FB4, 1H01, 1H05, 1H41, 1H6H, 1H6J, 1HC1, 1HE1, 1HF4, 1IDH, 1IHH, 1JE5, 1JF4, 20CD, 20D2, 20ID, 2278, 22F8, 239D, 23FD, 2478, 2708, 27FD, 293D, 29DD, 2A78, 2C0D, 2CF8, 2F0A, 2F9D, 2FAA, 2FDD, 2GFA, 2IF8, 3015, 301D, 301J, 3031, 3035, 305D, 30F1, 30FB, 30FD, 30HB, 30I2, 3105, 31C1, 329D, 330H, 336J, 33E1, 33EB, 33IH, 35CD, 363H, 390H, 39E5, 39ID, 3B0J, 3B6J, 3BC1, 3BEB, 3CC2, 3CF1, 3CFD, 3D32, 3E3B, 3E91, 3EF1, 3EFB, 3F05, 3FC1, 3FE1, 3FEB, 3H01, 3H0B, 3HC1, 3I3H, 3I9D, 404B, 4078, 40E1, 411B, 4131, 4171, 41FB, 4331, 4391, 43BB, 43FB, 43I2, 4432, 444B, 44E1, 477B, 4931, 4971, 499B, 49CG, 4A32, 4A4B, 4A78, 4A9G, 4B1B, 4B31, 4B3B, 4BFB, 4C78, 4E32, 4F1B, 4FB1, 5011, 509D, 50BD, 50DB, 50DD, 50EK, 50FD, 50H1, 50HJ, 50IK, 50JJ, 5101, 5105, 51E1, 530B, 53EB, 53IH, 550D, 550H, 55AB, 55BG, 55DG, 55E5, 55KH, 563H, 56JJ, 5701, 570H, 57EB, 57ID, 57IH, 590D, 59C1, 59CD, 5A3B, 5A7B, 5AGK, 5AIG, 5AKG, 5B0D, 5B9G, 5BCD, 5BGJ, 5BIH, 5C91, 5CD1, 5D0D, 5D0J, 5D6H, 5D6J, 5DC1, 5DG5, 5DIH, 5DJG, 5E0A, 5E11, 5E3B, 5E71, 5EAK, 5EEA, 5EKK, 5F0H, 5F9K, 5G0K, 5G9J, 5GF5, 5GIK, 5H0B, 5H5K, 5H9K, 5HE5, 5HID, 5I3H, 5I7D, 5IKG, 5J6J, 5JE1, 5KAG, 5KEK, 5KFH, 5KH5, 60FK, 63FH, 67JH, 69JH, 6E9K, 6F0K, 6F64, 6FJH, 6H04, 6HE4, 6HFJ, 6IF4, 6J9H, 6J9J, 7015, 7031, 703B, 7055, 705J, 7088, 709B, 709D, 709H, 70DJ, 70I8, 710J, 716J, 730B, 73C1, 73E5, 7501, 769H, 76FJ, 76JJ, 770D, 77E1, 77E5, 77EB, 7888, 78FH, 78H1, 78JH, 790H, 79CD, 79IH, 7B0J, 7B6H, 7C31, 7C91, 7C9D, 7E35, 7E75, 7E91, 7EF1, 7F01, 7F78, 7FCD, 7FH8, 7FIH, 7FJ8, 7H81, 7HC1, 7HF8, 7J6J, 7J78, 7JC1, 7JF8, 7JH8, 7JJ8, 809K, 80H8, 81H1, 81HH, 81J1, 839H, 83FH, 869K, 871H, 8731, 8771, 8788, 879J, 87F1, 87FJ, 87JH, 888H, 8911, 8931, 896K, 8A78, 8A9K, 8CC1, 8E9K, 8F3H, 8F9H, 8FF1, 8FHH, 8FJH, 8G9K, 8GJ8, 8H3J, 8H7H, 8H8K, 8HEK, 8HF1, 8HFJ, 8HH1, 8HJJ, 8HK8, 8J7J, 8J91, 8JC8, 8JFH, 8JJH, 8K78, 8K8H, 8KH8, 901J, 9035, 903B, 9071, 90CG, 90GG, 90JD, 9101, 91E5, 9301, 93E1, 93EB, 95CD, 95DG, 95FK, 967J, 969H, 96HH, 96HJ, 96JJ, 9705, 97C1, 9905, 995K, 999G, 99CD, 99EB, 99K5, 9B5G, 9BE5, 9BID, 9C0G, 9C11, 9C71, 9C9D, 9CEG, 9CGG, 9D05, 9D5G, 9DDG, 9DE5, 9E35, 9E3B, 9E71, 9EB5, 9EEG, 9EF1, 9EFB, 9F0J, 9F6H, 9F9K, 9FC1, 9FIH, 9FKH, 9G35, 9G95, 9GIK, 9GJJ, 9GKG, 9H0H, 9H5K, 9H6H, 9HKH, 9I6K, 9I9H, 9IDD, 9J6J, 9JEB, 9JGJ, 9JID, 9JJG, 9K35, 9K95, 9KEG, A33B, A3F5, A432, A478, A53B, A5IK, A788, A795, A878, A975, A99B, A9CG, A9GG, AAAB, AB44, AB55, ABFB, AC32, AC78, ACFA, AE32, AE5A, AE9K, AEEB, AEFA, AEFK, AF75, AFCA, AFCK, AFEA, AFFB, AG9G, AGFA, AI32, B064, B091, B09H, B0B1, B0D5, B0DH, B0EG, B0FB, B0FH, B0H5, B0HD, B114, B141, B14B, B1B4, B1E1, B411, B431, B444, B4B1, B59G, B5G5, B5JG, B67H, B67J, B69H, B6FH, B6FJ, B6HH, B6JH, B96H, B9DG, B9G5, BB14, BBB4, BC1D, BC31, BCDD, BD05, BD0H, BD6H, BDC1, BDE5, BDIH, BE0G, BE31, BEBB, BEE4, BEIG, BF0B, BF0H, BF6J, BFEB, BFIH, BG0G, BG55, BG95, BGG4, BGGG, BGIG, BGJ5, BH0B, BHEB, BIEG, BIGG, BIHH, BJ01, BJ0J, BJC1, BJF4, BJID, C041, C081, C09G, C0E1, C0FK, C104, C131, C171, C1C4, C1D1, C2F8, C2FA, C39D, C3F1, C478, C6FK, C7FD, C8C1, C90G, C911, C971, C99D, CAFK, CC78, CC81, CCE1, CCF8, CD02, CD31, CDFD, CE01, CE32, CF0A, CF48, CF64, CF68, CF6A, CF91, CFAK, CFDD, CFEA, CFEK, CIF4, CK32, CK9G, CKFA, D01D, D01H, D07B, D0DB, D0F1, D0FD, D0HD, D0HH, D10D, D10H, D16H, D1EB, D1IH, D23D, D3E1, D56H, D63J, D6CG, D6IG, D70B, D70D, D70J, D99G, D9GJ, D9ID, D9JG, DBC1, DBCD, DBE1, DBJG, DC11, DC9D, DCF1, DCFD, DCGG, DCIG, DD0H, DD6J, DDGJ, DDID, DDJG, DE31, DE75, DFC1, DFID, DFIH, DG22, DG6G, DG75, DG9J, DGCG, DGIG, DH6H, DHE5, DI6G, DICG, DIDH, DIFD, DIFH, DIGG, DJCD, DJDG, DJG5, E032, E05A, E0EB, E11B, E2F8, E31B, E331, E375, E395, E3C2, E3F1, E3F5, E3FB, E3I2, E53B, E555, E591, E59B, E5EG, E6F4, E755, E775, E89K, E915, E935, E93B, E96K, E9FB, EA32, EAEB, EAFK, EB44, EB95, ECFA, ECFK, EE32, EEEB, EEFK, EF08, EF35, EF48, EF55, EF64, EF68, EF75, EF95, EFB1, EFCA, EFCK, EFEA, EFIA, EFKK, EGF4, EIF4, EKF4, EKF8, EKFA, F008, F051, F075, F095, F09H, F0CA, F0CK, F0DD, F0FB, F0H1, F0IA, F0JJ, F105, F10B, F10D, F16J, F1B4, F1EB, F1ID, F1J4, F36H, F468, F47B, F491, F4C8, F501, F50B, F648, F66A, F69H, F705, F778, F864, F8F1, F90D, F99K, F9C1, F9HK, FA6A, FA75, FA7B, FA95, FAEA, FAFB, FB0D, FB14, FBCD, FBE1, FBJ4, FC1D, FC68, FC9D, FCAA, FCCK, FCD1, FCEA, FD6H, FDIH, FE0K, FE1B, FE64, FE95, 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====Additional known quasi-minimal primes (not necessarily the next)====

40&lt;sub&gt;47333&lt;/sub&gt;9G, CF&lt;sub&gt;479147&lt;/sub&gt;0K

===Base 22===

11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, IHH, IIJ, IJ7, IL7, J07, J55, J69, J8J, J99, J9J, JA5, JAH, JE7, JEH, JHH, JI9, JJ5, JJ9, JJH, JK7, K2J, K51, K5D, K75, K81, KA1, KB1, KB7, KBJ, KD1, KEJ, KG5, KIF, KJ5, KKD, KKJ, L0D, L47, L7F, L97, LAF, LD3, LD9, LDD, LEF, LGD, LI7, LJ7, LJJ, LLD, 104H, 10D5, 1205, 12B5, 140H, 1433, 144H, 14AH, 14B3, 16ED, 1AIH, 1B43, 1DD5, 1DDD, 1E6D, 1EGD, 1G05, 1GDD, 1GED, 1GGD, 1HB3, 1HHH, 1IAH, 200L, 2015, 2051, 20A1, 20DJ, 20GD, 20IL, 21B5, 21DD, 220D, 226D, 228D, 22B5, 22G5, 22K5, 22KD, 2555, 2557, 2581, 25C1, 26A1, 26B1, 2725, 2755, 2801, 2861, 288D, 28B1, 28KD, 2AA1, 2B25, 2B51, 2BB1, 2C81, 2D6D, 2DA1, 2DDJ, 2DGD, 2G0J, 2GB5, 2GDD, 2GGJ, 2I0L, 2I6L, 2ICL, 2J05, 2JK5, 2K07, 2K2D, 2K55, 2K6D, 2KB5, 2KI7, 2L2D, 2L8D, 2LK7, 302L, 30G3, 320L, 32IL, 332L, 33G3, 33G9, 36AF, 36EF, 382J, 388J, 39EL, 3AAF, 3BBJ, 3CG9, 3E2J, 3E6F, 3E6L, 3EEF, 3FAF, 3G69, 3GI3, 3GI9, 3IG9, 3LEL, 3LG3, 4025, 404H, 4063, 4075, 40AH, 40B5, 40B7, 40GH, 4225, 4363, 4447, 444H, 447H, 4487, 44B7, 44H7, 4525, 4555, 45B5, 4663, 4777, 47GH, 4807, 4B0H, 4BB5, 4BG3, 4EBH, 4G4H, 4GB3, 4HH3, 4I03, 4I63, 4IGH, 5069, 5077, 50KD, 5255, 52IJ, 5455, 5477, 5507, 5527, 556F, 557F, 5587, 56EF, 56FF, 56GD, 56KD, 5771, 57C1, 57EF, 5807, 580J, 589J, 58E7, 58IJ, 58K7, 5905, 5945, 5957, 5969, 5989, 598J, 5999, 59EJ, 59K7, 5AEF, 5B25, 5B55, 5BA5, 5BI7, 5BIJ, 5BK5, 5BK7, 5C21, 5EAF, 5EE7, 5F09, 5F6F, 5F95, 5FKD, 5G6D, 5G89, 5I09, 5I0F, 5I27, 5I6F, 5I89, 5IB7, 5IBJ, 5IFF, 5J77, 5K07, 5K25, 5K27, 5K6F, 5K87, 5KK7, 6013, 60A1, 60AF, 60EL, 6281, 62B1, 62KD, 63G9, 6403, 6643, 66EL, 66G9, 68KD, 69B3, 6A01, 6A0F, 6AAF, 6AFF, 6B21, 6B8D, 6BBD, 6BK1, 6D03, 6D43, 6D69, 6D6F, 6D93, 6D9F, 6E21, 6E81, 6E8D, 6EBD, 6ECL, 6ELL, 6FD9, 6GBD, 6GG9, 6IEF, 6K21, 6K8D, 6KBD, 6KE1, 6L43, 6LB3, 6LDF, 6LEL, 6LG9, 700F, 7027, 7057, 70EH, 70LF, 710H, 71G5, 7207, 7225, 7255, 727L, 7505, 7507, 755F, 75EF, 75F5, 75I7, 766F, 7681, 76CF, 7771, 7781, 77IL, 7861, 7A0H, 7A4H, 7BB5, 7C6F, 7EEH, 7EHL, 7FG5, 7G45, 7HEL, 7I0L, 7I27, 7IE7, 7IGH, 7K07, 7K61, 7K77, 7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, H0D1, H0HJ, H0LL, H1HH, H20D, H22D, H26D, H2I7, H2K7, H30L, H3KJ, H40H, H447, H4G3, H4H3, H4HH, H66L, H6ED, 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5IIIIIIIIIIIIIIIIIIIIIIIIIIF, D555555555555555555555550555, EEAAAAAAAAAAAAAAAAAAAAAAAAAF, HHHHHHHHHHHHHHHHHHHHHHHHEEBH, K66666666666666666666666666F, LLLLLLLLLLLLLLLLLLLLLLLLEEB7, D5555555555555555555555555A55, GGGGGGGGGGGGGGGGGGGGGGGGGGGG3, GIG0000000000000000000000000H, HH00000000000000000000000001H, K0000000000000000000000005KEF, 5BBBBBBBBBBBBBBBBBBBBBBBBBBBBD, HB0000000000000000000000000001, K000000000000000000000000505EF, L7777777777777777777777777772L, 2000000000000000000000000000CB1, C8CCCCCCCCCCCCCCCCCCCCCCCCCCCC9, IKKKKKKKKKKKKKKKKKKKKKKKKKKKKFF, JE0000000000000000000000000000J, K000000000000000000000000000261, A0000000000000000000000000004I4H, HD000000000000000000000000000001, K000000000000000000000000000EC01, K0FFFFFFFFFFFFFFFFFFFFFFFFFFFFCF, D0002222222222222222222222222222D, FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL2L, I700000000000000000000000000000GH, K00000000000000000000000000000E61, 20000000000000000000000000000000JJ, DD5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL0KE7, 44HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, F0000000000000000000000000000000000000000000000000000000L89, J000000000000000000000000000000000000000000000000000000JJIJ, K222222222222222222222222222222222222222222222222222222222DD, KCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCF, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKCF, D55KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, K000000000000000000000000000000000000000000000000000000000000077, 22222222222222222222222222222222222222222222222222222222222222222D, K000000000000000000000000000000000000000000000000000000000000008IJ, DFA00000000000000000000000000000000000000000000000000000000000000005, CC4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJE0J, 777777777777777777777777777777777777777777777777777777777777777777777777EK7, 6000000000000000000000000000000000000000000000000000000000000000000000000000000043, KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKB5, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEEH, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 50000000000000000000000000000000000000000000000000000000000000000000000000000000002C1, K0000000000000000000000000000000000000000000000000000000000000000000000000000000000055EF, H700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000K1, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0I7, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH, 5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, A400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, DKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, 4HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH, E0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000071, 7LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIL, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLK77, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEI7, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLIB3, I7G00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000H, 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77EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEK7, JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJKJ, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777772L, BKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK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KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK5

===Base 24===

15, 17, 1D, 1H, 1J, 1N, 25, 2B, 2D, 2J, 2N, 31, 37, 3B, 3H, 41, 45, 47, 4B, 4D, 4H, 57, 5B, 5H, 5J, 65, 67, 6D, 6J, 6N, 75, 7B, 7D, 7N, 81, 85, 87, 8J, 97, 9B, 9D, 9H, 9N, A1, AB, AH, AN, B5, B7, BD, BH, BJ, C5, CJ, CN, D1, D5, DJ, E1, EB, ED, EH, EN, F7, FD, FJ, FN, G5, GD, GH, H1, HB, HD, HN, I1, I7, IB, IH, J1, J5, J7, JB, JN, K7, KB, KJ, KN, L5, LH, LJ, MD, MJ, N5, NB, NH, NJ, 101, 10B, 111, 1F1, 1FB, 1GB, 1LB, 201, 221, 22H, 261, 271, 277, 28H, 2A7, 2C7, 2G7, 2H7, 2L1, 2L7, 2MH, 305, 30D, 30J, 33N, 34N, 35D, 35N, 38D, 395, 3A5, 3AJ, 3CD, 3DD, 3DN, 3E5, 3EJ, 3GJ, 3IJ, 3JJ, 3K5, 3KD, 3ND, 43N, 44N, 49J, 4EJ, 4GJ, 4GN, 4NN, 50N, 535, 54N, 551, 55N, 5C1, 5CD, 5E5, 5K1, 5KD, 5LN, 5M5, 5N1, 601, 60B, 61B, 66H, 68B, 691, 6CH, 6FH, 6GB, 6HH, 6MH, 70H, 70J, 711, 761, 771, 77H, 77J, 78H, 7C7, 7CH, 7FH, 7G7, 7H7, 7HH, 7IJ, 7JJ, 7K1, 7M1, 7M7, 80D, 82H, 83N, 88D, 88H, 8AD, 8CD, 8DB, 8DD, 8DH, 8DN, 8GB, 8KD, 8MB, 8MH, 905, 911, 921, 935, 955, 99J, 9AJ, 9G1, 9JJ, 9K5, 9L1, 9M5, A0J, A3J, A95, AA7, AD7, AE5, AG7, AGJ, AI5, AIJ, AJD, AL7, ALD, B01, B0N, B11, B61, B6B, B8N, B91, BIN, BL1, BLN, BNN, C1B, C21, C27, C2H, C3D, C61, C8H, C91, CA7, CB1, CBB, CC7, CCB, CCD, CDD, CFB, CG1, CGB, CK1, CL1, CMB, CMH, D0B, D3D, D3N, D4N, D6B, D6H, D7H, D8B, D8N, DAD, DCD, DCH, DDH, DDN, DG7, DGB, DID, DMN, DND, E05, E4J, EA7, EEJ, EF5, EGJ, EI5, EJJ, EM5, EM7, F01, F21, F51, F8H, F95, FC1, FF1, FFB, FKH, FM5, G0B, G0N, G11, G3N, G6B, G77, G7J, G8B, G8N, G91, GA7, GBB, GC7, GFB, GG1, GGJ, GGN, GK1, GL1, GLN, GMN, GN1, GNN, H0J, H2H, H3J, H4J, H77, HA5, HA7, HE5, HFH, HIJ, HJJ, HKH, HL7, HMH, I0N, I3D, I3J, I3N, I4N, I5D, I95, IA5, IAJ, IE5, IEJ, IF5, IGJ, IJD, IK5, IKD, J0D, J4J, J8D, JAD, JDH, JEJ, JFH, JHH, JKD, JMH, K35, K6H, KCD, KFH, KH5, KLD, KM1, L01, L0B, L0D, L0N, L61, L6B, L8D, LA7, LC7, LDD, LF1, LG7, LGB, LGN, LID, LK1, LKD, LL1, LLB, LLD, LMN, LNN, M0H, M11, M21, M4N, M71, M91, M95, MA5, MA7, MBN, MC7, MF1, MF5, MFB, MFH, MG7, MI5, MIN, ML1, ML7, MLB, MMH, N01, N21, N4N, N71, N8N, NC1, ND7, NE7, NG1, NID, NK1, NL7, NMN, NN7, 11CB, 11MB, 1291, 12G1, 16C1, 16CB, 16K1, 186B, 18CB, 19K1, 1BK1, 1C8B, 1K91, 1KC1, 1KL1, 1L21, 1LC1, 1LM1, 1M61, 1M8B, 1MG1, 206H, 20CH, 20M7, 21C1, 21M1, 2207, 260H, 26KH, 2991, 2C6H, 2CC1, 2CM1, 2F11, 2FHH, 2MC1, 2MK1, 2MM1, 308N, 30GN, 30IN, 30LN, 30MN, 333J, 33JD, 33LD, 343J, 344J, 35I5, 380N, 393J, 394J, 3A3D, 3FI5, 3IMN, 3J3D, 3JID, 3L3D, 3L8N, 3M0N, 3M55, 3NGN, 404J, 408N, 40LN, 434J, 44AJ, 4ILN, 4JAJ, 4L8N, 5091, 5095, 50F1, 50I5, 51L1, 5211, 5291, 52G1, 53ID, 53MN, 5595, 55AD, 56F1, 588N, 58MN, 58ND, 5961, 5991, 5A5D, 5AAD, 5F91, 5GF1, 5GIN, 5I05, 5I55, 5I8D, 5IDD, 5IDN, 5IIN, 5IMN, 5KI5, 5M61, 5M8N, 5N3N, 602H, 6211, 62F1, 62G1, 66C1, 66FB, 66M1, 66MB, 6B21, 6BM1, 6BMB, 6C6B, 6CF1, 6CLB, 6FG1, 6K21, 6K2H, 6KG1, 6KKH, 6L21, 6LCB, 6LM1, 6MB1, 6MBB, 6MG1, 6MK1, 7001, 7027, 7207, 726H, 739J, 793J, 79C1, 7A4J, 7A9J, 7AE7, 7C01, 7CC1, 7FL1, 7G21, 7G9J, 7GAJ, 7GC1, 7HGJ, 7J2H, 7J6H, 7MKH, 800B, 800H, 804N, 806H, 808N, 80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, NK3D, NK8D, 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5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, 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EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEG7, 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===Base 30===

11, 17, 1B, 1D, 1H, 1N, 1T, 21, 27, 2B, 2D, 2J, 2N, 2T, 37, 3B, 3D, 3H, 3J, 3N, 47, 4B, 4H, 4J, 4T, 51, 57, 5D, 5H, 5N, 5T, 61, 6B, 6D, 6H, 6J, 71, 7D, 7H, 7J, 7N, 7T, 81, 8B, 8H, 8N, 8T, 91, 97, 9B, 9D, 9N, A7, AB, AD, AH, B1, B7, BH, BJ, BN, BT, C7, CD, CJ, CN, CT, D7, DB, DJ, DT, E1, EB, ED, EJ, EN, ET, F7, FB, FD, FH, FT, G7, GB, GJ, GN, GT, HB, HD, I1, I7, IH, IN, IT, J1, J7, JH, JN, JT, K1, K7, KD, KH, KJ, L1, LB, LD, LH, LN, LT, M1, MD, MH, MN, N1, NB, NJ, NT, O7, OD, OJ, ON, P1, P7, PB, PJ, PN, Q7, QH, QT, R1, RB, RD, RH, RJ, RT, SD, SH, SJ, SN, T7, TB, TD, TH, 10J, 15J, 1IJ, 1JJ, 1LJ, 1MJ, 1QJ, 22H, 29H, 2EH, 2GH, 30T, 331, 33T, 36T, 39T, 3A1, 3C1, 3G1, 3KT, 3MT, 3OT, 3S1, 3T1, 40D, 40N, 431, 44D, 46N, 48D, 4AN, 4DD, 4DN, 4F1, 4FN, 4GD, 4ID, 4PD, 4S1, 50J, 58J, 59J, 5CB, 5FJ, 5IB, 5IJ, 5JB, 5MB, 5MJ, 5OB, 5SB, 607, 63T, 687, 6E7, 6KT, 6L7, 6M7, 6MT, 6NN, 6QN, 6R7, 6S7, 6ST, 6TT, 70B, 77B, 787, 7KB, 7M7, 7MB, 7SB, 807, 80D, 80J, 84D, 85J, 877, 88J, 89J, 8DD, 8FJ, 8ID, 8IJ, 8JJ, 8M7, 8MJ, 8PD, 8QD, 8R7, 90H, 93T, 95J, 99H, 99J, 9AJ, 9AT, 9EH, 9GH, 9HH, 9HJ, 9JJ, 9MJ, 9OH, 9OT, 9PH, 9ST, 9TT, A01, A0T, A1J, A31, A6N, A6T, AAJ, AAN, AFN, AKN, AKT, ALJ, AMJ, AMT, AO1, AOT, AQ1, AQN, ARN, AT1, B5B, BBD, BCB, BDD, BID, BOB, BPD, BQB, C31, C9H, CC1, CCB, CCH, CF1, CH1, CIB, CKB, CMB, D01, D0H, D2H, D41, D4D, D6N, D8D, D9H, DDH, DDN, DGH, DH1, DHH, DID, DKN, DNN, DO1, DOH, DQN, DS1, EEH, EHH, EM7, EOH, EPH, ER7, F0N, F31, F5J, F8J, F9J, FLJ, FO1, FQ1, FQN, FS1, G01, G4D, G8D, GF1, GGH, GO1, GOH, GQD, GS1, H07, H0J, H0T, H1J, H2H, H31, H4N, H8J, HA1, HAJ, HAT, HC1, HE7, HEH, HFN, HGH, HH7, HIJ, HJJ, HKN, HL7, HNH, HPH, HQN, HS1, HTJ, HTN, I4D, I5B, I5J, I8D, IAJ, IDD, IGD, IIJ, IKB, IMB, IMJ, IOB, IPD, IQD, J8J, J9J, JAJ, JBD, JCB, JFJ, JIJ, JJB, JJD, JLJ, JPD, JQB, K3T, K4N, KAT, KBB, KCB, KMB, KNN, KOB, KOT, KQN, KST, KTT, L0J, L5J, L67, LAJ, LJJ, LQJ, LR7, M0J, M3T, M5B, M7B, M87, M9J, MAT, MFJ, MIJ, MJJ, MKB, MMJ, MOT, MQJ, N07, N0H, N67, N6N, N87, NAN, NDH, NGD, NHH, NKN, NN7, NNH, NPH, NQD, NQN, NR7, O01, O0B, O0H, OAT, OC1, OCH, OEH, OF1, OH1, OKT, OMB, OOT, OPH, OQ1, OQB, OS1, OST, P3T, P6T, P9H, PCH, PDH, PEH, PGD, PHH, PHT, PID, PMT, PPT, PQD, PST, PTT, Q5B, Q6N, Q9J, QAJ, QBB, QBD, QC1, QDN, QFJ, QFN, QGD, QJB, QKN, QLJ, QMB, QND, QNN, QO1, QQ1, QQN, QSB, R4N, R77, R87, RAN, RKN, RM7, RNN, RR7, RS7, S0T, S41, S6T, S87, SBB, SC1, SM7, SMT, SOB, SQ1, SR7, SS1, STT, T0J, T6T, T9T, TAN, TFN, TKN, TKT, TNN, TO1, TOT, TPT, TQ1, TQN, TTN, 18AJ, 19FJ, 1AFJ, 1FAJ, 20CH, 20PH, 2C0H, 2CHH, 2COH, 2H0H, 2HCH, 2HHH, 2POH, 2PPH, 3001, 34Q1, 3F41, 3FF1, 3QF1, 3SPT, 3SST, 3TTT, 40A1, 4441, 44KN, 44QN, 4AC1, 4C01, 4CA1, 4DG1, 4GA1, 4GC1, 4NND, 4NRN, 4OA1, 4Q4N, 4QA1, 4QRN, 4RQN, 550B, 555B, 55KB, 5A5J, 5AJJ, 5BKB, 5JQJ, 5KKB, 5QQB, 5QQJ, 604N, 606T, 60FN, 60KN, 60OT, 60PT, 660N, 660T, 664N, 66AT, 66FN, 66KN, 66TN, 6A0N, 6A9T, 6AAT, 6FKN, 6FRN, 6K0N, 6KFN, 6KKN, 6O6T, 6PAT, 6RFN, 6RRN, 6T6N, 6TRN, 7067, 70R7, 75BB, 77R7, 7C5B, 7CQB, 7E67, 7IQB, 7O5B, 7OIB, 7QOB, 7R07, 7RE7, 8667, 88L7, 88S7, 8E87, 8EE7, 8EL7, 8J8D, 8LL7, 8LS7, 906T, 908J, 90FJ, 90QJ, 90TJ, 92CH, 966T, 99MT, 99PT, 9I0J, 9I8J, 9LFJ, 9LIJ, 9M6T, 9MKT, 9MMT, 9PKT, 9QQJ, 9TLJ, A04N, A0QJ, A3AT, A3ST, A4A1, A4G1, A5QJ, A8QJ, A90J, A9PT, AA3T, AA41, AAAT, AAF1, AAG1, AAPT, AC41, ACS1, AF0J, AFC1, AFIJ, AG41, AGA1, AGG1, AI8J, AIJJ, AIQJ, AJ5J, AJJJ, AQ8J, AQJJ, AQQJ, AS3T, AT4N, AT5J, ATST, B04D, B0QD, BBIB, BBKB, BI0B, BK0B, BKIB, BKSB, BMIB, BMSB, BQ4D, BQQD, BSMB, C00B, C05B, C0A1, C0BB, C0EH, C0GH, C0Q1, C0QB, C2OH, C2PH, C441, C4Q1, C50B, CB0B, CBSB, CEGH, CG2H, CGHH, CHHH, CHOH, COGH, COO1, COOB, COOH, CQ01, CQ0B, CQQB, CQS1, CS0B, D00N, D0PD, D44N, DA0N, DCEH, DCG1, DCQ1, DDC1, DDF1, DDPD, DDQ1, DECH, DF4N, DFA1, DFRN, DG0D, DGD1, DGQ1, DNEH, DPDD, DQA1, DQPD, DRFN, DRRN, E20H, E667, E767, E8L7, E8S7, ECGH, EG2H, EH67, EH77, ES67, F001, F01J, F0MJ, F0QJ, F44N, F4G1, F4RN, F6RN, FA0J, FAC1, FAF1, FAIJ, FC01, FCG1, FFAJ, FFC1, FFG1, FFIJ, FGA1, FJJJ, FJQJ, FKAN, FR6N, FRFN, G09H, G0EH, G0HH, G20H, G341, G3Q1, G4G1, G92H, GA41, GAC1, GC0H, GCEH, GCG1, GCPH, GD31, GDA1, GDG1, GDPD, GE2H, GE9H, GGG1, GGGD, GHCH, GHHH, GP0D, GP2H, GPPD, GQ31, GQA1, GQG1, H00N, H0CH, H3ST, H4O1, H667, H66N, H677, H69T, H99T, H9CH, H9KT, H9LJ, H9PT, H9QJ, HH6N, HHF1, HHFJ, HHH1, HHHT, HHLJ, HHOH, HHQ1, HHST, HK6T, HL9J, HLMJ, HMKT, HMMT, HNM7, HNRN, HOPT, HQ01, HQ5J, HR0N, HR6N, HS77, HSKT, HT01, I0BB, I0JD, I0JJ, I98J, I9LJ, I9QJ, ICBB, IF0J, IFFJ, IFQJ, IIBD, IJBB, IJID, IL8J, ILFJ, IQ8J, IQJJ, IQQJ, IS0B, ISQB, J00D, J05J, J0BB, J0MJ, J0OB, J4QD, J50B, J5QJ, JBKB, JBMB, JDGD, JDQD, JGDD, JI0D, JIIB, JISB, JKIB, JKKB, JM0B, JOIB, JQ4D, JQDD, JQID, JQJJ, JQMJ, JSIB, JSMB, K06N, K0KB, K0KT, K0MT, K0PT, K0SB, K5QB, K60T, K66T, K6AN, K6FN, K6PT, K6RN, K96T, K99T, KA0N, KF6N, KFKN, KI0B, KK0T, KK6N, KK6T, KKFN, KKKB, KKKT, KKQB, KKTN, KMMT, KQ0B, KQQB, KR0N, KS5B, L0M7, L8E7, L98J, LFFJ, LI8J, LIFJ, LL87, LL9J, LLIJ, LLM7, LM77, LML7, LMM7, M09T, M0E7, M0IB, M0L7, M0M7, M0TT, M55J, M5LJ, M60T, M69T, M707, M767, M777, M7E7, M7S7, M8LJ, M96T, M9KT, MC0B, MCOB, ME07, MIBB, MJ0B, MJBB, MJSB, MLLJ, MLS7, MM07, MM0T, MML7, MMOB, MMR7, MOBB, MOCB, MP0T, MP9T, MPKT, MQ0B, MQIB, MQOB, MS07, MSPT, MSSB, MT8J, MTTT, N00D, N04D, N04N, N0FN, N0ND, N0RN, N2OH, N92H, NCGH, ND0D, ND4N, NDPD, NEGH, NEL7, NF4N, NFRN, NGCH, NGEH, NHM7, NME7, NML7, NMM7, NN0D, NN4D, NN4N, NNDN, NNID, NNND, NNRN, NP4D, NSL7, NSS7, O00T, O03T, O0MT, O0PT, O2HH, O2OH, O341, O4A1, O4G1, O4O1, O5KB, O90T, OA41, OBSB, OC5B, OCOB, OG2H, OG31, OH9H, OH9T, OHHH, OHTT, OICB, OISB, OM0T, OM6T, OMPT, OMTT, OO2H, OO9H, OOA1, OOCB, OOGH, OOKB, OOO1, OSKB, OTMT, P00T, P04D, P08D, P09T, P0KT, P90T, PAAT, PKKT, PO0T, PP0D, PPPD, Q00B, Q0AN, Q0D1, Q0F1, Q0IB, Q0JJ, Q0MJ, Q0OB, Q55J, Q5QJ, Q88D, QA0N, QA41, QA4N, QDF1, QDPD, QF41, QFA1, QG31, QIQJ, QJ0J, QJ8D, QJDD, QJJJ, QJMJ, QJQD, QKKB, QKQB, QOKB, QOOB, QP0D, QPDD, QQ8D, QQID, QQJD, QQPD, QQQB, QS01, QSA1, R00N, R067, R06N, R0FN, R0L7, R60N, RE67, RFRN, RRFN, RRRN, S00B, S3AT, S3ST, S50B, S5QB, S99T, SAAT, SC0B, SE67, SGG1, SICB, SIQB, SK5B, SKIB, SKKT, SKPT, SKQB, SM0B, SMMB, SMSB, SOA1, SOG1, SPOT, SQCB, SQQB, SS0B, SS67, ST31, STG1, T00N, T03T, T041, T04N, T0AT, T0C1, T0MT, T0ST, T1AJ, T3F1, T4A1, T4RN, T5JJ, T66N, T8AJ, T9FJ, T9QJ, TA3T, TAA1, TAG1, TAIJ, TCS1, TFA1, TFAJ, TFF1, TFFJ, TFG1, TGA1, TI9J, TIJJ, TIQJ, TL8J, TL9J, TM0T, TM5J, TM8J, TMAJ, TMLJ, TMST, TSF1, TSG1, TT01, TT5J, TT8J, TTA1, TTC1, TTFJ, TTG1, TTLJ, TTMJ, 30441, 30O41, 30OO1, 34O41, 3AATT, 3ASAT, 3O441, 3Q041, 40G41, 40GQ1, 40O41, 40OG1, 40OO1, 44001, 440C1, 440G1, 4444N, 44CG1, 44GQ1, 4AAA1, 4COG1, 4DAA1, 4GQ41, 4KRRN, 4OGG1, 4Q041, 4QGG1, 5005B, 500BB, 500KB, 500QB, 50K0B, 50Q0B, 50QKB, 555AJ, 555LJ, 555QJ, 55J5J, 5B00B, 5B0BB, 5J55J, 5J5JJ, 5JJ5J, 5JJJJ, 5K00B, 5QJ5J, 5QK0B, 6006N, 6009T, 600AT, 6066N, 60R0N, 666N7, 6696T, 66OPT, 67767, 6900T, 696PT, 69P0T, 6F6AN, 6O9PT, 6OP0T, 6OP9T, 6P99T, 755QB, 77767, 7CBBB, 7IBIB, 7IIBB, 7IICB, 7QIIB, 8888D, 88E67, 8L887, 8SSE7, 9099T, 90KKT, 90KMT, 90MPT, 9690T, 96P9T, 990KT, 9990T, 9999T, 99K6T, 9FFFJ, 9HKMT, 9HMPT, 9ILLJ, 9KKMT, 9KKPT, 9M00T, 9T8QJ, 9TFIJ, A008J, A00NN, A055J, A0FJJ, A0I0J, A0I9J, A0J0J, A44C1, A555J, A9FFJ, A9QIJ, A9T8J, AFFA1, AFFFJ, AFFJJ, AFFQJ, AJ00J, AP99T, ASA9T, ASFF1, ASP9T, ASSAT, AT3TT, ATFQJ, ATQIJ, ATT9J, ATTQJ, B088D, B08GD, B0GGD, B0SKB, B8GGD, BG00D, BIISB, BISIB, BS0IB, BS0SB, BSSKB, BSSSB, C0041, C0G41, C0H0H, C0PPH, C0S01, C4001, C4AG1, C4OG1, C5BBB, CG00H, CGE0H, CGGA1, CGQ41, COSSB, CP20H, CPGPH, CPP2H, CPPOH, CPPPH, CS001, CS55B, CSSQB, D00GD, D0A4N, D0FAN, D0GDD, D0N0D, DAAA1, DAAC1, DAFG1, DDAA1, DDGGD, DFFAN, DFGC1, DGDGD, DGG31, DGGA1, DH0AN, DPPPH, DQDD1, DQGG1, E0L87, E88E7, EC00H, EC02H, EE867, EE887, EEL87, ELE87, F00IJ, F0441, F0AG1, F0F0J, F0F41, F0FF1, F0GG1, F4041, F44A1, F44C1, F4A41, F64KN, F6K6N, FAAA1, FAFQJ, FCAA1, FF041, FF0F1, FF64N, FF6KN, FFA41, FFA4N, FFF4N, FFFF1, FFFFN, FFKRN, FFQMJ, FIJ0J, FJ00J, FKKKN, FNNFN, FQQ0J, FQQMJ, G00DD, G00DH, G00PD, G02PH, G0DDD, G0PDD, GAAA1, GC2HH, GDDDD, GDGDD, GDPPH, GG0PD, GGCA1, GGCQ1, GGDD1, GH4Q1, GHHG1, GII0D, GQ441, H009H, H00G1, H00H1, H04Q1, H0ANN, H0H01, H0H9H, H0HG1, H0HO1, H0O41, H0OHH, H0QG1, H40G1, H4G41, H4GG1, H60AN, HAN0N, HF0G1, HFF41, HFFMJ, HFFQJ, HGQ41, HH3TT, HH401, HH441, HH4G1, HH55J, HH66T, HH6OT, HH96T, HHA0N, HHANN, HHC0H, HHCHH, HHH0N, HHHCH, HHHHN, HHHNN, HHKMT, HHO41, HHP9T, HHPKT, HHTMT, HHTT1, HK9MT, HKKMT, HLLFJ, HM66T, HM7R7, HMM67, HMM77, HMMM7, HMTST, HO9HT, HOO41, HOOOH, HOT3T, HOTT1, HPO9T, HSO3T, HTGG1, I000J, I009J, I00BD, I00QJ, I00SB, I08QJ, I0I0D, I0IQB, I0JIB, I0Q0B, I0SCB, I0SIB, I0SSB, IB00D, IBBSB, IBISB, ICSSB, II0CB, II0ID, II0SB, IIIBB, IIQ0B, IIQIB, IISIB, IJ0SB, IJJQJ, ILLLJ, IQ00J, IQC0B, IQIQB, IQQIB, ISSCB, J00MB, J08GD, J0IID, J0QQD, J55BB, J55JJ, J5J5J, J5JJJ, J88GD, JB0SB, JBB0B, JBBSB, JDDDD, JG0ID, JGG0D, JIIID, JJ0QJ, JJ5JJ, JQ08D, JQ0QJ, JQQ0D, JQQ5J, JS55B, JSK0B, JSS5B, K000T, K006T, K00TN, K0FAN, K505B, K6T0N, K9KMT, K9KPT, K9P0T, KIQIB, KK00N, KK05B, KK0AN, KK0KN, KKM9T, KS0IB, KS0QB, KSQIB, KTR6N, KTRRN, L0087, L08S7, LE087, LEE87, LL8LJ, LMEE7, LMSE7, LMSS7, M00CB, M00PT, M066T, M0CQB, M0K6T, M0MQB, M0MSB, M0QQB, M0SMB, M0SST, M666T, M900T, MBBSB, MEE77, MEL77, MELE7, MES77, MESE7, MESS7, MI00B, MIICB, MIISB, MIQCB, MK9PT, ML7L7, MLEL7, MLLE7, MLME7, MM677, MMBIB, MMBSB, MMCQB, MME77, MMEE7, MMICB, MMISB, MMK6T, MMKKT, MMM9T, MMMMT, MMMTT, MMQQB, MMSCB, MMSKT, MMSMB, MMSS7, MMTST, MOIIB, MOOIB, MOSIB, MQCQB, MR007, MR667, MRL07, MS7L7, MSEL7, MSK9T, MSL77, MSSL7, MT00T, MTMMT, N0DDD, N4NNN, N7LE7, N7S77, NE2CH, NE9CH, NEC2H, NEE77, NFFNN, NFNFN, NII0D, NL777, NL7L7, NLES7, NLLL7, NLS77, NOG9H, NRR0N, O6P9T, O9H6T, O9HPT, O9P9T, OCBBB, OG441, OGAG1, OGGA1, OHH6T, OHOOH, OKIIB, OKK5B, OKKIB, OO5BB, OOBIB, OOIIB, OSCSB, OT3TT, OT441, OTG41, OTGG1, P00PH, P0D0D, P0G2H, P0OGH, P0PGH, PA99T, PGP0H, POOOH, PP20H, PP88D, PPP0H, Q0001, Q000N, Q001J, Q00ID, Q00PD, Q044N, Q04G1, Q0I0D, Q0PPD, Q40G1, Q444N, Q44RN, Q4AA1, Q8QQJ, QAAS1, QAFF1, QAFG1, QASG1, QDGG1, QFGG1, QI00D, QIICB, QIIQB, QIQCB, QOIIB, QPP4D, QQ08J, QQ0CB, QQC0B, QQI0J, QQJ5J, QSFG1, R00E7, R0E07, R0NE7, R6F6N, R6FFN, REE07, RELE7, RFF6N, RL0E7, RLE07, RLEE7, RLLE7, RQR0N, RR0QN, RRQ0N, S03O1, S0AF1, S0AG1, S0O31, S0QIB, S0SIB, S30F1, S30O1, S7QIB, SA3PT, SAAA1, SAFG1, SASST, SCSQB, SF0G1, SFFF1, SI0SB, SISSB, SKSSB, SSCQB, SSCSB, SSMIB, SSPAT, SSSKT, SSSMB, SSSSB, STAF1, STF01, T0001, T0031, T0AF1, T0AS1, T0G31, T0R6N, T0T31, T0TF1, T3AAT, T40G1, T4CG1, T4G41, T5LLJ, T8LLJ, TA441, TA98J, TAFJJ, TAQ5J, TASST, TATAT, TC401, TCGG1, TFQIJ, TFQJJ, TFQMJ, TG441, TGC41, TI8LJ, TLLMJ, TMMMT, TMTMT, TQ8QJ, TSS3T, TT3AT, TT9IJ, TTAAT, TTQIJ, TTS31, TTS3T, TTT9J, TTTST, 20000H, 200OOH, 3440O1, 3TAAST, 404CQ1, 4KKKKN, 4KKKRN, 4QQQQD, 505BBB, 50BB0B, 6000AN, 6444RN, 66666N, 666O9T, 66999T, 669P9T, 66N777, 6A444N, 6FF66N, 6FFF6N, 6R666N, 766767, 77S677, 7IBBBB, 8888E7, 8LLLLJ, 8SSSL7, 9000MT, 90K90T, 90KP0T, 90M90T, 99000T, 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===Base 36===

11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 2UJ, 2WN, 2XN, 30Z, 337, 33B, 33H, 33P, 34H, 34P, 37D, 38Z, 39H, 3AB, 3AP, 3AZ, 3D7, 3DH, 3FD, 3FZ, 3HD, 3HH, 3KZ, 3L7, 3LZ, 3MB, 3O7, 3OZ, 3PD, 3PP, 3Q7, 3RH, 3S7, 3SZ, 3UP, 3UZ, 3XB, 40P, 43B, 43H, 46V, 48B, 48V, 49B, 4AP, 4BB, 4E1, 4FP, 4HB, 4HH, 4HP, 4IH, 4MB, 4OP, 4OV, 4PH, 4RH, 4VB, 4VH, 4WH, 4WP, 4WV, 4X1, 4XH, 55T, 565, 58Z, 5C5, 5E7, 5ET, 5EZ, 5IN, 5KT, 5L7, 5MP, 5NN, 5O5, 5O7, 5OP, 5R7, 5RT, 5RZ, 5ST, 5SZ, 5TN, 5TZ, 5W7, 5XN, 5YN, 61T, 625, 62J, 64V, 661, 66J, 691, 6AV, 6EV, 6IJ, 6J1, 6K5, 6O1, 6OT, 6QJ, 6QT, 6RV, 6SJ, 6T1, 6TT, 6U5, 6UV, 6VV, 6W5, 6Y1, 6YT, 72D, 737, 77D, 791, 7A1, 7AZ, 7D7, 7GD, 7I1, 7IN, 7LN, 7M7, 7MN, 7N7, 7NN, 7RN, 7RZ, 7WN, 7X7, 7Z1, 801, 80V, 82D, 83B, 841, 84H, 877, 881, 887, 88B, 88V, 88Z, 89H, 89V, 8A1, 8AB, 8B7, 8BH, 8D1, 8DH, 8EB, 8EV, 8GD, 8GZ, 8HD, 8IV, 8L7, 8LZ, 8M1, 8MB, 8MD, 8O7, 8OB, 8Q7, 8QD, 8RD, 8S7, 8SH, 8SZ, 8UZ, 8W7, 8WV, 8XV, 8Y1, 90H, 91J, 93B, 93H, 94V, 981, 98H, 99J, 9AH, 9BB, 9C5, 9IH, 9IV, 9JV, 9KH, 9M1, 9OB, 9QJ, 9R1, 9RH, 9SH, 9UJ, 9VB, 9VJ, 9W5, 9X1, 9Y1, 9YB, 9YJ, 9YV, A25, A3P, A3Z, A4H, A61, A81, A8B, AAB, AAH, ABB, AC5, AEZ, AHP, AK1, AKB, AKH, ALZ, AMB, AO5, AOZ, APH, AQP, AR1, ARV, ARZ, AUV, AV5, AVB, AVV, AWV, AX1, AXB, AZ1, B2J, B4B, B4V, B6V, B87, B8H, B9B, BAB, BAH, BBH, BBV, BE7, BEB, BGJ, BHB, BIJ, BJB, BJT, BLJ, BOB, BOT, BQ7, BRV, BS7, BTT, BTV, BVB, BVJ, BVT, BWV, BX7, BXH, BYH, BYV, C25, C2J, C2N, C55, C65, C6J, C95, CCN, CDD, CED, CF5, CFJ, CGD, CIN, CJD, CO5, CRN, CUJ, CXJ, CXN, D37, D3P, D41, D55, D5P, D6T, D77, D81, D9H, DA1, DCD, DD1, DD5, DDH, DDP, DE7, DEP, DF5, DFD, DG7, DHH, DI1, DK1, DK5, DKT, DO1, DOP, DP1, DPD, DQ7, DR7, DRH, DS7, DTT, DU1, DWD, DX7, DYH, E0P, E2D, E3Z, E41, E4P, E71, E81, E87, E8B, EAZ, EDP, EED, EEN, EEV, EFZ, EGT, EI1, EKZ, ELB, EMB, EMN, EN7, EO1, EOT, EPP, EPZ, EQ7, ERN, EST, ETN, ETV, EUP, EUZ, EVT, EWD, EWN, EX1, EYB, EYN, EZD, EZP, F2J, F3B, F3Z, F4P, F65, F8B, F8Z, FCJ, FD5, FEZ, FFB, FFD, FG5, FGD, FGZ, FIJ, FJ5, FJJ, FLZ, FPD, FQD, FSZ, FXB, G07, G0D, G0Z, G3D, G3Z, G55, G6T, G7D, G7Z, G87, G8Z, GA5, GCJ, GD7, GE7, GET, GG5, GG7, GGT, GGZ, GI7, GJD, GKZ, GL7, GLT, GQT, GRT, GS7, GST, GU5, GUZ, GW5, GXJ, GZ7, GZZ, H3D, H3H, H4H, H9B, HAH, HED, HGD, HHP, HIH, HKH, HKP, HLN, HNH, HOB, HOP, HQP, HRD, HRH, HRN, HSH, HWD, HWH, HWP, HYN, I41, I6J, I71, I7N, I87, I8H, I9H, I9J, IA1, IAV, IE1, IFJ, IG7, IHH, II1, IIH, IIV, IJV, IK1, IL7, ILJ, ILN, IM1, INN, IOV, IRH, IUV, IX1, IXH, IXV, IYJ, J2D, J4V, J55, J61, J6J, J8B, J8V, J95, J9J, J9V, JA5, JBB, JD5, JDT, JF5, JFJ, JGJ, JGT, JIV, JJ1, JJD, JJV, JK5, JKT, JLB, JLT, JMN, JN5, JNJ, JQJ, JQN, JQT, JRD, JTB, JVJ, JVN, JVV, JWN, JY1, JYJ, K0B, K25, K3P, K65, K81, K95, K9H, KAH, KC5, KEP, KEZ, KG5, KHP, KK1, KKT, KLB, KLP, KLZ, KM1, KMB, KMP, KOT, KP1, KQP, KR1, KRT, KRZ, KTT, KW5, KX1, KZB, L0N, L0P, L8Z, L9B, LAZ, LBJ, LEN, LET, LFB, LFZ, LG7, LGZ, LIJ, LJJ, LKB, LKP, LLB, LLP, LLT, LMB, LMN, LN7, LO7, LOT, LPZ, LQT, LRN, LTN, LTT, LWP, LX7, LYN, M01, M0P, M2D, M2N, M37, M3B, M41, M61, M77, M87, M8D, M91, ME1, ME7, MEB, MFB, MFP, MGD, MI7, MKB, MM7, MMN, MN7, MNN, MO7, MOB, MOP, MPP, MQP, MS7, MW7, MXN, MYN, N5N, N65, N87, N8D, N8H, N95, NCD, NCJ, NDH, NE7, NFJ, NG5, NG7, NGJ, NJ5, NN7, NND, NO5, NQD, NQJ, NRN, NU5, NWH, NWN, NXH, NXN, NYJ, O0Z, O25, O37, O3B, O3P, O3Z, O41, O4B, O5Z, O61, O71, O7Z, O81, OA5, OAP, OBB, OBV, OC5, OEZ, OF5, OG7, OKP, OKZ, OMB, OO5, OOZ, OP5, OPP, OQT, OR1, OR7, OS7, OST, OTB, OTZ, OU5, OVV, OW5, OX7, OXB, OXV, OYV, OZ5, OZ7, P01, P0D, P0N, P3P, P4H, P4P, P5N, P65, P6V, P8V, PD1, PED, PEZ, PHH, PHP, PI1, PIN, PLN, PLP, PLZ, POP, PP1, PPH, PPV, PQD, PQN, PRV, PV5, PVH, PVV, PWH, PWP, PX1, PXV, PYH, PYN, Q07, Q0P, Q2N, Q37, Q3P, Q6J, Q6T, Q7D, Q8D, Q9J, QCD, QCJ, QD7, QED, QFP, QGT, QI7, QIN, QJN, QLJ, QMD, QMN, QMP, QND, QNJ, QOT, QQJ, QTN, QWN, QYJ, QYT, R0V, R0Z, R37, R3H, R4H, R77, R7D, R7N, R7Z, R81, R8V, R91, RA1, RCD, RCN, RD1, RED, REV, REZ, RGT, RGZ, RHD, RIV, RKH, RKZ, RLN, RMD, RNH, RO7, RQN, RRD, RRT, RRZ, RS7, RSH, RT1, RTV, RU1, RUZ, RVN, RVT, RW7, RWH, RX7, RY1, S0J, S6J, S87, S9H, SAZ, SCJ, SET, SFJ, SG7, SGZ, SI7, SM7, SO7, SOT, SQJ, SQT, SRH, SSH, STT, SW7, SX7, SXH, SYH, T0N, T0Z, T1D, T1T, T6V, T8Z, TBB, TBV, TCN, TD1, TEV, TGT, TKT, TLB, TM1, TO1, TOB, TQN, TR1, TRD, TT1, TTB, TTN, TUZ, TVB, TVN, TVT, TWD, TWV, TXB, TXV, TYV, U2J, U3Z, U61, U8V, U95, UA1, UC5, UEP, UEZ, UFJ, UG5, UJ5, ULZ, UO5, UOP, UOZ, UQP, URV, URZ, UUJ, UW5, UWV, UXJ, UXV, UYJ, V0H, V25, V4V, V55, V6V, V9J, V9V, VBB, VBJ, VCJ, VET, VGJ, VHP, VIH, VIN, VJJ, VJN, VLT, VMP, VNJ, VOB, VP5, VQJ, VQT, VRT, VRV, VSH, VSJ, VST, VTB, VTN, VUP, VW5, VWN, VXH, VXN, VYB, W07, W25, W3D, W3H, W4P, W4V, W57, W6V, W7D, W8H, W95, W9H, WAH, WAV, WCN, WD7, WDD, WDH, WE7, WEN, WF5, WGD, WHH, WK5, WKH, WN7, WNN, WPP, WPV, WQP, WR7, WRD, WRH, WRN, WS7, WU5, WUP, WVN, WWH, WWP, WX7, WXH, WXN, WYN, WYV, X0J, X2J, X2N, X4B, X4H, X4V, X6J, X87, X8B, X91, X9B, XAV, XCN, XEB, XFB, XHB, XHH, XHN, XKB, XLJ, XNH, XO1, XQ7, XS7, XSH, XUJ, XWN, XWV, XX7, XXH, XXV, Y0N, Y1J, Y1T, Y2N, Y3H, Y61, Y8V, Y91, YBV, YCN, YEB, YFJ, YHB, YHN, YIH, YJN, YJT, YLJ, YLN, YMB, YMN, YNH, YOB, YOV, YRH, YTB, YTN, YTT, YTV, YVB, YVH, YWH, YWV, YXB, YY1, YYJ, YYT, YYV, Z01, Z3D, Z3Z, Z71, Z8B, ZB7, ZBB, ZD5, ZDD, ZDP, ZG7, ZGD, ZGZ, ZKB, ZLP, ZM1, ZO5, ZP1, ZPD, ZQD, ZU1, ZX1, ZXB, ZZD, 102J, 106T, 10ET, 10GJ, 10LJ, 10TD, 12FJ, 12LJ, 13ED, 13GD, 13MD, 192J, 1D0T, 1DET, 1E3D, 1EFD, 1F0D, 1FWD, 1G0J, 1GGJ, 1GIJ, 1GUJ, 1I0J, 1ISJ, 1JOT, 1JTT, 1M0D, 1M3D, 1MFD, 1MWD, 1OKT, 1OOT, 1Q3D, 1QDT, 1QKT, 1QTD, 1QWD, 1SLJ, 1TDT, 1TFD, 1TOT, 1TTD, 1TTT, 1UGJ, 1ULJ, 1WFD, 1WWD, 20CJ, 20FJ, 20G5, 20IN, 20JD, 20MD, 20RN, 20WD, 20YN, 2295, 25RN, 260J, 26LJ, 288D, 28DD, 28FD, 296J, 29G5, 29K5, 29LJ, 2C05, 2C0J, 2C0N, 2CJJ, 2CK5, 2CLJ, 2CSJ, 2D25, 2DC5, 2DDD, 2DW5, 2F0J, 2FDD, 2FMD, 2FSJ, 2G65, 2G95, 2GK5, 2I0N, 2JC5, 2JCJ, 2JWD, 2K05, 2L6J, 2L9J, 2N0N, 2NIN, 2NN5, 2NSJ, 2NW5, 2NYN, 2Q0N, 2QNN, 2R0N, 2RFD, 2RWD, 2SJJ, 2SXJ, 2SYJ, 2W05, 2WC5, 2WD5, 2WMD, 2XCJ, 2YCJ, 2YJJ, 2YRN, 2YSJ, 300D, 302D, 304B, 3077, 3087, 308D, 30BH, 30DD, 30FP, 30G7, 30GD, 30MD, 30QP, 30YH, 320D, 333D, 333Z, 33WD, 344B, 34FB, 373Z, 37E7, 37EZ, 37G7, 3807, 38BB, 38E7, 38G7, 38HB, 38I7, 38X7, 390B, 394B, 399B, 39EB, 39FB, 39KB, 3ASH, 3AWH, 3B8B, 3BG7, 3BLB, 3BR7, 3BW7, 3C8D, 3D0P, 3D2D, 3DKP, 3E3D, 3E77, 3EBB, 3EFB, 3EKB, 3EQP, 3ERD, 3ERZ, 3EZZ, 3F4B, 3FEP, 3G2D, 3GCD, 3GED, 3GM7, 3GR7, 3GWD, 3GZD, 3H0B, 3HEP, 3HKB, 3IAH, 3ISH, 3K4B, 3KBH, 3KEB, 3KKB, 3L4B, 3L8B, 3M3D, 3MEP, 3MG7, 3MWP, 3OFP, 3OKB, 3OMP, 3OOB, 3OYB, 3P8H, 3PXH, 3QDP, 3QQD, 3QRD, 3QWD, 3R3D, 3R3Z, 3R8D, 3RG7, 3RM7, 3RQD, 3RR7, 3RZ7, 3SIH, 3W0H, 3WED, 3WM7, 3WQD, 3WW7, 3WYH, 3X8H, 3XE7, 3XG7, 3XKH, 3Y0B, 3Y4B, 3Y8H, 3YAH, 3YSH, 3YYB, 3Z4B, 3ZFP, 404V, 4061, 40EB, 40I1, 40IV, 40KH, 40M1, 40R1, 40XB, 40XV, 43QP, 4441, 444H, 444P, 4481, 448H, 4491, 449H, 44EB, 44EV, 44K1, 44KB, 44KP, 44M1, 44R1, 44RV, 44SH, 44U1, 44VP, 44YB, 46I1, 46K1, 46M1, 480H, 488H, 48AH, 48I1, 48K1, 48U1, 4991, 49EV, 49I1, 49O1, 49UV, 4A0H, 4A41, 4A4B, 4A8H, 4A91, 4ALB, 4ASH, 4BXV, 4E4B, 4EEP, 4EFB, 4ELP, 4EUV, 4EXB, 4FOB, 4I01, 4I0V, 4I4V, 4I61, 4I81, 4I91, 4IEV, 4IY1, 4K0H, 4K91, 4KI1, 4KKH, 4KKP, 4KUP, 4L3P, 4L4P, 4LAB, 4LMP, 4MA1, 4MK1, 4MMP, 4MP1, 4MU1, 4O0B, 4OFB, 4OKB, 4OM1, 4OU1, 4OYB, 4P41, 4P81, 4PO1, 4PYV, 4QLP, 4QQP, 4RAV, 4RK1, 4RRV, 4S8H, 4U3P, 4U91, 4U9V, 4UIV, 4UUP, 4VAV, 4VLP, 4VVV, 4X0B, 4X0V, 4XAB, 4XBV, 4XVV, 4XYV, 4Y0V, 4Y41, 4Y4H, 4Y4V, 4Y81, 4YAV, 4YEV, 4YIV, 4YK1, 4YKB, 4YM1, 4YR1, 4YUV, 4YYB, 5025, 503Z, 5095, 50G5, 50PN, 50QN, 50WP, 50Z7, 5205, 52RN, 53FP, 53M7, 53PZ, 53X7, 53ZZ, 54PP, 550N, 553P, 554P, 5557, 555N, 557Z, 5595, 55EN, 55G5, 55GZ, 55LZ, 55OZ, 55QN, 55Z7, 572N, 577Z, 5787, 57CN, 57FZ, 57QN, 57S7, 57ZZ, 58I7, 58X7, 5955, 5A95, 5AEP, 5CMN, 5CWN, 5E2N, 5EAP, 5ECN, 5EKP, 5EQN, 5F95, 5FLP, 5FQP, 5G05, 5GK5, 5GQ7, 5I37, 5II7, 5IM7, 5L3P, 5LCN, 5LFP, 5LGT, 5LKZ, 5LUP, 5LZZ, 5M07, 5N55, 5NQ7, 5NX7, 5OAZ, 5OFZ, 5OLT, 5OOT, 5P25, 5P95, 5PG5, 5PGZ, 5PPN, 5PQP, 5PW5, 5Q0N, 5Q87, 5QCN, 5QKP, 5QN7, 5QPN, 5QPP, 5QQT, 5QTT, 5QWP, 5QX7, 5S37, 5SS7, 5T6T, 5TQT, 5U4P, 5UGZ, 5W0P, 5W2N, 5W5P, 5WLN, 5WP5, 5WQN, 5XM7, 5Y0T, 5YOT, 5Z25, 5Z4P, 5Z57, 5ZG5, 5ZLZ, 5ZQ7, 5ZWP, 6005, 600V, 601J, 6055, 606V, 60A1, 60C5, 60GT, 60JJ, 60LJ, 60M1, 60R1, 60VT, 60YV, 610J, 61FJ, 6401, 64K1, 6505, 665T, 6665, 66A5, 66F5, 66G5, 66GT, 66KT, 66V5, 66XV, 68OV, 68YV, 690J, 69G5, 69O5, 69XJ, 69XV, 6A55, 6AA1, 6AA5, 6AF5, 6C9J, 6CA5, 6CGJ, 6CYJ, 6EK1, 6ER1, 6FLJ, 6FXJ, 6G0J, 6G65, 6GGJ, 6GJ5, 6GLJ, 6I81, 6I9V, 6J65, 6J6V, 6JET, 6JST, 6JWV, 6K01, 6KET, 6KST, 6L0J, 6L6T, 6LFJ, 6LRT, 6LXJ, 6MA1, 6MM1, 6MX1, 6O8V, 6OV5, 6R0T, 6RET, 6RKT, 6S6T, 6T8V, 6U01, 6U9J, 6UE1, 6UGJ, 6UJJ, 6ULJ, 6UU1, 6V0T, 6VA5, 6VF5, 6VG5, 6VKT, 6VXJ, 6WWV, 6WXV, 6X41, 6X6V, 6XIV, 6XJV, 6XOV, 6Y0V, 6YCJ, 6YXV, 7001, 702N, 7041, 707N, 7081, 70GZ, 70I7, 70KZ, 70ND, 70O7, 70OZ, 70R7, 70U1, 70UZ, 70WD, 70YN, 71QD, 71WD, 720N, 72CN, 72YN, 733Z, 73ED, 73ZD, 7401, 7441, 74M1, 74O1, 74R1, 7641, 76K1, 76M1, 76R1, 76U1, 76X1, 773Z, 7741, 7771, 7781, 77FZ, 77G7, 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8KRH, 8KU1, 8KYH, 8KZ1, 8L4B, 8LBB, 8MG7, 8MR7, 8O4V, 8OI1, 8OK1, 8OOV, 8OX1, 8RAV, 8RHH, 8RUV, 8RX1, 8U4V, 8UK1, 8V4B, 8V9B, 8VHH, 8VRH, 8VVV, 8VYV, 8W0H, 8WFD, 8X07, 8XAH, 8XR1, 8XU1, 8XYB, 8Y6V, 8Y8H, 8YBB, 8YFB, 8YHH, 8YKB, 8YLB, 8YUV, 8YVV, 8YYH, 8Z1D, 8Z4B, 8Z8D, 8Z91, 8ZCD, 8ZEZ, 8ZFD, 8ZFZ, 8ZK1, 8ZO1, 8ZX7, 90A5, 90BJ, 90BV, 90EB, 90EV, 90GJ, 90IJ, 90J5, 90JB, 90LB, 90O1, 90OV, 90XB, 90XV, 9225, 92K5, 9441, 94BH, 94EB, 94KB, 94XB, 9505, 9601, 960V, 96A1, 96J5, 96K1, 96V5, 96WV, 96XV, 984B, 98AV, 98BV, 98FB, 994H, 998B, 998V, 999B, 99A1, 99A5, 99E1, 99G5, 99O1, 99O5, 99WH, 9A05, 9A55, 9A95, 9AA5, 9ABV, 9AE1, 9AEV, 9AI1, 9AO1, 9AOV, 9B0J, 9B0V, 9B8V, 9B9H, 9BCJ, 9BFJ, 9BSJ, 9BVV, 9CCJ, 9CLJ, 9CSJ, 9E01, 9E4B, 9EA1, 9EAB, 9EAV, 9EK1, 9F05, 9F25, 9FA5, 9FBJ, 9FFJ, 9FJB, 9FSJ, 9FXJ, 9G25, 9G2J, 9GIJ, 9GSJ, 9HFB, 9HLB, 9IIJ, 9IJ1, 9J01, 9J0B, 9J0J, 9J9B, 9JCJ, 9JE1, 9K05, 9K41, 9K55, 9K61, 9K8B, 9KA5, 9KI1, 9KK5, 9KU5, 9L2J, 9LCJ, 9O05, 9O0V, 9O95, 9OI1, 9OO1, 9OOV, 9OWV, 9RRV, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ

====Additional known quasi-minimal primes (not necessarily the next)====

P&lt;sub&gt;81993&lt;/sub&gt;SZ

==Unsolved families==

Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers &gt; base (''b'')) by covering congruence, algebraic factorization, or combine of them.

{|class=wikitable
|base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family
|-
|13||9{5}||(113×13&lt;sup&gt;''n''&lt;/sup&gt;−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|13||A{3}A||(41×13&lt;sup&gt;''n''+1&lt;/sup&gt;+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|16||{3}AF||(16&lt;sup&gt;''n''+2&lt;/sup&gt;+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1{7}||(23×17&lt;sup&gt;''n''&lt;/sup&gt;−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&amp;use=n&amp;n=1&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||1F{0}7||32×17&lt;sup&gt;''n''+1&lt;/sup&gt;+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||4{7}A||(71×17&lt;sup&gt;''n''+1&lt;/sup&gt;+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||51{0}D||86×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||70F{0}D||2038×17&lt;sup&gt;''n''+1&lt;/sup&gt;+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||8{B}9||(139×17&lt;sup&gt;''n''+1&lt;/sup&gt;−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||9{5}9||(149×17&lt;sup&gt;''n''+1&lt;/sup&gt;+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||95{F}||(2543×17&lt;sup&gt;''n''&lt;/sup&gt;−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||A{D}F||(173×17&lt;sup&gt;''n''+1&lt;/sup&gt;+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}B3||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+190||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||B{0}DB||11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+232||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2BE||(11×17&lt;sup&gt;''n''+3&lt;/sup&gt;−41579)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}2E||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;−2411)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}E9||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+773)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||{B}EE||(11×17&lt;sup&gt;''n''+2&lt;/sup&gt;+853)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||D0G{D}||(60381×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2860381*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||E9{B}||(3963×17&lt;sup&gt;''n''&lt;/sup&gt;−11)/16||29000||[http://factordb.com/index.php?query=%283963*17%5En-11%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||F1{9}||(4105×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16||1000000||[http://factordb.com/index.php?query=%284105*17%5En-9%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||FD0{D}||(72909×17&lt;sup&gt;''n''&lt;/sup&gt;−13)/16||29000||[http://factordb.com/index.php?query=%2872909*17%5En-13%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|17||G{7}F||(263×17&lt;sup&gt;''n''+1&lt;/sup&gt;+121)/16||29000||[http://factordb.com/index.php?query=%28263*17%5E%28n%2B1%29%2B121%29%2F16&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||2{7}9D||(47×21&lt;sup&gt;''n''+2&lt;/sup&gt;+953)/20||20000||[http://factordb.com/index.php?query=%2847*21%5E%28n%2B2%29%2B953%29%2F20&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||2F{C}A||(288×21&lt;sup&gt;''n''+1&lt;/sup&gt;−13)/5||20000||[http://factordb.com/index.php?query=%28288*21%5E%28n%2B1%29-13%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||4{3}B||(83×21&lt;sup&gt;''n''+1&lt;/sup&gt;+157)/20||20000||[http://factordb.com/index.php?query=%2883*21%5E%28n%2B1%29%2B157%29%2F20&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||5{0}DJ||5×21&lt;sup&gt;''n''+2&lt;/sup&gt;+292||20000||[http://factordb.com/index.php?query=5*21%5E%28n%2B2%29%2B292&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||{5}FEK||(21&lt;sup&gt;''n''+3&lt;/sup&gt;+18455)/4||20000||[http://factordb.com/index.php?query=%2821%5E%28n%2B3%29%2B18455%29%2F4&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|21||{7}ID||(7×21&lt;sup&gt;''n''+2&lt;/sup&gt;+4733)/20||20000||[http://factordb.com/index.php?query=%287*21%5E%28n%2B2%29%2B4733%29%2F20&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||7{K}Z||(53×36&lt;sup&gt;''n''+1&lt;/sup&gt;+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||B{0}EUV||11×36&lt;sup&gt;''n''+3&lt;/sup&gt;+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||HM{0}N||634×36&lt;sup&gt;''n''+1&lt;/sup&gt;+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||N{0}YYN||23×36&lt;sup&gt;''n''+3&lt;/sup&gt;+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||O{L}Z||(123×36&lt;sup&gt;''n''+1&lt;/sup&gt;+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|-
|36||S{0}8H||28×36&lt;sup&gt;''n''+2&lt;/sup&gt;+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=200&amp;format=1&amp;sent=Show]
|}

For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E

For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family B0{H}6H may or may not be quasi-minimal prime, since another unsolved family is B{H}6H, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D}

=== Base 17 ===

All searched to length 29000

* 1{7}
* 1F{0}7
* 4{7}A
* 51{0}D
* 70F{0}D
* 8{B}9
* 9{5}9
* 95{F}
* A{D}F
* B{0}B3
* B{0}DB
* {B}2BE
* {B}2E
* {B}E9
* {B}EE
* D0G{D}
* E9{B}
* F1{9}
* FD0{D}
* G{7}F

=== Base 21 ===

All searched to length 20000

* 2{7}9D
* 2F{C}A
* 4{3}B
* 5{0}DJ
* {5}FEK
* {7}ID
* 99{0}99H
* {9}0D
* {9}D
* B3{0}EB
* B9{0}E5
* B{D}B
* B{H}6H
* DH{D}
* F{9}D
* {F}35
* G{0}FK
* H{0}7771
* H{D}
* {J}6J

=== Base 36 ===

All searched to length 20000

* 7{K}Z
* B{0}EUV
* HM{0}N
* N{0}YYN
* O{L}Z
* S{0}8H

==Primality certificates for the proven primes &gt; 10&lt;sup&gt;299&lt;/sup&gt;==

See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]]

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime
|-
|9||149||76&lt;sub&gt;329&lt;/sub&gt;2||(31×9&lt;sup&gt;330&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642]
|-
|9||150||27&lt;sub&gt;686&lt;/sub&gt;07||(23×9&lt;sup&gt;688&lt;/sup&gt;−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486]
|-
|9||151||30&lt;sub&gt;1158&lt;/sub&gt;11||3×9&lt;sup&gt;1160&lt;/sup&gt;+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423]
|-
|11||1065||A&lt;sub&gt;713&lt;/sub&gt;58||11&lt;sup&gt;715&lt;/sup&gt;−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487]
|-
|11||1066||7&lt;sub&gt;759&lt;/sub&gt;44||(7×11&lt;sup&gt;761&lt;/sup&gt;−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840]
|-
|11||1067||557&lt;sub&gt;1011&lt;/sub&gt;||(607×11&lt;sup&gt;1011&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522]
|-
|13||3165||50&lt;sub&gt;270&lt;/sub&gt;44||5×13&lt;sup&gt;272&lt;/sup&gt;+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005]
|-
|13||3166||9&lt;sub&gt;271&lt;/sub&gt;095||(3×13&lt;sup&gt;274&lt;/sup&gt;−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654]
|-
|13||3167||10&lt;sub&gt;286&lt;/sub&gt;7771||13&lt;sup&gt;290&lt;/sup&gt;+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633]
|-
|13||3168||9&lt;sub&gt;308&lt;/sub&gt;1||(3×13&lt;sup&gt;309&lt;/sup&gt;−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=308&amp;c0=-&amp;EN= 13&lt;sup&gt;308&lt;/sup&gt;−1]
|-
|13||3169||B&lt;sub&gt;341&lt;/sub&gt;C4||(11×13&lt;sup&gt;343&lt;/sup&gt;+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618]
|-
|13||3170||8B&lt;sub&gt;343&lt;/sub&gt;||(107×13&lt;sup&gt;343&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736]
|-
|13||3171||710&lt;sub&gt;371&lt;/sub&gt;111||92×13&lt;sup&gt;374&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609]
|-
|13||3172||75&lt;sub&gt;375&lt;/sub&gt;7||(89×13&lt;sup&gt;376&lt;/sup&gt;+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596]
|-
|13||3173||9B0&lt;sub&gt;391&lt;/sub&gt;9||128×13&lt;sup&gt;392&lt;/sup&gt;+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790]
|-
|13||3174||7B0B&lt;sub&gt;397&lt;/sub&gt;||(15923×13&lt;sup&gt;397&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574]
|-
|13||3175||10&lt;sub&gt;414&lt;/sub&gt;93||13&lt;sup&gt;416&lt;/sup&gt;+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240]
|-
|13||3176||81010&lt;sub&gt;415&lt;/sub&gt;1||17746×13&lt;sup&gt;416&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3177||8110&lt;sub&gt;435&lt;/sub&gt;1||1366×13&lt;sup&gt;436&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3178||B7&lt;sub&gt;486&lt;/sub&gt;||(139×13&lt;sup&gt;486&lt;/sup&gt;−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892]
|-
|13||3179||B&lt;sub&gt;563&lt;/sub&gt;C||(11×13&lt;sup&gt;564&lt;/sup&gt;+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=564&amp;c0=-&amp;EN= 13&lt;sup&gt;564&lt;/sup&gt;−1]
|-
|13||3180||1B&lt;sub&gt;576&lt;/sub&gt;||(23×13&lt;sup&gt;576&lt;/sup&gt;−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&amp;Exp=576&amp;c0=-&amp;EN= 13&lt;sup&gt;576&lt;/sup&gt;−1]
|-
|13||3181||80&lt;sub&gt;693&lt;/sub&gt;87||8×13&lt;sup&gt;695&lt;/sup&gt;+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532]
|-
|13||3182||CC5&lt;sub&gt;713&lt;/sub&gt;||(2021×13&lt;sup&gt;713&lt;/sup&gt;−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353]
|-
|13||3183||B&lt;sub&gt;834&lt;/sub&gt;74||(11×13&lt;sup&gt;836&lt;/sup&gt;−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871]
|-
|13||3184||9&lt;sub&gt;968&lt;/sub&gt;B||(3×13&lt;sup&gt;969&lt;/sup&gt;+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244]
|-
|13||3185||10&lt;sub&gt;1295&lt;/sub&gt;181||13&lt;sup&gt;1298&lt;/sup&gt;+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013]
|-
|13||3186||9&lt;sub&gt;1362&lt;/sub&gt;5||(3×13&lt;sup&gt;1363&lt;/sup&gt;−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776]
|-
|13||3187||7&lt;sub&gt;1504&lt;/sub&gt;1||(7×13&lt;sup&gt;1505&lt;/sup&gt;−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755]
|-
|13||3188||930&lt;sub&gt;1551&lt;/sub&gt;1||120×13&lt;sup&gt;1552&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3189||720&lt;sub&gt;2297&lt;/sub&gt;2||93×13&lt;sup&gt;2298&lt;/sup&gt;+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910]
|-
|13||3190||1770&lt;sub&gt;2703&lt;/sub&gt;17||267×13&lt;sup&gt;2705&lt;/sup&gt;+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825]
|-
|13||3191||390&lt;sub&gt;6266&lt;/sub&gt;1||48×13&lt;sup&gt;6267&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|13||3192||B0&lt;sub&gt;6540&lt;/sub&gt;BBA||11×13&lt;sup&gt;6543&lt;/sup&gt;+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906]
|-
|13||3193||C&lt;sub&gt;10631&lt;/sub&gt;92||13&lt;sup&gt;10633&lt;/sup&gt;−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750]
|-
|14||649||34D&lt;sub&gt;708&lt;/sub&gt;||47×14&lt;sup&gt;708&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|14||650||4D&lt;sub&gt;19698&lt;/sub&gt;||5×14&lt;sup&gt;19698&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2328||880&lt;sub&gt;246&lt;/sub&gt;7||136×16&lt;sup&gt;247&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is &lt; 10&lt;sup&gt;299&lt;/sup&gt;
|-
|16||2329||D4&lt;sub&gt;263&lt;/sub&gt;D||(199×16&lt;sup&gt;264&lt;/sup&gt;+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238]
|-
|16||2330||E0&lt;sub&gt;261&lt;/sub&gt;4DD||14×16&lt;sup&gt;264&lt;/sup&gt;+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352]
|-
|16||2331||8C0&lt;sub&gt;290&lt;/sub&gt;ED||140×16&lt;sup&gt;292&lt;/sup&gt;+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307]
|-
|16||2332||DA&lt;sub&gt;305&lt;/sub&gt;5||(41×16&lt;sup&gt;306&lt;/sup&gt;−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284]
|-
|16||2333||CE80&lt;sub&gt;422&lt;/sub&gt;D||3304×16&lt;sup&gt;423&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257]
|-
|16||2334||5F&lt;sub&gt;544&lt;/sub&gt;6F||6×16&lt;sup&gt;546&lt;/sup&gt;−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967]
|-
|16||2335||88F&lt;sub&gt;545&lt;/sub&gt;||137×16&lt;sup&gt;545&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|16||2336||BE0&lt;sub&gt;792&lt;/sub&gt;BB||190×16&lt;sup&gt;794&lt;/sup&gt;+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938]
|-
|16||2337||D9&lt;sub&gt;1052&lt;/sub&gt;||(68×16&lt;sup&gt;1052&lt;/sup&gt;−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020]
|-
|16||2338||FAF&lt;sub&gt;1062&lt;/sub&gt;45||251×16&lt;sup&gt;1064&lt;/sup&gt;−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610]
|-
|16||2339||F8&lt;sub&gt;1517&lt;/sub&gt;F||(233×16&lt;sup&gt;1518&lt;/sup&gt;+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824]
|-
|16||2340||20&lt;sub&gt;1713&lt;/sub&gt;321||2×16&lt;sup&gt;1716&lt;/sup&gt;+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735]
|-
|16||2341||300F&lt;sub&gt;1960&lt;/sub&gt;AF||769×16&lt;sup&gt;1962&lt;/sup&gt;−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750]
|-
|16||2342||90&lt;sub&gt;3542&lt;/sub&gt;91||9×16&lt;sup&gt;3544&lt;/sup&gt;+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191]
|-
|16||2343||5BC&lt;sub&gt;3700&lt;/sub&gt;D||(459×16&lt;sup&gt;3701&lt;/sup&gt;+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322]
|-
|16||2344||D0B&lt;sub&gt;17804&lt;/sub&gt;||(3131×16&lt;sup&gt;17804&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511]
|-
|18||547||80&lt;sub&gt;298&lt;/sub&gt;B||8×18&lt;sup&gt;299&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is an algebraic factor of ''N''+1, factordb entry of 6×18&lt;sup&gt;99&lt;/sup&gt;+1 is [http://factordb.com/index.php?id=1100000000900149167]
|-
|18||548||H&lt;sub&gt;766&lt;/sub&gt;FH||18&lt;sup&gt;768&lt;/sup&gt;−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490]
|-
|18||549||C0&lt;sub&gt;6268&lt;/sub&gt;C5||12×18&lt;sup&gt;6270&lt;/sup&gt;+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437]
|-
|20||3301||H&lt;sub&gt;247&lt;/sub&gt;A0H||(17×20&lt;sup&gt;250&lt;/sup&gt;−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619]
|-
|20||3302||7&lt;sub&gt;249&lt;/sub&gt;A7||(7×20&lt;sup&gt;251&lt;/sup&gt;+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602]
|-
|20||3303||J7&lt;sub&gt;270&lt;/sub&gt;||(368×20&lt;sup&gt;270&lt;/sup&gt;−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462]
|-
|20||3304||J&lt;sub&gt;330&lt;/sub&gt;CCC7||20&lt;sup&gt;334&lt;/sup&gt;−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572]
|-
|20||3305||40&lt;sub&gt;387&lt;/sub&gt;404B||4×20&lt;sup&gt;391&lt;/sup&gt;+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563]
|-
|20||3306||EC0&lt;sub&gt;429&lt;/sub&gt;7||292×20&lt;sup&gt;430&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702]
|-
|20||3307||G&lt;sub&gt;447&lt;/sub&gt;99||(16×20&lt;sup&gt;449&lt;/sup&gt;−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753]
|-
|20||3308||3A&lt;sub&gt;527&lt;/sub&gt;3||(67×20&lt;sup&gt;528&lt;/sup&gt;−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531]
|-
|20||3309||E&lt;sub&gt;566&lt;/sub&gt;C7||(14×20&lt;sup&gt;568&lt;/sup&gt;−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516]
|-
|20||3310||JCJ&lt;sub&gt;629&lt;/sub&gt;||393×20&lt;sup&gt;629&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|20||3311||J&lt;sub&gt;655&lt;/sub&gt;05J||20&lt;sup&gt;658&lt;/sup&gt;−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052]
|-
|20||3312||50&lt;sub&gt;1163&lt;/sub&gt;AJ||5×20&lt;sup&gt;1165&lt;/sup&gt;+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412]
|-
|20||3313||CD&lt;sub&gt;2449&lt;/sub&gt;||(241×20&lt;sup&gt;2449&lt;/sup&gt;−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915]
|-
|20||3314||G0&lt;sub&gt;6269&lt;/sub&gt;D||16×20&lt;sup&gt;6270&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457]
|-
|22||7984||I7G0&lt;sub&gt;254&lt;/sub&gt;H||8882×22&lt;sup&gt;255&lt;/sup&gt;+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788]
|-
|22||7985||D0&lt;sub&gt;255&lt;/sub&gt;5EEF||13×22&lt;sup&gt;259&lt;/sup&gt;+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932]
|-
|22||7986||IK&lt;sub&gt;322&lt;/sub&gt;F||(398×22&lt;sup&gt;323&lt;/sup&gt;−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145]
|-
|22||7987||C0&lt;sub&gt;340&lt;/sub&gt;G9||12×22&lt;sup&gt;342&lt;/sup&gt;+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159]
|-
|22||7988||77E&lt;sub&gt;348&lt;/sub&gt;K7||(485×22&lt;sup&gt;350&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779]
|-
|22||7989||J&lt;sub&gt;379&lt;/sub&gt;KJ||(19×22&lt;sup&gt;381&lt;/sup&gt;+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027]
|-
|22||7990||J&lt;sub&gt;388&lt;/sub&gt;EJ||(19×22&lt;sup&gt;390&lt;/sup&gt;−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729]
|-
|22||7991||DJ&lt;sub&gt;400&lt;/sub&gt;||(292×22&lt;sup&gt;400&lt;/sup&gt;−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110]
|-
|22||7992||E&lt;sub&gt;404&lt;/sub&gt;K7||(2×22&lt;sup&gt;406&lt;/sup&gt;+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298]
|-
|22||7993||66F&lt;sub&gt;453&lt;/sub&gt;B3||(971×22&lt;sup&gt;455&lt;/sup&gt;−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809]
|-
|22||7994||L0&lt;sub&gt;454&lt;/sub&gt;B63||21×22&lt;sup&gt;457&lt;/sup&gt;+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331]
|-
|22||7995||L&lt;sub&gt;483&lt;/sub&gt;G3||22&lt;sup&gt;485&lt;/sup&gt;−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730]
|-
|22||7996||E60&lt;sub&gt;496&lt;/sub&gt;L||314×22&lt;sup&gt;497&lt;/sup&gt;+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239]
|-
|22||7997||I&lt;sub&gt;626&lt;/sub&gt;AF||(6×22&lt;sup&gt;628&lt;/sup&gt;−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334]
|-
|22||7998||K0&lt;sub&gt;760&lt;/sub&gt;EC1||20×22&lt;sup&gt;763&lt;/sup&gt;+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415]
|-
|22||7999||J0&lt;sub&gt;767&lt;/sub&gt;IGGJ||19×22&lt;sup&gt;771&lt;/sup&gt;+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567]
|-
|22||8000||7&lt;sub&gt;959&lt;/sub&gt;K7||(22&lt;sup&gt;961&lt;/sup&gt;+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817]
|-
|22||8001||L&lt;sub&gt;2385&lt;/sub&gt;KE7||22&lt;sup&gt;2388&lt;/sup&gt;−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774]
|-
|22||8002||7&lt;sub&gt;3815&lt;/sub&gt;2L||(22&lt;sup&gt;3817&lt;/sup&gt;−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839]
|-
|24||3400||I0&lt;sub&gt;241&lt;/sub&gt;I5||18×24&lt;sup&gt;243&lt;/sup&gt;+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037]
|-
|24||3401||D0&lt;sub&gt;259&lt;/sub&gt;KKD||13×24&lt;sup&gt;262&lt;/sup&gt;+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725]
|-
|24||3402||C7&lt;sub&gt;298&lt;/sub&gt;||(283×24&lt;sup&gt;298&lt;/sup&gt;−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235]
|-
|24||3403||20&lt;sub&gt;313&lt;/sub&gt;7||2×24&lt;sup&gt;314&lt;/sup&gt;+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241]
|-
|24||3404||BC0&lt;sub&gt;331&lt;/sub&gt;B||276×24&lt;sup&gt;332&lt;/sup&gt;+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842]
|-
|24||3405||N&lt;sub&gt;2644&lt;/sub&gt;LLN||24&lt;sup&gt;2647&lt;/sup&gt;−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089]
|-
|24||3406||D&lt;sub&gt;2698&lt;/sub&gt;LD||(13×24&lt;sup&gt;2700&lt;/sup&gt;+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876]
|-
|24||3407||A0&lt;sub&gt;2951&lt;/sub&gt;8ID||10×24&lt;sup&gt;2954&lt;/sup&gt;+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654]
|-
|24||3408||88N&lt;sub&gt;5951&lt;/sub&gt;||201×24&lt;sup&gt;5951&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|-
|24||3409||N00N&lt;sub&gt;8129&lt;/sub&gt;LN||13249×24&lt;sup&gt;8131&lt;/sup&gt;−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606]
|-
|30||2613||AN&lt;sub&gt;206&lt;/sub&gt;||(313×30&lt;sup&gt;206&lt;/sup&gt;−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073]
|-
|30||2614||M&lt;sub&gt;241&lt;/sub&gt;QB||(22×30&lt;sup&gt;243&lt;/sup&gt;+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295]
|-
|30||2615||M0&lt;sub&gt;547&lt;/sub&gt;SS7||22×30&lt;sup&gt;550&lt;/sup&gt;+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988]
|-
|30||2616||C0&lt;sub&gt;1022&lt;/sub&gt;1||12×30&lt;sup&gt;1023&lt;/sup&gt;+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored
|-
|30||2617||5&lt;sub&gt;4882&lt;/sub&gt;J||(5×30&lt;sup&gt;4883&lt;/sup&gt;+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423]
|-
|30||2619||OT&lt;sub&gt;34205&lt;/sub&gt;||25×30&lt;sup&gt;34205&lt;/sup&gt;−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored
|}

==Unproven PRPs==

{|class=wikitable
|base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+''c'')/''d'') form)||factordb entry of this PRP
|-
|11||1068||57&lt;sub&gt;62668&lt;/sub&gt;||(57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10||[http://factordb.com/index.php?id=1100000003573679860]
|-
|13||3194||C5&lt;sub&gt;23755&lt;/sub&gt;C||(149×13&lt;sup&gt;23756&lt;/sup&gt;+79)/12||[http://factordb.com/index.php?id=1100000003590647776]
|-
|13||3195||80&lt;sub&gt;32017&lt;/sub&gt;111||8×13&lt;sup&gt;32020&lt;/sup&gt;+183||[http://factordb.com/index.php?id=1100000000490878060]
|-
|16||2345||DB&lt;sub&gt;32234&lt;/sub&gt;||(206×16&lt;sup&gt;32234&lt;/sup&gt;−11)/15||[http://factordb.com/index.php?id=1100000002383583629]
|-
|16||2346||4&lt;sub&gt;72785&lt;/sub&gt;DD||(4×16&lt;sup&gt;72787&lt;/sup&gt;+2291)/15||[http://factordb.com/index.php?id=1100000003615909841]
|-
|22||8003||BK&lt;sub&gt;22001&lt;/sub&gt;5||(251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21||[http://factordb.com/index.php?id=1100000003594696838]
|-
|30||2618||I0&lt;sub&gt;24608&lt;/sub&gt;D||18×30&lt;sup&gt;24609&lt;/sup&gt;+13||[http://factordb.com/index.php?id=1100000003593967511]
|}

All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10&lt;sup&gt;16&lt;/sup&gt;. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]])

==Proof==

===Base 2===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.

===Base 3===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (2,1), (2,2)

* Case (1,1):
** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes)
*** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime.

===Base 4===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (2,1), (2,3), (3,1), (3,3)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.

===Base 5===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4)

* Case (1,1):
** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes)
*** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime.
* Case (1,2):
** 12 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes)
*** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes)
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes)
***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb])
***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes)
**** The smallest prime of the form 1{4}4 is 14444.
**** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,2):
** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes)
*** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes)
**** For the 3{0,3}1 family, we can separate this family to four families:
***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01.
****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime.
***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031.
***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime.
****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes.
******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001.
***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31.
****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime.
**** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime.
* Case (3,4):
** 34 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes)
*** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes)
**** The smallest prime of the form 4{4}1 is 44441.
**** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime.
* Case (4,2):
** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime.

===Base 6===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes)
*** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes)
**** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 4{0}41 is 40041
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes)
*** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.

===Base 7===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

* Case (1,1):
** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes)
*** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {}
**** If there are &gt;=3 number of 1 and 5 in the {}:
***** If there is 111 in the {}, then we have the prime 11111
***** If there is 115 in the {}, then the prime 115 is a subsequence
***** If there is 151 in the {}, then the prime 115 is a subsequence
***** If there is 155 in the {}, then the prime 155 is a subsequence
***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime
***** If there is 515 in the {}, then the prime 515 is a subsequence
***** If there is 551 in the {}, then the prime 551 is a subsequence
***** If there is 555 in the {}, then the prime 551 is a subsequence
**** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1
***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1
****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1
******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21
******** The smallest prime of the form 11{0}21 is 1100021
******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime
******** The smallest prime of the form 1001{0}21 is 100121
********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families
***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1
****** Since 1051 is prime, we only need to consider the family 15{0}1
******* The smallest prime of the form 15{0}1 is 150001
* Case (1,2):
** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes)
*** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {}
**** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime.
**** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime.
**** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2
***** Since 1022 and 1202 are primes, we only need to consider the number 122
****** 122 is not prime.
**** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime.
* Case (1,3):
** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3
**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
**** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime.
* Case (1,4):
** 14 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime.
* Case (1,6):
** 16 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime.
* Case (2,2):
** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,4):
** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime.
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,6):
** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (3,1):
** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes)
*** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes)
**** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's)
***** For the 3{0,1}1 family:
****** If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
****** If there are no 1's between (3,1), then the form will be 3{0}1
******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime.
****** If there are no 0's between (3,1), then the form will be 3{1}1
******* The smallest prime of the form 3{1}1 is 31111
****** If there are exactly 1 0's between (3,1), then there must be &lt;3 1's between (3,1), or 31111 will be a subsequence.
******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime.
******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011
******** Neither 3101 nor 3011 is prime.
******* If there are no 1's between (3,1), then the number must be 301
******** 301 is not prime.
***** For the 3{3}3{0,1}1 family:
****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence.
****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1
******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence.
******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime.
******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime.
******* If there are no 1's between (33,1), then the form will be 33{0}1
******** The smallest prime of the form 33{0}1 is 33001
****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1
******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence.
******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301
******** The smallest prime of the form 3{3}301 is 33333301
******* If there are no 0's between (3{3}3,1), then the form is 3{3}31
******** The smallest prime of the form 3{3}31 is 33333333333333331
**** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1
***** Numbers containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1,3}3{0,1,3}1
******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes)
******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes)
********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes)
********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence.
********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime.
********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1
*********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime.
********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime.
***** Numbers not containing 3 between (3{0,1}5,1):
****** The form is 3{0,1}5{0,1}1
******* If there are &gt;=2 0's and &gt;=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1
******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime.
******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1
******** If there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111
********* None of 31151, 31511, 35111 are primes.
******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 351
********* 351 is not prime.
******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are &gt;=3 1's between (3,1), then 31111 will be a subsequence.
******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111
********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes.
********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime.
******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1
********* Neither 3051 nor 3501 is prime.
* Case (3,2):
** 32 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes)
*** If there are &gt;=2 5's in {}, then 553 will be a subsequence.
*** If there are no 5's in {}, then the family will be 3{0,3,6}3
**** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
*** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3
**** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes)
***** The smallest prime of the form 3{0}53 is 300053
* Case (3,4):
** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes)
*** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes)
**** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes)
**** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes)
* Case (3,5):
** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes)
*** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence.
*** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence.
*** If there are at least two 1's in {}, then 115 will be a subsequence.
*** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5
**** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime.
*** If there is no 1's in {}, then the family will be 3{0,4,5}5
**** If there are at least to 4's in {}, then 344 and 445 will be subsequences.
**** If there is no 4's in {}, then the family will be 3{0,5}5
***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5
****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime.
****** The smallest prime of the form 3{5}5 is 35555
**** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5
***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes)
****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime.
* Case (3,6):
** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,2):
** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,4):
** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes)
*** If there is no 5's in {}, then the family will be 4{0,2,4,6}4
**** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4
**** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes)
***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence.
***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4
****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes)
***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4
****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4
******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes)
* Case (4,5):
** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
*** If there are at least two 5's in {}, then 4555 will be a subsequence.
*** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there is no 5's in {}, then the family will be 4{0}5
**** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime.
* Case (4,6):
** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime.
* Case (5,1):
** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 3's in {}, then 533 will be a subsequence.
*** If there is no 3's in {}, then the family will be 5{0,1}1
**** Since 5011 is prime, we only need to consider the family 5{1}{0}1
***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes)
****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime.
****** The smallest prime of the form 51{0}1 is 5100000001
****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime.
****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime.
*** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1
**** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence.
***** Thus we only need to consider the family 5{0}3{0,1}1
****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime.
****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime.
****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence.
****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1
******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0)
******** The smallest prime of the form 5311{0}1 is 531101
* Case (5,2):
** 52 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
*** If there are at least two 1's in {}, then 113 will be a subsequence.
*** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
*** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime.
* Case (5,4):
** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes)
*** If there are no 5's in {}, then the family will be 5{0,3}4
**** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime.
*** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence.
*** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
*** If there are at least two 5's in {}, then 5554 will be a subsequence.
* Case (5,5):
** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes)
*** If there are no 4's in {}, then the family will be 5{0,5}5
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
*** If there are no 5's in {}, then the family will be 5{0,4}5
**** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime.
*** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence.
* Case (5,6):
** 56 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,2):
** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime.
* Case (6,3):
** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,4):
** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4
**** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime.
*** If there are at least three 3's in {}, then 3334 will be a subsequence.
*** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4
**** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence.
**** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence.
**** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence.
**** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence.
**** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence.
**** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence.
**** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence.
**** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence.
**** Thus the number can only be 634
***** 634 is not prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,6):
** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes)
*** If there is no 3's in {}, then the family will be 6{0,2,4,6}6
**** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime.
*** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6
**** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime.
*** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence.
*** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence.

===Base 8===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7)

* Case (1,1):
** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes)
**** 171 is not prime.
**** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 &gt; 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 &gt; 1, this factorization is nontrivial), thus cannot be prime.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes)
*** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime)
* Case (2,1):
** 21 is prime, and thus the only minimal prime in this family.
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,5):
** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes)
*** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes)
*** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes)
**** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime.
**** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes)
***** The smallest prime of the form 33{4}1 is 3344441
***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime.
**** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes)
***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime.
***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes)
****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes.
* Case (3,3):
** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes)
*** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes)
**** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime)
**** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461
***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes)
****** The smallest prime of the form 4{4}41 is 444444441
****** The smallest prime of the form 4{4}641 is 444641
***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes)
****** The smallest prime of the form 4{4}411 is 444444441
****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes)
***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461
****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime)
**** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes)
***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime)
***** The smallest prime of the form 4{4}641 is 444641
* Case (4,3):
** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes)
*** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes)
**** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime.
**** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime.
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes)
*** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes)
**** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb])
**** The smallest prime of the form 4{4}77 is 4444477
**** The smallest prime of the form 4{7}7 is 47777
**** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime)
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes)
*** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes)
**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
**** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes)
***** 500525 is not prime.
***** The smallest prime of the form 5{5}25 is 555555555555525
***** The smallest prime of the form 5{5}025 is 55555025
***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes)
***** The smallest prime of the form 5{5}0525 is 5550525
***** The smallest prime of the form 5{5}00525 is 5500525
***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes)
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence)
**** The form is 6{0,1,7}4{0,1,7}1
***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1
****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411
******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41
******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41
********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.
********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime.
********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime.
******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411
******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime)
*** Numbers not containing 4:
**** The form is 6{0,1,7}1
***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1
****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime.
****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1
******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1
******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.
******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime)
* Case (6,3):
** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.
* Case (6,5):
** 65 is prime, and thus the only minimal prime in this family.
* Case (6,7):
** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes)
*** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them
**** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17
***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime.
**** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77
***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime.
**** If there is neither 1 nor 7 between them, then the form is 6{0,4}7
***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes)
****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime.
****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime.
* Case (7,1):
** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes)
*** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes)
**** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes)
***** The smallest prime of the form 7{7}1 is 7777777777771
***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime)
***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime)
***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime.
***** The smallest prime of the form 7{7}4444441 is 77774444441
****** Since this prime has just 4 7's, we only need to consider the families with &lt;=3 7's
******* The smallest prime of the form 7{4}1 is 744444441
******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime.
******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes)
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.

===Base 10===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,3):
** 13 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,9):
** 19 is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 2{0}21 is 20021
**** The smallest prime of the form 22{0}1 is 22000001
* Case (2,3):
** 23 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes)
*** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes)
**** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime.
**** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime.
* Case (2,9):
** 29 is prime, and thus the only minimal prime in this family.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,3):
** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,9):
** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime.
* Case (4,1):
** 41 is prime, and thus the only minimal prime in this family.
* Case (4,3):
** 43 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** 47 is prime, and thus the only minimal prime in this family.
* Case (4,9):
** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime.
* Case (5,1):
** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes)
**** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes)
***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 5{5}1 is 555555555551
**** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581
***** 581 is not prime.
* Case (5,3):
** 53 is prime, and thus the only minimal prime in this family.
* Case (5,7):
** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes)
**** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime.
**** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027
**** The smallest prime of the form 52{0}7 is 5200007
* Case (5,9):
** 59 is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,3):
** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,9):
** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49
***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49
****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes)
******* The smallest prime of the form 6{6}49 is 666649
******** Since this prime has just 4 6's, we only need to consider the families with &lt;=3 6's
********* The smallest prime of the form 6{0}49 is 60000049
********* The smallest prime of the form 66{0}49 is 66000049
********* The smallest prime of the form 666{0}49 is 66600049
* Case (7,1):
** 71 is prime, and thus the only minimal prime in this family.
* Case (7,3):
** 73 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,9):
** 79 is prime, and thus the only minimal prime in this family.
* Case (8,1):
** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes)
**** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes)
***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime.
***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime)
***** The smallest prime of the form 80{5}1 is 80555551
* Case (8,3):
** 83 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime.
* Case (8,9):
** 89 is prime, and thus the only minimal prime in this family.
* Case (9,1):
** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes)
*** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes)
**** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021
***** None of 921, 9201, 9021 are primes.
**** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801
***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
**** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801
***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime.
* Case (9,3):
** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes)
*** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime.
* Case (9,7):
** 97 is prime, and thus the only minimal prime in this family.
* Case (9,9):
** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes)
*** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes)
**** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime.
**** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9
***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes)
****** The smallest prime of the form 94{6}9 is 946669

===Base 12===

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B)

* Case (1,1):
** 11 is prime, and thus the only minimal prime in this family.
* Case (1,5):
** 15 is prime, and thus the only minimal prime in this family.
* Case (1,7):
** 17 is prime, and thus the only minimal prime in this family.
* Case (1,B):
** 1B is prime, and thus the only minimal prime in this family.
* Case (2,1):
** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes)
*** The smallest prime of the form 2{0}1 is 2001
* Case (2,5):
** 25 is prime, and thus the only minimal prime in this family.
* Case (2,7):
** 27 is prime, and thus the only minimal prime in this family.
* Case (2,B):
** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes)
**** None of 20B, 22B, 29B, 209B, 220B are primes.
* Case (3,1):
** 31 is prime, and thus the only minimal prime in this family.
* Case (3,5):
** 35 is prime, and thus the only minimal prime in this family.
* Case (3,7):
** 37 is prime, and thus the only minimal prime in this family.
* Case (3,B):
** 3B is prime, and thus the only minimal prime in this family.
* Case (4,1):
** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes)
**** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form 44{A}1 is 44AAA1
* Case (4,5):
** 45 is prime, and thus the only minimal prime in this family.
* Case (4,7):
** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes)
**** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077
* Case (4,B):
** 4B is prime, and thus the only minimal prime in this family.
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family.
* Case (5,5):
** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes)
*** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime.
* Case (5,7):
** 57 is prime, and thus the only minimal prime in this family.
* Case (5,B):
** 5B is prime, and thus the only minimal prime in this family.
* Case (6,1):
** 61 is prime, and thus the only minimal prime in this family.
* Case (6,5):
** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes)
**** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime.
**** The smallest prime of the form 6{0}A5 is 600A5
* Case (6,7):
** 67 is prime, and thus the only minimal prime in this family.
* Case (6,B):
** 6B is prime, and thus the only minimal prime in this family.
* Case (7,1):
** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes)
*** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1
**** None of 741, 7B1, 74B1 are primes.
* Case (7,5):
** 75 is prime, and thus the only minimal prime in this family.
* Case (7,7):
** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.
* Case (7,B):
** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
*** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes)
**** The smallest prime of the form 7{9}B is 7999B
**** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes)
**** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime)
* Case (8,1):
** 81 is prime, and thus the only minimal prime in this family.
* Case (8,5):
** 85 is prime, and thus the only minimal prime in this family.
* Case (8,7):
** 87 is prime, and thus the only minimal prime in this family.
* Case (8,B):
** 8B is prime, and thus the only minimal prime in this family.
* Case (9,1):
** 91 is prime, and thus the only minimal prime in this family.
* Case (9,5):
** 95 is prime, and thus the only minimal prime in this family.
* Case (9,7):
** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes)
**** None of 947, 977, 997, 9477, 9977 are primes.
* Case (9,B):
** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes)
*** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes)
**** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes)
***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime)
***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime)
**** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes)
***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes)
****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime.
****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime)
****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime.
**** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B
***** None of 99B, 999B, 979B, 9799B, 9979B are primes.
* Case (A,1):
** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
*** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes)
**** For the A{A}{0}1 family:
***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form AA{0}1 is AA000001
***** The smallest prime of the form AAA{0}1 is AAA0001
***** The smallest prime of the form AAAA{0}1 is AAAA1
****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc.
**** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime.
* Case (A,5):
** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes)
*** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes)
**** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime.
**** The smallest prime of the form A{0}65 is A00065
* Case (A,7):
** A7 is prime, and thus the only minimal prime in this family.
* Case (A,B):
** AB is prime, and thus the only minimal prime in this family.
* Case (B,1):
** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes)
*** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes)
**** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1
***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1
****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime)
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1
****** The smallest prime of the form B{A}01 is BAA01
****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
**** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1
***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1
****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1
******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1
******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes.
****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1
******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime)
***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes)
****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B04{A}1 is B04A1
****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime)
****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime)
**** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1
***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1
****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1
******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime.
******* The smallest prime of the form B{B}AA1 is BBBAA1
****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1
******* The smallest prime of the form B{B}AA1 is BBBAA1
**** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes)
***** The smallest prime of the form B{B}B1 is BBBB1
***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime.
***** The smallest prime of the form B{B}BAA1 is BBBAA1
* Case (B,5):
** B5 is prime, and thus the only minimal prime in this family.
* Case (B,7):
** B7 is prime, and thus the only minimal prime in this family.
* Case (B,B):
** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes)
*** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B
**** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B
***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime.
***** For the B{0,B}9B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes)
******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B
******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 &gt; 1, 10^n+5 ≥ 10^1+5 = 15 &gt; 1, this factorization is nontrivial), thus cannot be prime.
***** For the B{0,B}99B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes)
******* The smallest prime of the form B{0}99B is B00099B
******* The smallest prime of the form B{B}99B is BBBBBB99B
***** For the B{0,B}999B family:
****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes)
******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes)
******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime)

== Examples of families which can be ruled out as contain no primes &gt; ''b'' ==

It is not known if this problem is solvable:

Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''?

It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime &gt; ''b'' or not. We use two different heuristic strategies to show that such families contain no primes &gt; ''b''.

In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;.

Examples of first strategy: (we can show that the corresponding numbers are &gt; all elements in ''S'', if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial)

* In base 10, all numbers of the form 4{6}9 are divisible by 7
* In base 6, all numbers of the form 4{0}1 are divisible by 5
* In base 15, all numbers of the form 9{6}8 are divisible by 11
* In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5}
* In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3}
* In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5}
* In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13}
* In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17}
* In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13}
* In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13}

Examples of second strategy: (we can show that both factors are &gt; 1, if ''n'' makes corresponding numbers &gt; ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial)

* In base 9, all numbers of the form {1} factored as difference of squares
* In base 8, all numbers of the form 1{0}1 factored as sum of cubes
* In base 9, all numbers of the form 3{8} factored as difference of squares
* In base 16, all numbers of the form 8{F} factored as difference of squares
* In base 16, all numbers of the form {F}7 factored as difference of squares
* In base 9, all numbers of the form 3{1} factored as difference of squares
* In base 16, all numbers of the form {4}1 factored as difference of squares
* In base 16, all numbers of the form 1{5} factored as difference of squares
* In base 16, all numbers of the from {C}D factored as ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;
* In base 16, all numbers of the form B{4}1 factored as difference of squares

Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are &gt; all elements in S, and for the part of the second strategy, both factors are &gt; 1, if n makes corresponding numbers &gt; b, thus these factorizations are nontrivial)

* In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd
* In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even
* In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even

== Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes &gt; ''b'' ==

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-powers factorization

=== 1{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 1{0}3 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== 1{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 1{0}5 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 5: Finite covering set {5}

=== 1{0}6 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== 1{0}7 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 7: Finite covering set {7}

=== 1{0}z ===

(none)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

(none)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {1} ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with ''r''&gt;1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2))

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' == 0 mod 2: Finite covering set {2}

=== 1{2} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 1{3} ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361)
* ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865)

=== 1{4} ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization

=== 1{z} ===

(none)

=== 2{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}

=== 2{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) ===

* ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

=== {2}1 ===

* ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49)

=== 2{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 3{0}2 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 3{0}4 ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {3}1 ===

* ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121)

=== 3{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 4{0}1 ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt;

=== 4{0}3 ===

* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 7: Finite covering set {7}

=== {4}1 ===

* ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225)

=== 4{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 5{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 5{z} ===

* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}
* ''b'' = 6''m''&lt;sup&gt;2&lt;/sup&gt; with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014)

=== 6{0}1 ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== 6{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}

=== 7{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}

=== 7{z} ===

* ''b'' == 1 mod 7: Finite covering set {7}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993)
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Difference-of-cubes factorization

=== 8{0}1 ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 20 mod 21: Finite covering set {3, 7}
* ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022)
* ''b'' = 467: Finite covering set {3, 5, 7, 19, 37}
* ''b'' = 722: Finite covering set {3, 5, 13, 73, 109}
* ''b'' = ''m''&lt;sup&gt;3&lt;/sup&gt;: Sum-of-cubes factorization
* ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2

=== 8{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== 9{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}

=== 9{z} ===

* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{0}1 ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 32 mod 33: Finite covering set {3, 11}

=== A{z} ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{0}1 ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}

=== B{z} ===

* ''b'' == 1 mod 11: Finite covering set {11}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 307: Finite covering set {5, 11, 29}
* ''b'' = 901: Finite covering set {7, 11, 13, 19}

=== C{0}1 ===

* ''b'' == 1 mod 13: Finite covering set {13}
* ''b'' == 142 mod 143: Finite covering set {11, 13}
* ''b'' = 296, 901: Finite covering set {7, 11, 13, 19}
* ''b'' = 562, 828, 900: Finite covering set {7, 13, 19}
* ''b'' = 563: Finite covering set {5, 7, 13, 19, 29}
* ''b'' = 597: Finite covering set {5, 13, 29}

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

(none)

=== {y}z ===

(none)

=== z{0}1 ===

(none)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

* ''b'' = ''m''&lt;sup&gt;''r''&lt;/sup&gt; with odd ''r''&gt;1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3))
* ''b'' = 4''m''&lt;sup&gt;4&lt;/sup&gt;: Aurifeuillian factorization of ''x''&lt;sup&gt;4&lt;/sup&gt;+4''y''&lt;sup&gt;4&lt;/sup&gt; (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}t ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 0 mod 7: Finite covering set {7}

=== {z}u ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}
* ''b'' == 1 mod 5: Finite covering set {5}
* ''b'' == 34 mod 35: Finite covering set {5, 7}

=== {z}v ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 5: Finite covering set {5}

=== {z}w ===

* ''b'' == 0 mod 2: Finite covering set {2}
* ''b'' == 1 mod 3: Finite covering set {3}
* ''b'' == 14 mod 15: Finite covering set {3, 5}
* ''b'' = ''m''&lt;sup&gt;2&lt;/sup&gt;: Difference-of-squares factorization
* ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

=== {z}x ===

* ''b'' == 1 mod 2: Finite covering set {2}
* ''b'' == 0 mod 3: Finite covering set {3}

=== {z}y ===

* ''b'' == 0 mod 2: Finite covering set {2}

== Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) ==

Format: base (length)

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

=== 1{0}1 ===

(none)

=== 1{0}2 ===

(none)

=== 1{0}3 ===

(none)

=== 1{0}4 ===

53 (13403)

113 (10647)

=== 1{0}z ===

113 (20089)

123 (64371)

=== 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

(none)

=== 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) ===

208 (26682)

607 (11032)

828 (19659)

=== 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) ===

201 (31276)

222 (52727)

227 (36323)

327 (135983)

425 (11231)

710 (24112)

717 (37508)

719 (13420)

=== {1} ===

152 (270217)

184 (16703)

200 (17807)

311 (36497)

326 (26713)

331 (25033)

371 (15527)

485 (99523)

629 (32233)

649 (43987)

670 (18617)

684 (22573)

691 (62903)

693 (41189)

731 (15427)

752 (32833)

872 (10093)

932 (20431)

=== {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) ===

(none)

=== 1{z} ===

107 (21911)

170 (166429)

278 (43909)

303 (40175)

383 (20957)

515 (58467)

522 (62289)

578 (129469)

590 (15527)

647 (21577)

662 (16591)

698 (127559)

704 (62035)

845 (39407)

938 (40423)

969 (24097)

989 (26869)

=== 2{0}1 ===

101 (192276)

206 (46206)

218 (333926)

236 (161230)

257 (12184)

305 (16808)

467 (126776)

578 (44166)

626 (174204)

695 (94626)

752 (26164)

788 (72918)

869 (49150)

887 (27772)

899 (15732)

932 (13644)

=== 2{z} ===

432 (16003)

=== 3{0}1 ===

912 (132174)

=== 3{z} ===

72 (1119850)

212 (34414)

218 (23050)

270 (89662)

303 (198358)

312 (51566)

422 (21738)

480 (93610)

513 (38032)

527 (46074)

566 (23874)

650 (498102)

686 (16584)

758 (15574)

783 (12508)

800 (33838)

921 (98668)

947 (10056)

=== 4{0}1 ===

107 (32587)

227 (13347)

257 (160423)

355 (10990)

410 (144079)

440 (56087)

452 (14155)

482 (30691)

542 (15983)

579 (67776)

608 (20707)

635 (11723)

650 (96223)

679 (69450)

737 (269303)

740 (58043)

789 (149140)

797 (468703)

920 (103687)

934 (101404)

962 (84235)

=== 4{z} ===

14 (19699)

68 (13575)

254 (15451)

800 (20509)

=== 5{0}1 ===

326 (400786)

350 (20392)

554 (10630)

662 (13390)

926 (40036)

=== 5{z} ===

258 (212135)

272 (148427)

299 (64898)

307 (26263)

354 (25566)

433 (283919)

635 (36163)

678 (40859)

692 (45447)

719 (20552)

768 (70214)

857 (23083)

867 (61411)

972 (36703)

=== 6{0}1 ===

108 (16318)

129 (16797)

409 (369833)

522 (52604)

587 (24120)

643 (164916)

762 (11152)

789 (27297)

986 (21634)

=== 6{z} ===

68 (25396)

332 (15222)

338 (42868)

362 (146342)

488 (33164)

566 (164828)

980 (50878)

986 (12506)

1016 (23336)

=== 7{0}1 ===

398 (17473)

1004 (54849)

=== 7{z} ===

97 (192336)

170 (15423)

194 (38361)

202 (155772)

282 (21413)

283 (164769)

332 (13205)

412 (29792)

560 (19905)

639 (10668)

655 (53009)

811 (31784)

814 (17366)

866 (108591)

908 (61797)

962 (31841)

992 (10605)

997 (15815)

=== 8{0}1 ===

23 (119216)

53 (227184)

158 (123476)

254 (67716)

320 (52004)

410 (279992)

425 (94662)

513 (19076)

518 (11768)

596 (148446)

641 (87702)

684 (23387)

695 (39626)

785 (900326)

788 (11408)

893 (86772)

908 (243440)

920 (107822)

962 (47222)

998 (81240)

1013 (43872)

=== 8{z} ===

138 (35686)

412 (12154)

788 (11326)

990 (23032)

=== 9{0}1 ===

248 (39511)

592 (96870)

=== 9{z} ===

431 (43574)

446 (152028)

458 (126262)

599 (11776)

846 (12781)

=== A{0}1 ===

173 (264235)

198 (47665)

311 (314807)

341 (106009)

449 (18507)

492 (42843)

605 (12395)

708 (17563)

710 (31039)

743 (285479)

744 (137056)

786 (68169)

800 (15105)

802 (149320)

879 (25004)

929 (13065)

977 (125873)

986 (48279)

1004 (10645)

=== A{z} ===

368 (10867)

488 (10231)

534 (80328)

662 (13307)

978 (14066)

=== B{0}1 ===

710 (15272)

740 (33520)

878 (227482)

=== B{z} ===

153 (21660)

186 (112718)

439 (18752)

593 (16064)

602 (36518)

707 (10573)

717 (67707)

=== C{0}1 ===

68 (656922)

219 (29231)

230 (94751)

312 (21163)

334 (83334)

353 (20262)

359 (61295)

457 (10024)

481 (45941)

501 (20140)

593 (42779)

600 (11242)

604 (17371)

641 (26422)

700 (91953)

887 (13961)

919 (45359)

923 (64365)

992 (10300)

=== {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) ===

(none)

=== {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) ===

(none)

=== #{z} (for even bases b, # = b/2−1) ===

(none)

=== y{z} ===

38 (136212)

83 (21496)

113 (286644)

188 (13508)

401 (103670)

417 (21003)

458 (46900)

494 (21580)

518 (129372)

527 (65822)

602 (17644)

608 (36228)

638 (74528)

663 (47557)

723 (24536)

758 (50564)

833 (12220)

904 (13430)

938 (50008)

950 (16248)

=== z{0}1 ===

202 (46774)

251 (102979)

272 (16681)

297 (14314)

298 (60671)

326 (64757)

347 (69661)

363 (142877)

452 (71941)

543 (10042)

564 (38065)

634 (84823)

788 (13541)

869 (12289)

890 (37377)

953 (60995)

1004 (29685)

=== {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) ===

53 (21942)

124 (16426)

175 (31626)

188 (22036)

316 (48538)

365 (25578)

373 (24006)

434 (10090)

530 (11086)

545 (12346)

560 (15072)

596 (12762)

701 (12576)

706 (10656)

821 (13536)

833 (17116)

966 (14820)

983 (11272)

=== {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) ===

(none)

=== {z}1 ===

(none)

=== {z}y ===

317 (13896)

== Bases 2≤''b''≤1024 which have these families as unsolved families ==

Unsolved families are families which are neither primes (&gt;''b'') found nor can be ruled out as contain no primes &gt; ''b''

(using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. &quot;z&quot; means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit &quot;7&quot; is only interpretable for bases ≥8, and digit &quot;u&quot; (means ''b''−6) is only interpretable for bases ≥7)

1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608)

1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000)

1{0}3: 646, 718, 998 (length limit: 2000)

1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000)

1{0}5

1{0}6

1{0}7

1{0}8

1{0}9

1{0}A

1{0}B

1{0}C

1{0}D

1{0}E

1{0}F

1{0}G

1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000)

1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000)

10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000)

11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000)

{1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000)

{1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000)

11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z})

{1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000)

{1}z

1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000)

1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995

1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000)

1{z}: 581, 992, 1019 (length limit: ≥100000)

2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000)

2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000)

2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000)

{2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000)

2{z}: 588, 972 (length limit: ≥100000)

3{0}1: 718 (length limit: ≥100000)

3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000)

3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000)

{3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000)

3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000)

4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000)

4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000)

{4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000)

4{z}: 338, 998 (length limit: ≥100000)

5{0}1: 308, 512, 824 (length limit: ≥100000)

5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000)

6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000)

6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000)

7{0}1: (none)

7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000)

8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000)

8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000)

9{0}1: 724, 884 (length limit: ≥100000)

9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000)

A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000)

A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000)

B{0}1: 560, 770, 968 (length limit: ≥100000)

B{z}: 263, 615, 912, 978 (length limit: ≥100000)

C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000)

C{z}

D{0}1

D{z}

E{0}1

E{z}

F{0}1

F{z}

G{0}1

{&amp;#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000)

{&amp;#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143)

&amp;#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000)

${&amp;#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2)

x{z}

y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000)

z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000)

{y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000)

{z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326)

zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z})

{z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000)

{z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1)

{z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000)

{z}k

{z}l

{z}m

{z}n

{z}o

{z}p

{z}q

{z}r

{z}s

{z}t

{z}u

{z}v

{z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000)

{z}x: (none)

{z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000)

== List of lengths for quasi-minimal primes in some simple families ==

[https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024]

NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases &lt;=7, family {z}x in bases &lt;=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime &gt; base) in this base)

RC: this family can be proven to only contain composite numbers (only count numbers &gt; base)

unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers &gt; base)

Background color: red for title (bases or families), green for length &gt; 10000, orange for 2500 &lt; length ≤ 10000, white for length ≤ 2500, cyan for &quot;RC&quot;, pink for &quot;NB&quot;, yellow for &quot;unknown&quot;.

Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families.

== References ==

* [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem]

* [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are &gt; ''b'' for bases ''b''≤16]

* [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary]

* [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia]

* [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime&gt;base is not required) in The Prime Puzzles &amp; Problems Connection]

* [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles &amp; Problems Connection]

* [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16]

* [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime&gt;base is not required]

* [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime&gt;base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29]

* [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime&gt;base is not required, search limits of lengths: 10000 for all b]

* [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime&gt;base is not required]

* [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime&gt;base is not required]

* [http://www.prothsearch.com/sierp.html the Sierpinski problem]

* [http://www.prothsearch.com/rieselprob.html the Riesel problem]

* [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem]

* [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&amp;lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &amp;gt; b]

* [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030]

* [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100]

* [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem]

* [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b]

* [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b&gt;k]

* [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024]

* [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project]

* [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project]

* [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count]

* [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty]

* [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers]

* [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares]

* [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software]

* [https://sourceforge.net/projects/openpfgw/ pfgw software]

* [http://jpenne.free.fr/index2.html llr software]

* [http://www.ellipsa.eu/public/primo/primo.html PRIMO software]

* [https://primes.utm.edu/prove/index.html website for primality proving]

* [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!]

* [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime&gt;base is not required]

* [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime&gt;base is not required]

* [https://primes.utm.edu/primes/lists/all.txt top proven primes]

* [http://www.primenumbers.net/prptop/prptop.php top PRPs]

* [http://factordb.com online factor database, including many primes which are minimal primes in a small base]</text>
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==Motivation==
# [[Academic help-seeking]] - What are the barriers and enablers of AHS and how can AHS be fostered? - [[User:MyUserName|MyUserName]]
# [[/Academic self-regulation/]] - What is academic self-regulation, why does it matter, and how can it be fostered? - [[U3216563]]
# [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? - [[User:MyUserName|MyUserName]]
# [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it? - [[User:MyUserName|MyUserName]]
# [[/Antidepressants and motivation/]] - What are the effects of popular antidepressants on motivation? - [[User:U3222363|U3222363]]
# [[/Approach motivation/]] - What is approach motivation and how does it lead to behaviour? - [[User:U3189370|U3189370]]
# [[/Behavioural economics and motivation/]] - What aspects of motivation theory are useful in behavioural economics? - [[User:U3141987|U3141987]]
# [[/Behavioural model of health services/]] - What is the BMHS and how can it be used? - SoSilverLibby
# [[/Beneficence as a psychological need/]] - What is beneficence and what are its implications as a psychological need? - [[User:MyUserName|CaitlinEmc]]
# [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention? - [[User:MyUserName|MyUserName]]
# [[/Choice overload/]] - What is choice overload? What is the optimal amount of choice? - [[UserGeorgiaFairweather|GeorgiaFairweather]]
# [[/Chunking and goal pursuit/]] - How does chunking affect goal pursuit? - [[User:MyUserName|MyUserName]]
# [[/Cognitive entrenchment/]] - What is cognitive entrenchment and how can it be avoided? - [[User:MyUserName|MyUserName]]
# [[/Climate change helplessness/]] - How does learned helpless impact motivation to engage in behaviours to limit climate change? - [[User:MyUserName|MyUserName]]
# [[/Closeness communication bias/]] - What is the CCB, why does it occur, and how can it be overcome? - [[User:MyUserName|MyUserName]]
# [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirably outcomes? - [[User:MyUserName|MyUserName]]
# [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories? - [[User:KingMob221|KingMob221]]
# [[/Construal level theory/]] - What is construal level theory and how can it be applied? - [[User:MyUserName|MyUserName]]
# [[/Courage motivation/]] - What is courage, what motivates courage, and how can courage be enhanced? -[[User:Hanarose123|Hanarose123]]
# [[/Death drive/]] - What is the death drive and how can it be negotiated? - [[User:U3086459|U3086459]]
# [[Motivation and emotion/Book/2022/Drugs-violence nexus and motivation Atu3202070|Drugs-violence nexus and motivation]] - What is the role of motivation in the drugs-violence nexus? -  [[Atu3202070|Atu3202070]]
# [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD? - [[User:MyUserName|MyUserName]]
# [[/Episodic memory and planning/]] - What role does episodic memory play in planning? - [[User:MyUserName|U3246310]]
# [[/Equity theory/]] - What is equity theory and how can it be applied? - [[EKS2001|EKS2001]]
# [[/ERG theory/]] - What is Alderfer's ERG theory? - [[User:MyUserName|MyUserName]]
# [[/Frame of reference and motivation/]] - How does frame of reference affect motivation? - [[User:MyUserName|MyUserName]]
# [[/Freedom and motivation/]] - What is the effect of freedom on motivation? - [[Cedevlin9|Cedevlin9]]
# [[/Fully functioning person/]] - What is a FFP and how can full functioning be developed? - [[User:Sebastian Armstrong|Sebastian Armstrong]]
# [[/Functional fixedness/]] - What is functional fixedness and how can it be overcome? - [[User:U3214117|U3214117]]
# [[/Functional imagery training/]] - What is FIT and how can it be applied? - [[User:MyUserName|MyUserName]]
# [[/Gamification and work motivation/]] - How can gamification enhance work motivation? - [[U3211125|U3211125]]
# [[/Giving up goals/]] - When should we give up goals and when should we persist? - [[User:MyUserName|U3161584]]
# [[/Green prescription motivation/]] - What motivates green prescription compliance? - [[User:Earthxangel|Earthxangel]]
# [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour? - [[SoSilverLibby]]
# [[/Help-seeking among boys/]] - What are the barriers to help-seeking for boys and what motivates them to seek help? - [[User:BradMcGrath|BradMcGrath]]
# [[/Hidden costs of reward/]] - What are the hidden costs of motivating by reward? - [[User:MyUserName|MyUserName]]
# [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis, what is the evidence, and how does it help to understand drug addiction? - [[U3218292|U3218292]]
# [[/Honesty motivation/]] - What motivates honesty? - [[User:U3200859|U3200859]]
# [[/Humour, leadership, and work/]] - What role does humour play in effective leadership in the workplace? - [[User:U3210264|U3210264]] 
# [[/IKEA effect/]] - What is the IKEA effect and how can it be applied? - [[U3216963|U]]3216963
# [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? - [[User:MyUserName|MyUserName]]
# [[/Kindness motivation/]] - What motivates kindness? - [[User:Clairelogan|MyUserName]]
# [[/Motivational music and exercise/]] - How can music be used to help motivate exercise? - [[User:MyUserName|U3183466]]
# [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]]
# [[/Nucleus accumbens and motivation/]] - What role does the nucleus accumbens play in motivation? - [[User:U3213250|U3213250]]
# [[/Perfectionism/]] - What motivates perfectionism? Is perfectionism good or bad? How can it be managed? - [[User:AEMOR|AEMOR]]
# [[/Physiological needs/]] - How do human's physiological needs affect motivation? - [[U3203655]]
# [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19? - [[User:U3200956|U3200956]]
# [[/Relative deprivation and motivation/]] - What is the effect of relative deprivation on motivation? - [[User:U3191574 (PHP)|U3191574 (PHP)]]
# [[/Retrospective regret/]] - What is the motivational role of retrospective regret? - [[User:Will-U3214082|Will-U3214082]]
# [[/Revenge motivation/]] - What motivates revenge and how does it affect us? - [[User:U3216654|U3216654]]
# [[/Self-efficacy and academic achievement/]] - What role does self-efficacy play in academic achievement? - [[User:U943292|U943292]]a
# [[/Self-efficacy and achievement/]] - What role does self-efficacy play in achievement outcomes? - [[User:U3216513mt|U3216513mt]]
# [[/Sexual harassment at work motivation/]] - What motivates sexual harassment at work and what can be done about it? - [[User:U3037979|U3037979]]
# [[/Signature strengths/]] - What are signature strengths and how can they be applied? - [[User:MyUserName|MyUserName]]
# [[/Social cure/]] - What is the social cure and how can it be applied? - [[User:U3215976|U3215976]]
# [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? - [[User:MyUserName|MyUserName]]
# [[/Stretch goals/]] - What are stretch goals? Do they work? - [[User:MyUserName|MyUserName]]
# [[/Sublimation/]] - What is sublimation and how can it be fostered? - [[User:MyUserName|MyUserName]]
# [[/Survival needs and motivation/]] - What are survival needs and how do they influence motivation? - [[User:U3148161|U3148161]]
# [[/Task initiation/]] - What are the challenges with task initiation and how to get get started? - [[User:MyUserName|MyUserName]]
# [[/Theoretical domains framework/]] - What is the TDF and how can be used to guide behaviour change? - [[User:MyUserName|MyUserName]]
# [[/Time and motivation/]] - What is the effect of time on motivation? - [[User:Lturner2311|Lturner2311]]
# [[/Time management/]] - How can one's time be managed effectively? - [[User:CNK.20|CNK.20]]
# [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - [[User:U3207458|U3207458]]
# [[/Uncertainty avoidance/]] - What is uncertainty avoidance, why does it occur, and what are its consequences? - [[User:Franklin Brightt|Franklin Brightt]]
# [[/Urgency bias and productivity/]] - What is the impact of urgency bias on productivity and what can be done about it? - U3055143
# [[/Vocational identity/]] - What is vocational identity and how does it develop? - [[User:MyUserName|MyUserName]]
# [[/Volunteer tourism motivation/]] - What motivates volunteer tourism? - [[User:U962051|U962051]]
# [[/Wanting and liking/]] - What are the similarities and differences between wanting and liking, and what are the implications? - [[User:MyUserName|MyUserName]]
# [[/Work breaks, well-being, and productivity/]] - How do work breaks affect well-being and productivity? - [[User:MyUserName|U3215603]]
# [[/Work and flow/]] - What characteristics of work can produce flow and how can flow at work be fostered? - [[User:U3213441|U3213441]]

==Emotion==
# [[/Animal emotion/]] - What is the emotional experience of animals? - [[U3216502]]
# [[/Attributions and emotion/]] - How do attributions affect emotion? - [[User:MyUserName|MyUserName]]
# [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? - [[User:U3186959|U3186959]] 
# [[/Benzodiazepines and emotion/]] - What are the effects of benzodiazepines on emotion? - [[User:FulaAjeo22|FulaAjeo22]]
# [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? - [[User:MyUserName|MyUserName]]
# [[/Burnout/]] - What is burnout and how can be it be managed and prevented? - [[User:U3211603|U3211603]]
# [[/Cognitive dissonance reduction/]] - What strategies do people use to reduce cognitive dissonance and how effective are they? - [[User:Tatjurate|Tatjurate]]
# [[/Colonisation and emotion in Australia/]] - What are the emotional responses to colonisation in Australia? - [[User:Micabaker1|Micabaker1]]
# [[/Compassion/]] - What is compassion, what are its pros and cons, and how can it be fostered? - u3203545 
# [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? - [[User:MyUserName|MyUserName]]
# [[/Contempt/]] - What is contempt, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]]
# [[/Core emotions/]] - What are the core emotions and what is their function?  U3203140
# [[/Creative arts and trauma/]] - How can creative arts help in dealing with trauma? - [[MyUserName|SashaBrooksby]]
# [[/Cultural influences on shame, guilt, and pride/]] - How does culture influence shame, guilt, and pride? - [[User:MyUserName|MyUserName]]
# [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? - [[User:MyUserName|MyUserName]]
# [[/Difficult conversations and emotion/]] - What communication and emotional skills are needed to successfully negotiate difficult conversations? - [[User:MyUserName|MyUserName]]
# [[/Disappointment/]] - What is disappointment, what causes disappointment, and how can disappointment be managed? - [[User:U3216256|U3216256]]
# [[/DMT and spirituality/]] - How can DMT facilitate spiritual experiences? - [[DenniseSoleymani]]
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? - [[User:MyUserName|MyUserName]]
# [[/Ecological grief/]] - What is ecological grief and what can be done about it? - [[User:Brewerjr|Brewerjr]]
# [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress? - [[User:MyUserName|MyUserName]]
# [[/Embarrassment/]] - What is embarrassment, what causes it, and how can it be managed? - u3190353
# [[/Emotional intelligence training/]] - How can emotional intelligence be trained? - [[User:MyUserName|MyUserName]]
# [[/Emotion knowledge/]] - What is emotion knowledge and how can it be developed? - [[User:MyUserName|MyUserName]]
# [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? - u3230861
# [[/Endocannabinoid system and emotion/]] - What is the role of the endocannabinoid system in emotion? - [[User:RWilliams12|Rwilliams12]]
# [[/Environmental grief/]] - What is eco-grief, its causes and consequences, and what can be done? - [[User:Gabrielle Eagling|Gabrielle Eagling]]
# [[/Exercise and endocannabinoids/]] - What is the relationship between exercise and the endocannabinoid system? - [[User:MyUserName|MyUserName]]
# [[/Expressive suppression and emotion regulation/]] - What is the role of expressive suppression in emotion regulation? - [[User:MyUserName|MyUserName]]
# [[/Fairness and emotion/]] - What is the relation between fairness and emotion? - [[User:MyUserName|MyUserName]]
# [[/Fatigue and emotion/]] - What is the effect of fatigue on emotion and what can be done about it? - [[User:Lewis.Kusk|Lewis.Kusk]]
# [[/Fear/]] - What is fear, what causes it, and how can it be managed? - [[User:Icantchooseone|Icantchooseone]]
# [[/Fear of working out/]] - What is FOWO and how can it be overcome? - [[User:MyUserName|MyUserName]]
# [[/Fundamental attribution error and emotion/]] - What is the relationship between the FAE and emotion? - [[User:MyUserName|MyUserName]]
# [[/Gratitude and subjective wellbeing/]] - What is the relationship between gratitude and subjective wellbeing? - [[User:MyUserName|MyUserName]]
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]]
# [[/Heart rate variability and emotion regulation/]] - What is the relationship between HRV and emotion regulation? - [[User:MyUserName|MyUserName]]
# [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? - [[User:MyUserName|MyUserName]]
# [[/Humility/]] - What is humility, what causes it, and is it desirable? - [[User:MyUserName|MyUserName]]
# [[/Hypomania and emotion/]] - What are the emotional characteristics of hypomania? - [[User:Alec.cortez|Alec.cortez]]
# [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided? - [[User:MyUserName|MyUserName]]
# [[Indigenous Australian emotionality]] - In what ways is emotionality experienced by Indigenous Australian people? - [[User:MyUserName|MyUserName]]
# [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness? - [[User:MyUserName|MyUserName]]
# [[/Inspiration/]] - What is inspiration, what causes it, what are its consequences, and how can it be fostered? - [[User:MyUserName|u3227354]]
# [[/Insular cortex and emotion/]] - What role does the insular cortex play in emotion? - [[User:U3190094|U3190094]]
# [[/Interoception and emotion/]] - What is the relationship between interoception and emotion? - [[User:MyUserName|MyUserName]]
# [[/Kama muta/]] - What is kama muta, what are its effects, and how can it be fostered? - [[User:U3183521|U3183521]]
# [[/Linguistic relativism and emotion/]] - What is the role of linguistic relativism in emotion? - [[User:U3119310|U3119310]]
# [[/Menstrual cycle mood disorders/]] - What causes menstrual cycle mood disorders and how can they be managed? - [[User:MyUserName|U3217109]]
# [[/Mindfulness and creativity/]] - How can mindfulness enhance creativity? - [[CaityDcr1603]]
# [[/Mindful self-care/]] - What is mindful self-care, why does it matter, and how can it be developed? - [[User:clairelogan|clairelogan]]
# [[/Mixed emotions/]] - What are mixed emotions, what causes them, and how can they be managed? - [[User:MyUserName|MyUserName]]
# [[/Mudita/]] - What is mudita and how can it be developed? - [[User:MyUserName|MyUserName]]
# [[/Natural disasters and emotion/]] - How do people respond emotionally to natural disasters and how can they be supported? -[[User:U3148366_Chris|U3148366_Chris]]
# [[/Nature therapy/]] - What is nature therapy and how can it be applied? - [[User:MyUserName|MyUserName]]
# [[/Narcissism and emotion/]] - What is the relationship between narcissism and emotion? - [[User:A Super Villain|A Super Villain]]
# [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy? - [[User:MyUserName|MyUserName]]
# [[/Needle fear/]] - How does needle fear develop, what are its consequences, and what can be done about it? - [[User:U3166273|U3166273]]
# [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? - [[User:MyUserName|MyUserName]]
# [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? - [[User:JorjaFive|JorjaFive]]
# [[/Psychological distress/]] - What is PD, what are the main types, and how can they be managed? - [[User:U3190773|U3190773]]
# [[/Psychological trauma/]] - What causes psychological trauma, what are the consequences, and how can people recover from psychological trauma? - [[User:U3210431|U3210431]]
# [[/Psilocybin assisted psychotherapy/]] - How can psilocybin be used to assist psychotherapy? - [[User:U3083720|U3083720]]
# [[/Rational compassion/]] - What is rational compassion and how can it be cultivated? - [[User:MyUserName|MyUserName]]
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? - [[User:MyUserName|MyUserName]]
# [[/Religiosity and coping/]] - What is the relationship between religiosity and coping? - [[User:MyUserName|MyUserName]]
# [[/Resentment/]] - What is resentment, what causes it, and what are its consequences? - [[User:U3216389|U3216389]]
# [[/Risk-as-feelings/]] - What is the emotional experience of risk and how does it influence decision-making and behaviour? - [[User:MyUserName|MyUserName]]
# [[/Self-esteem and culture/]] - What are the cultural influences on self-esteem? - [[User:MyUserName|MyUserName]]
# [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? - U3200902 
# [[/Social media and suicide prevention/]] - How can social media be used to help prevent suicide? - [[JaimeTegan|JaimeTegan]]
# [[/Sorry business/]] - What is sorry business and what role does it play in Indigenous communities in Australia? - Isaacem13
# [[/Stress control mindset/]] - What is a SCM, why does it matter, and how can it be cultivated? - [[User:MyUserName|MyUserName]]
# [[/Suffering as emotion/]] - What is the emotional experience of suffering and how can people cope with suffering? - [[User:MyUserName|MyUserName]]
# [[/Telemental health/]] - What are the pros and cons of TMH and what are the key ingredients for effective TMH practices? - [[User:MyUserName|MyUserName]]
# [[/Topophilia/]] - What is topophilia, how does it develop, and what are the psychological impacts? - [[User:RSPMeredith|RSPMeredith]]
# [[/Triumph/]] - What is triumph, what causes it, and how can it be managed? - [[User:Bill.miosge|Bill.miosge]]
# [[/Unemployment and mental health/]]: What is the relationship between unemployment and mental health? - [[User:MyUserName|MyUserName]]
# [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? - [[User:MyUserName|Ana028]]
# [[/Wave metaphor for emotion/]] - In what respects is an ocean wave a helpful metaphor for understanding human emotions? - [[jamieepiper]]
# [[/Window of tolerance/]] - What is the window of tolerance and how this concept be used? - [[User:U3223109|U3223109]]
# [[/Workplace mental health training/]] - What is WMHT, what techniques are used, and what are the impacts? - [[ArtOfHappiness]]
# [[/Zoom fatigue/]] - What is Zoom fatigue, what causes it, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]]

==Motivation and emotion==
# [[/Financial investing, motivation, and emotion/]] - What role does motivation and emotion play in financial investing? - [[U3217287|U3217287]]
# [[/Hostage negotiation, motivation, and emotion/]] - What role does motivation and emotion play in hostage negotiation? - [[User:U3213549|U3213549]]
# [[/Money priming, motivation, and emotion/]] - What is the effect of money priming on motivation and emotion? - [[User:MyUserName|MyUserName]]
# [[/Motivational dimensional model of affect/]] - What is the motivational dimensional model of affect and what are its implications? - [[User:MyUserName|MyUserName]]
# [[/Napping, motivation, and emotion/]] - What are the motivational and emotional effects of napping? - [[User:MyUserName|MyUserName]]
# [[/Overchoice, emotion, and motivation/]] - What are the emotional and motivational effects of overchoice? - [[User:MyUserName|MyUserName]]
# [[/Patience and impatience/]] - What are the psychological causes and consequences of patience and impatience? - [[User:MyUserName|MyUserName]]
# [[/Reward system, motivation, and emotion/]] - What role does the reward system play in motivation and emotion? - [[User:U3162201|U3162201]]

[[Category:Motivation and emotion/Book/2022]]</text>
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      <comment>/* Motivation */</comment>
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      <text bytes="21226" xml:space="preserve">{{/Banner}}
==Motivation==
# [[Academic help-seeking]] - What are the barriers and enablers of AHS and how can AHS be fostered? - [[User:MyUserName|MyUserName]]
# [[/Academic self-regulation/]] - What is academic self-regulation, why does it matter, and how can it be fostered? - [[U3216563]]
# [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? - [[User:MyUserName|MyUserName]]
# [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it? - [[User:MyUserName|MyUserName]]
# [[/Antidepressants and motivation/]] - What are the effects of popular antidepressants on motivation? - [[User:U3222363|U3222363]]
# [[/Approach motivation/]] - What is approach motivation and how does it lead to behaviour? - [[User:U3189370|U3189370]]
# [[/Behavioural economics and motivation/]] - What aspects of motivation theory are useful in behavioural economics? - [[User:U3141987|U3141987]]
# [[/Behavioural model of health services/]] - What is the BMHS and how can it be used? - SoSilverLibby
# [[/Beneficence as a psychological need/]] - What is beneficence and what are its implications as a psychological need? - [[User:MyUserName|CaitlinEmc]]
# [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention? - [[User:MyUserName|MyUserName]]
# [[/Choice overload/]] - What is choice overload? What is the optimal amount of choice? - [[UserGeorgiaFairweather|GeorgiaFairweather]]
# [[/Chunking and goal pursuit/]] - How does chunking affect goal pursuit? - [[User:MyUserName|MyUserName]]
# [[/Cognitive entrenchment/]] - What is cognitive entrenchment and how can it be avoided? - [[User:MyUserName|MyUserName]]
# [[/Climate change helplessness/]] - How does learned helpless impact motivation to engage in behaviours to limit climate change? - [[User:MyUserName|MyUserName]]
# [[/Closeness communication bias/]] - What is the CCB, why does it occur, and how can it be overcome? - [[User:MyUserName|MyUserName]]
# [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirably outcomes? - [[User:MyUserName|MyUserName]]
# [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories? - [[User:KingMob221|KingMob221]]
# [[/Construal level theory/]] - What is construal level theory and how can it be applied? - [[User:MyUserName|MyUserName]]
# [[/Courage motivation/]] - What is courage, what motivates courage, and how can courage be enhanced? -[[User:Hanarose123|Hanarose123]]
# [[/Death drive/]] - What is the death drive and how can it be negotiated? - [[User:U3086459|U3086459]]
# [[Motivation and emotion/Book/2022/Drugs-violence nexus and motivation Atu3202070|Drugs-violence nexus and motivation]] - What is the role of motivation in the drugs-violence nexus? -  [[Atu3202070|Atu3202070]]
# [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD? - [[User:MyUserName|MyUserName]]
# [[/Episodic memory and planning/]] - What role does episodic memory play in planning? - [[User:MyUserName|U3246310]]
# [[/Equity theory/]] - What is equity theory and how can it be applied? - [[EKS2001|EKS2001]]
# [[/ERG theory/]] - What is Alderfer's ERG theory? - [[User:MyUserName|MyUserName]]
# [[/Frame of reference and motivation/]] - How does frame of reference affect motivation? - [[User:MyUserName|MyUserName]]
# [[/Freedom and motivation/]] - What is the effect of freedom on motivation? - [[Cedevlin9|Cedevlin9]]
# [[/Fully functioning person/]] - What is a FFP and how can full functioning be developed? - [[User:Sebastian Armstrong|Sebastian Armstrong]]
# [[/Functional fixedness/]] - What is functional fixedness and how can it be overcome? - [[User:U3214117|U3214117]]
# [[/Functional imagery training/]] - What is FIT and how can it be applied? - [[User:MyUserName|MyUserName]]
# [[/Gamification and work motivation/]] - How can gamification enhance work motivation? - [[U3211125|U3211125]]
# [[/Giving up goals/]] - When should we give up goals and when should we persist? - [[User:MyUserName|U3161584]]
# [[/Green prescription motivation/]] - What motivates green prescription compliance? - [[User:Earthxangel|Earthxangel]]
# [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour? - [[SoSilverLibby]]
# [[/Help-seeking among boys/]] - What are the barriers to help-seeking for boys and what motivates them to seek help? - [[User:BradMcGrath|BradMcGrath]]
# [[/Hidden costs of reward/]] - What are the hidden costs of motivating by reward? - [[User:MyUserName|MyUserName]]
# [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis, what is the evidence, and how does it help to understand drug addiction? - [[U3218292|U3218292]]
# [[/Honesty motivation/]] - What motivates honesty? - [[User:U3200859|U3200859]]
# [[/Humour, leadership, and work/]] - What role does humour play in effective leadership in the workplace? - [[User:U3210264|U3210264]] 
# [[/IKEA effect/]] - What is the IKEA effect and how can it be applied? - [[U3216963|U]]3216963
# [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? - [[User:MyUserName|MyUserName]]
# [[/Kindness motivation/]] - What motivates kindness? - u3205429
# [[/Motivational music and exercise/]] - How can music be used to help motivate exercise? - [[User:MyUserName|U3183466]]
# [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]]
# [[/Nucleus accumbens and motivation/]] - What role does the nucleus accumbens play in motivation? - [[User:U3213250|U3213250]]
# [[/Perfectionism/]] - What motivates perfectionism? Is perfectionism good or bad? How can it be managed? - [[User:AEMOR|AEMOR]]
# [[/Physiological needs/]] - How do human's physiological needs affect motivation? - [[U3203655]]
# [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19? - [[User:U3200956|U3200956]]
# [[/Relative deprivation and motivation/]] - What is the effect of relative deprivation on motivation? - [[User:U3191574 (PHP)|U3191574 (PHP)]]
# [[/Retrospective regret/]] - What is the motivational role of retrospective regret? - [[User:Will-U3214082|Will-U3214082]]
# [[/Revenge motivation/]] - What motivates revenge and how does it affect us? - [[User:U3216654|U3216654]]
# [[/Self-efficacy and academic achievement/]] - What role does self-efficacy play in academic achievement? - [[User:U943292|U943292]]a
# [[/Self-efficacy and achievement/]] - What role does self-efficacy play in achievement outcomes? - [[User:U3216513mt|U3216513mt]]
# [[/Sexual harassment at work motivation/]] - What motivates sexual harassment at work and what can be done about it? - [[User:U3037979|U3037979]]
# [[/Signature strengths/]] - What are signature strengths and how can they be applied? - [[User:MyUserName|MyUserName]]
# [[/Social cure/]] - What is the social cure and how can it be applied? - [[User:U3215976|U3215976]]
# [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? - [[User:MyUserName|MyUserName]]
# [[/Stretch goals/]] - What are stretch goals? Do they work? - [[User:MyUserName|MyUserName]]
# [[/Sublimation/]] - What is sublimation and how can it be fostered? - [[User:MyUserName|MyUserName]]
# [[/Survival needs and motivation/]] - What are survival needs and how do they influence motivation? - [[User:U3148161|U3148161]]
# [[/Task initiation/]] - What are the challenges with task initiation and how to get get started? - [[User:MyUserName|MyUserName]]
# [[/Theoretical domains framework/]] - What is the TDF and how can be used to guide behaviour change? - [[User:MyUserName|MyUserName]]
# [[/Time and motivation/]] - What is the effect of time on motivation? - [[User:Lturner2311|Lturner2311]]
# [[/Time management/]] - How can one's time be managed effectively? - [[User:CNK.20|CNK.20]]
# [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - [[User:U3207458|U3207458]]
# [[/Uncertainty avoidance/]] - What is uncertainty avoidance, why does it occur, and what are its consequences? - [[User:Franklin Brightt|Franklin Brightt]]
# [[/Urgency bias and productivity/]] - What is the impact of urgency bias on productivity and what can be done about it? - U3055143
# [[/Vocational identity/]] - What is vocational identity and how does it develop? - [[User:MyUserName|MyUserName]]
# [[/Volunteer tourism motivation/]] - What motivates volunteer tourism? - [[User:U962051|U962051]]
# [[/Wanting and liking/]] - What are the similarities and differences between wanting and liking, and what are the implications? - [[User:MyUserName|MyUserName]]
# [[/Work breaks, well-being, and productivity/]] - How do work breaks affect well-being and productivity? - [[User:MyUserName|U3215603]]
# [[/Work and flow/]] - What characteristics of work can produce flow and how can flow at work be fostered? - [[User:U3213441|U3213441]]

==Emotion==
# [[/Animal emotion/]] - What is the emotional experience of animals? - [[U3216502]]
# [[/Attributions and emotion/]] - How do attributions affect emotion? - [[User:MyUserName|MyUserName]]
# [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? - [[User:U3186959|U3186959]] 
# [[/Benzodiazepines and emotion/]] - What are the effects of benzodiazepines on emotion? - [[User:FulaAjeo22|FulaAjeo22]]
# [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? - [[User:MyUserName|MyUserName]]
# [[/Burnout/]] - What is burnout and how can be it be managed and prevented? - [[User:U3211603|U3211603]]
# [[/Cognitive dissonance reduction/]] - What strategies do people use to reduce cognitive dissonance and how effective are they? - [[User:Tatjurate|Tatjurate]]
# [[/Colonisation and emotion in Australia/]] - What are the emotional responses to colonisation in Australia? - [[User:Micabaker1|Micabaker1]]
# [[/Compassion/]] - What is compassion, what are its pros and cons, and how can it be fostered? - u3203545 
# [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? - [[User:MyUserName|MyUserName]]
# [[/Contempt/]] - What is contempt, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]]
# [[/Core emotions/]] - What are the core emotions and what is their function?  U3203140
# [[/Creative arts and trauma/]] - How can creative arts help in dealing with trauma? - [[MyUserName|SashaBrooksby]]
# [[/Cultural influences on shame, guilt, and pride/]] - How does culture influence shame, guilt, and pride? - [[User:MyUserName|MyUserName]]
# [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? - [[User:MyUserName|MyUserName]]
# [[/Difficult conversations and emotion/]] - What communication and emotional skills are needed to successfully negotiate difficult conversations? - [[User:MyUserName|MyUserName]]
# [[/Disappointment/]] - What is disappointment, what causes disappointment, and how can disappointment be managed? - [[User:U3216256|U3216256]]
# [[/DMT and spirituality/]] - How can DMT facilitate spiritual experiences? - [[DenniseSoleymani]]
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? - [[User:MyUserName|MyUserName]]
# [[/Ecological grief/]] - What is ecological grief and what can be done about it? - [[User:Brewerjr|Brewerjr]]
# [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress? - [[User:MyUserName|MyUserName]]
# [[/Embarrassment/]] - What is embarrassment, what causes it, and how can it be managed? - u3190353
# [[/Emotional intelligence training/]] - How can emotional intelligence be trained? - [[User:MyUserName|MyUserName]]
# [[/Emotion knowledge/]] - What is emotion knowledge and how can it be developed? - [[User:MyUserName|MyUserName]]
# [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? - u3230861
# [[/Endocannabinoid system and emotion/]] - What is the role of the endocannabinoid system in emotion? - [[User:RWilliams12|Rwilliams12]]
# [[/Environmental grief/]] - What is eco-grief, its causes and consequences, and what can be done? - [[User:Gabrielle Eagling|Gabrielle Eagling]]
# [[/Exercise and endocannabinoids/]] - What is the relationship between exercise and the endocannabinoid system? - [[User:MyUserName|MyUserName]]
# [[/Expressive suppression and emotion regulation/]] - What is the role of expressive suppression in emotion regulation? - [[User:MyUserName|MyUserName]]
# [[/Fairness and emotion/]] - What is the relation between fairness and emotion? - [[User:MyUserName|MyUserName]]
# [[/Fatigue and emotion/]] - What is the effect of fatigue on emotion and what can be done about it? - [[User:Lewis.Kusk|Lewis.Kusk]]
# [[/Fear/]] - What is fear, what causes it, and how can it be managed? - [[User:Icantchooseone|Icantchooseone]]
# [[/Fear of working out/]] - What is FOWO and how can it be overcome? - [[User:MyUserName|MyUserName]]
# [[/Fundamental attribution error and emotion/]] - What is the relationship between the FAE and emotion? - [[User:MyUserName|MyUserName]]
# [[/Gratitude and subjective wellbeing/]] - What is the relationship between gratitude and subjective wellbeing? - [[User:MyUserName|MyUserName]]
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]]
# [[/Heart rate variability and emotion regulation/]] - What is the relationship between HRV and emotion regulation? - [[User:MyUserName|MyUserName]]
# [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? - [[User:MyUserName|MyUserName]]
# [[/Humility/]] - What is humility, what causes it, and is it desirable? - [[User:MyUserName|MyUserName]]
# [[/Hypomania and emotion/]] - What are the emotional characteristics of hypomania? - [[User:Alec.cortez|Alec.cortez]]
# [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided? - [[User:MyUserName|MyUserName]]
# [[Indigenous Australian emotionality]] - In what ways is emotionality experienced by Indigenous Australian people? - [[User:MyUserName|MyUserName]]
# [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness? - [[User:MyUserName|MyUserName]]
# [[/Inspiration/]] - What is inspiration, what causes it, what are its consequences, and how can it be fostered? - [[User:MyUserName|u3227354]]
# [[/Insular cortex and emotion/]] - What role does the insular cortex play in emotion? - [[User:U3190094|U3190094]]
# [[/Interoception and emotion/]] - What is the relationship between interoception and emotion? - [[User:MyUserName|MyUserName]]
# [[/Kama muta/]] - What is kama muta, what are its effects, and how can it be fostered? - [[User:U3183521|U3183521]]
# [[/Linguistic relativism and emotion/]] - What is the role of linguistic relativism in emotion? - [[User:U3119310|U3119310]]
# [[/Menstrual cycle mood disorders/]] - What causes menstrual cycle mood disorders and how can they be managed? - [[User:MyUserName|U3217109]]
# [[/Mindfulness and creativity/]] - How can mindfulness enhance creativity? - [[CaityDcr1603]]
# [[/Mindful self-care/]] - What is mindful self-care, why does it matter, and how can it be developed? - [[User:clairelogan|clairelogan]]
# [[/Mixed emotions/]] - What are mixed emotions, what causes them, and how can they be managed? - [[User:MyUserName|MyUserName]]
# [[/Mudita/]] - What is mudita and how can it be developed? - [[User:MyUserName|MyUserName]]
# [[/Natural disasters and emotion/]] - How do people respond emotionally to natural disasters and how can they be supported? -[[User:U3148366_Chris|U3148366_Chris]]
# [[/Nature therapy/]] - What is nature therapy and how can it be applied? - [[User:MyUserName|MyUserName]]
# [[/Narcissism and emotion/]] - What is the relationship between narcissism and emotion? - [[User:A Super Villain|A Super Villain]]
# [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy? - [[User:MyUserName|MyUserName]]
# [[/Needle fear/]] - How does needle fear develop, what are its consequences, and what can be done about it? - [[User:U3166273|U3166273]]
# [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? - [[User:MyUserName|MyUserName]]
# [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? - [[User:JorjaFive|JorjaFive]]
# [[/Psychological distress/]] - What is PD, what are the main types, and how can they be managed? - [[User:U3190773|U3190773]]
# [[/Psychological trauma/]] - What causes psychological trauma, what are the consequences, and how can people recover from psychological trauma? - [[User:U3210431|U3210431]]
# [[/Psilocybin assisted psychotherapy/]] - How can psilocybin be used to assist psychotherapy? - [[User:U3083720|U3083720]]
# [[/Rational compassion/]] - What is rational compassion and how can it be cultivated? - [[User:MyUserName|MyUserName]]
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? - [[User:MyUserName|MyUserName]]
# [[/Religiosity and coping/]] - What is the relationship between religiosity and coping? - [[User:MyUserName|MyUserName]]
# [[/Resentment/]] - What is resentment, what causes it, and what are its consequences? - [[User:U3216389|U3216389]]
# [[/Risk-as-feelings/]] - What is the emotional experience of risk and how does it influence decision-making and behaviour? - [[User:MyUserName|MyUserName]]
# [[/Self-esteem and culture/]] - What are the cultural influences on self-esteem? - [[User:MyUserName|MyUserName]]
# [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? - U3200902 
# [[/Social media and suicide prevention/]] - How can social media be used to help prevent suicide? - [[JaimeTegan|JaimeTegan]]
# [[/Sorry business/]] - What is sorry business and what role does it play in Indigenous communities in Australia? - Isaacem13
# [[/Stress control mindset/]] - What is a SCM, why does it matter, and how can it be cultivated? - [[User:MyUserName|MyUserName]]
# [[/Suffering as emotion/]] - What is the emotional experience of suffering and how can people cope with suffering? - [[User:MyUserName|MyUserName]]
# [[/Telemental health/]] - What are the pros and cons of TMH and what are the key ingredients for effective TMH practices? - [[User:MyUserName|MyUserName]]
# [[/Topophilia/]] - What is topophilia, how does it develop, and what are the psychological impacts? - [[User:RSPMeredith|RSPMeredith]]
# [[/Triumph/]] - What is triumph, what causes it, and how can it be managed? - [[User:Bill.miosge|Bill.miosge]]
# [[/Unemployment and mental health/]]: What is the relationship between unemployment and mental health? - [[User:MyUserName|MyUserName]]
# [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? - [[User:MyUserName|Ana028]]
# [[/Wave metaphor for emotion/]] - In what respects is an ocean wave a helpful metaphor for understanding human emotions? - [[jamieepiper]]
# [[/Window of tolerance/]] - What is the window of tolerance and how this concept be used? - [[User:U3223109|U3223109]]
# [[/Workplace mental health training/]] - What is WMHT, what techniques are used, and what are the impacts? - [[ArtOfHappiness]]
# [[/Zoom fatigue/]] - What is Zoom fatigue, what causes it, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]]

==Motivation and emotion==
# [[/Financial investing, motivation, and emotion/]] - What role does motivation and emotion play in financial investing? - [[U3217287|U3217287]]
# [[/Hostage negotiation, motivation, and emotion/]] - What role does motivation and emotion play in hostage negotiation? - [[User:U3213549|U3213549]]
# [[/Money priming, motivation, and emotion/]] - What is the effect of money priming on motivation and emotion? - [[User:MyUserName|MyUserName]]
# [[/Motivational dimensional model of affect/]] - What is the motivational dimensional model of affect and what are its implications? - [[User:MyUserName|MyUserName]]
# [[/Napping, motivation, and emotion/]] - What are the motivational and emotional effects of napping? - [[User:MyUserName|MyUserName]]
# [[/Overchoice, emotion, and motivation/]] - What are the emotional and motivational effects of overchoice? - [[User:MyUserName|MyUserName]]
# [[/Patience and impatience/]] - What are the psychological causes and consequences of patience and impatience? - [[User:MyUserName|MyUserName]]
# [[/Reward system, motivation, and emotion/]] - What role does the reward system play in motivation and emotion? - [[User:U3162201|U3162201]]

[[Category:Motivation and emotion/Book/2022]]</text>
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==Motivation==
# [[Academic help-seeking]] - What are the barriers and enablers of AHS and how can AHS be fostered? - [[User:MyUserName|MyUserName]]
# [[/Academic self-regulation/]] - What is academic self-regulation, why does it matter, and how can it be fostered? - [[U3216563]]
# [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? - [[User:MyUserName|MyUserName]]
# [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it? - [[User:MyUserName|MyUserName]]
# [[/Antidepressants and motivation/]] - What are the effects of popular antidepressants on motivation? - [[User:U3222363|U3222363]]
# [[/Approach motivation/]] - What is approach motivation and how does it lead to behaviour? - [[User:U3189370|U3189370]]
# [[/Behavioural economics and motivation/]] - What aspects of motivation theory are useful in behavioural economics? - [[User:U3141987|U3141987]]
# [[/Behavioural model of health services/]] - What is the BMHS and how can it be used? - SoSilverLibby
# [[/Beneficence as a psychological need/]] - What is beneficence and what are its implications as a psychological need? - [[User:MyUserName|CaitlinEmc]]
# [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention? - [[User:MyUserName|MyUserName]]
# [[/Choice overload/]] - What is choice overload? What is the optimal amount of choice? - [[UserGeorgiaFairweather|GeorgiaFairweather]]
# [[/Chunking and goal pursuit/]] - How does chunking affect goal pursuit? - [[User:MyUserName|MyUserName]]
# [[/Cognitive entrenchment/]] - What is cognitive entrenchment and how can it be avoided? - [[User:MyUserName|MyUserName]]
# [[/Climate change helplessness/]] - How does learned helpless impact motivation to engage in behaviours to limit climate change? - [[User:MyUserName|MyUserName]]
# [[/Closeness communication bias/]] - What is the CCB, why does it occur, and how can it be overcome? - [[User:MyUserName|MyUserName]]
# [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirably outcomes? - [[User:MyUserName|MyUserName]]
# [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories? - [[User:KingMob221|KingMob221]]
# [[/Construal level theory/]] - What is construal level theory and how can it be applied? - [[User:MyUserName|MyUserName]]
# [[/Courage motivation/]] - What is courage, what motivates courage, and how can courage be enhanced? -[[User:Hanarose123|Hanarose123]]
# [[/Death drive/]] - What is the death drive and how can it be negotiated? - [[User:U3086459|U3086459]]
# [[Motivation and emotion/Book/2022/Drugs-violence nexus and motivation Atu3202070|Drugs-violence nexus and motivation]] - What is the role of motivation in the drugs-violence nexus? -  [[Atu3202070|Atu3202070]]
# [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD? - [[User:MyUserName|MyUserName]]
# [[/Episodic memory and planning/]] - What role does episodic memory play in planning? - [[User:MyUserName|U3246310]]
# [[/Equity theory/]] - What is equity theory and how can it be applied? - [[EKS2001|EKS2001]]
# [[/ERG theory/]] - What is Alderfer's ERG theory? - [[User:MyUserName|MyUserName]]
# [[/Frame of reference and motivation/]] - How does frame of reference affect motivation? - [[User:MyUserName|MyUserName]]
# [[/Freedom and motivation/]] - What is the effect of freedom on motivation? - [[Cedevlin9|Cedevlin9]]
# [[/Fully functioning person/]] - What is a FFP and how can full functioning be developed? - [[User:Sebastian Armstrong|Sebastian Armstrong]]
# [[/Functional fixedness/]] - What is functional fixedness and how can it be overcome? - [[User:U3214117|U3214117]]
# [[/Functional imagery training/]] - What is FIT and how can it be applied? - [[User:MyUserName|MyUserName]]
# [[/Gamification and work motivation/]] - How can gamification enhance work motivation? - [[U3211125|U3211125]]
# [[/Giving up goals/]] - When should we give up goals and when should we persist? - [[User:MyUserName|U3161584]]
# [[/Green prescription motivation/]] - What motivates green prescription compliance? - [[User:Earthxangel|Earthxangel]]
# [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour? - [[SoSilverLibby]]
# [[/Help-seeking among boys/]] - What are the barriers to help-seeking for boys and what motivates them to seek help? - [[User:BradMcGrath|BradMcGrath]]
# [[/Hidden costs of reward/]] - What are the hidden costs of motivating by reward? - [[User:MyUserName|MyUserName]]
# [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis, what is the evidence, and how does it help to understand drug addiction? - [[U3218292|U3218292]]
# [[/Honesty motivation/]] - What motivates honesty? - [[User:U3200859|U3200859]]
# [[/Humour, leadership, and work/]] - What role does humour play in effective leadership in the workplace? - [[User:U3210264|U3210264]] 
# [[/IKEA effect/]] - What is the IKEA effect and how can it be applied? - [[U3216963|U]]3216963
# [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? - [[User:MyUserName|MyUserName]]
# [[/Kindness motivation/]] - What motivates kindness? - u3205429
# [[/Motivational music and exercise/]] - How can music be used to help motivate exercise? - [[User:MyUserName|U3183466]]
# [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]]
# [[/Nucleus accumbens and motivation/]] - What role does the nucleus accumbens play in motivation? - [[User:U3213250|U3213250]]
# [[/Perfectionism/]] - What motivates perfectionism? Is perfectionism good or bad? How can it be managed? - [[User:AEMOR|AEMOR]]
# [[/Physiological needs/]] - How do human's physiological needs affect motivation? - [[U3203655]]
# [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19? - [[User:U3200956|U3200956]]
# [[/Relative deprivation and motivation/]] - What is the effect of relative deprivation on motivation? - [[User:U3191574 (PHP)|U3191574 (PHP)]]
# [[/Retrospective regret/]] - What is the motivational role of retrospective regret? - [[User:Will-U3214082|Will-U3214082]]
# [[/Revenge motivation/]] - What motivates revenge and how does it affect us? - [[User:U3216654|U3216654]]
# [[/Self-efficacy and academic achievement/]] - What role does self-efficacy play in academic achievement? - [[User:U943292|U943292]]a
# [[/Self-efficacy and achievement/]] - What role does self-efficacy play in achievement outcomes? - [[User:U3216513mt|U3216513mt]]
# [[/Sexual harassment at work motivation/]] - What motivates sexual harassment at work and what can be done about it? - [[User:U3037979|U3037979]]
# [[/Signature strengths/]] - What are signature strengths and how can they be applied? - [[User:MyUserName|MyUserName]]
# [[/Social cure/]] - What is the social cure and how can it be applied? - [[User:U3215976|U3215976]]
# [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? - [[User:MyUserName|MyUserName]]
# [[/Stretch goals/]] - What are stretch goals? Do they work? - [[User:MyUserName|MyUserName]]
# [[/Sublimation/]] - What is sublimation and how can it be fostered? - [[User:MyUserName|MyUserName]]
# [[/Survival needs and motivation/]] - What are survival needs and how do they influence motivation? - [[User:U3148161|U3148161]]
# [[/Task initiation/]] - What are the challenges with task initiation and how to get get started? - [[User:MyUserName|MyUserName]]
# [[/Theoretical domains framework/]] - What is the TDF and how can be used to guide behaviour change? - [[User:MyUserName|MyUserName]]
# [[/Time and motivation/]] - What is the effect of time on motivation? - [[User:Lturner2311|Lturner2311]]
# [[/Time management/]] - How can one's time be managed effectively? - [[User:CNK.20|CNK.20]]
# [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - [[User:U3207458|U3207458]]
# [[/Uncertainty avoidance/]] - What is uncertainty avoidance, why does it occur, and what are its consequences? - [[User:Franklin Brightt|Franklin Brightt]]
# [[/Urgency bias and productivity/]] - What is the impact of urgency bias on productivity and what can be done about it? - U3055143
# [[/Vocational identity/]] - What is vocational identity and how does it develop? - [[User:MyUserName|MyUserName]]
# [[/Volunteer tourism motivation/]] - What motivates volunteer tourism? - [[User:U962051|U962051]]
# [[/Wanting and liking/]] - What are the similarities and differences between wanting and liking, and what are the implications? - [[User:MyUserName|MyUserName]]
# [[/Work breaks, well-being, and productivity/]] - How do work breaks affect well-being and productivity? - [[User:MyUserName|U3215603]]
# [[/Work and flow/]] - What characteristics of work can produce flow and how can flow at work be fostered? - [[User:U3213441|U3213441]]

==Emotion==
# [[/Animal emotion/]] - What is the emotional experience of animals? - [[U3216502]]
# [[/Attributions and emotion/]] - How do attributions affect emotion? - [[User:MyUserName|MyUserName]]
# [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? - [[User:U3186959|U3186959]] 
# [[/Benzodiazepines and emotion/]] - What are the effects of benzodiazepines on emotion? - [[User:FulaAjeo22|FulaAjeo22]]
# [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? - [[User:MyUserName|MyUserName]]
# [[/Burnout/]] - What is burnout and how can be it be managed and prevented? - [[User:U3211603|U3211603]]
# [[/Cognitive dissonance reduction/]] - What strategies do people use to reduce cognitive dissonance and how effective are they? - [[User:Tatjurate|Tatjurate]]
# [[/Colonisation and emotion in Australia/]] - What are the emotional responses to colonisation in Australia? - [[User:Micabaker1|Micabaker1]]
# [[/Compassion/]] - What is compassion, what are its pros and cons, and how can it be fostered? - u3203545 
# [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? - [[User:MyUserName|MyUserName]]
# [[/Contempt/]] - What is contempt, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]]
# [[/Core emotions/]] - What are the core emotions and what is their function?  U3203140
# [[/Creative arts and trauma/]] - How can creative arts help in dealing with trauma? - [[MyUserName|SashaBrooksby]]
# [[/Cultural influences on shame, guilt, and pride/]] - How does culture influence shame, guilt, and pride? - [[User:MyUserName|MyUserName]]
# [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? - [[User:MyUserName|MyUserName]]
# [[/Difficult conversations and emotion/]] - What communication and emotional skills are needed to successfully negotiate difficult conversations? - [[User:MyUserName|MyUserName]]
# [[/Disappointment/]] - What is disappointment, what causes disappointment, and how can disappointment be managed? - [[User:U3216256|U3216256]]
# [[/DMT and spirituality/]] - How can DMT facilitate spiritual experiences? - [[DenniseSoleymani]]
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? - [[User:MyUserName|MyUserName]]
# [[/Ecological grief/]] - What is ecological grief and what can be done about it? - [[User:Brewerjr|Brewerjr]]
# [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress? - [[User:MyUserName|MyUserName]]
# [[/Embarrassment/]] - What is embarrassment, what causes it, and how can it be managed? - u3190353
# [[/Emotional intelligence training/]] - How can emotional intelligence be trained? - [[User:MyUserName|MyUserName]]
# [[/Emotion knowledge/]] - What is emotion knowledge and how can it be developed? - [[User:MyUserName|MyUserName]]
# [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? - u3230861
# [[/Endocannabinoid system and emotion/]] - What is the role of the endocannabinoid system in emotion? - [[User:RWilliams12|Rwilliams12]]
# [[/Environmental grief/]] - What is eco-grief, its causes and consequences, and what can be done? - [[User:Gabrielle Eagling|Gabrielle Eagling]]
# [[/Exercise and endocannabinoids/]] - What is the relationship between exercise and the endocannabinoid system? - [[User:MyUserName|MyUserName]]
# [[/Expressive suppression and emotion regulation/]] - What is the role of expressive suppression in emotion regulation? - [[User:MyUserName|MyUserName]]
# [[/Fairness and emotion/]] - What is the relation between fairness and emotion? - [[User:MyUserName|MyUserName]]
# [[/Fatigue and emotion/]] - What is the effect of fatigue on emotion and what can be done about it? - [[User:Lewis.Kusk|Lewis.Kusk]]
# [[/Fear/]] - What is fear, what causes it, and how can it be managed? - [[User:Icantchooseone|Icantchooseone]]
# [[/Fear of working out/]] - What is FOWO and how can it be overcome? - [[User:MyUserName|MyUserName]]
# [[/Fundamental attribution error and emotion/]] - What is the relationship between the FAE and emotion? - [[User:MyUserName|MyUserName]]
# [[/Gratitude and subjective wellbeing/]] - What is the relationship between gratitude and subjective wellbeing? - [[User:MyUserName|MyUserName]]
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]]
# [[/Heart rate variability and emotion regulation/]] - What is the relationship between HRV and emotion regulation? - [[User:MyUserName|MyUserName]]
# [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? - [[User:MyUserName|MyUserName]]
# [[/Humility/]] - What is humility, what causes it, and is it desirable? - [[User:MyUserName|MyUserName]]
# [[/Hypomania and emotion/]] - What are the emotional characteristics of hypomania? - [[User:Alec.cortez|Alec.cortez]]
# [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided? - [[User:MyUserName|MyUserName]]
# [[Indigenous Australian emotionality]] - In what ways is emotionality experienced by Indigenous Australian people? - [[User:MyUserName|MyUserName]]
# [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness? - [[User:MyUserName|MyUserName]]
# [[/Inspiration/]] - What is inspiration, what causes it, what are its consequences, and how can it be fostered? - [[User:MyUserName|u3227354]]
# [[/Insular cortex and emotion/]] - What role does the insular cortex play in emotion? - [[User:U3190094|U3190094]]
# [[/Interoception and emotion/]] - What is the relationship between interoception and emotion? - [[User:MyUserName|MyUserName]]
# [[/Kama muta/]] - What is kama muta, what are its effects, and how can it be fostered? - [[User:U3183521|U3183521]]
# [[/Linguistic relativism and emotion/]] - What is the role of linguistic relativism in emotion? - [[User:U3119310|U3119310]]
# [[/Menstrual cycle mood disorders/]] - What causes menstrual cycle mood disorders and how can they be managed? - [[User:MyUserName|U3217109]]
# [[/Mindfulness and creativity/]] - How can mindfulness enhance creativity? - [[CaityDcr1603]]
# [[/Mindful self-care/]] - What is mindful self-care, why does it matter, and how can it be developed? - [[User:clairelogan|clairelogan]]
# [[/Mixed emotions/]] - What are mixed emotions, what causes them, and how can they be managed? - [[User:MyUserName|MyUserName]]
# [[/Mudita/]] - What is mudita and how can it be developed? - [[User:MyUserName|MyUserName]]
# [[/Natural disasters and emotion/]] - How do people respond emotionally to natural disasters and how can they be supported? -[[User:U3148366_Chris|U3148366_Chris]]
# [[/Nature therapy/]] - What is nature therapy and how can it be applied? - [[User:MyUserName|MyUserName]]
# [[/Narcissism and emotion/]] - What is the relationship between narcissism and emotion? - [[User:A Super Villain|A Super Villain]]
# [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy? - [[User:MyUserName|MyUserName]]
# [[/Needle fear/]] - How does needle fear develop, what are its consequences, and what can be done about it? - [[User:U3166273|U3166273]]
# [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? - [[User:MyUserName|MyUserName]]
# [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? - [[User:JorjaFive|JorjaFive]]
# [[/Psychological distress/]] - What is PD, what are the main types, and how can they be managed? - [[User:U3190773|U3190773]]
# [[/Psychological trauma/]] - What causes psychological trauma, what are the consequences, and how can people recover from psychological trauma? - [[User:U3210431|U3210431]]
# [[/Psilocybin assisted psychotherapy/]] - How can psilocybin be used to assist psychotherapy? - [[User:U3083720|U3083720]]
# [[/Rational compassion/]] - What is rational compassion and how can it be cultivated? - [[User:MyUserName|MyUserName]]
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? - [[User:MyUserName|MyUserName]]
# [[/Religiosity and coping/]] - What is the relationship between religiosity and coping? - [[User:MyUserName|MyUserName]]
# [[/Resentment/]] - What is resentment, what causes it, and what are its consequences? - [[User:U3216389|U3216389]]
# [[/Risk-as-feelings/]] - What is the emotional experience of risk and how does it influence decision-making and behaviour? - [[User:MyUserName|MyUserName]]
# [[/Self-esteem and culture/]] - What are the cultural influences on self-esteem? - [[User:MyUserName|MyUserName]]
# [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? - U3200902 
# [[/Social media and suicide prevention/]] - How can social media be used to help prevent suicide? - [[JaimeTegan|JaimeTegan]]
# [[/Sorry business/]] - What is sorry business and what role does it play in Indigenous communities in Australia? - Isaacem13
# [[/Stress control mindset/]] - What is a SCM, why does it matter, and how can it be cultivated? - [[User:MyUserName|MyUserName]]
# [[/Suffering as emotion/]] - What is the emotional experience of suffering and how can people cope with suffering? - [[User:Brookewin|Brookewin]]
# [[/Telemental health/]] - What are the pros and cons of TMH and what are the key ingredients for effective TMH practices? - [[User:MyUserName|MyUserName]]
# [[/Topophilia/]] - What is topophilia, how does it develop, and what are the psychological impacts? - [[User:RSPMeredith|RSPMeredith]]
# [[/Triumph/]] - What is triumph, what causes it, and how can it be managed? - [[User:Bill.miosge|Bill.miosge]]
# [[/Unemployment and mental health/]]: What is the relationship between unemployment and mental health? - [[User:MyUserName|MyUserName]]
# [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? - [[User:MyUserName|Ana028]]
# [[/Wave metaphor for emotion/]] - In what respects is an ocean wave a helpful metaphor for understanding human emotions? - [[jamieepiper]]
# [[/Window of tolerance/]] - What is the window of tolerance and how this concept be used? - [[User:U3223109|U3223109]]
# [[/Workplace mental health training/]] - What is WMHT, what techniques are used, and what are the impacts? - [[ArtOfHappiness]]
# [[/Zoom fatigue/]] - What is Zoom fatigue, what causes it, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]]

==Motivation and emotion==
# [[/Financial investing, motivation, and emotion/]] - What role does motivation and emotion play in financial investing? - [[U3217287|U3217287]]
# [[/Hostage negotiation, motivation, and emotion/]] - What role does motivation and emotion play in hostage negotiation? - [[User:U3213549|U3213549]]
# [[/Money priming, motivation, and emotion/]] - What is the effect of money priming on motivation and emotion? - [[User:MyUserName|MyUserName]]
# [[/Motivational dimensional model of affect/]] - What is the motivational dimensional model of affect and what are its implications? - [[User:MyUserName|MyUserName]]
# [[/Napping, motivation, and emotion/]] - What are the motivational and emotional effects of napping? - [[User:MyUserName|MyUserName]]
# [[/Overchoice, emotion, and motivation/]] - What are the emotional and motivational effects of overchoice? - [[User:MyUserName|MyUserName]]
# [[/Patience and impatience/]] - What are the psychological causes and consequences of patience and impatience? - [[User:MyUserName|MyUserName]]
# [[/Reward system, motivation, and emotion/]] - What role does the reward system play in motivation and emotion? - [[User:U3162201|U3162201]]

[[Category:Motivation and emotion/Book/2022]]</text>
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      <text bytes="429" xml:space="preserve">{{User:Jtwsaddress42/People}}
'''[[User:Jtwsaddress42/Bibliography_F|&lt;big&gt;F&lt;/big&gt;]]'''
{{User:Jtwsaddress42/People/Faraday,_Michael}}
{{User:Jtwsaddress42/People/Fermi,_Enrico}}
{{User:Jtwsaddress42/People/Feynman,_Richard Phillips}}
{{User:Jtwsaddress42/People/Freud, Sigmund}}
{{User:Jtwsaddress42/People/Fuller, R. Buckminster}}
{{User:Jtwsaddress42/Includes/Notes_&amp;_Citations}}
{{User:Jtwsaddress42/Navigation/Footer Navbar}}</text>
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    <title>User:Jtwsaddress42/People/Darwin, Charles</title>
    <ns>2</ns>
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      <comment>/* Darwin, Charles (1809 - 1882) */</comment>
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=== [[w:Charles_Darwin|Darwin, Charles (1809 - 1882)]] ===
&lt;hr /&gt;
[[File:Charles-darwin-portrait-svg.svg|thumb|Charles Darwin (1809 – 1882)]]
[[File:Origin of Species 1859 title page.jpg|thumb|Origin of Species 1859 (title page)]]
'''Notable Accomplishments'''
* The Theory of Natural Selection
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 480px
|{{User:Jtwsaddress42/Gallery/The Darwinian Revolution}}
|}&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Darwin,_Charles}}
&lt;br /&gt;&lt;hr /&gt;
'''Web Resources'''
* [http://darwin-online.org.uk/Freeman_intro.html Darwin Online]
* [https://www.darwinproject.ac.uk/ Darwin Correspondence Project]
* [https://www.khanacademy.org/science/ap-biology/natural-selection/natural-selection-ap/a/darwin-evolution-natural-selection Khan Academy - Darwin, evolution, &amp; natural selection]
{{RoundBoxBottom}}
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=== [[w:Charles_Darwin|Darwin, Charles (1809 - 1882)]] ===
&lt;hr /&gt;
[[File:Charles-darwin-portrait-svg.svg|thumb|Charles Darwin (1809 – 1882)]]
[[File:Origin of Species 1859 title page.jpg|thumb|Origin of Species 1859 (title page)]]
'''Notable Accomplishments'''
* The Theory of Natural Selection
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|{{User:Jtwsaddress42/Gallery/The Darwinian Revolution}}
|}&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Darwin,_Charles}}
&lt;br /&gt;&lt;hr /&gt;
'''Web Resources'''
* [http://darwin-online.org.uk/Freeman_intro.html Darwin Online]
* [https://www.darwinproject.ac.uk/ Darwin Correspondence Project]
* [https://www.khanacademy.org/science/ap-biology/natural-selection/natural-selection-ap/a/darwin-evolution-natural-selection Khan Academy - Darwin, evolution, &amp; natural selection]
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    <title>User:Jtwsaddress42/People/Faraday, Michael</title>
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=== [[w:Michael_Faraday|Faraday, Michael (1791 - 1867)]] ===
&lt;hr /&gt;
[[File:M Faraday Th Phillips oil 1842.jpg|thumb|Michael Faraday (1791 - 1867){{efn|portrait by Thomas Phillips (oil, 1842)}}]]
[[File:Michael Faraday signature.svg|thumb|Signature]]
'''Notable Accomplishments'''
* Magnetic Field
* Electromagnetic Induction
* Electrolysis
&lt;br /&gt;&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Faraday, Michael}}
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 480px
|{{User:Jtwsaddress42/Gallery/Michael Faraday}}
&lt;br /&gt;
|}
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=== [[w:Michael_Faraday|Faraday, Michael (1791 - 1867)]] ===
&lt;hr /&gt;
[[File:M Faraday Th Phillips oil 1842.jpg|thumb|Michael Faraday (1791 - 1867){{efn|portrait by Thomas Phillips (oil, 1842)}}]]
[[File:Michael Faraday signature.svg|thumb|Signature]]
'''Notable Accomplishments'''
* Magnetic Field
* Electromagnetic Induction
* Electrolysis
&lt;br /&gt;&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Faraday, Michael}}
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|{{User:Jtwsaddress42/Gallery/Michael Faraday}}
&lt;br /&gt;
|}
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=== [[w:Michael_Faraday|Faraday, Michael (1791 - 1867)]] ===
&lt;hr /&gt;
[[File:M Faraday Th Phillips oil 1842.jpg|thumb|Michael Faraday (1791 - 1867){{efn|portrait by Thomas Phillips (oil, 1842)}}]]
[[File:Michael Faraday signature.svg|thumb|Signature]]
'''Notable Accomplishments'''
* Magnetic Field
* Electromagnetic Induction
* Electrolysis
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|{{User:Jtwsaddress42/Gallery/Michael Faraday}}
&lt;br /&gt;
|}
&lt;br /&gt;&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Faraday, Michael}}
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=== [[w:Michael_Faraday|Faraday, Michael (1791 - 1867)]] ===
&lt;hr /&gt;
[[File:M Faraday Th Phillips oil 1842.jpg|thumb|Michael Faraday (1791 - 1867){{efn|portrait by Thomas Phillips (oil, 1842)}}]]
[[File:Michael Faraday signature.svg|thumb|Signature]]
'''Notable Accomplishments'''
* Magnetic Field
* Electromagnetic Induction
* Electrolysis
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|{{User:Jtwsaddress42/Gallery/Michael Faraday}}
&lt;br /&gt;
|}&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Faraday, Michael}}
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    <title>User:Jtwsaddress42/People/Fermi, Enrico</title>
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      <comment>/* Fermi, Enrico (1901 - 1954) */</comment>
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=== [[w:Enrico_Fermi|Fermi, Enrico (1901 - 1954)]] ===
&lt;hr /&gt;
[[File:Enrico Fermi 1943-49.jpg|thumb|Enrico Fermi (1901 - 1954)]]
[[File:Beta-minus Decay.svg|thumb|Fermi and Pauli postulated that a neutrino (&lt;math&gt;\bar{\nu}_e&lt;/math&gt;) would be emitted during Beta Decay.]]
'''Notable Accomplishments'''
* [https://www.nobelprize.org/prizes/physics/1938/fermi/facts/ The Nobel Prize in Physics 1938] - &quot;for his demonstrations of the existence of new radioactive elements produced by neutron irradiation, and for his related discovery of nuclear reactions brought about by slow neutrons.&quot;
&lt;br /&gt;&lt;hr /&gt;
{| align= center
|'''Stagg Field Reactor'''
''Chicago Pile-1, the first nuclear reactor to achieve a self-sustaining chain reaction''&lt;br /&gt;
[[File:Stagg Field reactor.jpg|640px|Stagg Field reactor]]
&lt;br/&gt;
|}
&lt;br /&gt;&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Fermi,_Enrico}}
{{RoundBoxBottom}}
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|{{Gallery
 | title =  ''The Darwinian Revolution''
 | width = 100
 | height = 100
 | align= center
  | File:Darwin Tree 1837.png | alt1= Transmutation diagram | ''Transmutation Diagram''.{{efn| Taken from Charles Darwin's ''Notebook B - On the Transmutation of Species'' (1837).&lt;br /&gt;This is Darwin's first diagram showing his thoughts on evolutionary diversification.  Gerald Edelman's ''Neural Darwinism'' would be published 150 years later - a fitting tribute to the enduring power of Darwin's ideas.  Although many will claim that this is the first evolutionary tree diagram, it is more of a bush, if anything, and Darwin's initial instincts are prescient.}}{{sfn|Darwin|1837}}
 | File:Darwins scheme showing evolution of the species. Wellcome L0051068.jpg | alt2= Darwins scheme showing evolution of the species. | Evolution Of The Species.{{efn|Wellcome L0051068}}{{sfn|Darwin|1859}}
 | File:Charles Darwin, Journal of Researches... Wellcome L0026712.jpg| alt3= Charles_Darwin,_Journal_of_Researches... |Galapagos Finches.{{efn|Four types of finch: 1: Geospiza magnirostris; 2. Geospiza fortis; 3. Geospiza parvula; 4. Gerthidea olivacea. (Charles Darwin, Journal of Researches... Wellcome_L0026712)}}
 | File:Illustration The Expression of Emotions in Man and Animals by Charles Darwin.jpg | alt4= The Expression of Emotions in Man and Animals
| The Expression of Emotions in Man and Animals{{sfn|Darwin|1872}}
}}
|}</text>
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  <page>
    <title>User:Atcovi/German Notes/Kapital 5/Notes</title>
    <ns>2</ns>
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      <comment>/* Two way prepositions */</comment>
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      <text bytes="6366" xml:space="preserve">== Kapital 5 - 2/6/2022 ==
- '''Weak verbs''': ge- + verb stem + -t/-et (gearbeitet, geregnet, gehört)&lt;br&gt;
--verbs ending with &quot;-ieren&quot; don't have the &quot;ge-&quot;

* Review &quot;strong&quot;, &quot;weak&quot; and &quot;mixed&quot; verbs
* https://chatterbug.com/grammar/german/perfect-tense-questions - &quot;Perfekt&quot;

=== &quot;Strong&quot;/&quot;Weak&quot;/&quot;Mixed&quot; Verbs ===
- https://www.vocabulix.com/german/strong-weak-verbs.shtml

=== Accusative Pronouns ===
-accusacative verbs: https://en.wikibooks.org/wiki/German/Grammar/Transitive_verbs#The_accusative_pronouns
{| class=&quot;wikitable&quot;
|+
!English
!Deutsch
|-
|me
|'''mich'''
|-
|you (informal)
|'''dich'''
|-
|you (formal)
|'''Sie'''
|-
|him
|'''ihn'''
|-
|her
|'''sie'''
|-
|it
|'''es'''
|-
|us
|'''uns'''
|-
|you all (informal)
|'''euch'''
|-
|you all (formal)
|'''Sie'''
|-
|them
|'''sie'''
|}
====Examples====
{| class=&quot;wikitable&quot;
|+
!Deutsch
!Deutsch (accusativ)
|-
|Ich kaufe den Apfel
|Ich kaufe '''''ihn''''' [masculine &quot;it&quot;]
|-
|Ich höre das Mäuschen
|Ich höre '''''es''''' [neutral &quot;it&quot;]
|-
|Ich sehe die Blume
|Ich sehe '''''sie''''' [feminine &quot;it&quot;]
|}

=== Dative Pronouns ===
accusative pronoun &gt; dative pronoun

==== Review of what dative means ====
''What makes a dative case''? Indirect object/&quot;to whom?&quot; (Ich gebe '''dem alten Mann''' eine Fahrkarte; &quot;dem alten Mann&quot; is the indirect object, while the &quot;Fahrkarte&quot; is the direct job that is being acted on. You'd ask yourself, in this case: &quot;Who was the direct object (eine Fahrkarte) given [verb] to?&quot; - The man)

-dative pronouns: https://en.wikibooks.org/wiki/German/Grammar/Ditransitive_verbs#The_dative_pronouns
{| class=&quot;wikitable&quot;
|+
!English
!Deutsch
|-
|I
|'''mir'''
|-
|You (informal)
|'''dir'''
|-
|You (formal)
|'''Ihnen'''
|-
|Him
|'''ihm'''
|-
|Her
|'''ihr'''
|-
|It
|'''ihm'''
|-
|We
|'''uns'''
|-
|You (plural)
|'''euch'''
|-
|Them
|'''ihnen'''
|}

====== Overview ======
- https://german.net/exercises/cases/accusative-dative/

=== Vocabulary (Feiern &amp; Party)===
Es ist die Zeit Feste zu feiern!
*Ich '''feiere'''
*Du '''feierst'''
*Er/Sie/Es '''feiert'''
*Wir '''feieren'''
*Ihr '''feiert'''
*Sie '''feieren'''

==== Feste ====
{| class=&quot;wikitable&quot;
|+
!English
!Deutsch
|-
|The holiday
|Der Feiertag
|-
|The card
|Die Karte
|-
|The party
|Die Party
|-
|The room
|Das Zimmer
|-
|To toast
|anstoßen (stößt... an)
|-
|To recieve
|bekommen
|-
|To invite
|einladen (lädt... an)
|-
|To celebrate
|feiern
|-
|To throw a party
|eine Party geben
|-
|To [not] have fun
|(keinen) Spaß haben
|-
|To give [a gift]
|schenken
|-
|To surprise
|Überraschen
|-
|Congratulations!
|Herlizchen Glückwunsch
|}

==== Besondere Anlässe ====
{| class=&quot;wikitable&quot;
|+
!English
!Deutsch
|-
|The marriage
|Die Ehe
|-
|The newlywed
|Der/Die Frischvermählte
|-
|The birth
|Die Geburt
|-
|The birthday
|Der Geburtstag
|-
|The wedding
|Die Hochzeit
|-
|The wedding anniversary
|Der Hochzeitstag
|-
|The anniversary
|Der Jahrestag
|-
|The New Year's Eve
|Das/Der Silvester
|-
|The Christmas
|Das Weihnachten
|-
|To [go into] retirement
|in Rente gehen
|-
|To [make] graduate [happen]
|einen Abschluss machen
|}

==== Ausdrücke ====
{| class=&quot;wikitable&quot;
|+
!English
!Deutsch
|-
|The friendship
|Die Freündschaft
|-
|The happiness
|Das Glück
|-
|The kiss
|Der Kuss
|-
|The love
|Die Liebe
|}

=== Vocabulary (das Kleidung) ===
{| class=&quot;wikitable&quot;
|+
!Deutsch
!Englisch
|-
|Der Hut, -e
|hat
|-
|Der Badeanzug, -e
|bathing suit
|-
|Teuer
|expensive
|-
|Die Krawatte
|tie
|-
|Der Gürtel
|belt
|-
|Das Kleid
|dress
|-
|Das Hemd
|shirt
|-
|Die kurze Hose
|shorts
|-
|Der Turnschuh
|sneakers
|-
|Er trägt einen Anzug
|He's wearing a suit
|-
|Die Handtasche
|purse
|-
|gelb
|yellow
|-
|grün
|green
|-
|lila
|purple
|-
|rosa
|pink
|-
|grau
|gray
|-
|schwarz
|black
|-
|weiß
|white
|-
|braun
|brown
|-
|blau
|blue
|-
|rot
|red
|-
|orange
|orange
|-
|die Sonnenbrille, -n
|sunglasses
|-
|die Mütze, -n
|cap
|-
|die Halskette, -n
|necklace
|-
|der Schal, -s
|scarf
|-
|Es ist günstig
|It's cheap
|-
|die Jacke, -n
|jacket
|-
|der Rock, -e
|skirt
|-
|die Hose, -n
|pants
|-
|der Schuh, -e
|shoe
|}

== Wissen vs. Kennen ==

* '''Wissen''' - To know as a fact (I know that her name is... I know her address is...)
* '''Kennen''' - To be familiar with/acquainted with (Familiar with people or places)

- https://languageposters.com/pages/german-verbs-wissen-conjugation

-https://www.vocabulix.com/conjugation/query_german.html?conjugation3,kennen,kennen,_german_present

''Are you familiar with the subject? If not, then use wissen.''

=== Examples ===

==== Wissen ====

* die Adresse von Amy
* ihren Familiennamen
* die Antwort
* viel ueber Mode
* die Uhrzeit

==== Kennen ====

* das Anne Frank-Haus in Amsterdam
* deinen Freundin nicht
* Maries Eltern
* ein gutes chinesisches Restaurant
* Wien gut

== Das Perfekt: The conversational past ==
Ich kaufe ein Brot in der Bäckerei. 

* Decide: sein or haben? '''Sein''' is a change of a physical condition/location (Point A --&gt; Point B), while '''haben''' is for everything else. 
* Conguate based on decision. 
* Pay attention to if the verb is weak, strong, or mixed.

== Two way prepositions ==
Either destination/movement towards a destination (accusative) or location/no movement towards a destination (dative). TIP: Use '''wohin''' ''(where to)'' with accusative as it talks about a movement towards a destination. Use '''wo''' with dative as it talks about a location/the item in question already being there.

* '''An''' [at, on (vertical)] ''(Ich haenge das Bild an die Wand [accusative]) or (Das Bild haengt an der Wand)''
* '''Auf''' [on, on top of (horizontal)] ''(Stell deine Schuhe nicht auf den Tisch)''
* '''Zwischen''' [between]
* '''Neben''' [next]
* '''Hinter''' [behind]
* '''Über''' [above, over]
* '''Unter''' [under] ''(Ich trage ein T-Shirt unter dem Pullover)''
* '''In''' [in, into] ''(William geht in die Stadt; Was macht er in der Stadt?) or (Ich laufe jeden Tag in die Schule; Ich lerne in der Bibliothek)'' 
* '''Vor''' [in front of]
{{info|
Common ''Wo''/''Wohin'' verbs, use dative/accusative after these verbs
*'''liegen''' --&gt; to lie
*'''sitzen''' --&gt; to sit
*'''stehen''' --&gt; to stand
*'''hängen''' --&gt; to hang}}

'''Tips'''
* A movement verb = accusative

=== Contractions ===

* an das --&gt; '''ans'''
* auf das --&gt; '''aufs'''
* in das --&gt; '''ins'''
* an dem --&gt; '''am'''
* in dem --&gt; '''im'''

[[Category:Atcovi's Work]]</text>
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    </revision>
    <revision>
      <id>2411960</id>
      <parentid>2411959</parentid>
      <timestamp>2022-08-04T22:13:10Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Kennen */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6379" xml:space="preserve">== Kapital 5 - 2/6/2022 ==
- '''Weak verbs''': ge- + verb stem + -t/-et (gearbeitet, geregnet, gehört)&lt;br&gt;
--verbs ending with &quot;-ieren&quot; don't have the &quot;ge-&quot;

* Review &quot;strong&quot;, &quot;weak&quot; and &quot;mixed&quot; verbs
* https://chatterbug.com/grammar/german/perfect-tense-questions - &quot;Perfekt&quot;

=== &quot;Strong&quot;/&quot;Weak&quot;/&quot;Mixed&quot; Verbs ===
- https://www.vocabulix.com/german/strong-weak-verbs.shtml

=== Accusative Pronouns ===
-accusacative verbs: https://en.wikibooks.org/wiki/German/Grammar/Transitive_verbs#The_accusative_pronouns
{| class=&quot;wikitable&quot;
|+
!English
!Deutsch
|-
|me
|'''mich'''
|-
|you (informal)
|'''dich'''
|-
|you (formal)
|'''Sie'''
|-
|him
|'''ihn'''
|-
|her
|'''sie'''
|-
|it
|'''es'''
|-
|us
|'''uns'''
|-
|you all (informal)
|'''euch'''
|-
|you all (formal)
|'''Sie'''
|-
|them
|'''sie'''
|}
====Examples====
{| class=&quot;wikitable&quot;
|+
!Deutsch
!Deutsch (accusativ)
|-
|Ich kaufe den Apfel
|Ich kaufe '''''ihn''''' [masculine &quot;it&quot;]
|-
|Ich höre das Mäuschen
|Ich höre '''''es''''' [neutral &quot;it&quot;]
|-
|Ich sehe die Blume
|Ich sehe '''''sie''''' [feminine &quot;it&quot;]
|}

=== Dative Pronouns ===
accusative pronoun &gt; dative pronoun

==== Review of what dative means ====
''What makes a dative case''? Indirect object/&quot;to whom?&quot; (Ich gebe '''dem alten Mann''' eine Fahrkarte; &quot;dem alten Mann&quot; is the indirect object, while the &quot;Fahrkarte&quot; is the direct job that is being acted on. You'd ask yourself, in this case: &quot;Who was the direct object (eine Fahrkarte) given [verb] to?&quot; - The man)

-dative pronouns: https://en.wikibooks.org/wiki/German/Grammar/Ditransitive_verbs#The_dative_pronouns
{| class=&quot;wikitable&quot;
|+
!English
!Deutsch
|-
|I
|'''mir'''
|-
|You (informal)
|'''dir'''
|-
|You (formal)
|'''Ihnen'''
|-
|Him
|'''ihm'''
|-
|Her
|'''ihr'''
|-
|It
|'''ihm'''
|-
|We
|'''uns'''
|-
|You (plural)
|'''euch'''
|-
|Them
|'''ihnen'''
|}

====== Overview ======
- https://german.net/exercises/cases/accusative-dative/

=== Vocabulary (Feiern &amp; Party)===
Es ist die Zeit Feste zu feiern!
*Ich '''feiere'''
*Du '''feierst'''
*Er/Sie/Es '''feiert'''
*Wir '''feieren'''
*Ihr '''feiert'''
*Sie '''feieren'''

==== Feste ====
{| class=&quot;wikitable&quot;
|+
!English
!Deutsch
|-
|The holiday
|Der Feiertag
|-
|The card
|Die Karte
|-
|The party
|Die Party
|-
|The room
|Das Zimmer
|-
|To toast
|anstoßen (stößt... an)
|-
|To recieve
|bekommen
|-
|To invite
|einladen (lädt... an)
|-
|To celebrate
|feiern
|-
|To throw a party
|eine Party geben
|-
|To [not] have fun
|(keinen) Spaß haben
|-
|To give [a gift]
|schenken
|-
|To surprise
|Überraschen
|-
|Congratulations!
|Herlizchen Glückwunsch
|}

==== Besondere Anlässe ====
{| class=&quot;wikitable&quot;
|+
!English
!Deutsch
|-
|The marriage
|Die Ehe
|-
|The newlywed
|Der/Die Frischvermählte
|-
|The birth
|Die Geburt
|-
|The birthday
|Der Geburtstag
|-
|The wedding
|Die Hochzeit
|-
|The wedding anniversary
|Der Hochzeitstag
|-
|The anniversary
|Der Jahrestag
|-
|The New Year's Eve
|Das/Der Silvester
|-
|The Christmas
|Das Weihnachten
|-
|To [go into] retirement
|in Rente gehen
|-
|To [make] graduate [happen]
|einen Abschluss machen
|}

==== Ausdrücke ====
{| class=&quot;wikitable&quot;
|+
!English
!Deutsch
|-
|The friendship
|Die Freündschaft
|-
|The happiness
|Das Glück
|-
|The kiss
|Der Kuss
|-
|The love
|Die Liebe
|}

=== Vocabulary (das Kleidung) ===
{| class=&quot;wikitable&quot;
|+
!Deutsch
!Englisch
|-
|Der Hut, -e
|hat
|-
|Der Badeanzug, -e
|bathing suit
|-
|Teuer
|expensive
|-
|Die Krawatte
|tie
|-
|Der Gürtel
|belt
|-
|Das Kleid
|dress
|-
|Das Hemd
|shirt
|-
|Die kurze Hose
|shorts
|-
|Der Turnschuh
|sneakers
|-
|Er trägt einen Anzug
|He's wearing a suit
|-
|Die Handtasche
|purse
|-
|gelb
|yellow
|-
|grün
|green
|-
|lila
|purple
|-
|rosa
|pink
|-
|grau
|gray
|-
|schwarz
|black
|-
|weiß
|white
|-
|braun
|brown
|-
|blau
|blue
|-
|rot
|red
|-
|orange
|orange
|-
|die Sonnenbrille, -n
|sunglasses
|-
|die Mütze, -n
|cap
|-
|die Halskette, -n
|necklace
|-
|der Schal, -s
|scarf
|-
|Es ist günstig
|It's cheap
|-
|die Jacke, -n
|jacket
|-
|der Rock, -e
|skirt
|-
|die Hose, -n
|pants
|-
|der Schuh, -e
|shoe
|}

== Wissen vs. Kennen ==

* '''Wissen''' - To know as a fact (I know that her name is... I know her address is...)
* '''Kennen''' - To be familiar with/acquainted with (Familiar with people or places)

- https://languageposters.com/pages/german-verbs-wissen-conjugation

-https://www.vocabulix.com/conjugation/query_german.html?conjugation3,kennen,kennen,_german_present

''Are you familiar with the subject? If not, then use wissen.''

=== Examples ===

==== Wissen ====

* die Adresse von Amy
* ihren Familiennamen
* die Antwort
* viel ueber Mode
* die Uhrzeit

==== Kennen ====

* das Anne Frank-Haus in Amsterdam
* deinen Freundin nicht
* Maries Eltern
* ein gutes chinesisches Restaurant
* Wien gut
* eine Stadt

== Das Perfekt: The conversational past ==
Ich kaufe ein Brot in der Bäckerei. 

* Decide: sein or haben? '''Sein''' is a change of a physical condition/location (Point A --&gt; Point B), while '''haben''' is for everything else. 
* Conguate based on decision. 
* Pay attention to if the verb is weak, strong, or mixed.

== Two way prepositions ==
Either destination/movement towards a destination (accusative) or location/no movement towards a destination (dative). TIP: Use '''wohin''' ''(where to)'' with accusative as it talks about a movement towards a destination. Use '''wo''' with dative as it talks about a location/the item in question already being there.

* '''An''' [at, on (vertical)] ''(Ich haenge das Bild an die Wand [accusative]) or (Das Bild haengt an der Wand)''
* '''Auf''' [on, on top of (horizontal)] ''(Stell deine Schuhe nicht auf den Tisch)''
* '''Zwischen''' [between]
* '''Neben''' [next]
* '''Hinter''' [behind]
* '''Über''' [above, over]
* '''Unter''' [under] ''(Ich trage ein T-Shirt unter dem Pullover)''
* '''In''' [in, into] ''(William geht in die Stadt; Was macht er in der Stadt?) or (Ich laufe jeden Tag in die Schule; Ich lerne in der Bibliothek)'' 
* '''Vor''' [in front of]
{{info|
Common ''Wo''/''Wohin'' verbs, use dative/accusative after these verbs
*'''liegen''' --&gt; to lie
*'''sitzen''' --&gt; to sit
*'''stehen''' --&gt; to stand
*'''hängen''' --&gt; to hang}}

'''Tips'''
* A movement verb = accusative

=== Contractions ===

* an das --&gt; '''ans'''
* auf das --&gt; '''aufs'''
* in das --&gt; '''ins'''
* an dem --&gt; '''am'''
* in dem --&gt; '''im'''

[[Category:Atcovi's Work]]</text>
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  </page>
  <page>
    <title>Eventmath/Lesson plans/Comparing streaming service pay rates to artists</title>
    <ns>0</ns>
    <id>282299</id>
    <revision>
      <id>2411871</id>
      <parentid>2410941</parentid>
      <timestamp>2022-08-04T17:36:45Z</timestamp>
      <contributor>
        <username>Professorbrendan</username>
        <id>2917898</id>
      </contributor>
      <comment>/* Writing alt-text */ added link to POET image description training tool</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21027" xml:space="preserve">&lt;!-- DON'T MESS WITH THIS BIT --&gt;
{{Eventmath draft header}}
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   If you see unfamiliar code, look for an explanation in a comment like this.
2. Heads up: you'll find a comment at the bottom with simple instructions on adding &quot;category tags,&quot; 
   which will help people find this lesson plan. Please follow them :)
--&gt;
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{{Eventmath lesson plan overview
&lt;!--EDIT BELOW THIS LINE ONLY--&gt;
&lt;!-- The summary below is displayed as a guide for visitors and contributors. You can publish without it, but it's required for removal from draft status.--&gt;

|assumed-knowledge    = &lt;!--Below, please briefly describe any mathematical skills or background knowledge students should have before completing this lesson.--&gt;

Students only need familiarity with basic arithmetic.

|activities           = &lt;!--Below, please briefly summarize what students will do during this lesson and what they will learn.--&gt;

Students will consider a Tweet discussing streaming service pay rates for artists, and then they will create a graphical representation of the data, as well as alt-text for sharing that graph on social media.

|class-time           = &lt;!--Below, please type one of the following, exactly as it appears: 
                         00-15 minutes, 15-30 minutes, 30-45 minutes, 45-60 minutes, 60-90 minutes, more than 90 minutes.
                         Your best guess is fine! It can be changed later.--&gt;
15-30 minutes
                     
&lt;!--Next, you'll provide all citation data for the source (article/social media post/video).

Article example:
| source-website      = Huffington Post
| source-title        = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It’s Time To End The Electoral College.
| source-date         = 2016-11-10
| source-url          = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764
| source-archive-url  = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764
| source-archive-date = 2021-10-28

Post example:
| source-website      = Twitter
| source-title        = Remember: Germany is supposed to be a SUCCESS STORY.
| source-date         = 2020-12-22
| source-url          = https://twitter.com/Neoavatara/status/1341581271991013376
| source-archive-url  = https://web.archive.org/web/20201223064324/https://twitter.com/Neoavatara/status/1341581271991013376
| source-archive-date = 2021-02-16

Notes:
1. To ensure lesson plans can be sorted by date, please enter all dates in the format YYYY-MM-DD,
   including leading zeros: January is 01, not 1, for example. 

2. If you cannot find a date, use n.d. as the date (no date).

3. To ensure that the article/post will remain accessible for educators and students...
   a. Archive it at https://archive.org/web/ (use the &quot;Save Page Now&quot; form to generate a permanent URL).
   b. Paste that URL here as the source-archive-url.
   c. Enter the date you archived the source as the source-archive date.

4. Linking to a website helps the site. If the lesson plan is debunking disinformation, you may 
   avoid spreading disinformation by omitting the url of the original site entirely. In that case, 
   use the URL of the archived version as the value of source-url, and omit values for archive-url and archive-date.
--&gt;

| source-website      = Twitter
| source-title        = Would you like another reason to switch to a music service that is NOT Spotify?
| source-date         = 2022-02-07
| source-url          = https://twitter.com/zoecello/status/1490817801552027651
| source-archive-url  = https://web.archive.org/web/20220305162020/https://twitter.com/zoecello/status/1490817801552027651
| source-archive-date = 2022-03-05
}}
&lt;/div&gt;
&lt;div style=&quot;float:left; width:20%;&quot;&gt;
[[File:Eventmath_logo_bitmap_graphic.png|250px]]
&lt;/div&gt;
&lt;/div&gt;
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&lt;!--INSTRUCTIONS FOR THE BODY OF THE LESSON PLAN
1. Note that &quot;==My heading==&quot; is a level 2 section heading, &quot;===My subheading===&quot; is level 3, etc. Level 1 is reserved for the title.
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3. Check out the &quot;Advanced&quot; and &quot;Help&quot; menus at the top of the editor if you don't know how to format lists, etc.
--&gt;
==Activities==
The main idea is to help students make sense of very small/large numbers and compare them using appropriate units of measurement.

===Initial discussion===
Provide students with the text or a screenshot of the original Tweet and ask them to quickly answer some questions as if they were just seeing it in their timeline and trying to make sense of it. Consider using polling software to gather answers and display the results, or maybe give a &quot;pop quiz&quot; and then ask students to swap papers with their neighbor to compare ideas.

For convenience, the text of the original Tweet is quoted below, along with some possible questions to ask about it:

{{Quote box
 |quote  = Would you like another reason to switch to a music service that is NOT Spotify? 
 
This is what services paid me on average per stream in 2021

Amazon - $0.0107463

Tidal - $0.007981

Youtube - $0.007795

Apple - $0.006040

Deezer - $0.0045861

Spotify - $0.003661

(I am my label fyi)
 |author = [https://en.wikipedia.org/wiki/Zo%C3%AB_Keating Zoë Keating]
 |source = https://twitter.com/zoecello/status/1490817801552027651
}}

#Based on the information shown, how many streams would this artist need (on average) to earn $1 on the Amazon platform? What about for Apple? And for Spotify?
#What would 1 million streams pay on Amazon? What about Apple? And Spotify?
#How many Spotify streams are needed in 1 year to amount to the same pay as $15/hour for 40 hours/week?
#How do the pay rates of Amazon and Apple compare? Write a sentence that expresses a numerical comparison between those two. Then, do the same for Apple and Spotify.

(I had success with posing these questions to students on paper as a mock &quot;quiz&quot; for a couple of minutes, and then asking them to compare answers in groups of 2 or 3 to see if any questions arise before the main task.)

===Main task: create a graphical representation of the data===
The central activity of the lesson asks students to present the given information in a graphical form instead, and then write appropriate alt-text for sharing the graph as an image on social media.

====Units: $/stream or streams/$====
The original Tweet expresses the pay rates in the units of '''dollars per stream.''' While you discuss the solutions to the &quot;quiz&quot; questions, steer the discussion towards the idea that perhaps '''streams per dollar''' might be a more useful unit, for the sake of comparing the various platforms. From the point of view of an artist, it's probably much easier to compare the numerical values of &quot;How many streams are needed to earn $1?&quot;, as opposed to comparing the tiny decimal numbers expressed in the original Tweet. Or, as hinted at by one of the possible questions above, students may want to express everything in the units of '''yearly streams needed to make annual minimum wage,''' which would essentially be a multiple of the '''streams per dollar''' unit.

For your convenience, here is a table with the information from the original Tweet, as well as the corresponding value expressed in the '''Streams / $''' unit. (To obtain the '''yearly streams needed to make annual minimum wage''' value or something like it, simply multiply the '''Streams / $''' value by the annual wage you want to consider.) Aligning decimal points makes the sizes of the numbers easier to compare. 

{| class=&quot;wikitable sortable&quot;
|+ 
|-
! Platform !! $ / Stream !! Streams / $
|-
| Amazon || $0.0107463 || style=&quot;text-align:right;&quot; | 93.1
|-
| Tidal || $0.007981 || style=&quot;text-align:right;&quot; | 125.3
|-
| YouTube || $0.007795 || style=&quot;text-align:right;&quot; | 128.3
|-
| Apple || $0.006040 || style=&quot;text-align:right;&quot; | 165.6
|-
| Deezer || $0.0045861 || style=&quot;text-align:right;&quot; | 218.1
|-
| Spotify || $0.003661 || style=&quot;text-align:right;&quot; | 273.1
|}


====Graphical representation of the data====
You might also steer the discussion to considering whether the original Tweet is engaging or informative for a reader scrolling through their timeline. For example, you might ask students to reflect on what they would do if they encountered this Tweet in their feed or someone shared it with them. How would they process this information? What calculations, if any, would they do to make sense of these numbers? Through this discussion, students may realize that a visual data display, such as a bar graph, could be a more engaging way of presenting the underlying information.

Ask students to resume discussion in their small groups and make a &quot;mock up&quot; of a graphical representation of this data. In other words, just ask them to think about the key features of their data display -- the labels and scales on the axes, the order of the categories, a title, etc. -- and make a quick sketch of what it might look like. They may also decide whether they want to use $/stream or streams/$ (or something else) as the unit of measurement.

After a short while, reconvene the class-wide discussion so that students can learn from each other's ideas. Then, each group can create a more polished version of their graph during class time or as homework, depending on the classroom context.

====Writing alt-text====
Whether the students create a graph in class or for homework, they should also write appropriate alt-text for sharing that graph as an image on social media. Here is a prompt you might use on a worksheet or assignment description:

&lt;blockquote&gt;Create ALT text for the graphic so that it's accessible to more people (e.g. people with visual or cognitive impairments, people who are searching with a search engine, or people for whom the image fails to load). [https://help.twitter.com/en/using-twitter/picture-descriptions According to the Twitter Help Center], &quot;Good image descriptions are concise and descriptive, helping people understand what’s happening in an image.&quot; If the graphic conveys certain insights better than the numbers alone, an image description might convey those same insights verbally.&lt;/blockquote&gt;

You may also consider sharing the [https://poet.diagramcenter.org/about.html POET Training Tool] with students, or assigning them [https://poet.diagramcenter.org/how.html some practice examples from that site] about how to write image descriptions.

===Concluding discussion===
Finally, have the students share their creations with each other and discuss them! Perhaps you create a slideshow of each submission to view together during class time, or perhaps the students must share and comment on each other's submission on an online discussion forum, or... Whatever you do, it's important for students to see each other's work and share feedback. This will emphasize the point that there is no &quot;right answer&quot; here, and that there are many decisions to be made when sharing numerical information visually and in writing.

Here are some questions you might use to spark some discussion. You might even use these as prompts for a follow-up survey or short writing assignment, too.
* Look back at the original Tweet and then at your graphical representation of the data. How do they compare and contrast? If you encountered one of them for the first time in your social media feed, how might you react and process the information?
* Which unit of measurement did you find most helpful when thinking about this topic: $/stream or streams/$? Or, did you find it helpful to go back and forth between the two?
* Based on your work in this lesson, what advice do you have for yourself and your peers about how to process numerical information, especially when it involves small numbers (like the original Tweet)?
* Reflect on the discussions you had with your group and the decisions you made regarding the alt-text for your image. Were there any challenges you faced in translating the data display into words? Did you gain any new insights into this topic or the underlying data by creating the graph and its alt-text?
* 

==Assignments==
# '''Homework prompt:''' Find another example of an artist sharing their average pay rate for at least one streaming service. Express any information you find using both the '''Streams / $''' and '''$ / Stream''' units, and compare/contrast with the original source for this lesson.
# '''Homework prompt:''' If you have an account on one of the streaming services considered in this activity, identify an artist you have listened to a lot. How many songs have you streamed from them in the last month? Year? (Perhaps you can find this information in your user profile or stats, or perhaps you'll need to do some estimation.) Using the rates provided in the original Tweet here, how much have you personally contributed to that artist's earnings on that platform this month/year? (But then, how certain can we be that your artist has the same pay rate per stream?)
# '''Essay prompt:''' One issue raised by the original Tweet is that different artists likely have different pay rates per stream because of contracts with record labels. Do some research into this issue and share what you discover. Are the pay rates per stream shared in this lesson likely representative of &quot;most artists&quot; or not?
# '''Essay prompt:''' Investigate how, for example, Spotify actually decides how much to pay an artist. It is not quite as simple as &quot;(# of streams) X ($ rate per stream).&quot; [https://www.headphonesty.com/2021/11/how-much-does-spotify-pay-per-stream/ Do some research] and share at least two new ideas you discover about what goes into deciding the amount an artist is paid.

==Resources==

===Background===
There is almost no prerequisite mathematical knowledge for this lesson plan. However, the activities and ensuing discussion may bring up issues of social justice and information/media literacy. In case it is helpful for you, here is some information about those ideas.
====Artists and labor====
* There is an [https://www.nytimes.com/2021/05/07/arts/music/streaming-music-payments.html ongoing conversation about pay rates of streaming services]. Some [https://www.nytimes.com/2022/01/26/arts/music/spotify-neil-young-joe-rogan.html prominent musicians recently removed their catalogs from Spotify]. Initially, this was done in protest of how the platform spreads misinformation (especially through podcasts), but some have also cited the poor compensation for working musicians, as well. There have even been protests by musicians and songwriters [https://www.latimes.com/entertainment-arts/music/story/2022-03-01/spotify-protest-songwriters-royalty-rates at various Spotify corporate offices], including the Union of Musicians and Allied Workers’s [https://www.unionofmusicians.org/justice-at-spotify “Justice at Spotify Day of Action.”]
* Students may notice that the original Tweet begins and ends with some comments about the ''context'' for these numbers. Consider preempting the discussion by explicitly asking students to respond to these prompts:
** Why did the Tweet start with ''&quot;Would you like another reason to switch to a music service that is NOT Spotify?&quot;''? What framing/context does that provide for the rest of the information? How might you have interpreted this information differently if that sentence were not included?
** Why did the Tweet end with ''&quot;(I am my label fyi)&quot;'' at the end? What additional contextual information does that provide, and how might it change our interpretation of the quantitative information in the Tweet? (Answer: Artists typically split the profits from streaming services with their record label, according to whatever contract they've signed. In this case, Zoë Keating operating as her own label means that she does not have to split profits in that way. Accordingly, an artist who is ''not'' their own label would likely have noticeably ''smaller'' pay rates.)
** Does the Tweet specify where this information came from, or how the numbers were calculated? (Answer: Yes, she addresses this in [https://twitter.com/zoecello/status/1491081258629095430 a reply to her own Tweet] and implies that she took her annual earnings from each platform and divided by the number of streams she earned on that platform. This further emphasizes the idea that different artists may obtain different values from these calculations.)
====Alt-text for images====
Below are some guides about alt-text for accessibility. You may consider sharing some relevant portions with students as part of the main task of this lesson plan, or else ask students to read portions of these articles as part of their assignment.
* [https://webaim.org/techniques/alttext/ WebAIM article on alt text]
* [https://accessibility.umn.edu/what-you-can-do/start-7-core-skills/alternative-text University of Minnesota guide to alt text (with examples for charts and graphs)]
* [https://accessibility.huit.harvard.edu/describe-content-images Advice from a Harvard University guide on digital accessibility]

===Explorations===
''You're welcome to share references for additional learning and exploration, such as links to other articles, videos, spreadsheets, or computer code. When an open-access substitute is unavailable, links to paywalled sites are acceptable in this section.''

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* I used an early version of this lesson plan with my class for liberal-arts students as an activity on numeracy in social media. The students were surprised to learn just how little artists are paid on these platforms and found the ensuing discussion helpful and engaging. I was also pleased to find that students actually ''wanted'' to do calculations with the numbers in the Tweet to better understand what the data is telling us. I have not yet tried the &quot;create a graph and alt-text&quot; task, but I'm looking forward to doing that next semester! -- [[User:Professorbrendan|Professorbrendan]] ([[User talk:Professorbrendan|discuss]] • [[Special:Contributions/Professorbrendan|contribs]]) 16:41, 13 July 2022 (UTC)
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  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
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        <id>21186</id>
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      <comment>/* Handling Series of Data */</comment>
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      <text bytes="9802" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

=== Handling Repetition ===
* Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]])
* Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]])

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Arrays ([[Media:C04.Series1.Array.1.A.20220803.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Pointers ([[Media:C04.Series2.Pointer.1.A.20180726.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.Series3.ArrayPointer.1.A.20220803.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20220418.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.11.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Applications ==== 
* Applications of Arrays ([[Media:C04.Series1App.Array.1.A.20220803.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.Series7.AppPoint.1.A.20200424.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.Series3App.ArrayPointer.1.A.2022024.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming]]

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    <title>Workings of gcc and ld in plain view</title>
    <ns>0</ns>
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      <id>2412077</id>
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      <timestamp>2022-08-05T09:05:38Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Workings of the GNU Compiler */</comment>
      <model>wikitext</model>
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      <text bytes="2369" xml:space="preserve">=== Workings of the GNU Compiler ===
* Overview ([[Media:Overview.20200211.pdf |pdf]])
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])
* Conditions ([[Media:Condition.20220804.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])
&lt;/br&gt;

=== Workings of the GNU Linker ===

==== Overview ====

* Static Linking Overview ([[Media:Link.1.StaticOverview.20181120.pdf |pdf]])

* Dynamic Linking Overview ([[Media:Link.2.DynamicOverview.20181120.pdf |pdf]])

==== Linking Process ====

* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])

* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])

* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])

* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])

* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]])

* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]])

* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])

==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])

==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])

* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])

* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])


&lt;/br&gt;
go to [ [[C programming in plain view]] ]

[[Category:C programming]]</text>
      <sha1>jefph7nlycesf84yy5dbl000nfg23vp</sha1>
    </revision>
    <revision>
      <id>2412079</id>
      <parentid>2412077</parentid>
      <timestamp>2022-08-05T09:16:27Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Workings of the GNU Compiler */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2369" xml:space="preserve">=== Workings of the GNU Compiler ===
* Overview ([[Media:Overview.20200211.pdf |pdf]])
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])
* Conditions ([[Media:Condition.20220805.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])
&lt;/br&gt;

=== Workings of the GNU Linker ===

==== Overview ====

* Static Linking Overview ([[Media:Link.1.StaticOverview.20181120.pdf |pdf]])

* Dynamic Linking Overview ([[Media:Link.2.DynamicOverview.20181120.pdf |pdf]])

==== Linking Process ====

* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])

* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])

* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])

* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])

* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]])

* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]])

* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])

==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])

==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])

* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])

* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])


&lt;/br&gt;
go to [ [[C programming in plain view]] ]

[[Category:C programming]]</text>
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      <contributor>
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      <comment>/* Mircéa Éliade (1907 - 1986) */</comment>
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=== [[w:Mircea_Eliade|Mircéa Éliade (1907 - 1986)]] ===
&lt;hr /&gt;
[[File:Mircea Eliade young.jpg|thumb|Éliade, Mircéa (1907 - 1986)]]
'''Notable Accomplishments'''
* The History of Alchemy, Shamanism, and other Religious Modes
&lt;br /&gt;&lt;hr /&gt;
{{User:Jtwsaddress42/Gallery/Mircéa Éliade}}
&lt;br /&gt;&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Éliade, Mircéa}}
{{RoundBoxBottom}}
&lt;hr /&gt;</text>
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|{{Gallery
 | title =  ''Michael Faraday''
 | width = 100
 | height = 100
 | align= center
 | Michael Faraday - AUTHOR OF CHEMICAL MANIPULATION.png | alt1= Michael Faraday - AUTHOR OF CHEMICAL MANIPULATION | Chemical Manipulation
 | Faraday Michael Christmas lecture detail.jpg | alt2= Faraday Christmas Lecture | Faraday Christmas Lecture
 | Faraday disk generator.jpg | alt3= Faraday disk generator | Faraday disk generator
 | Michael Faraday Apprentice plaque.jpg | alt4= Apprentice plaque | Plaque Apprentice plaque
 | Faraday-Daniell.PNG | alt5= English chemists Michael Faraday and John Frederic Daniell | English chemists Michael Faraday and John Frederic Daniell
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|{{Gallery
 | title =  ''Michael Faraday''
 | width = 100
 | height = 100
 | align= center
 | Michael Faraday - AUTHOR OF CHEMICAL MANIPULATION.png | alt1= Michael Faraday - AUTHOR OF CHEMICAL MANIPULATION | Chemical Manipulation
 | Faraday Michael Christmas lecture detail.jpg | alt2= Faraday Christmas Lecture | Faraday Christmas Lecture
 | Faraday disk generator.jpg | alt3= Faraday disk generator | Faraday disk generator
| Faraday-Daniell.PNG | alt4= English chemists Michael Faraday and John Frederic Daniell | English chemists Michael Faraday and John Frederic Daniell
 }}
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  <page>
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    <revision>
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      <contributor>
        <username>Jtwsaddress42</username>
        <id>234843</id>
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      <comment>/* Chomsky, Noam (1929 - ) */</comment>
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=== [[w:Noam Chomsky|Chomsky, Noam (1929 - )]] ===
&lt;hr /&gt;
[[File:Noam Chomsky portrait 2017 retouched.png|thumb|Noam Chomsky 2017]]
[[File:Noam Chomsky signature.svg|thumb]Signature]
[[File:Syntactic Structures Front Cover (1957 first edition).jpg|thumb|Syntactic Structures Front Cover (1957 first edition)]]
[[File:Current Issues in Linguistic Theory Front Cover (1964 first edition).jpg|thumb|Current Issues in Linguistic Theory Front Cover (1964 first edition)]]
'''Notable Accomplishments'''
* Father of Modern Linguistics
* Critique of Skinnerian Behaviorism
* Critique of U.S. Foreign and Domestic Policy
* Socialist-Libertarian Anarchism
&lt;br /&gt;&lt;hr /&gt;
{{User:Jtwsaddress42/Quotes/Chomsky, Noam 2005a}}
&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Chomsky, Noam}}
&lt;hr /&gt;
Buckley Jr. et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Buckley Jr. &amp; Chomsky}}
Hauser et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Hauser, Marc D.}}
Herman et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Herman, Edward S.}}
Lenneberg et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Lenneberg, Eric H.}}
Stemmer et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Stemmer, Brigitte}}
&lt;hr /&gt;
{| align= center | width= 480px
|{{User:Jtwsaddress42/Gallery/Chomskian Linguistics}}
&lt;br /&gt;
|}
&lt;hr /&gt;
{{User:Jtwsaddress42/Quotes/Chomsky, Noam 1967a}}
&lt;hr /&gt;
'''Related'''
{{User:Jtwsaddress42/Bibliography/Barsky, Robert F.}}
{{User:Jtwsaddress42/Bibliography/Calvin &amp; Bickerton}}
{{User:Jtwsaddress42/Bibliography/Collier, Peter}}
{{RoundBoxBottom}}
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=== [[w:Noam Chomsky|Chomsky, Noam (1929 - )]] ===
&lt;hr /&gt;
[[File:Noam Chomsky portrait 2017 retouched.png|thumb|Noam Chomsky 2017]]
[[File:Noam Chomsky signature.svg|thumb Signature]]
[[File:Syntactic Structures Front Cover (1957 first edition).jpg|thumb|Syntactic Structures Front Cover (1957 first edition)]]
[[File:Current Issues in Linguistic Theory Front Cover (1964 first edition).jpg|thumb|Current Issues in Linguistic Theory Front Cover (1964 first edition)]]
'''Notable Accomplishments'''
* Father of Modern Linguistics
* Critique of Skinnerian Behaviorism
* Critique of U.S. Foreign and Domestic Policy
* Socialist-Libertarian Anarchism
&lt;br /&gt;&lt;hr /&gt;
{{User:Jtwsaddress42/Quotes/Chomsky, Noam 2005a}}
&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Chomsky, Noam}}
&lt;hr /&gt;
Buckley Jr. et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Buckley Jr. &amp; Chomsky}}
Hauser et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Hauser, Marc D.}}
Herman et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Herman, Edward S.}}
Lenneberg et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Lenneberg, Eric H.}}
Stemmer et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Stemmer, Brigitte}}
&lt;hr /&gt;
{| align= center | width= 640px
|{{User:Jtwsaddress42/Gallery/Chomskian Linguistics}}
&lt;br /&gt;
|}
&lt;hr /&gt;
{{User:Jtwsaddress42/Quotes/Chomsky, Noam 1967a}}
&lt;hr /&gt;
'''Related'''
{{User:Jtwsaddress42/Bibliography/Barsky, Robert F.}}
{{User:Jtwsaddress42/Bibliography/Calvin &amp; Bickerton}}
{{User:Jtwsaddress42/Bibliography/Collier, Peter}}
{{RoundBoxBottom}}
&lt;hr /&gt;</text>
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        <username>Jtwsaddress42</username>
        <id>234843</id>
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      <comment>/* Chomsky, Noam (1929 - ) */</comment>
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=== [[w:Noam Chomsky|Chomsky, Noam (1929 - )]] ===
&lt;hr /&gt;
[[File:Noam Chomsky portrait 2017 retouched.png|thumb|Noam Chomsky 2017]]
[[File:Noam Chomsky signature.svg|thumb|Signature]]
[[File:Syntactic Structures Front Cover (1957 first edition).jpg|thumb|Syntactic Structures Front Cover (1957 first edition)]]
[[File:Current Issues in Linguistic Theory Front Cover (1964 first edition).jpg|thumb|Current Issues in Linguistic Theory Front Cover (1964 first edition)]]
'''Notable Accomplishments'''
* Father of Modern Linguistics
* Critique of Skinnerian Behaviorism
* Critique of U.S. Foreign and Domestic Policy
* Socialist-Libertarian Anarchism
&lt;br /&gt;&lt;hr /&gt;
{{User:Jtwsaddress42/Quotes/Chomsky, Noam 2005a}}
&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Chomsky, Noam}}
&lt;hr /&gt;
Buckley Jr. et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Buckley Jr. &amp; Chomsky}}
Hauser et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Hauser, Marc D.}}
Herman et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Herman, Edward S.}}
Lenneberg et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Lenneberg, Eric H.}}
Stemmer et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Stemmer, Brigitte}}
&lt;hr /&gt;
{| align= center | width= 640px
|{{User:Jtwsaddress42/Gallery/Chomskian Linguistics}}
&lt;br /&gt;
|}
&lt;hr /&gt;
{{User:Jtwsaddress42/Quotes/Chomsky, Noam 1967a}}
&lt;hr /&gt;
'''Related'''
{{User:Jtwsaddress42/Bibliography/Barsky, Robert F.}}
{{User:Jtwsaddress42/Bibliography/Calvin &amp; Bickerton}}
{{User:Jtwsaddress42/Bibliography/Collier, Peter}}
{{RoundBoxBottom}}
&lt;hr /&gt;</text>
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=== [[w:Noam Chomsky|Chomsky, Noam (1929 - )]] ===
&lt;hr /&gt;
[[File:Noam Chomsky portrait 2017 retouched.png|thumb|Noam Chomsky 2017]]
[[File:Noam Chomsky signature.svg|thumb|Signature]]
[[File:Syntactic Structures Front Cover (1957 first edition).jpg|thumb|Syntactic Structures Front Cover (1957 first edition)]]
[[File:Current Issues in Linguistic Theory Front Cover (1964 first edition).jpg|thumb|Current Issues in Linguistic Theory Front Cover (1964 first edition)]]
'''Notable Accomplishments'''
* Father of Modern Linguistics
* Critique of Skinnerian Behaviorism
* Critique of U.S. Foreign and Domestic Policy
* Socialist-Libertarian Anarchism
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|{{User:Jtwsaddress42/Gallery/Chomskian Linguistics}}
&lt;br /&gt;
|}&lt;hr /&gt;
{{User:Jtwsaddress42/Quotes/Chomsky, Noam 2005a}}
&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Chomsky, Noam}}
&lt;hr /&gt;
Buckley Jr. et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Buckley Jr. &amp; Chomsky}}
Hauser et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Hauser, Marc D.}}
Herman et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Herman, Edward S.}}
Lenneberg et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Lenneberg, Eric H.}}
Stemmer et al.&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Stemmer, Brigitte}}
&lt;br /&gt;&lt;hr /&gt;
{{User:Jtwsaddress42/Quotes/Chomsky, Noam 1967a}}
&lt;hr /&gt;
'''Related'''
{{User:Jtwsaddress42/Bibliography/Barsky, Robert F.}}
{{User:Jtwsaddress42/Bibliography/Calvin &amp; Bickerton}}
{{User:Jtwsaddress42/Bibliography/Collier, Peter}}
{{RoundBoxBottom}}
&lt;hr /&gt;</text>
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    <ns>2</ns>
    <id>285577</id>
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      <id>2412051</id>
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      <timestamp>2022-08-05T05:16:18Z</timestamp>
      <contributor>
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        <id>234843</id>
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      <text bytes="593" xml:space="preserve">{| align= center | style=&quot;font-size:85%&quot;
|{{Gallery
 | title =  ''Chomskian Linguistics''
 | width = 100
 | height = 100
 | align= center
 | File:Chomsky-Syntactic-Structures-Grammar-Model.jpg | alt1= Chomsky's Syntactic Structures Grammar Model | Chomsky's Syntactic Structures Grammar Model
 | File:Tree Diagram for Chomsky's Sentence.png | alt2= Tree Diagram for Chomsky's Sentence | Tree Diagram for Chomsky's Sentence
 | File:Aspects Grammar Model.jpg | alt3= Aspects Grammar Model | Aspects Grammar Model
 | File:Chomsky-hierarchy.svg | alt4= Chomsky-hierarchy | Chomsky-hierarchy
 }}
|}</text>
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  </page>
  <page>
    <title>Social Victorians/Terminology</title>
    <ns>0</ns>
    <id>285723</id>
    <revision>
      <id>2411950</id>
      <parentid>2411418</parentid>
      <timestamp>2022-08-04T21:26:58Z</timestamp>
      <contributor>
        <username>Scogdill</username>
        <id>1331941</id>
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      <text bytes="335" xml:space="preserve">Especially with respect to fashion, the newspapers at the end of the 19th century in the UK often used specialized terminology. These definitions were to provide a sense of what someone in the late 19th century might have meant by the term.

=== Corsage ===

=== Fancy-dress Ball ===

=== Girdle ===

=== Selesia ===

=== Stomacher ===</text>
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  <page>
    <title>User:Jtwsaddress42/People/Axelrod, Julius</title>
    <ns>2</ns>
    <id>285834</id>
    <revision>
      <id>2412061</id>
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      <timestamp>2022-08-05T06:09:48Z</timestamp>
      <contributor>
        <username>Jtwsaddress42</username>
        <id>234843</id>
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      <comment>/* Axelrod, Julius (1912 - 2004) */</comment>
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=== [[w:Julius Axelrod|Axelrod, Julius (1912 - 2004)]] ===
&lt;hr/&gt;
[[File:Axelrod01.jpg|thumb|Julius Axelrod (1912 - 2004)]]
[[File:Julius Axelrod 1970.jpg|thumb|Julius Axelrod 1970]]
[[File:Julius Axelrod.jpg|thumb|]]
'''Notable Accomplishments'''
* [https://www.nobelprize.org/prizes/medicine/1970/axelrod/facts/ The Nobel Prize in Physiology or Medicine 1970] -shared with Sir Bernard Katz and Ulf von Euler “for their discoveries concerning the humoral transmittors in the nerve terminals and the mechanism for their storage, release and inactivation.”
* Catecholamine Neurotransmitters &amp; their Metabolism
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|'''Biosynthesis of the Catecholamine Neurotransmitters'''
[[File:Biosynthese catecholaminen.png|640px|Biosynthese catecholaminen]]
&lt;br /&gt;
|}&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Axelrod, Julius}}
&lt;br /&gt;&lt;hr /&gt;
{| align= center | witdh= 640px
|'''Julius Axelrod at the Blackboard'''
[[File:Axelrod.jpg|640px|Julius Axelrod at the Blackboard]]
&lt;br /&gt;
|}
{{RoundBoxBottom}}
&lt;hr /&gt;</text>
      <sha1>3twalqg0h1des6hzb8cr73e431p44w4</sha1>
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  <page>
    <title>User:Jtwsaddress42/People/Baltimore, David</title>
    <ns>2</ns>
    <id>285837</id>
    <revision>
      <id>2412060</id>
      <parentid>2410609</parentid>
      <timestamp>2022-08-05T06:07:43Z</timestamp>
      <contributor>
        <username>Jtwsaddress42</username>
        <id>234843</id>
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      <comment>/* Baltimore, David (1938 - ) */</comment>
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=== [[w:David Baltimore|Baltimore, David (1938 - )]] ===
&lt;hr/&gt;
[[File:David Baltimore at Caltech in 2021 A 05.jpg|thumb|David Baltimore at Caltech (2021)]]
[[File:David Baltimore NIH.jpg|thumb|David Baltimore NIH]]
[[File:VirusBaltimoreClassification.svg|thumb|VirusBaltimoreClassification]]
'''Notable Accomplishments'''
* [https://www.nobelprize.org/prizes/medicine/1975/baltimore/facts/ The Nobel Prize in Physiology or Medicine 1976] -shared with Renato Dulbecco and Howard Martin Temin “for their discoveries concerning the interaction between tumour viruses and the genetic material of the cell.”
* Reverse Transcriptase
* Baltimore Viral Classification System
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|'''Baltimore Viral Classification System'''
[[File:Virus-en.gif|640px|Virus-en]]
&lt;br /&gt;
|}&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Baltimore, David}}
{{RoundBoxBottom}}
&lt;hr /&gt;</text>
      <sha1>c1n3i41y12skh35jcxelrcj524nxvhe</sha1>
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  <page>
    <title>User:Jtwsaddress42/People/Bassler, Bonnie</title>
    <ns>2</ns>
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      <contributor>
        <username>Jtwsaddress42</username>
        <id>234843</id>
      </contributor>
      <comment>/* Bassler, Bonnie Lynn (1962 - ) */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="611" xml:space="preserve">{{RoundBoxTop|theme=3}}
=== [[w:Bonnie Bassler|Bassler, Bonnie Lynn (1962 - )]] ===
&lt;hr/&gt;
[[File:Bonnie Bassler.jpg|thumb|Bonnie Bassler]]
[[File:PDB 1tjy EBI.jpg|thumb|LsrB with R-THMF ligand]]
'''Notable Accomplishments'''
* Quorum Sensing in Bacteria
&lt;br /&gt;&lt;hr /&gt;
{| align=center | width= 640px
|'''Quorum Sensing in Bacteria'''
{{User:Jtwsaddress42/Gallery/Bonnie L. Bassler - Quorum Sensing}}
&lt;br /&gt;
|}&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Bassler, Bonnie}}
&lt;br /&gt;&lt;hr /&gt;
Rutherford et al.&lt;br /&gt;&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Rutherford, Steven T.}}
{{RoundBoxBottom}}
&lt;hr /&gt;</text>
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=== [[w:Bonnie Bassler|Bassler, Bonnie Lynn (1962 - )]] ===
&lt;hr/&gt;
[[File:Bonnie Bassler.jpg|thumb|Bonnie Bassler]]
[[File:PDB 1tjy EBI.jpg|thumb|LsrB with R-THMF ligand]]
'''Notable Accomplishments'''
* Quorum Sensing in Bacteria
&lt;br /&gt;&lt;hr /&gt;
{| align=center | width= 640px
|{{User:Jtwsaddress42/Gallery/Bonnie L. Bassler - Quorum Sensing}}
|}&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Bassler, Bonnie}}
&lt;br /&gt;&lt;hr /&gt;
Rutherford et al.&lt;br /&gt;&lt;hr /&gt;
{{User:Jtwsaddress42/Bibliography/Rutherford, Steven T.}}
{{RoundBoxBottom}}
&lt;hr /&gt;</text>
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|{{Gallery
 | title =  ''Quorum Sensing''
 | width = 100
 | height = 100
 | align= center
 | File:Intra-Species Communication.jpg|alt1= Intra-Species Communication via N-acyl-homoserine lactones|Intra-Species Communication via N-acyl-homoserine lactones
 | File:Inter-Species Communication.jpg|alt2= Inter-Species Communication via 4,5-Dihydroxypentane-2,3-dione|Inter-Species Communication via 4,5-Dihydroxypentane-2,3-dione
 | File:TPDhydratesJanda.png|alt3= 4,5-Dihydroxypentane-2,3-dione in equilibrium with its cyclic acetal forms|4,5-Dihydroxypentane-2,3-dione in equilibrium with its cyclic acetal forms R-DHMF &amp; S-SHMF
 | File:AI-2.png|alt4= Autoinducer-2 for the LuxPQ quorum sensing elements|Auto Inducer-2 for the LuxPQ is a furanosyl borate diester of 4,5-Dihydroxypentane-2,3-dione
}}
|}</text>
      <sha1>kdnih4oq34kvrie8zx35e07r3jq9zx1</sha1>
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  </page>
  <page>
    <title>User talk:SotanScholar</title>
    <ns>3</ns>
    <id>285887</id>
    <revision>
      <id>2411994</id>
      <parentid>2411125</parentid>
      <timestamp>2022-08-05T00:49:43Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>/* Portals */ new section</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3368" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] SotanScholar!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 22:52, 2 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}

== Portals ==

I wouldn't put too much effort into portal development. Portals don't appear in regular searches. They are intended to be used to organize users rather than content. We don't have many topical areas that have enough users that they need to coordinate their efforts. It's probably better to spend time improving main space resources. See [[Wikiversity:Statistics]] for the resources that draw the most interest. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:49, 5 August 2022 (UTC)</text>
      <sha1>gbtawrtu5tx8b401ipt90a3y5i7i813</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2022/Disappointment</title>
    <ns>0</ns>
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    <revision>
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      <text bytes="7841" xml:space="preserve">{{title|Disappointment&lt;br&gt;What is disappointment, what causes disappointment, and how can disappointment be managed?}}
{{MECR3|1=https://yourlinkgoeshere.com}}
__TOC__

==Overview==
You are underway {{smile}}!

This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]].

At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark.

This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help.

Small paragraph(ish) on:
 - What is disappointment?
 - Why is disappointment important?
 - what emotion theories/research is there?
 - example/case study/etc. of how to manage disappointment (same scenario throughout)


{{RoundBoxTop|theme=13}}
'''Focus questions:'''
* What is disappointment?
* What causes disappointment? 
* How can disappointment be managed?
{{RoundBoxBottom}}

==What is disappointment?==

* Key content: disappointment is ... (reference) - discuss what emotions are and the difference between simple and complex emotions
* Key content: disappointment is different from ... (reference)
* Key content: something else about disappointment (reference)
* Can have more content

==What causes disappointment?==

* Key content: why people experience disappointment (reference)
* Key content: what causes disappointment - go into physiology/chemistry/etc. (reference)
* key content: contexts? eg. disappointment in self vs in external thing??? (reference)
* can have more content

==How can disappointment be managed?==

* key content: research on managing disappointment (reference)
* key content: how to mitigate/reduce disappointment (reference)
* key content: something else (reference)
* can have more content


If using sub-headings, must use 2-5.


==Learning features==
What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter.

===Case studies===
Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]].

===Boxes===
Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include:
* Focus questions
* Case studies or examples
* Quiz questions
* Take-home messages
{{RoundBoxTop|theme=3}}
;Feature box example
* Shaded background
* Coloured border
{{RoundBoxBottom}}

===Figures===
[[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]]

Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1).

===Links===
Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?).

===Tables===
Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2).

Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted.

===Quizzes===
Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia.

Here are some simple quiz questions which could be adapted. Choose the correct answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{Quizzes are an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long quizzes are a good idea:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

To learn about different types of quiz questions, see [[Help:Quiz|Quiz]].

==Conclusion==
The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic.
{{tip|Suggestions for this section:
* What is the answer to the question in the sub-title (based on psychological theory and research)?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==

* [[wikipedia:Disappointment|Disappointment]] (Wikipedia)

* [[Motivation and emotion/Book/2016/Regret|Regret]] (Book chapter, 2016)
{{tip|Suggestions for this section:
* Present in alphabetical order.
* Include the source in parentheses.
}}

==References==
List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example:
{{Hanging indent|1=

Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8

Buckholtz, J. W., &amp; Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006

Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., &amp; Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x
}}
{{tip|Suggestions for this section:
* Important aspects for APA style include:
** Wrap the set of references in the hanging indent template. Using &quot;Edit source&quot;: &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
** Author surname, followed by a comma, then author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case except first letter and proper names, ending in a full-stop.
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop.
** Provide the full doi as a URL and working hyperlink
* Common mistakes include:
** incorrect capitalisation
** incorrect italicisation
** providing a &quot;retrieved from&quot; date (not part of APA 7th ed. style).
** citing sources that weren't actually read or consulted
}}

==External links==
* [https://www.youtube.com/watch?v=8KgUFMN7aJQ The value of disappointment] (TEDxPCC) 
{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Present in alphabetical order
* Include the source in parentheses after the link
}}
[[Category:{{#titleparts:{{PAGENAME}}|3}}]]</text>
      <sha1>7cfywfd8i0phcuf9hb9rj4pskywjjlm</sha1>
    </revision>
  </page>
  <page>
    <title>Beat (acoustics)</title>
    <ns>0</ns>
    <id>286028</id>
    <revision>
      <id>2411816</id>
      <parentid>2411813</parentid>
      <timestamp>2022-08-04T12:18:18Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1958" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.&lt;sup&gt;(from [[Wikipedia:Beat (acoustics)]])&lt;/sup&gt;

The fundamentals of this topic are well known and taught at some level in virtually every introductory high school or college course.  Yet, anything beyond what is shown in the figure to the left or an equation&lt;ref&gt;https://www.omnicalculator.com/physics/beat-frequency&lt;/ref&gt; like,

:&lt;math&gt;f_\text{beat}=f_1-f_2\,&lt;/math&gt;

are never pursued, and for a good reason: The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  But if you enjoy [https://www.pinterest.com/pin/164662930098107644/ &quot;messing around&quot;] with mathematical ideas, you might find this interesting.

&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;

==Links==
* Using beats to tune a violin: https://www.violinist.com/discussion/archive/18551/
* Perfect fifth as a 3:2 rhythm pattern (demonstration) https://www.youtube.com/watch?v=4C5xWmuuQOU
* Physics education at hyperphysics: http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html


==Footnotes==</text>
      <sha1>aole9sjlxfl0raysgxr4u6ew6hpl1vg</sha1>
    </revision>
    <revision>
      <id>2411841</id>
      <parentid>2411816</parentid>
      <timestamp>2022-08-04T14:50:50Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Links */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2000" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.&lt;sup&gt;(from [[Wikipedia:Beat (acoustics)]])&lt;/sup&gt;

The fundamentals of this topic are well known and taught at some level in virtually every introductory high school or college course.  Yet, anything beyond what is shown in the figure to the left or an equation&lt;ref&gt;https://www.omnicalculator.com/physics/beat-frequency&lt;/ref&gt; like,

:&lt;math&gt;f_\text{beat}=f_1-f_2\,&lt;/math&gt;

are never pursued, and for a good reason: The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  But if you enjoy [https://www.pinterest.com/pin/164662930098107644/ &quot;messing around&quot;] with mathematical ideas, you might find this interesting.

&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;

==Links==
* Using beats to tune a violin: https://www.violinist.com/discussion/archive/18551/
* Perfect fifth as a 3:2 rhythm pattern (demonstration) https://www.youtube.com/watch?v=4C5xWmuuQOU
* Physics education at hyperphysics: http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html
* Music and the brain (Scientific American)

==Footnotes==</text>
      <sha1>sqi9o5tnyp4k30xarkouq3kfu2ur5v9</sha1>
    </revision>
    <revision>
      <id>2411842</id>
      <parentid>2411841</parentid>
      <timestamp>2022-08-04T15:00:15Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2262" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are well known and taught at some level in virtually every introductory high school or college course.  Yet, anything beyond what is shown in the figure to the left or an equationlike,

:&lt;math&gt;f_\text{beat}=f_1-f_2\,&lt;/math&gt;

are never pursued, and for a good reason: The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  But if you enjoy [https://www.pinterest.com/pin/164662930098107644/ &quot;messing around&quot;] with mathematical ideas, you might find this interesting.


==Links==
Basics
* [[Wikipedia:Beat (acoustics)]]): The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* Physics education at hyperphysics: http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html
* [https://www.omnicalculator.com/physics/beat-frequency Omnicalculator's beat frequency demonstration] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

Beyond the basics


* Using beats to tune a violin: https://www.violinist.com/discussion/archive/18551/
* Perfect fifth as a 3:2 rhythm pattern (demonstration) https://www.youtube.com/watch?v=4C5xWmuuQOU

==Footnotes==</text>
      <sha1>ppcj8xc6x3tab47cch34e6ag3lmzb25</sha1>
    </revision>
    <revision>
      <id>2411843</id>
      <parentid>2411842</parentid>
      <timestamp>2022-08-04T15:01:17Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Links */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2174" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are well known and taught at some level in virtually every introductory high school or college course.  Yet, anything beyond what is shown in the figure to the left or an equationlike,

:&lt;math&gt;f_\text{beat}=f_1-f_2\,&lt;/math&gt;

are never pursued, and for a good reason: The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  But if you enjoy [https://www.pinterest.com/pin/164662930098107644/ &quot;messing around&quot;] with mathematical ideas, you might find this interesting.


==Links==
Basic
* [[Wikipedia:Beat (acoustics)]]): The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* Physics education at hyperphysics: http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html
* [https://www.omnicalculator.com/physics/beat-frequency Omnicalculator's beat frequency demonstration] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

Advanced or otherwise different

* Using beats to tune a violin: https://www.violinist.com/discussion/archive/18551/

==Footnotes==</text>
      <sha1>t4c3ek62udtfsvofo2vstzip9eyjlls</sha1>
    </revision>
    <revision>
      <id>2411844</id>
      <parentid>2411843</parentid>
      <timestamp>2022-08-04T15:02:07Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Links */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2188" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are well known and taught at some level in virtually every introductory high school or college course.  Yet, anything beyond what is shown in the figure to the left or an equationlike,

:&lt;math&gt;f_\text{beat}=f_1-f_2\,&lt;/math&gt;

are never pursued, and for a good reason: The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  But if you enjoy [https://www.pinterest.com/pin/164662930098107644/ &quot;messing around&quot;] with mathematical ideas, you might find this interesting.


==Links==
'''Basic'''
* [[Wikipedia:Beat (acoustics)]]): The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* Physics education at hyperphysics: http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html
* [https://www.omnicalculator.com/physics/beat-frequency Omnicalculator's beat frequency demonstration] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* Using beats to tune a violin: https://www.violinist.com/discussion/archive/18551/

==Footnotes==</text>
      <sha1>9ke8xyg34vvvtipv8trh9avszu27u4i</sha1>
    </revision>
    <revision>
      <id>2411845</id>
      <parentid>2411844</parentid>
      <timestamp>2022-08-04T15:02:27Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Links */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2187" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are well known and taught at some level in virtually every introductory high school or college course.  Yet, anything beyond what is shown in the figure to the left or an equationlike,

:&lt;math&gt;f_\text{beat}=f_1-f_2\,&lt;/math&gt;

are never pursued, and for a good reason: The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  But if you enjoy [https://www.pinterest.com/pin/164662930098107644/ &quot;messing around&quot;] with mathematical ideas, you might find this interesting.


==Links==
'''Basic'''
* [[Wikipedia:Beat (acoustics)]]: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* Physics education at hyperphysics: http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html
* [https://www.omnicalculator.com/physics/beat-frequency Omnicalculator's beat frequency demonstration] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* Using beats to tune a violin: https://www.violinist.com/discussion/archive/18551/

==Footnotes==</text>
      <sha1>79m6sjtyhrwx8yw20hgr7c5fdt1pkps</sha1>
    </revision>
    <revision>
      <id>2411846</id>
      <parentid>2411845</parentid>
      <timestamp>2022-08-04T15:04:22Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Links */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2218" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are well known and taught at some level in virtually every introductory high school or college course.  Yet, anything beyond what is shown in the figure to the left or an equationlike,

:&lt;math&gt;f_\text{beat}=f_1-f_2\,&lt;/math&gt;

are never pursued, and for a good reason: The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  But if you enjoy [https://www.pinterest.com/pin/164662930098107644/ &quot;messing around&quot;] with mathematical ideas, you might find this interesting.


==Links==
'''Basic'''
* [[Wikipedia:Beat (acoustics)]]: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* Physics education at hyperphysics: http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html
* [https://www.omnicalculator.com/physics/beat-frequency Omnicalculator's beat frequency demonstration] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* [https://www.violinist.com/discussion/archive/18551/ violinist.com] discussion on using beats to tune a violin
[

==Footnotes==</text>
      <sha1>nukxv7vo1yhmcqb0087ms3v5h9g928s</sha1>
    </revision>
    <revision>
      <id>2411847</id>
      <parentid>2411846</parentid>
      <timestamp>2022-08-04T15:04:56Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Links */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2225" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are well known and taught at some level in virtually every introductory high school or college course.  Yet, anything beyond what is shown in the figure to the left or an equationlike,

:&lt;math&gt;f_\text{beat}=f_1-f_2\,&lt;/math&gt;

are never pursued, and for a good reason: The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  But if you enjoy [https://www.pinterest.com/pin/164662930098107644/ &quot;messing around&quot;] with mathematical ideas, you might find this interesting.


==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* Physics education at hyperphysics: http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html
* [https://www.omnicalculator.com/physics/beat-frequency Omnicalculator's beat frequency demonstration] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* [https://www.violinist.com/discussion/archive/18551/ violinist.com] discussion on using beats to tune a violin
[

==Footnotes==</text>
      <sha1>q7eo7jzvfyi2v9lxzjyv4h3uewgholu</sha1>
    </revision>
    <revision>
      <id>2411848</id>
      <parentid>2411847</parentid>
      <timestamp>2022-08-04T15:06:08Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Links */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2224" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are well known and taught at some level in virtually every introductory high school or college course.  Yet, anything beyond what is shown in the figure to the left or an equationlike,

:&lt;math&gt;f_\text{beat}=f_1-f_2\,&lt;/math&gt;

are never pursued, and for a good reason: The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  But if you enjoy [https://www.pinterest.com/pin/164662930098107644/ &quot;messing around&quot;] with mathematical ideas, you might find this interesting.


==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat]&lt;/u&gt;
* [https://www.omnicalculator.com/physics/beat-frequency Omnicalculator's beat frequency demonstration] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* [https://www.violinist.com/discussion/archive/18551/ violinist.com] discussion on using beats to tune a violin
[

==Footnotes==</text>
      <sha1>mn7q8aqu9tpemneby7uwo1nwyhzgz0m</sha1>
    </revision>
    <revision>
      <id>2411849</id>
      <parentid>2411848</parentid>
      <timestamp>2022-08-04T15:07:13Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Links */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2263" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are well known and taught at some level in virtually every introductory high school or college course.  Yet, anything beyond what is shown in the figure to the left or an equationlike,

:&lt;math&gt;f_\text{beat}=f_1-f_2\,&lt;/math&gt;

are never pursued, and for a good reason: The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  But if you enjoy [https://www.pinterest.com/pin/164662930098107644/ &quot;messing around&quot;] with mathematical ideas, you might find this interesting.


==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency Omnicalculator's beat frequency demonstration] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>a2wuymgq6xq8ymnagtwdxi2rpx7scnc</sha1>
    </revision>
    <revision>
      <id>2411850</id>
      <parentid>2411849</parentid>
      <timestamp>2022-08-04T15:07:30Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Links */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2263" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are well known and taught at some level in virtually every introductory high school or college course.  Yet, anything beyond what is shown in the figure to the left or an equationlike,

:&lt;math&gt;f_\text{beat}=f_1-f_2\,&lt;/math&gt;

are never pursued, and for a good reason: The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  But if you enjoy [https://www.pinterest.com/pin/164662930098107644/ &quot;messing around&quot;] with mathematical ideas, you might find this interesting.


==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency Omnicalculator's beat frequency demonstration] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>pr7wjt1aptj9bwztxey9o11vzszlzc8</sha1>
    </revision>
    <revision>
      <id>2411851</id>
      <parentid>2411850</parentid>
      <timestamp>2022-08-04T15:08:06Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Links */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2270" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are well known and taught at some level in virtually every introductory high school or college course.  Yet, anything beyond what is shown in the figure to the left or an equationlike,

:&lt;math&gt;f_\text{beat}=f_1-f_2\,&lt;/math&gt;

are never pursued, and for a good reason: The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  But if you enjoy [https://www.pinterest.com/pin/164662930098107644/ &quot;messing around&quot;] with mathematical ideas, you might find this interesting.


==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>opbnwtjiiqu0who5uorqqx8d5ooi0ei</sha1>
    </revision>
    <revision>
      <id>2411852</id>
      <parentid>2411851</parentid>
      <timestamp>2022-08-04T15:32:07Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2479" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are well known and taught at some level in virtually every introductory high school or college course.  Yet, anything beyond what is shown in the figure to the left or an equationlike,

:&lt;math&gt;f_\text{beat}=f_1-f_2\,&lt;/math&gt;

are never pursued, and for a good reason: The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  This Wikiversity resource is not so much aimed at answering these difficult questions, but rather to give students the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] them.  This attempt to understand how beats in musical harmony can generate a variety of learning projects involving mathematics and computer programming.

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>eo2axu5mys0v361rq9yac2lg487paap</sha1>
    </revision>
    <revision>
      <id>2411853</id>
      <parentid>2411852</parentid>
      <timestamp>2022-08-04T15:34:14Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2502" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are [[w:Beat (acoustics)|well known]] and taught at some level in virtually every introductory high school or college course.  Yet, anything beyond what is shown in the figure to the left or an equationlike,

:&lt;math&gt;f_\text{beat}=f_1-f_2\,&lt;/math&gt;

are never pursued, and for a good reason: The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  This Wikiversity resource is not so much aimed at answering these difficult questions, but rather to give students the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] them.  This attempt to understand how beats in musical harmony can generate a variety of learning projects involving mathematics and computer programming.

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>f3btaibdolwcr7g7vnfsny06mdi50el</sha1>
    </revision>
    <revision>
      <id>2411854</id>
      <parentid>2411853</parentid>
      <timestamp>2022-08-04T15:41:14Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2498" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt;f_\text{beat}=f_1-f_2\.&lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  This Wikiversity resource is not so much aimed at answering these difficult questions, but rather to give students the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] them.  This attempt to understand how beats in musical harmony can generate a variety of learning projects involving mathematics and computer programming.

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>dv013o99fez395c9s1p4kz8y7zrl5mn</sha1>
    </revision>
    <revision>
      <id>2411855</id>
      <parentid>2411854</parentid>
      <timestamp>2022-08-04T15:41:36Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2499" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt; f_\text{beat}=f_1-f_2\.&lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  This Wikiversity resource is not so much aimed at answering these difficult questions, but rather to give students the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] them.  This attempt to understand how beats in musical harmony can generate a variety of learning projects involving mathematics and computer programming.

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>bwvpo4tl3ab8n00jhpwkdrryos18xsl</sha1>
    </revision>
    <revision>
      <id>2411856</id>
      <parentid>2411855</parentid>
      <timestamp>2022-08-04T15:41:55Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2540" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt; f_\text{beat}=f_1-f_2\.&lt;/math&gt;


:&lt;math&gt; f_\text{beat}=f_1-f_2\.&lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  This Wikiversity resource is not so much aimed at answering these difficult questions, but rather to give students the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] them.  This attempt to understand how beats in musical harmony can generate a variety of learning projects involving mathematics and computer programming.

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>5xo7au0cczyfs1d2xx0g396ibxdggqq</sha1>
    </revision>
    <revision>
      <id>2411857</id>
      <parentid>2411856</parentid>
      <timestamp>2022-08-04T15:42:14Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2530" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt; f_\text{beat}=f_1-f_2\.&lt;/math&gt;


:&lt;math&gt; f_\text{beat}&lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  This Wikiversity resource is not so much aimed at answering these difficult questions, but rather to give students the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] them.  This attempt to understand how beats in musical harmony can generate a variety of learning projects involving mathematics and computer programming.

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>q0uc2tbstgk6z5ptxfaw0syqc5t0gcc</sha1>
    </revision>
    <revision>
      <id>2411858</id>
      <parentid>2411857</parentid>
      <timestamp>2022-08-04T15:42:40Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2542" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt; f_\text{beat}=f_1-f_2\.&lt;/math&gt;


:&lt;math&gt; f_\text{beat} = f_1-f_2\.&lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  This Wikiversity resource is not so much aimed at answering these difficult questions, but rather to give students the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] them.  This attempt to understand how beats in musical harmony can generate a variety of learning projects involving mathematics and computer programming.

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>pdsq94su77ulox66b3pezo7g2rbpz6z</sha1>
    </revision>
    <revision>
      <id>2411859</id>
      <parentid>2411858</parentid>
      <timestamp>2022-08-04T15:42:49Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2533" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt; f_\text{beat}=f_1-f_2\.&lt;/math&gt;


:&lt;math&gt; f_\text{beat} = &lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  This Wikiversity resource is not so much aimed at answering these difficult questions, but rather to give students the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] them.  This attempt to understand how beats in musical harmony can generate a variety of learning projects involving mathematics and computer programming.

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>m24a8kxiva1orm29sa960wk3h49l8wi</sha1>
    </revision>
    <revision>
      <id>2411860</id>
      <parentid>2411859</parentid>
      <timestamp>2022-08-04T15:43:24Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2543" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt; f_\text{beat}=f_1-f_2\.&lt;/math&gt;


:&lt;math&gt; f_\text{beat} = f_1 - f_1.&lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  This Wikiversity resource is not so much aimed at answering these difficult questions, but rather to give students the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] them.  This attempt to understand how beats in musical harmony can generate a variety of learning projects involving mathematics and computer programming.

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>21f921tv8r48o479f0frb0p29egn7ty</sha1>
    </revision>
    <revision>
      <id>2411861</id>
      <parentid>2411860</parentid>
      <timestamp>2022-08-04T15:43:35Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2502" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt; f_\text{beat} = f_1 - f_1.&lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. The topic is apparently poorly understood, even by the experts.  If your intent is to understand anything beyond what is routinely taught on this subject, I suggest you pursue a different topic because some things are likely to be never understood.  This Wikiversity resource is not so much aimed at answering these difficult questions, but rather to give students the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] them.  This attempt to understand how beats in musical harmony can generate a variety of learning projects involving mathematics and computer programming.

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>lsr6u7k82zq8koc9va99cgup7dnn8nm</sha1>
    </revision>
    <revision>
      <id>2411862</id>
      <parentid>2411861</parentid>
      <timestamp>2022-08-04T16:30:24Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2254" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt; f_\text{beat} = f_1 - f_1.&lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. Much about how humans perceive beats in musical intervals is poorly understood.  This Wikiversity resource is not so much aimed at answering difficult questions, but to offer students the the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] in a way that enhances one's understanding of mathematics and computer programming.

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>df92psz76etr7z94bqnvsqyb8jdenj5</sha1>
    </revision>
    <revision>
      <id>2412091</id>
      <parentid>2411862</parentid>
      <timestamp>2022-08-05T11:51:26Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Links */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2355" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt; f_\text{beat} = f_1 - f_1.&lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. Much about how humans perceive beats in musical intervals is poorly understood.  This Wikiversity resource is not so much aimed at answering difficult questions, but to offer students the the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] in a way that enhances one's understanding of mathematics and computer programming.

==Fourier analysis==
&lt;math&gt;\frac{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2}  - i \pi \delta(X)&lt;/math&gt;
==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>gow7qtc2cg1nf3bln8vbhzb9tkkg89b</sha1>
    </revision>
    <revision>
      <id>2412092</id>
      <parentid>2412091</parentid>
      <timestamp>2022-08-05T11:52:01Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Fourier analysis */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2359" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt; f_\text{beat} = f_1 - f_1.&lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. Much about how humans perceive beats in musical intervals is poorly understood.  This Wikiversity resource is not so much aimed at answering difficult questions, but to offer students the the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] in a way that enhances one's understanding of mathematics and computer programming.

==Fourier analysis==
&lt;math&gt;\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2}  - i \pi \delta(X)&lt;/math&gt;

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>mycsxgoo6ou6095gzfefx1czjdyz8yw</sha1>
    </revision>
    <revision>
      <id>2412094</id>
      <parentid>2412092</parentid>
      <timestamp>2022-08-05T11:53:34Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Fourier analysis */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2457" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt; f_\text{beat} = f_1 - f_1.&lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. Much about how humans perceive beats in musical intervals is poorly understood.  This Wikiversity resource is not so much aimed at answering difficult questions, but to offer students the the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] in a way that enhances one's understanding of mathematics and computer programming.

==Fourier analysis==
&lt;math&gt;\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2}  - i \pi \delta(X)&lt;/math&gt;

&lt;math&gt;\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2}  - i \frac {i\epsilon}{X^2+ \epsilon^2}

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>m1shwza74ypcnpozo85r8fo3f64moou</sha1>
    </revision>
    <revision>
      <id>2412095</id>
      <parentid>2412094</parentid>
      <timestamp>2022-08-05T11:53:48Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Fourier analysis */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2464" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt; f_\text{beat} = f_1 - f_1.&lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. Much about how humans perceive beats in musical intervals is poorly understood.  This Wikiversity resource is not so much aimed at answering difficult questions, but to offer students the the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] in a way that enhances one's understanding of mathematics and computer programming.

==Fourier analysis==
&lt;math&gt;\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2}  - i \pi \delta(X)&lt;/math&gt;

&lt;math&gt;\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2}  - i \frac {i\epsilon}{X^2+ \epsilon^2}&lt;/math&gt;

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>qifdjiohu94zvnfrzz7ckwefdp68pd8</sha1>
    </revision>
    <revision>
      <id>2412096</id>
      <parentid>2412095</parentid>
      <timestamp>2022-08-05T11:54:35Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Fourier analysis */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2463" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt; f_\text{beat} = f_1 - f_1.&lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. Much about how humans perceive beats in musical intervals is poorly understood.  This Wikiversity resource is not so much aimed at answering difficult questions, but to offer students the the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] in a way that enhances one's understanding of mathematics and computer programming.

==Fourier analysis==
&lt;math&gt;\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2}  - i \pi \delta(X)&lt;/math&gt;

&lt;math&gt;\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2}  -  \frac {i\epsilon}{X^2+ \epsilon^2}&lt;/math&gt;

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>7nq9nyolrb8ooz3swu2y1cm32eg6not</sha1>
    </revision>
    <revision>
      <id>2412097</id>
      <parentid>2412096</parentid>
      <timestamp>2022-08-05T11:57:36Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Fourier analysis */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2525" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt; f_\text{beat} = f_1 - f_1.&lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. Much about how humans perceive beats in musical intervals is poorly understood.  This Wikiversity resource is not so much aimed at answering difficult questions, but to offer students the the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] in a way that enhances one's understanding of mathematics and computer programming.

==Fourier analysis==
&lt;math&gt;\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2}  - i \pi \delta(X)&lt;/math&gt;

&lt;math&gt;\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2}  -  \frac {i\epsilon}{X^2+ \epsilon^2}&lt;/math&gt;

&lt;math&gt;\frac 1 X = \text{pp} \frac 1 x -i\pi \delta(X)&lt;/math&gt;

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>49hbd5zqd15exme7h4gwda50ovndsfr</sha1>
    </revision>
    <revision>
      <id>2412098</id>
      <parentid>2412097</parentid>
      <timestamp>2022-08-05T11:58:22Z</timestamp>
      <contributor>
        <username>Guy vandegrift</username>
        <id>813252</id>
      </contributor>
      <comment>/* Fourier analysis */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2525" xml:space="preserve">[[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]]
A '''beat''' is an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies.

The fundamentals of this topic are taught at some level in virtually every introductory high school or college course.  [[w:Beat (acoustics)|Wikipedia]] features the figure shown and equations like,

:&lt;math&gt; f_\text{beat} = f_1 - f_1.&lt;/math&gt;

But topics beyond these fundamentals are rarely pursued, and for good reason. Much about how humans perceive beats in musical intervals is poorly understood.  This Wikiversity resource is not so much aimed at answering difficult questions, but to offer students the the opportunity to [https://www.pinterest.com/pin/164662930098107644/ &quot;mess around&quot;] in a way that enhances one's understanding of mathematics and computer programming.

==Fourier analysis==
&lt;math&gt;\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2}  - i \pi \delta(X)&lt;/math&gt;

&lt;math&gt;\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2}  -  \frac {i\epsilon}{X^2+ \epsilon^2}&lt;/math&gt;

&lt;math&gt;\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)&lt;/math&gt;

==Links==
'''Basic'''
* &lt;u&gt;[[Wikipedia:Beat (acoustics)]]&lt;/u&gt;: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.&lt;ref&gt;{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}&lt;/ref&gt;&lt;ref&gt;Note the inserted footnote templates regarding &quot;verification&quot; and &quot;original research?&quot; at [[w:special:permalink/1102305761]]. &lt;/ref&gt;
* &lt;u&gt;[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]&lt;/u&gt; parallels the Wikipedia article.
* [https://www.omnicalculator.com/physics/beat-frequency &lt;u&gt;Omnicalculator's beat frequency demonstration&lt;/u&gt;] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music.

'''Advanced (or otherwise different)'''

* &lt;u&gt;[https://www.violinist.com/discussion/archive/18551/ Violinist.com]&lt;/u&gt; discussion on using beats to tune a violin

==Footnotes==</text>
      <sha1>8jr8cgyro4wmea4bz2jb3vsslpdueij</sha1>
    </revision>
  </page>
  <page>
    <title>Talk:Design engineering</title>
    <ns>1</ns>
    <id>286030</id>
    <revision>
      <id>2411831</id>
      <timestamp>2022-08-04T13:58:58Z</timestamp>
      <contributor>
        <username>Ramosunya</username>
        <id>2947663</id>
      </contributor>
      <comment>New resource with &quot;Designing cake&quot;</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14" xml:space="preserve">Designing cake</text>
      <sha1>plu9ibrggvi3nroj36biltde1x6o9af</sha1>
    </revision>
  </page>
  <page>
    <title>File:Laurent.6.Application.6A.20220803.pdf</title>
    <ns>6</ns>
    <id>286031</id>
    <revision>
      <id>2411834</id>
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  <page>
    <title>JCCAP FDF/2022</title>
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      <comment>created page</comment>
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      <text bytes="41" xml:space="preserve">This is landing page for 2022 FDF content</text>
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      </contributor>
      <comment>added addresses</comment>
      <model>wikitext</model>
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      <text bytes="500" xml:space="preserve">== '''Addresses''' ==

=== '''''Future Directions Address 1: ''''' ===
'''Presented by Dr. Jessica Schleider, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here.

=== '''''Future Directions Address 2: ''''' ===
'''Presented by Dr. Riana Anderson, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here

=== '''''Future Directions Address 3: ''''' ===
'''Presented by Dr. Miya Barnett, Ph.D.'''

====Description====</text>
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      <id>2411876</id>
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      <comment>/* Addresses */ added names of addresses</comment>
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      <text bytes="719" xml:space="preserve">== '''Addresses''' ==

=== '''''Future Directions Address 1: Future Directions in Research on Brief, Scalable Interventions''''' ===
'''Presented by Dr. Jessica Schleider, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here.

=== '''''Future Directions Address 2: Future Directions in Research on Racism ''''' ===
'''Presented by Dr. Riana Anderson, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here

=== '''''Future Directions Address 3: Future Directions in Research on Lay Health Workers''''' ===
'''Presented by Dr. Miya Barnett, Ph.D.'''

====Description====
Watch the YouTube video recording of the address enter link here.</text>
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    </revision>
    <revision>
      <id>2411877</id>
      <parentid>2411876</parentid>
      <timestamp>2022-08-04T18:09:54Z</timestamp>
      <contributor>
        <username>Ncharamut</username>
        <id>2824970</id>
      </contributor>
      <comment>added workshop template</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2524" xml:space="preserve">== '''Addresses''' ==

=== '''''Future Directions Address 1: Future Directions in Research on Brief, Scalable Interventions''''' ===
'''Presented by Dr. Jessica Schleider, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here.

=== '''''Future Directions Address 2: Future Directions in Research on Racism ''''' ===
'''Presented by Dr. Riana Anderson, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here

=== '''''Future Directions Address 3: Future Directions in Research on Lay Health Workers''''' ===
'''Presented by Dr. Miya Barnett, Ph.D.'''

====Description====
Watch the YouTube video recording of the address enter link here.

== '''Workshops''' ==

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.</text>
      <sha1>p32a8j17vyjgc6222n5tpdly19dtjwu</sha1>
    </revision>
    <revision>
      <id>2411878</id>
      <parentid>2411877</parentid>
      <timestamp>2022-08-04T18:13:20Z</timestamp>
      <contributor>
        <username>Ncharamut</username>
        <id>2824970</id>
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      <comment>/* Selecting Mentors when Applying to Doctoral Programs */ updated first 2 workshops</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2494" xml:space="preserve">== '''Addresses''' ==

=== '''''Future Directions Address 1: Future Directions in Research on Brief, Scalable Interventions''''' ===
'''Presented by Dr. Jessica Schleider, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here.

=== '''''Future Directions Address 2: Future Directions in Research on Racism ''''' ===
'''Presented by Dr. Riana Anderson, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here

=== '''''Future Directions Address 3: Future Directions in Research on Lay Health Workers''''' ===
'''Presented by Dr. Miya Barnett, Ph.D.'''

====Description====
Watch the YouTube video recording of the address enter link here.

== '''Workshops''' ==

=== '''''Diversity and Inclusion''''' ===
'''Presented by Cassandra Boness, Ph.D. and Chardée Galán, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Building a Lab''''' ===
'''Presented by Christine Cha, Ph.D. and Kate Humphreys, Ph.D.''' 

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Selecting Mentors when Applying to Doctoral Programs''''' ===
'''Presented by Dr. Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.</text>
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    </revision>
    <revision>
      <id>2411879</id>
      <parentid>2411878</parentid>
      <timestamp>2022-08-04T18:17:29Z</timestamp>
      <contributor>
        <username>Ncharamut</username>
        <id>2824970</id>
      </contributor>
      <comment>/* Selecting Mentors when Applying to Doctoral Programs */ updated names and presenters for workshops</comment>
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      <text bytes="2309" xml:space="preserve">== '''Addresses''' ==

=== '''''Future Directions Address 1: Future Directions in Research on Brief, Scalable Interventions''''' ===
'''Presented by Dr. Jessica Schleider, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here.

=== '''''Future Directions Address 2: Future Directions in Research on Racism ''''' ===
'''Presented by Dr. Riana Anderson, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here

=== '''''Future Directions Address 3: Future Directions in Research on Lay Health Workers''''' ===
'''Presented by Dr. Miya Barnett, Ph.D.'''

====Description====
Watch the YouTube video recording of the address enter link here.

== '''Workshops''' ==

=== '''''Diversity and Inclusion''''' ===
'''Presented by Cassandra Boness, Ph.D. and Chardée Galán, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Building a Lab''''' ===
'''Presented by Christine Cha, Ph.D. and Kate Humphreys, Ph.D.''' 

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Outlook on Academia''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Diversity and Inclusion''''' ===
'''Presented by Omar Gudiño, Ph.D. and Amy West, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''APA Advocacy Training''''' ===
'''Presented by Parfaite-Claude, Kobor, &amp; Sharpe, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Writing Mechanics''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Mentorship Gone Wrong''''' ===
'''Presented by Deb Drabick, Ph.D. and Sarah Racz, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Taking a Leadership Role''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Participatory Action Framework''''' ===
'''Presented by Judith Scott, Ph.D. and Yoyo Yau, Ph.D'''

===== Description =====

Watch the YouTube recording of the workshop here.</text>
      <sha1>0boiyx0xgk5r3bbklymlwen63wyftq5</sha1>
    </revision>
    <revision>
      <id>2411884</id>
      <parentid>2411879</parentid>
      <timestamp>2022-08-04T18:24:47Z</timestamp>
      <contributor>
        <username>Ncharamut</username>
        <id>2824970</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2315" xml:space="preserve">Lead

== '''Addresses''' ==

=== '''''Future Directions Address 1: Future Directions in Research on Brief, Scalable Interventions''''' ===
'''Presented by Dr. Jessica Schleider, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here.

=== '''''Future Directions Address 2: Future Directions in Research on Racism ''''' ===
'''Presented by Dr. Riana Anderson, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here

=== '''''Future Directions Address 3: Future Directions in Research on Lay Health Workers''''' ===
'''Presented by Dr. Miya Barnett, Ph.D.'''

====Description====
Watch the YouTube video recording of the address enter link here.

== '''Workshops''' ==

=== '''''Diversity and Inclusion''''' ===
'''Presented by Cassandra Boness, Ph.D. and Chardée Galán, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Building a Lab''''' ===
'''Presented by Christine Cha, Ph.D. and Kate Humphreys, Ph.D.''' 

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Outlook on Academia''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Diversity and Inclusion''''' ===
'''Presented by Omar Gudiño, Ph.D. and Amy West, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''APA Advocacy Training''''' ===
'''Presented by Parfaite-Claude, Kobor, &amp; Sharpe, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Writing Mechanics''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Mentorship Gone Wrong''''' ===
'''Presented by Deb Drabick, Ph.D. and Sarah Racz, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Taking a Leadership Role''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Participatory Action Framework''''' ===
'''Presented by Judith Scott, Ph.D. and Yoyo Yau, Ph.D'''

===== Description =====

Watch the YouTube recording of the workshop here.</text>
      <sha1>7ffcddl0hy9r767ga5eem0s5a4l0mjw</sha1>
    </revision>
    <revision>
      <id>2411885</id>
      <parentid>2411884</parentid>
      <timestamp>2022-08-04T18:26:39Z</timestamp>
      <contributor>
        <username>Ncharamut</username>
        <id>2824970</id>
      </contributor>
      <comment>added lead</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2723" xml:space="preserve">This year's Forum looks a little bit different. Instead of in past years where the conference was held over the course of 2 to 3 days, this year's programming will be held in the form of a webinar series spanning from July to December. While we have all address and workshops listed, the descriptions and YouTube links will be updated as the events occur, so stay tuned for full descriptions of the programming. 

== '''Addresses''' ==

=== '''''Future Directions Address 1: Future Directions in Research on Brief, Scalable Interventions''''' ===
'''Presented by Dr. Jessica Schleider, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here.

=== '''''Future Directions Address 2: Future Directions in Research on Racism ''''' ===
'''Presented by Dr. Riana Anderson, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here

=== '''''Future Directions Address 3: Future Directions in Research on Lay Health Workers''''' ===
'''Presented by Dr. Miya Barnett, Ph.D.'''

====Description====
Watch the YouTube video recording of the address enter link here.

== '''Workshops''' ==

=== '''''Diversity and Inclusion''''' ===
'''Presented by Cassandra Boness, Ph.D. and Chardée Galán, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Building a Lab''''' ===
'''Presented by Christine Cha, Ph.D. and Kate Humphreys, Ph.D.''' 

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Outlook on Academia''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Diversity and Inclusion''''' ===
'''Presented by Omar Gudiño, Ph.D. and Amy West, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''APA Advocacy Training''''' ===
'''Presented by Parfaite-Claude, Kobor, &amp; Sharpe, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Writing Mechanics''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Mentorship Gone Wrong''''' ===
'''Presented by Deb Drabick, Ph.D. and Sarah Racz, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Taking a Leadership Role''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Participatory Action Framework''''' ===
'''Presented by Judith Scott, Ph.D. and Yoyo Yau, Ph.D'''

===== Description =====

Watch the YouTube recording of the workshop here.</text>
      <sha1>aue3gcl43jv7snu1eys489tkb9fy55g</sha1>
    </revision>
    <revision>
      <id>2411918</id>
      <parentid>2411885</parentid>
      <timestamp>2022-08-04T20:40:24Z</timestamp>
      <contributor>
        <username>Magnolia321</username>
        <id>2933848</id>
      </contributor>
      <comment>adding template</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2745" xml:space="preserve">{{underconstruction}}
This year's Forum looks a little bit different. Instead of in past years where the conference was held over the course of 2 to 3 days, this year's programming will be held in the form of a webinar series spanning from July to December. While we have all address and workshops listed, the descriptions and YouTube links will be updated as the events occur, so stay tuned for full descriptions of the programming. 

== '''Addresses''' ==

=== '''''Future Directions Address 1: Future Directions in Research on Brief, Scalable Interventions''''' ===
'''Presented by Dr. Jessica Schleider, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here.

=== '''''Future Directions Address 2: Future Directions in Research on Racism ''''' ===
'''Presented by Dr. Riana Anderson, Ph.D.'''

====Description====

Watch the YouTube video recording of the address enter link here

=== '''''Future Directions Address 3: Future Directions in Research on Lay Health Workers''''' ===
'''Presented by Dr. Miya Barnett, Ph.D.'''

====Description====
Watch the YouTube video recording of the address enter link here.

== '''Workshops''' ==

=== '''''Diversity and Inclusion''''' ===
'''Presented by Cassandra Boness, Ph.D. and Chardée Galán, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Building a Lab''''' ===
'''Presented by Christine Cha, Ph.D. and Kate Humphreys, Ph.D.''' 

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Outlook on Academia''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Diversity and Inclusion''''' ===
'''Presented by Omar Gudiño, Ph.D. and Amy West, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''APA Advocacy Training''''' ===
'''Presented by Parfaite-Claude, Kobor, &amp; Sharpe, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Writing Mechanics''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Mentorship Gone Wrong''''' ===
'''Presented by Deb Drabick, Ph.D. and Sarah Racz, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Taking a Leadership Role''''' ===
'''Presented by Andres De Los Reyes, Ph.D.'''

===== Description =====

Watch the YouTube recording of the workshop here.

=== '''''Participatory Action Framework''''' ===
'''Presented by Judith Scott, Ph.D. and Yoyo Yau, Ph.D'''

===== Description =====

Watch the YouTube recording of the workshop here.</text>
      <sha1>0xnj1xrffzob7m37axkafo3gmx4tjyg</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:HassanAlsamara</title>
    <ns>3</ns>
    <id>286037</id>
    <revision>
      <id>2411887</id>
      <timestamp>2022-08-04T19:52:29Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2769" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] HassanAlsamara!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>r3s5epxttc9m2atjcxd125so64m6ywa</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:U3227354</title>
    <ns>3</ns>
    <id>286038</id>
    <revision>
      <id>2411888</id>
      <timestamp>2022-08-04T19:52:29Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2763" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3227354!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>q374e6mb9in0r0majbodnrkurfc9vqq</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:U3217109</title>
    <ns>3</ns>
    <id>286039</id>
    <revision>
      <id>2411889</id>
      <timestamp>2022-08-04T19:52:30Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2763" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3217109!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>eq1apddk3fe186naiksyl2iftphq34s</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:U3246310</title>
    <ns>3</ns>
    <id>286040</id>
    <revision>
      <id>2411890</id>
      <timestamp>2022-08-04T19:52:31Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2763" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3246310!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>bjqvqymkny9dwdt03suzxse14hv392z</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:SoSilverLibby</title>
    <ns>3</ns>
    <id>286041</id>
    <revision>
      <id>2411891</id>
      <timestamp>2022-08-04T19:52:31Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2768" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] SoSilverLibby!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>o6pbqch8cftqaz9ffvs2b9r32ojhxhu</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:U3230861</title>
    <ns>3</ns>
    <id>286042</id>
    <revision>
      <id>2411892</id>
      <timestamp>2022-08-04T19:52:32Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2763" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3230861!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>p1sfm588wutcx19ptyphcxun37kvyzg</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:U3183466</title>
    <ns>3</ns>
    <id>286043</id>
    <revision>
      <id>2411893</id>
      <timestamp>2022-08-04T19:52:33Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2763" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3183466!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>bp2im5m3ab98l27nagfn11110lph150</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:U3190094</title>
    <ns>3</ns>
    <id>286044</id>
    <revision>
      <id>2411894</id>
      <timestamp>2022-08-04T19:52:33Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2763" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3190094!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>irerpfz2j1fj9aga1irec35uuf08ttg</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:U3207458</title>
    <ns>3</ns>
    <id>286045</id>
    <revision>
      <id>2411895</id>
      <timestamp>2022-08-04T19:52:34Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2763" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3207458!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>lpamqq0e80emtcvjy4md54c9jizgahg</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:U3216502</title>
    <ns>3</ns>
    <id>286046</id>
    <revision>
      <id>2411896</id>
      <timestamp>2022-08-04T19:52:35Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2763" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3216502!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>owzabkpacqw7p2f5ou40cv5312capk5</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Jamieepiper</title>
    <ns>3</ns>
    <id>286047</id>
    <revision>
      <id>2411897</id>
      <timestamp>2022-08-04T19:52:35Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2766" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Jamieepiper!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>dhvbv2skrzmn6w0vtv4e96n07nep0wz</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:U3216963</title>
    <ns>3</ns>
    <id>286048</id>
    <revision>
      <id>2411898</id>
      <timestamp>2022-08-04T19:52:36Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2763" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3216963!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>pkw5dv0rt7wuk16o36t01xuu512kko6</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Bill.miosge</title>
    <ns>3</ns>
    <id>286049</id>
    <revision>
      <id>2411899</id>
      <timestamp>2022-08-04T19:52:37Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2766" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Bill.miosge!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>iw0czr47ln1ksezw65eijj7onog5asd</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:U3215603</title>
    <ns>3</ns>
    <id>286050</id>
    <revision>
      <id>2411900</id>
      <timestamp>2022-08-04T19:52:37Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2763" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3215603!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>gbsz38hwrq3ke4znwsak1kxwtkod7e5</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Isaacem13</title>
    <ns>3</ns>
    <id>286051</id>
    <revision>
      <id>2411901</id>
      <timestamp>2022-08-04T19:52:38Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2764" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Isaacem13!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>eyy00kftoej6qtmuan9aaqb20sffhnb</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:U3166273</title>
    <ns>3</ns>
    <id>286052</id>
    <revision>
      <id>2411902</id>
      <timestamp>2022-08-04T19:52:38Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2763" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3166273!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>2iognb35xsibdep5qv4lz1j4frqvo26</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:U3190353</title>
    <ns>3</ns>
    <id>286053</id>
    <revision>
      <id>2411903</id>
      <timestamp>2022-08-04T19:52:39Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2763" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3190353!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>gbck9hx2cuhfygeq8e7zee8hlru550x</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Cedevlin9</title>
    <ns>3</ns>
    <id>286054</id>
    <revision>
      <id>2411904</id>
      <timestamp>2022-08-04T19:52:40Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2764" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Cedevlin9!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>t5iiof9w7gd1a153lerxj406p3r8yfj</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:U3203545</title>
    <ns>3</ns>
    <id>286055</id>
    <revision>
      <id>2411905</id>
      <timestamp>2022-08-04T19:52:40Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2763" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3203545!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>ti08fcddfebx3lplfuqipqya24c07eu</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:SashaBrooksby</title>
    <ns>3</ns>
    <id>286056</id>
    <revision>
      <id>2411906</id>
      <timestamp>2022-08-04T19:52:41Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2768" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] SashaBrooksby!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:52, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>kmc3lrz7ebsyvgu1sirl8tmgw9bcdy0</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Matshediso cedrick</title>
    <ns>3</ns>
    <id>286057</id>
    <revision>
      <id>2411908</id>
      <timestamp>2022-08-04T19:54:30Z</timestamp>
      <contributor>
        <username>Dave Braunschweig</username>
        <id>426084</id>
      </contributor>
      <comment>Welcome</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2773" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Matshediso cedrick!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:54, 4 August 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>hoiyahc4jg89472u0mivvfgzp8nsqon</sha1>
    </revision>
  </page>
  <page>
    <title>School:Religious Studies</title>
    <ns>100</ns>
    <id>286058</id>
    <revision>
      <id>2411910</id>
      <timestamp>2022-08-04T20:24:42Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>School of Religious Studies</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7019" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

Religious Studies, also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''7 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for Religious Studies.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.


==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>n2gs4kbf1lbb46d7ihvrw0qh97cjb3n</sha1>
    </revision>
    <revision>
      <id>2411919</id>
      <parentid>2411910</parentid>
      <timestamp>2022-08-04T20:40:37Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* How can I get involved? */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7018" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

Religious Studies, also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''7 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for Religious Studies.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School Talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>eoke4dxuis74me4e0i2bkab8mq4fokz</sha1>
    </revision>
    <revision>
      <id>2411920</id>
      <parentid>2411919</parentid>
      <timestamp>2022-08-04T20:41:46Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* How can I get involved? */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7011" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

Religious Studies, also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''7 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for Religious Studies.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[Talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>4t99qrfdy0fexe35m68jqrfdkfpasav</sha1>
    </revision>
    <revision>
      <id>2411921</id>
      <parentid>2411920</parentid>
      <timestamp>2022-08-04T20:42:34Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* How can I get involved? */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7017" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

Religious Studies, also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''7 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for Religious Studies.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[Talk:Religious Studies|discussion]] page, or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>0u07loisly9s28vi7mpzii6g0ja9764</sha1>
    </revision>
    <revision>
      <id>2411922</id>
      <parentid>2411921</parentid>
      <timestamp>2022-08-04T20:44:56Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* How can I get involved? */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7017" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

Religious Studies, also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''7 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for Religious Studies.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[talk:Religious Studies|discussion]] page, or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>1jubi3refs4mqub974lf7781l3225di</sha1>
    </revision>
    <revision>
      <id>2411923</id>
      <parentid>2411922</parentid>
      <timestamp>2022-08-04T20:50:34Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* How can I get involved? */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7018" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

Religious Studies, also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''7 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for Religious Studies.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>3omhbgtjty4q470lat9debvass4mx2i</sha1>
    </revision>
    <revision>
      <id>2411925</id>
      <parentid>2411923</parentid>
      <timestamp>2022-08-04T20:52:43Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7024" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''7 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for Religious Studies.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>pxvikpdpnlwhul5w90iyc03ouxmc95q</sha1>
    </revision>
    <revision>
      <id>2411926</id>
      <parentid>2411925</parentid>
      <timestamp>2022-08-04T20:53:48Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* Things you can do! */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7035" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''7 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for [[Portal:Religious_studies]].
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>soefgl9ngb730x00oxhbce4h1hder5t</sha1>
    </revision>
    <revision>
      <id>2411927</id>
      <parentid>2411926</parentid>
      <timestamp>2022-08-04T20:54:01Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* Things you can do! */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7035" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''7 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for [[Portal:Religious Studies]].
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>02taqh3h2zicbohq5uh9h5137wt7j51</sha1>
    </revision>
    <revision>
      <id>2411928</id>
      <parentid>2411927</parentid>
      <timestamp>2022-08-04T20:54:22Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* Things you can do! */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7035" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''7 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for [[Portal:Religious studies]].
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>s1oauqdedpqiblj6pv3bvypejxd4z0w</sha1>
    </revision>
    <revision>
      <id>2411929</id>
      <parentid>2411928</parentid>
      <timestamp>2022-08-04T20:55:01Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* Things you can do! */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7061" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''7 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for [[Portal:Religious studies]]. Perhaps fix spelling too.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>mgxmk9xb4h9jgour5setb4y73n5wj08</sha1>
    </revision>
    <revision>
      <id>2411930</id>
      <parentid>2411929</parentid>
      <timestamp>2022-08-04T20:58:30Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7065" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''7 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for [[Portal:Religious studies]]. Perhaps fix spelling too.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>t1e4xo50ge5bnr79xgt5n2grem1qsm5</sha1>
    </revision>
    <revision>
      <id>2411931</id>
      <parentid>2411930</parentid>
      <timestamp>2022-08-04T21:00:59Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* Active participants */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7103" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''7 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for [[Portal:Religious studies]]. Perhaps fix spelling too.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* [[SotanScholar]] as of Aug 3, 2022.
* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>2e2css3xmw0u3s8fnkg4g8flv1a5fsx</sha1>
    </revision>
    <revision>
      <id>2411932</id>
      <parentid>2411931</parentid>
      <timestamp>2022-08-04T21:01:08Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* School news */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7103" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''4 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for [[Portal:Religious studies]]. Perhaps fix spelling too.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* [[SotanScholar]] as of Aug 3, 2022.
* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>e2cpzmmcmml87f42agnxute01h4dsv0</sha1>
    </revision>
    <revision>
      <id>2411935</id>
      <parentid>2411932</parentid>
      <timestamp>2022-08-04T21:05:39Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* Active participants */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7108" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''4 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for [[Portal:Religious studies]]. Perhaps fix spelling too.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* [[user:SotanScholar]] as of Aug 3, 2022.
* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>reesl6mrj58nvgxsge7p7k5bfd1dl1q</sha1>
    </revision>
    <revision>
      <id>2411936</id>
      <parentid>2411935</parentid>
      <timestamp>2022-08-04T21:05:54Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* Active participants */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7121" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (Theology) founded! 
* '''4 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for [[Portal:Religious studies]]. Perhaps fix spelling too.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* [[user:SotanScholar|SotanScholar]] as of Aug 3, 2022.
* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>g1vqodukpr6pxyls2qqwp5x9x8n821m</sha1>
    </revision>
    <revision>
      <id>2411937</id>
      <parentid>2411936</parentid>
      <timestamp>2022-08-04T21:06:26Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* School news */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7127" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page.''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (under Theology) founded! 
* '''4 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for [[Portal:Religious studies]]. Perhaps fix spelling too.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* [[user:SotanScholar|SotanScholar]] as of Aug 3, 2022.
* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>a2l7ckj0jggek0n90k8e1e4bgofios9</sha1>
    </revision>
    <revision>
      <id>2411938</id>
      <parentid>2411937</parentid>
      <timestamp>2022-08-04T21:07:45Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* Areas of Study */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7188" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page. (this note will go away when things start getting linked up)''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (under Theology) founded! 
* '''4 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for [[Portal:Religious studies]]. Perhaps fix spelling too.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* [[user:SotanScholar|SotanScholar]] as of Aug 3, 2022.
* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>28seg9j7x7jlbl558s1vbts176w5a2v</sha1>
    </revision>
    <revision>
      <id>2411956</id>
      <parentid>2411938</parentid>
      <timestamp>2022-08-04T21:48:25Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* How can I get involved? */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7188" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|175px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page. (this note will go away when things start getting linked up)''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (under Theology) founded! 
* '''4 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for [[Portal:Religious studies]]. Perhaps fix spelling too.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* [[user:SotanScholar|SotanScholar]] as of Aug 3, 2022.
* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious_Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious_Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
      <sha1>inzz87v0899d8g4ytmpsa4vhr72t4om</sha1>
    </revision>
    <revision>
      <id>2411981</id>
      <parentid>2411956</parentid>
      <timestamp>2022-08-04T22:57:28Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7188" xml:space="preserve">'''&lt;div align=&quot;center&quot;&gt;{{font|size=x-large|Welcome to the School of Religious Studies}}&lt;/div&gt;'''

&lt;div align=&quot;center&quot;&gt;''part of the [[Portal:Interdisciplinary Studies|Faculty of Interdisciplinary Studies]] and the [[School:Theology|School of Theology]]''&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''The School of Religious Studies is for multi-disciplinary and secular study of religion.''&lt;/div&gt;
[[Image:Religions 4x5.png|thumb|275px|Symbols of religions]]

'''Religious Studies''', also known as the study of religion, is an academic field devoted to research into religious beliefs, behaviors, and institutions. It describes, compares, interprets, and explains religion, emphasizing systematic, historically based, and cross-cultural perspectives.

While [[Theology]] attempts to understand the transcendent or supernatural according to traditional religious accounts, religious studies takes a more scientific and objective approach independent of any particular religious viewpoint. Religious Studies thus draws upon multiple academic disciplines and methodologies including anthropology, sociology, psychology, philosophy, and history of religion.

The '''School of Religious Studies''' works in close cooperation with the [[School:Theology|School of Theology]].


==Areas of Study==
Currently the School of Religious Studies is composed of the following areas of studies.

{{Multicol}}

'''Western Religions'''
* Abrahamic Religions
** Judaism
** Christianisty
** Islam
** Baháʼí
** ''Others''
* Iranian Religions
** Zoroastrianism
** Yazidism
{{Multicol-break}}

'''Eastern Religions'''
* East Asian
** Taoism
** Shinto
*Indian
** Hinduism
** Buddhism
** Sikhism
** ''Others''
{{Multicol-break}}

'''Indigenous Religions'''
* Maori
* Native American
* Tibeto-Burmese
* African
* Ethnic
{{Multicol-break}}

'''New Movement Religions'''
** Jediism
** Neoshamanism
** Scientology
** Spiritualism
* Modern Paganism
** Druidry
** Heathenry
** Wicca
{{Multicol-break}}

'''Historic Religions'''
* Pre-historic
* Inca
* Germanic
* Mesopotamian
** Sumerian/Babylonian
* Mithraism
* Semitic

{{Multicol-end}}


''These Areas of Study (Department and Divisions) links will be added when course and topic pages are found or created. The majority of divisions will be found on the Department page. (this note will go away when things start getting linked up)''

==Core Curriculum==
In order to form a basic understanding of religious studies, it is recommended that students visit the following courses below. After which it would benefit students and researchers to further thier studies by exploring the wikimedia resources below before considering a specialised Area of Study.

{{Multicol}}
'''Introductory'''
* [[w:Religious Studies|Religious Studies]]
* [[w:History of Religion|History of Religion]]
* [[w:Anthropology of Religion|Anthropology of Religion]]
* [[w:Sociology of Religion|Sociology of Religion]]
* [[w:Psychology of Religion|Psychology of Religion]]
{{Multicol-break}}

'''Methodology'''
* [[w:Phenomenology of Religion|Phenomenology of Religion]]
* [[Political and historical methodology|Historical Methodology]]

{{Multicol-end}}

== Courses ==

{{Multicol}}
'''Active'''
{{Multicol-break}}

'''Planned'''
{{Multicol-break}}

'''Requested'''

{{Multicol-end}}


== Wikimedia Resources ==
{{Div col}}
* [[w:Religious Studies|Religious Studies]] article at Wikipedia
* [[b:Subject:Subject:Religion|Related Religion]] textbooks at Wikibooks
* [[c:Category:Religious_studies|Religious Studies]] media at Wikimedia Commons
{{Div col end}}

''Note:  This list is far from complete.  Feel free to add to them.''


{{Multicol}}
==School news==
* '''9 September 2006''' - Original Division (under Theology) founded! 
* '''4 August 2022''' - Division converted to a School!

==Things you can do!==
* Establish new box style Portal page for [[Portal:Religious studies]]. Perhaps fix spelling too.
* Clean up Religion topics through research.
* Create new Courses of study!

{{Multicol-break}}

==Active participants==
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed).

* [[user:SotanScholar|SotanScholar]] as of Aug 3, 2022.
* Since 27 August 2014‎ with [[Draft:Creationism|Creationism]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:47, 9 May 2018 (UTC).
* [[user:dchmelik|dchmelik]] up to 26 November 2017!
* [[user:DErnestWachter|DErnestWachter]]

==Inactive participants==
* [[user:davymast|davymast]]
* [[user:vectoreq|vectoreq]]
* [[user:vanguard|vanguard]]
* [[user:melpomenon|melpomenon]]
{{Multicol-end}}

==How can I get involved?==

====As a learner====
Are you new to Wikiversity or the School of [[w:Religious Studies|Religious Studies]]? You have come across an important resource in the development of interactive [[Open educational resources|Open Educational Resources]]. You are welcome here as a student browsing for resources to use in your studies. You may like to to sign up for a specific course. Either way you can use this area of cyberspace to keep notes while you study - but remember, they are open for other people to read! (You can create pages linked to your user page by adding &quot;/subspacename&quot; after your username when starting a new page). Please also use the discussion pages to give feed back about courses whether you like them or were disappointed. 
We also encourage more active involvement. You can help in the development of courses, adding to the bibliography, putting in links to wikipedia (use &quot;w:&quot; at the beginning of the box), correcting typos etc.
If you are a newcomer, please see [[Help:The original tour for newcomers|this]] quick tour to help you familiarise yourself with wikiversity.

====As a teacher/facilitator====
We also encourage teachers to use wikiversity as a place to develop and share their teaching materials. For more general information, please look [[Help:Creating educational content at Wikiversity|here]].

====As a researcher====
In addition to Wikiversity's learning mission in various educational sectors, Wikiversity also hosts research. Please see [[Portal:Research]] for more information.

====As part of a Learning Community====
We also function as a learning community, sharing general maintenance tasks, discussing problems that arise and developing collaborative projects
We keep a special [[School:Religious_Studies/Tasks|page]] of things that need to be done for people who just want to jump in!

*[[School:Religious_Studies/Tasks|Get involved!]]

Have any questions? Leave them on the School's [[School talk:Religious Studies|talk page]], or contact one of the active particpants.

==See also==
{{Div col}}
* [[School:Theology|School of Theology]]
* [[Portal:Interdisciplinary Studies]]
* [[Portal:Humanities]]
* [[Peace studies|Peace Studies]]
* [[Portal:Comparative_religion|Comparative Religion]]
{{Div col end}}


== External links ==


[[Category:Wikiversity schools]]
[[Category:Religious studies ]]</text>
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      <comment>/* Welcome */ new section</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="259" xml:space="preserve">== Welcome ==

I'll do my best to get answer for questions here, or someone else will. There is a lot to do. [[User:SotanScholar|SotanScholar]] ([[User talk:SotanScholar|discuss]] • [[Special:Contributions/SotanScholar|contribs]]) 20:52, 4 August 2022 (UTC)</text>
      <sha1>cyoygeybouvx42j7bp86m94q7av6h6b</sha1>
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    <revision>
      <id>2411987</id>
      <parentid>2411924</parentid>
      <timestamp>2022-08-04T23:07:41Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2904" xml:space="preserve">== Welcome ==

I'll do my best to get answer for questions here, or someone else will. There is a lot to do. [[User:SotanScholar|SotanScholar]] ([[User talk:SotanScholar|discuss]] • [[Special:Contributions/SotanScholar|contribs]]) 20:52, 4 August 2022 (UTC)


=============== OLDER CONVERSATIONS FROM DIVISION ==================================================================

== Religious &amp; Mystical Geography ==
What on earth is 'Religious &amp; Mystical Geography?'  It sounds like it can go in 'Comparative Religion.'  Anyone saying religion is not mystical does not know; mysticisim is appropriate for advanced or elective Comparative Religion, which already has epistemology and metaphysics.--[[User:Dchmelik|Dchmelik]] 22:18, 22 March 2008 (UTC)

== Remove Department of Christian Studies ==
hey, I was sprucing up a few things over in the [[School:Theology|School of Theology]] when I noticed that the Division of Christian Studies is in both that school &amp;&amp; religious studies.  Judging by the content, I think that the Division of Christian Studies tends to focus on adherents to Christianity, not the unbiased study of it.  So I wanted to suggest removing it from this division, and replacing it with a comparable NPOV department. (sorry, 4got to sign) --[[User:Opensourcejunkie|Opensourcejunkie]] 03:38, 25 March 2008 (UTC)

:You may as well say that about any single religious dept..  Perhaps having the dept. of Comparative Religion is enough.--[[User:Dchmelik|Dchmelik]] 22:17, 22 March 2008 (UTC)

:: yeah, that and perhaps the dept. of Inter-Religious Conflict Studies - only those things that have to do with NPOV study of religions.  That's how I see the division, anyway. --[[User:Opensourcejunkie|Opensourcejunkie]] 03:42, 25 March 2008 (UTC)

::: I strongly disagree. The secular study of any religion is just as needed as the secular study of religious structural composition (Comparative Religion) and as a conflict study area. And why is there no sociology of religion present? The sociological, social-anthropological and similar approaches are hugely influential in the academic study of religious phenomena. You can't study a religion as a scientist of religion with a POV - then you're a theologian of that religion (which is noble and all respect to those guys) - but we, as scientists doing research on this (and in this forum, communicating current research on the topic) need to adapt a ''methodological atheism'', albeit not necessarily a personal one (Cf. Weber, Berger). I thus suggest the following structure (based in part on the BA curriculum in the Science of Religion at University of Copenhagen):
::: *Sociology of Religion
::: * Comparative Religion
::: * Cognitive approaches to Religion
::: * Theory and Method in the Study of Religion
::: * Area studies: Christianity, Islam, Judaism, Hinduism, etc. [[User:Melpomenon|Melpomenon]] 22:49, 27 October 2009 (UTC)</text>
      <sha1>timi5nvqlhigtty76upcp0acl54t9j4</sha1>
    </revision>
    <revision>
      <id>2411988</id>
      <parentid>2411987</parentid>
      <timestamp>2022-08-04T23:08:08Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2912" xml:space="preserve">== Welcome ==

I'll do my best to get answer for questions here, or someone else will. There is a lot to do. [[User:SotanScholar|SotanScholar]] ([[User talk:SotanScholar|discuss]] • [[Special:Contributions/SotanScholar|contribs]]) 20:52, 4 August 2022 (UTC)


=============== OLDER CONVERSATIONS FROM PREVIOUS VERSION ==================================================================

== Religious &amp; Mystical Geography ==
What on earth is 'Religious &amp; Mystical Geography?'  It sounds like it can go in 'Comparative Religion.'  Anyone saying religion is not mystical does not know; mysticisim is appropriate for advanced or elective Comparative Religion, which already has epistemology and metaphysics.--[[User:Dchmelik|Dchmelik]] 22:18, 22 March 2008 (UTC)

== Remove Department of Christian Studies ==
hey, I was sprucing up a few things over in the [[School:Theology|School of Theology]] when I noticed that the Division of Christian Studies is in both that school &amp;&amp; religious studies.  Judging by the content, I think that the Division of Christian Studies tends to focus on adherents to Christianity, not the unbiased study of it.  So I wanted to suggest removing it from this division, and replacing it with a comparable NPOV department. (sorry, 4got to sign) --[[User:Opensourcejunkie|Opensourcejunkie]] 03:38, 25 March 2008 (UTC)

:You may as well say that about any single religious dept..  Perhaps having the dept. of Comparative Religion is enough.--[[User:Dchmelik|Dchmelik]] 22:17, 22 March 2008 (UTC)

:: yeah, that and perhaps the dept. of Inter-Religious Conflict Studies - only those things that have to do with NPOV study of religions.  That's how I see the division, anyway. --[[User:Opensourcejunkie|Opensourcejunkie]] 03:42, 25 March 2008 (UTC)

::: I strongly disagree. The secular study of any religion is just as needed as the secular study of religious structural composition (Comparative Religion) and as a conflict study area. And why is there no sociology of religion present? The sociological, social-anthropological and similar approaches are hugely influential in the academic study of religious phenomena. You can't study a religion as a scientist of religion with a POV - then you're a theologian of that religion (which is noble and all respect to those guys) - but we, as scientists doing research on this (and in this forum, communicating current research on the topic) need to adapt a ''methodological atheism'', albeit not necessarily a personal one (Cf. Weber, Berger). I thus suggest the following structure (based in part on the BA curriculum in the Science of Religion at University of Copenhagen):
::: *Sociology of Religion
::: * Comparative Religion
::: * Cognitive approaches to Religion
::: * Theory and Method in the Study of Religion
::: * Area studies: Christianity, Islam, Judaism, Hinduism, etc. [[User:Melpomenon|Melpomenon]] 22:49, 27 October 2009 (UTC)</text>
      <sha1>n73mydsirzctrtxnwob6gv96t0ruzrz</sha1>
    </revision>
    <revision>
      <id>2411989</id>
      <parentid>2411988</parentid>
      <timestamp>2022-08-04T23:08:34Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2951" xml:space="preserve">== Welcome ==

I'll do my best to get answer for questions here, or someone else will. There is a lot to do. [[User:SotanScholar|SotanScholar]] ([[User talk:SotanScholar|discuss]] • [[Special:Contributions/SotanScholar|contribs]]) 20:52, 4 August 2022 (UTC)


=============== OLDER CONVERSATIONS FROM PREVIOUS VERSION ==================================================================
(back when the School was a Division)


== Religious &amp; Mystical Geography ==
What on earth is 'Religious &amp; Mystical Geography?'  It sounds like it can go in 'Comparative Religion.'  Anyone saying religion is not mystical does not know; mysticisim is appropriate for advanced or elective Comparative Religion, which already has epistemology and metaphysics.--[[User:Dchmelik|Dchmelik]] 22:18, 22 March 2008 (UTC)

== Remove Department of Christian Studies ==
hey, I was sprucing up a few things over in the [[School:Theology|School of Theology]] when I noticed that the Division of Christian Studies is in both that school &amp;&amp; religious studies.  Judging by the content, I think that the Division of Christian Studies tends to focus on adherents to Christianity, not the unbiased study of it.  So I wanted to suggest removing it from this division, and replacing it with a comparable NPOV department. (sorry, 4got to sign) --[[User:Opensourcejunkie|Opensourcejunkie]] 03:38, 25 March 2008 (UTC)

:You may as well say that about any single religious dept..  Perhaps having the dept. of Comparative Religion is enough.--[[User:Dchmelik|Dchmelik]] 22:17, 22 March 2008 (UTC)

:: yeah, that and perhaps the dept. of Inter-Religious Conflict Studies - only those things that have to do with NPOV study of religions.  That's how I see the division, anyway. --[[User:Opensourcejunkie|Opensourcejunkie]] 03:42, 25 March 2008 (UTC)

::: I strongly disagree. The secular study of any religion is just as needed as the secular study of religious structural composition (Comparative Religion) and as a conflict study area. And why is there no sociology of religion present? The sociological, social-anthropological and similar approaches are hugely influential in the academic study of religious phenomena. You can't study a religion as a scientist of religion with a POV - then you're a theologian of that religion (which is noble and all respect to those guys) - but we, as scientists doing research on this (and in this forum, communicating current research on the topic) need to adapt a ''methodological atheism'', albeit not necessarily a personal one (Cf. Weber, Berger). I thus suggest the following structure (based in part on the BA curriculum in the Science of Religion at University of Copenhagen):
::: *Sociology of Religion
::: * Comparative Religion
::: * Cognitive approaches to Religion
::: * Theory and Method in the Study of Religion
::: * Area studies: Christianity, Islam, Judaism, Hinduism, etc. [[User:Melpomenon|Melpomenon]] 22:49, 27 October 2009 (UTC)</text>
      <sha1>4eh58pgmw5u8adolyhzsz13gzrzq42e</sha1>
    </revision>
    <revision>
      <id>2411990</id>
      <parentid>2411989</parentid>
      <timestamp>2022-08-04T23:08:52Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2953" xml:space="preserve">== Welcome ==

I'll do my best to get answer for questions here, or someone else will. There is a lot to do. [[User:SotanScholar|SotanScholar]] ([[User talk:SotanScholar|discuss]] • [[Special:Contributions/SotanScholar|contribs]]) 20:52, 4 August 2022 (UTC)


=============== OLDER CONVERSATIONS FROM PREVIOUS VERSION ==================================================================
(back when the School was a Division)


== Religious &amp; Mystical Geography ==
What on earth is 'Religious &amp; Mystical Geography?'  It sounds like it can go in 'Comparative Religion.'  Anyone saying religion is not mystical does not know; mysticisim is appropriate for advanced or elective Comparative Religion, which already has epistemology and metaphysics.--[[User:Dchmelik|Dchmelik]] 22:18, 22 March 2008 (UTC)

== Remove Department of Christian Studies ==
hey, I was sprucing up a few things over in the [[School:Theology|School of Theology]] when I noticed that the Division of Christian Studies is in both that school &amp;&amp; religious studies.  Judging by the content, I think that the Division of Christian Studies tends to focus on adherents to Christianity, not the unbiased study of it.  So I wanted to suggest removing it from this division, and replacing it with a comparable NPOV department. (sorry, 4got to sign) --[[User:Opensourcejunkie|Opensourcejunkie]] 03:38, 25 March 2008 (UTC)

:You may as well say that about any single religious dept..  Perhaps having the dept. of Comparative Religion is enough.--[[User:Dchmelik|Dchmelik]] 22:17, 22 March 2008 (UTC)

:: yeah, that and perhaps the dept. of Inter-Religious Conflict Studies - only those things that have to do with NPOV study of religions.  That's how I see the division, anyway. --[[User:Opensourcejunkie|Opensourcejunkie]] 03:42, 25 March 2008 (UTC)

::: I strongly disagree. The secular study of any religion is just as needed as the secular study of religious structural composition (Comparative Religion) and as a conflict study area. And why is there no sociology of religion present? The sociological, social-anthropological and similar approaches are hugely influential in the academic study of religious phenomena. You can't study a religion as a scientist of religion with a POV - then you're a theologian of that religion (which is noble and all respect to those guys) - but we, as scientists doing research on this (and in this forum, communicating current research on the topic) need to adapt a ''methodological atheism'', albeit not necessarily a personal one (Cf. Weber, Berger). I thus suggest the following structure (based in part on the BA curriculum in the Science of Religion at University of Copenhagen):
::: *Sociology of Religion
::: * Comparative Religion
::: * Cognitive approaches to Religion
::: * Theory and Method in the Study of Religion
::: * Area studies: Christianity, Islam, Judaism, Hinduism, etc. [[User:Melpomenon|Melpomenon]] 22:49, 27 October 2009 (UTC)</text>
      <sha1>ef3r9a6s9vvhtatzm77vi4orotpyy8p</sha1>
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    <revision>
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        <username>Dave Braunschweig</username>
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      <comment>/* Welcome */ Reply</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3386" xml:space="preserve">== Welcome ==

I'll do my best to get answer for questions here, or someone else will. There is a lot to do. [[User:SotanScholar|SotanScholar]] ([[User talk:SotanScholar|discuss]] • [[Special:Contributions/SotanScholar|contribs]]) 20:52, 4 August 2022 (UTC)

:@[[User:SotanScholar|SotanScholar]] Where did the previous content come from? Due to CC-BY-SA licensing, you can't just copy it. It has to have a reference to the original source, preferably by including the authors' own contributions in the edit history. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:55, 5 August 2022 (UTC)


=============== OLDER CONVERSATIONS FROM PREVIOUS VERSION ==================================================================
(back when the School was a Division)


== Religious &amp; Mystical Geography ==
What on earth is 'Religious &amp; Mystical Geography?'  It sounds like it can go in 'Comparative Religion.'  Anyone saying religion is not mystical does not know; mysticisim is appropriate for advanced or elective Comparative Religion, which already has epistemology and metaphysics.--[[User:Dchmelik|Dchmelik]] 22:18, 22 March 2008 (UTC)

== Remove Department of Christian Studies ==
hey, I was sprucing up a few things over in the [[School:Theology|School of Theology]] when I noticed that the Division of Christian Studies is in both that school &amp;&amp; religious studies.  Judging by the content, I think that the Division of Christian Studies tends to focus on adherents to Christianity, not the unbiased study of it.  So I wanted to suggest removing it from this division, and replacing it with a comparable NPOV department. (sorry, 4got to sign) --[[User:Opensourcejunkie|Opensourcejunkie]] 03:38, 25 March 2008 (UTC)

:You may as well say that about any single religious dept..  Perhaps having the dept. of Comparative Religion is enough.--[[User:Dchmelik|Dchmelik]] 22:17, 22 March 2008 (UTC)

:: yeah, that and perhaps the dept. of Inter-Religious Conflict Studies - only those things that have to do with NPOV study of religions.  That's how I see the division, anyway. --[[User:Opensourcejunkie|Opensourcejunkie]] 03:42, 25 March 2008 (UTC)

::: I strongly disagree. The secular study of any religion is just as needed as the secular study of religious structural composition (Comparative Religion) and as a conflict study area. And why is there no sociology of religion present? The sociological, social-anthropological and similar approaches are hugely influential in the academic study of religious phenomena. You can't study a religion as a scientist of religion with a POV - then you're a theologian of that religion (which is noble and all respect to those guys) - but we, as scientists doing research on this (and in this forum, communicating current research on the topic) need to adapt a ''methodological atheism'', albeit not necessarily a personal one (Cf. Weber, Berger). I thus suggest the following structure (based in part on the BA curriculum in the Science of Religion at University of Copenhagen):
::: *Sociology of Religion
::: * Comparative Religion
::: * Cognitive approaches to Religion
::: * Theory and Method in the Study of Religion
::: * Area studies: Christianity, Islam, Judaism, Hinduism, etc. [[User:Melpomenon|Melpomenon]] 22:49, 27 October 2009 (UTC)</text>
      <sha1>pmhteyxqipol2yy65le5ach41qnux70</sha1>
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  <page>
    <title>User:SotanScholar</title>
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      <comment>New resource with &quot;I'm working on setting up my user page. Please bear with my.  Specializing in Religious Studies.&quot;</comment>
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      <text bytes="96" xml:space="preserve">I'm working on setting up my user page. Please bear with my.

Specializing in Religious Studies.</text>
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  </page>
  <page>
    <title>School:Religious Studies/Tasks</title>
    <ns>100</ns>
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    <revision>
      <id>2411962</id>
      <timestamp>2022-08-04T22:21:10Z</timestamp>
      <contributor>
        <username>SotanScholar</username>
        <id>2947466</id>
      </contributor>
      <comment>/* Religious Studies School Task Page */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="938" xml:space="preserve">&lt;noinclude&gt;Welcome to the [[School:History|School of Religious Studies]]'s Task page!  This page is designed as a simple place where our Departments (Areas of Studies) can list things that they need help on, so that you can dig in and get involved contributing to Wikiversity. If any of the following sound like things you'd be interested in doing, jump right in! [[Wikiversity:Be bold|Be bold]] !&lt;/noinclude&gt;

== All Departments ==
With the [[Wikiversity:Browse|Browse]] system in place at Wikiversity, Categorization is important.  Please be sure to add any categories your for Area of Study where appropriate.
&lt;br&gt;


=== Core Curriculum ===

=== Course Lists ===


&lt;br&gt;
== Departmental Task Pages ==
Each Department will have tasks listed below. If nothing listed, then no tasks exist.
&lt;br&gt;


=== Western Religions ===

=== Eastern Religions ===

=== Indigenous Religions ===

=== New Movement Religions ===

=== Historic Religions ===</text>
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  <page>
    <title>Evidence-based assessment/Anorexia nervosa (assessment portfolio)/extended version</title>
    <ns>0</ns>
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      <contributor>
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      <comment>Added information to extended page</comment>
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      <text bytes="42999" xml:space="preserve">&lt;noinclude&gt;{{Helping Give Away Psychological Science Banner}}&lt;/noinclude&gt;

{{medical disclaimer}}
{{:{{ROOTPAGENAME}}/Sidebar}}

==[[Evidence based assessment/Portfolio template/What is a &quot;portfolio&quot;|'''What is a &quot;portfolio&quot;?''']]==
For background information on what assessment portfolios are, click the link in the heading above.
Does this page feel like too much information? Click [[Evidence-based assessment/Anorexia nervosa (assessment portfolio)|here]] for the condensed version.

== [[Evidence based assessment/Preparation phase|'''Preparation phase''']] ==
=== Diagnostic criteria for anorexia nervosa ===
{{blockquotetop}}
'''ICD-11 Criteria'''

Anorexia Nervosa is characterized by significantly low body weight, which is less than minimal normal/expected weight for the individual’s height, sex, age and developmental stage (body mass index (BMI) less than 18.5 kg/m2 in adults and BMI-for-age under fifth percentile in children and adolescents) that is not due to another health condition or to the unavailability of food. Low body weight is accompanied by a persistent pattern of behaviors to prevent restoration of normal weight, which may include behaviors aimed at reducing energy intake (restricted eating), purging behaviors (e.g., self-induced vomiting, misuse of laxatives), and behaviors aimed at increasing energy expenditure (e.g., excessive exercise), typically associated with a fear of weight gain. Low body weight or shape is central to the person's self-evaluation or is inaccurately perceived to be normal or even excessive.

'''Changes in DSM-5'''

* The diagnostic criteria for anorexia nervosa changed slightly from [[DSM-IV]] to [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders#DSM-IV-TR_.282000.29|DSM-5]]. Summaries are available [http://www.dsm5.org/Documents/changes%20from%20dsm-iv-tr%20to%20dsm-5.pdf here] and [[w:DSM-5|here]].
{{blockquotebottom}}

=== Base rates of anorexia nervosa in different populations and clinical settings ===
This section describes the demographic setting of the population(s) sampled, base rates of diagnosis such as prevalence rates, country/region sampled, and the diagnostic method that was used. Using this information, clinicians will be able to anchor the rate of adolescent depression that they are likely to see in their clinical practice.

* '''''To see prevalence rates across multiple disorders,''''' [[Evidence based assessment/Preparation phase#Base rates for transdiagnostic comparison|'''''click here.''''']]

{| class=&quot;wikitable&quot;
|-
! Setting (Reference) 
! Base Rate
! Demography
! Diagnostic Method
|-
|Non-clinical: Population-based (NCS-R)&lt;ref&gt;Hudson, James I.; Hiripi, Eva; Pope, Harrison G.; Kessler, Ronald C. (2007-02-01). &quot;The prevalence and correlates of eating disorders in the National Comorbidity Survey Replication&quot;.''Biological Psychiatry'' '''61''' (3): 348–358. doi:10.1016/j.biopsych.2006.03.040. ISSN 0006-3223. PMC 1892232. &lt;nowiki&gt;PMID 16815322&lt;/nowiki&gt;&lt;/ref&gt;
|.9% Female, .3% Male
|Nationally representative US sample of adults
|World Health Organization-Clinical International Diagnostic Interview (WHO-CIDI)
|-
|Non-clinical: Population-based (NCS-A)&lt;ref&gt;Swanson, Sonja A.; Crow, Scott J.; Le Grange, Daniel; Swendsen, Joel; Merikangas, Kathleen R. (2011-07-01). &quot;Prevalence and correlates of eating disorders in adolescents. Results from the national comorbidity survey replication adolescent supplement&quot;. ''Archives of General Psychiatry'' '''68''' (7): 714–723. doi:10.1001/archgenpsychiatry.2011.22. ISSN 1538-3636.&lt;nowiki&gt;PMID 21383252&lt;/nowiki&gt;&lt;/ref&gt;
|.3% Female, .3% Male
|Nationally representative US sample of adolescents
|WHO-CIDI
|-
|Non-clinical: US Population-based prevalence (Adolescent Brain Cognitive Development (ABCD) study&lt;ref&gt;{{Cite journal|last=Rozzell|first=Kaitlin|last2=Moon|first2=Da Yeoun|last3=Klimek|first3=Patrycja|last4=Brown|first4=Tiffany|last5=Blashill|first5=Aaron J.|date=2019-01-01|title=Prevalence of Eating Disorders Among US Children Aged 9 to 10 Years: Data From the Adolescent Brain Cognitive Development (ABCD) Study|url=http://archpedi.jamanetwork.com/article.aspx?doi=10.1001/jamapediatrics.2018.3678|journal=JAMA Pediatrics|language=en|volume=173|issue=1|pages=100|doi=10.1001/jamapediatrics.2018.3678|issn=2168-6203|pmc=PMC6583451|pmid=30476983}}&lt;/ref&gt;
|0.1%, no gender differences
|Nationally representative US sample of 9- and 10-year old children
|DSM-5 using Kiddie Schedule for Affective Disorders and Schizophrenia (KSADS)
|-
|Non-clinical: Latinos in US&lt;ref&gt;Alegria, Margarita; Woo, Meghan; Cao, Zhun; Torres, Maria; Meng, Xiao-li; Striegel-Moore, Ruth (2007-11-01). &quot;Prevalence and correlates of eating disorders in Latinos in the United States&quot;.''The International Journal of Eating Disorders''. 40 Suppl: S15–21. doi:10.1002/eat.20406. ISSN 0276-3478. PMC 2680162. &lt;nowiki&gt;PMID 17584870&lt;/nowiki&gt;&lt;/ref&gt;
|.12% Female, .03% Male
|Latino Households in the US
|WHO-CIDI
|-
|Non-clinical: African Americans and Caribbean Blacks in the US (NSAL)&lt;ref&gt;Taylor, Jacquelyn Y.; Caldwell, Cleopatra Howard; Baser, Raymond E.; Faison, Nakesha; Jackson, James S. (2007-11-01). &quot;Prevalence of eating disorders among Blacks in the National Survey of American Life&quot;. ''The International Journal of Eating Disorders''. 40 Suppl: S10–14. doi:10.1002/eat.20451. ISSN 0276-3478. PMC 2882704. &lt;nowiki&gt;PMID 17879287&lt;/nowiki&gt;&lt;/ref&gt;
|.14% Female, .2% Male
|National probability sample of adult and adolescent African Americans and Caribbean Blacks
|WHO-CIDI
|-
|Non-clinical: National Latino and Asian American Study (NLAAS)
|.12% Female, .05% Male
|Asian American adults in US households
|WHO-CIDI
|-
| Non-clinical; college students&lt;ref&gt;Tyler ID. A true picture of eating disorders among African American women: a review of literature. ABNF J. 2003;14(3):73-4.&lt;/ref&gt; || .0%  || US African American college females || Eating Disorder Diagnostic Questionnaire (EDD-Q)
|-
| Non-clinical; Adolescents&lt;ref&gt;Stice, E., Becker, C. B., &amp; Yokum, S. (2013). Eating disorder prevention: Current evidence-base and future directions. ''Int. J. Eat. Disord. International Journal of Eating Disorders,'' ''46''(5), 478-485.&lt;/ref&gt; || .8% || US Female Adolescents || Eating Disorder Diagnosis Interview (EDDI)
|-
| Non-clinical; student-athletes&lt;ref&gt;Johnson C, Powers PS, Dick R. Athletes and eating disorders: the national collegiate athletic association study. Int J Eat Disord 1999;26:179e88.&lt;/ref&gt; || .0% || US Division-I Varsity Student Athletes || Eating Disorder Inventory-2 (EDI-2)
|-
| Non-clinical; Military&lt;ref&gt;McNulty, PAF. (2001). Prevalence and contributing factors of eating disorder behaviors in active duty service women in the Army, Navy, Air Force and Marines. Military Medicine, 166(1), 53-58. &lt;/ref&gt; || 1.1% || Active duty females in US Army, Navy, 
Airforce, and Marines 
|| EDI-2
|-
|Non-clinical; Commonwealth of Virginia Mid-Atlantic Twin Registry (MATR)&lt;ref&gt;Kendler KS, Walters EE, Neale MC, Kessler R, Heath A, Eaves L. The structure of genetic and environmental risk factors for six major psychiatric disorders in women. Archives of general psychiatry. 1995;52:374–383.&lt;/ref&gt;
|1.62% (narrow), 3.70% (broad)
|US Caucasian female same-sex twins
|Structured Clinical Interview for DSM Disorders (SCID)
|-
|Non-clinical; Health Omnibus Survey (HOS)
|.46% (3 months; combined)
|South Australian older adolescents and adults
|Eating Disorder Examination (EDE)
|-
|Military&lt;ref&gt;{{Cite journal|title=Diagnosed eating disorders in the U.S. Military: a nine year review|url=http://www.ncbi.nlm.nih.gov/pubmed/18821361|journal=Eating Disorders|date=2008-12-01|issn=1532-530X|pmid=18821361|pages=363–377|volume=16|issue=5|doi=10.1080/10640260802370523|first=Amanda J.|last=Antczak|first2=Teresa L.|last2=Brininger}}&lt;/ref&gt;
|.04% (combined)
|US Military
|ICD codes from electronic records
|-
|Non-clinical; Military
|.02% (7 years) Female, 
0.0% (7 years) Males
|US Military Academy cadets
|[https://www.eat-26.com/ Eating Attitudes Test- 26 items] (EAT-26)&lt;ref&gt;{{Cite web|url=https://www.eat-26.com/|title=EAT-26: Eating Attitudes Test &amp; Eating Disorder Testing – Use the EAT-26 to help you determine if you need to speak to a mental health professional to get help for an eating disorder.|last=admin|language=en|access-date=2022-05-30}}&lt;/ref&gt;
|-
|Non-clinical; Military
|1.1% (current &amp; past) Female
|US Navy female nurses
|DSM-III
|-
|Non-clinical; Military
|.04% Female, .005% Males
|US veterans
|ICD-9-CM
|-
|Non-clinical; Military
|2.5% Males
|Active duty males in US Navy
|N/A
|-
|Clinical; Collaborative Study on the Genetics of Alcoholism (COGA)&lt;ref&gt;{{Cite journal|title=Anorexia nervosa and bulimia nervosa in alcohol-dependent men and women and their relatives|url=http://www.ncbi.nlm.nih.gov/pubmed/8540597|journal=The American Journal of Psychiatry|date=1996-01-01|issn=0002-953X|pmid=8540597|pages=74–82|volume=153|issue=1|doi=10.1176/ajp.153.1.74|first=M. A.|last=Schuckit|first2=J. E.|last2=Tipp|first3=R. M.|last3=Anthenelli|first4=K. K.|last4=Bucholz|first5=V. M.|last5=Hesselbrock|first6=J. I.|last6=Nurnberger}}&lt;/ref&gt;
|1.41% (lifetime) Female,
.00% (lifetime) Male
|US alcohol-dependent adults from San Diego, St. Lois, Iowa City, Farmington, New York, &amp; Indianapolis
|SSAGA
|-
|Non-clinical; healthcare members
|.0269% (current)
|Healthcare provider records
|
|-
|Non-clinical; high school students
|. 00% (point), .45% (lifetime) Female
.00% (point), .00% (lifetime) Males
|US high school students in west central Oregon
|DSM-III-R&lt;sub&gt;4&lt;/sub&gt;
|-
|Clinical; substance users
|.4% (lifetime), .3% (current) Female
.4% (lifetime), .3% (current) Males
|Canadian treatment-seeking substance users
|DIS&lt;sub&gt;9&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Europe'''
|-
|Non-clinical; adolescents
|.3% Female
|Adolescent females residing in Navarra, Spain
|EAT-40&lt;sub&gt;3&lt;/sub&gt;
|-
|Non-clinical; adolescents
|.7% (lifetime) Female, .2% Male
|Adolescents in secondary schools in Sør-Trøndelag, County in Norway
|SEDs&lt;sub&gt;10&lt;/sub&gt;
|-
|Non-clinical; adolescents
|.7% (point; age 15), 1.8% (lifetime, age 15), .00% (point, age 18), 2.6% (lifetime, age 18), .9% (3 years) Female
.00% (point &amp; lifetime) Males
|Adolescents in a comprehensive school in Ostrobothnia district in Finland
|RAB-T&lt;sub&gt;11&lt;/sub&gt;&amp;nbsp;&amp; RAB-R&lt;sub&gt;12&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Australia'''
|-
|Non-clinical; adolescents
|.00% (full), 1.8% (partial) Female
|Adolescent females residing in Victoria, Australia
|BET&lt;sub&gt;13&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''Central &amp; South America'''
|-
|Non-clinical; college students
|.00% Female
|Mexican first &amp; second year college females
|EAT-40&lt;sub&gt;3&lt;/sub&gt;
|-
| colspan=&quot;4&quot; style=&quot;font-size:110%; text-align:center;&quot; |'''East Asia'''
|-
|Clinical; eating disorder patients
|.53% Female
|Adolescent and adult Japanese patients at a university hospital
|DSM-III-R&lt;sub&gt;4&lt;/sub&gt;
|-
|Non-clinical; Korean Epidemiologic Catchment Area (KECA) Study
|.1% (lifetime), .1% (12 months) Female
.2% (lifetime), .00% (12 months)

Males
|Korean adults
|K-CIDI&lt;sub&gt;15&lt;/sub&gt;&amp;nbsp;2.1
|-
|Centers participating in the Collaborative Study on the Genetics of  Alcoholism in San Diego; St. Louis; Iowa City; Farmington, CN; New York; &amp; Indianapolis
|1.41% Females
|Alcohol-dependent adults
|Semi-Structured Assessment for the Genetics of Alcoholism; criteria based on DSM-III-R
|}

==[[Evidence based assessment/Prediction phase|'''Prediction phase''']]==

=== Psychometric properties of screening instruments for anorexia nervosa ===
The following section contains a list of screening and diagnostic instruments for anorexia nerova. The section includes administration information, psychometric data, and PDFs or links to the screenings. 
* Screenings are used as part of the [[Evidence based assessment/Prediction phase|prediction phase]] of assessment; for more information on interpretation of this data, or how screenings fit in to the assessment process, click [[Evidence based assessment/Prediction phase|here.]]

* ''For a list of more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Psychometric_properties_of_common_screening_instruments '''click here.''']''

{| class=&quot;wikitable&quot;
|-
! Measure 
!Format (Reporter)
!Age Range
!Administration/
Completion Time
!Inter-rater reliability
!Test-retest reliability
!Construct validity
!Content validity
!Highly recommended
|-
|Structured Clinical Interview Diagnosis for DSM-IV (SCID-IV) Module H

* [https://www.columbiapsychiatry.org/research/research-areas/services-policy-and-law/structured-clinical-interview-dsm-disorders-scid-1 SCID-IV (for purchase)]
* [[wikipedia:Structured_Clinical_Interview_for_DSM-IV|Wiki Page]]
|Interview
(Patient)
|
|
|G
|L
|A
|E
|X
|-
|Eating Disorder Examination (EDE)

*OSF
*[[wikipedia:Eating_Disorder_Examination_Interview|Wiki Page]]
|Interview
(Patient)
|
|45min-1.25hrs
|E
|A
|A
|E
|X
|-
|[http://www.corc.uk.net/media/1273/ede-q_quesionnaire.pdf EDE-Q (Eating Disorder Examination Questionnaire)]
*OSF
*[[wikipedia:Eating_Disorder_Examination_Interview|Wiki Page]]
|Questionnaire

(Patient)
|
|15-20 min
|NA
|L
|U
|E
|
|-
|[http://www.ori.org/files/Static%20Page%20Files/EDDSDSM-5_10_14.pdf EDDS (Eating Disorder Diagnosis Scale)]
*OSF
*[[wikipedia:Eating_Disorder_Diagnostic_Scale|Wiki Page]]
|Questionnaire 
(Patient)
|
|10-15 min
|NA
|A
|A
|G
|X
|}
&lt;ref&gt;Hunsley, J., &amp; Mash, E. J. (2008). Guide to Assessments that Work. Cary, NC, USA: Oxford University Press, USA. Retrieved from &lt;nowiki&gt;http://www.ebrary.com&lt;/nowiki&gt;&lt;/ref&gt;'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Likelihood ratios and AUCs of screening instruments for anorexia nervosa===
* ''For a list of the likelihood ratios for more broadly reaching screening instruments, [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prediction_phase&amp;wteswitched=1#Likelihood_ratios_and_AUCs_of_common_screening_instruments '''click here.''']''
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
|-
! &lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Mintz|first=L. B.|last2=O'Halloran|first2=M. S.|date=June 2000|title=The Eating Attitudes Test: validation with DSM-IV eating disorder criteria|url=https://www.ncbi.nlm.nih.gov/pubmed/10900574|journal=Journal of Personality Assessment|volume=74|issue=3|pages=489–503|doi=10.1207/S15327752JPA7403_11|issn=0022-3891|pmid=10900574}}&lt;/ref&gt;Screening Measure (Primary Reference)
! AUC
! DiLR+ (Score)
! DiLR- (Score)
!Clinical Generalizability
!Download
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''Biological &amp; Physical Measures'''
|
|-
|Serum leptin level&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Föcker|first=M.|last2=Timmesfeld|first2=N.|last3=Scherag|first3=S.|last4=Bühren|first4=K.|last5=Langkamp|first5=M.|last6=Dempfle|first6=A.|last7=Sheridan|first7=E. M.|last8=Zwaan|first8=M. de|last9=Fleischhaker|first9=C.|date=2011-04-01|title=Screening for anorexia nervosa via measurement of serum leptin levels|url=https://link.springer.com/article/10.1007/s00702-010-0551-z|journal=Journal of Neural Transmission|language=en|volume=118|issue=4|pages=571–578|doi=10.1007/s00702-010-0551-z|issn=0300-9564}}&lt;/ref&gt;
|0.984 (N=139)
|14.72 (&lt;2.31)
|0.10 (2.31+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
|Serum leptin standard deviation score (SDS)&lt;ref name=&quot;:7&quot; /&gt;
|0.939 (N=139)
|5.89 (&lt;-0.45)
|0.09 (-0.45+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
|Body Mass Index (BMI)&lt;ref name=&quot;:7&quot; /&gt;
|0.936 (N=139)
|5.89 (&lt;17.10)
|0.11 (17.10+)
|Adolescent and adult patients in the acute phase of AN according to the DSM-IV and no AN pretreatment versus healthy lean female volunteer university students.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''REDS-C&lt;sub&gt;1&lt;/sub&gt;'''
|
|-
|REDS-C&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Boraska|first=V.|last2=Franklin|first2=C. S.|last3=Floyd|first3=J. a. B.|last4=Thornton|first4=L. M.|last5=Huckins|first5=L. M.|last6=Southam|first6=L.|last7=Rayner|first7=N. W.|last8=Tachmazidou|first8=I.|last9=Klump|first9=K. L.|date=October 2014|title=A genome-wide association study of anorexia nervosa|url=https://www.ncbi.nlm.nih.gov/pubmed/24514567|journal=Molecular Psychiatry|volume=19|issue=10|pages=1085–1094|doi=10.1038/mp.2013.187|issn=1476-5578|pmc=PMC4325090|pmid=24514567}}&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt;
|0.658 (N=333), 0.624 (N=236, &gt; 13 years old), 0.772 (N=97, ≤ 13 years old)
| --
| --
|Children and adolescents (8-18 years old) at a clinic being treated for eating and weight concerns. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDI&lt;sub&gt;2&lt;/sub&gt;'''
|
|-
|EDI-2 &lt;ref name=&quot;:3&quot;&gt;Garner, D. M. (1991). ''Manual for Eating Disorder Inventory.'' Odessa, FL: Psychological Assessment Resources, Inc.&lt;/ref&gt;&lt;ref name=&quot;:8&quot; /&gt; 
|0.556 (N=77)
| --
| --
|Children and adolescents (10-18 years old) at a clinic being treated for eating and weight concerns. Eating disorders were not separated.
|
|-
|EDI-2 - Drive for Thinness Sub-scale&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Segura-Garcia|first=Cristina|last2=Ramacciotti|first2=Carla|last3=Rania|first3=Marianna|last4=Aloi|first4=Matteo|last5=Caroleo|first5=Mariarita|last6=Bruni|first6=Antonella|last7=Gazzarrini|first7=Denise|last8=Sinopoli|first8=Flora|last9=De Fazio|first9=Pasquale|date=June 2015|title=The prevalence of orthorexia nervosa among eating disorder patients after treatment|url=https://www.ncbi.nlm.nih.gov/pubmed/25543324|journal=Eating and weight disorders: EWD|volume=20|issue=2|pages=161–166|doi=10.1007/s40519-014-0171-y|issn=1590-1262|pmid=25543324}}&lt;/ref&gt; 
|0.97 (N=92)
|29.39 (14+)
|0.274 (DT≤14)
|Italian individuals with a DSM-IV-TR eating disorder versus individuals at risk for eating disorders. Eating disorders were not separated.
|
|-
|EDI-3 - Drive for Thinness Subscale (Garner, 2004)

Lehmann et al., 2013
|.903 (N=1298)
| --
| --
|&amp;nbsp;
|
|-
|EDI-3 - Eating Disorder Risk Composite &lt;ref name=&quot;:9&quot; /&gt;(EDRC) (Garner, 2004)
|0.942 (N=92)
|52.08 (EDRC 75+, adults; 90+ adolescents)
|0.379 (EDRC≤75, adults; ≤90 adolescents)
|Italian individuals with a DSM-IV-TR eating disorder versus individuals at risk for eating disorders. Eating disorders were not separated.
|
|-
|EDI-3 - Interoceptive Deficits Subscale &lt;ref name=&quot;:10&quot;&gt;{{Cite journal|last=Clausen|first=Loa|last2=Rosenvinge|first2=Jan H.|last3=Friborg|first3=Oddgeir|last4=Rokkedal|first4=Kristian|date=2011-03-01|title=Validating the Eating Disorder Inventory-3 (EDI-3): A Comparison Between 561 Female Eating Disorders Patients and 878 Females from the General Population|url=https://link.springer.com/article/10.1007/s10862-010-9207-4|journal=Journal of Psychopathology and Behavioral Assessment|language=en|volume=33|issue=1|pages=101–110|doi=10.1007/s10862-010-9207-4|issn=0882-2689}}&lt;/ref&gt;(Garner, 2004)
|.911 (N=2561)
|4.0 (9+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interoceptive Deficits Subscale (Garner, 2004)

Lehmann et al., 2013
|.901 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Asceticism Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.886 (N=2561)
|6.5 (9+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Asceticism Subscale (Garner, 2004)

Lehmann et al., 2013
|.902 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Low Self-Esteem Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.884 (N=2561)
|5.8 (10+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited fm an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Low Self-Esteem Subscale (Garner, 2004)

Lehmann et al., 2013
|.906 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Personal Alienation Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.88 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Personal Alienation Subscale (Garner, 2004)

Lehmann et al., 2013
|.899 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Emotional Dysregulation Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.81 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Emotional Dysregulation Subscale (Garner, 2004)

Lehmann et al., 2013
|.779 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Interpersonal Alienation Subscale &lt;ref name=&quot;:10&quot; /&gt;(Garner, 2004)
|.79 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interpersonal Alienation Subscale (Garner, 2004)

Lehmann et al., 2013
|.743 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Perfectionism Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.79 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Perfectionism Subscale (Garner, 2004)

Lehmann et al., 2013
|.768 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Maturity Fears Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.77 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Maturity Fears Subscale (Garner, 2004)

Lehmann et al., 2013
|.678 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Bulimia Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.76 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Bulimia Subscale (Garner, 2004)

Lehmann et al., 2013
|.776 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Interpersonal Insecurity Subscale &lt;ref name=&quot;:10&quot; /&gt;(Garner, 2004)
|.76 (N=2561)
| --
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Interpersonal Insecurity Subscale (Garner, 2004)

Lehmann et al., 2013
|.697 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
|EDI-3 - Body Dissatisfaction Subscale&lt;ref name=&quot;:10&quot; /&gt; (Garner, 2004)
|.722 (N=2561)
|1.7 (15+)
| --
|Danish females with a DSM-IV eating disorder diagnosis recruited from an eating disorder center versus non-clinical control individuals from the Danish Civil Registration system.
|
|-
|EDI-3 - Body Dissatisfaction Subscale (Garner, 2004)

Lehmann et al., 2013
|.849 (N=1298)
| --
| --
|Dutch patients recruited from eating disorder specialized clinics versus undergraduate female psychology students from a Dutch university. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EAT&lt;sub&gt;3&lt;/sub&gt;'''
|
|-
|EAT-26&amp;nbsp;&lt;ref name=&quot;:4&quot; /&gt;&lt;ref&gt;{{Cite journal|title = The eating attitudes test: psychometric features and clinical correlates|url = http://www.ncbi.nlm.nih.gov/pubmed/6961471|journal = Psychological Medicine|date = 1982-11-01|issn = 0033-2917|pmid = 6961471|pages = 871–878|volume = 12|issue = 4|first = D. M.|last = Garner|first2 = M. P.|last2 = Olmsted|first3 = Y.|last3 = Bohr|first4 = P. E.|last4 = Garfinkel}}&lt;/ref&gt;
|.90 (N=129)
|12.83 (20+)
|.24 (&lt;20)
|Low-moderate: College women with no eating disorder versus college women with a DSM-IV eating disorder. Eating disorders were not separated.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDE-Q&lt;sub&gt;4&lt;/sub&gt;'''
|
|-
|EDE-Q&amp;nbsp;&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|last=Aardoom|first=Jiska J.|last2=Dingemans|first2=Alexandra E.|last3=Slof Op't Landt|first3=Margarita C. T.|last4=Van Furth|first4=Eric F.|date=December 2012|title=Norms and discriminative validity of the Eating Disorder Examination Questionnaire (EDE-Q)|url=https://www.ncbi.nlm.nih.gov/pubmed/23121779|journal=Eating Behaviors|volume=13|issue=4|pages=305–309|doi=10.1016/j.eatbeh.2012.09.002|issn=1873-7358|pmid=23121779}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|title = Assessment of eating disorders: interview or self-report questionnaire?|url = http://www.ncbi.nlm.nih.gov/pubmed/7866415|journal = The International Journal of Eating Disorders|date = 1994-12-01|issn = 0276-3478|pmid = 7866415|pages = 363–370|volume = 16|issue = 4|first = C. G.|last = Fairburn|first2 = S. J.|last2 = Beglin}}&lt;/ref&gt;
|.96 (N=1170)
| --
| --
|Moderate: Dutch treatment-seeking females meeting DSM-IV criteria for an eating disorder versus female adult general population sample recruited through advertisements and personal contacts. Eating disorders were not separated.
|
|-
|EDE-Q&amp;nbsp;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Fairburn|first=C. G.|last2=Beglin|first2=S. J.|date=December 1994|title=Assessment of eating disorders: interview or self-report questionnaire?|url=https://www.ncbi.nlm.nih.gov/pubmed/7866415|journal=The International Journal of Eating Disorders|volume=16|issue=4|pages=363–370|issn=0276-3478|pmid=7866415}}&lt;/ref&gt;
| --
|6.57 (2.3+)
|0.09 (&lt;2.3)
|Moderate: “Clinically significant eating disorder” from a community sample versus female adultsindividuals without “clinically significant eating disorder” from same sample. Eating disorders were not separated.
|
|-
|EDE-Q &lt;ref name=&quot;:1&quot; /&gt;&lt;ref&gt;Rø, Ø., Reas, D. L., &amp; Stedal, K. (2015). Eating Disorder Examination Questionnaire (EDE‐Q) in Norwegian Adults: Discrimination between Female Controls and Eating Disorder Patients. ''European Eating Disorders Review'', ''23''(5), 408-412.&lt;/ref&gt;
|0.89 (N=2465)
| --
| --
|Norwegian adult females with an eating disorder recruited from eating disorder specialist centers versus controls determined using the DSM-IV.
|
|-
| colspan=&quot;5&quot; style=&quot;font-size:110%; text-align:center;&quot; span |'''EDQ-O&lt;sub&gt;5&lt;/sub&gt;'''
|
|-
|EDQ-Q&amp;nbsp;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;
|0.72 (N=134)
|1.00 (all criteria met)
|0.92 (≥1 criteria not met)
|Dutch adults recruited from an eating disorder specialist center determined using the DSM-IV-TR.
|
|}

=== Interpreting anorexia nervosa screening measure scores ===
* For information on interpreting screening measure scores, click [[Evidence based assessment/Prediction phase#Interpreting screening measure scores|here.]]
* Also see the page on [https://en.wikipedia.org/wiki/Likelihood_ratios_in_diagnostic_testing likelihood ratios in diagnostic testing] for more information.

==[[Evidence based assessment/Prescription phase|'''Prescription phase''']]==
===Gold standard diagnostic interviews===

* For a list of broad reaching diagnostic interviews sortable by disorder with PDFs (if applicable), [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Prescription_phase&amp;wteswitched=1#Common_Diagnostic_Interviews click here.]

===Recommended diagnostic interviews for anorexia nervosa===
{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
! colspan=&quot;10&quot; |Diagnostic instruments for anorexia nervosa
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
! Interrater Reliability
! Test-Retest Reliability
! Construct Validity
! Content Validity
! Highly Recommended
!Free and Accessible Measures
|-
| [https://dawba.info/a0.html Development and Well-Being Assessment (DAWBA)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
| Structured; child/adolescent and parent versions
| 11-18
|10-20 minutes for the eating disorder module
|
|
|
|
|
|
*
|-
|[https://www.phenxtoolkit.org/protocols/view/230101 Children's Eating Disorder Examination- Child (ChEDE)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
|Semistructured; child version
|8-16 
|45-75 minutes; mean 1 hour
|E
|
|
|
|
|
|-
|[https://www.credo-oxford.com/7.2.html Eating Disorder Examination (EDE)] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt; &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/314222270|title=A guide to assessments that work|date=2008|publisher=Oxford University Press|others=John Hunsley, Eric J. Mash|isbn=0-19-804245-0|location=New York|oclc=314222270}}&lt;/ref&gt;
|Semistructured; adolescent version
|16+
|45-75 minutes
|E
|A
|A
|E
|X
|
|-
|[https://mccaed.slam.nhs.uk/wp-content/uploads/2022/02/PARDI-Self-14-V.2.1-Oct-2020.pdf Pica, ARFID, and Rumination Disorder Interview] &lt;ref&gt;{{Cite book|url=https://www.worldcat.org/oclc/1160972551|title=Assessment of disorders in childhood and adolescence|date=2020|others=Eric Arden Youngstrom, Mitchell J. Prinstein, Eric J. Mash, Russell A. Barkley|isbn=978-1-4625-4365-6|edition=Fifth edition|location=New York|oclc=1160972551}}&lt;/ref&gt;
|Semistructured; multi-informant and different versions
|2-22
|Mean of 39 minutes
|G (ARFID diagnosis)
|
|
|
|
|
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

===Severity interviews for anorexia nervosa===
{| class=&quot;wikitable&quot; border=&quot;1&quot;
|-
! Measure
! Format (Reporter)
! Age Range
! Administration/
Completion Time
!Free and Accessible Measures
|-
| Placeholder example (CDRS-R)
| Structured Interview&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Mayes|first=Taryn L.|last2=Bernstein|first2=Ira H.|last3=Haley|first3=Charlotte L.|last4=Kennard|first4=Betsy D.|last5=Emslie|first5=Graham J.|date=2010-12|title=Psychometric Properties of the Children's Depression Rating Scale–Revised in Adolescents|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3003451/|journal=Journal of Child and Adolescent Psychopharmacology|volume=20|issue=6|pages=513–516|doi=10.1089/cap.2010.0063|issn=1044-5463|pmc=PMC3003451|pmid=21186970}}&lt;/ref&gt;
| 6-12
| 15-20 minutes
|
* Link to purchase [https://www.wpspublish.com/store/p/2703/cdrs-r-children-s-depression-rating-scale-revised#purchase-product CDRS-R]
*[http://www.opapc.com/uploads/documents/CDRS-R.pdf PDF] (excerpt)
|}
'''Note:''' '''L''' = Less than adequate; '''A''' = Adequate; '''G''' = Good; '''E''' = Excellent; '''U''' = Unavailable; '''NA''' = Not applicable

==[[Evidence based assessment/Process phase|'''Process phase''']]==
The following section contains a brief overview of treatment options for anorexia nervosa and a list of process and outcome measures for anorexia nervosa. The section includes benchmarks based on published norms for several outcome and severity measures, as well as information about commonly used process measures. Process and outcome measures are used as part of the [[Evidence based assessment/Process phase|process phase]] of assessment. For more information on the differences between process and outcome measures, see the page on the [[Evidence based assessment/Process phase|process phase of assessment]].

===Process measures===
Body weight is commonly monitored by clinicians throughout the AN treatment process as helping individuals regain and maintain a healthy weight is a primary treatment goal for AN. Many treatment centers have policies prohibiting patients from seeing their weight.

Motivational Stages of Change may be used to monitor individuals’ readiness to take action against eating disorder behaviors. It has demonstrated predictive validity in a sample of female adolescents attending eating disorder treatment groups.&lt;ref&gt;{{Cite journal|last=Gusella|first=Joanne|last2=Butler|first2=Gordon|last3=Nichols|first3=Laura|last4=Bird|first4=Debbie|date=2003-01-01|title=A brief questionnaire to assess readiness to change in adolescents with eating disorders: its applications to group therapy|url=http://onlinelibrary.wiley.com/doi/10.1002/erv.481/abstract|journal=European Eating Disorders Review|language=en|volume=11|issue=1|pages=58–71|doi=10.1002/erv.481|issn=1099-0968}}&lt;/ref&gt; See Appendix E.

=== Outcome and severity measures ===
This table includes clinically significant benchmarks for anorexia nervosa specific outcome measures
* Information on how to interpret this table can be [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase found here].
* Additionally, these [[Evidence based assessment/Vignettes|vignettes]] might be helpful resources for understanding appropriate adaptation of outcome measures in practice.
*''&lt;u&gt;For clinically significant change benchmarks for the CBCL, YSR, and TRF total, externalizing, internalizing, and attention benchmarks,&lt;/u&gt;'' [https://en.wikiversity.org/w/index.php?title=Evidence_based_assessment/Process_phase&amp;wteswitched=1#Clinically_significant_change_benchmarks_for_widely-used_outcome_measures see here.]

{| class=&quot;wikitable sortable&quot; border=&quot;1&quot;
| colspan=&quot;8&quot; |'''Clinically significant change benchmarks with common instruments for anorexia nervosa'''
|-
| colspan=&quot;8&quot;  span style=&quot;font-size:110%; text-align:center;&quot; | &lt;b&gt; Benchmarks Based on Published Norms&lt;/b&gt;
|-
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; |&lt;b&gt; Measure&lt;/b&gt;
| rowspan=&quot;2&quot; style=&quot;text-align:center;font-size:130%;&quot; | &lt;b&gt; Subscale&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:130%&quot; width=&quot;300&quot; | &lt;b&gt; Cut-off scores&lt;/b&gt;
| colspan=&quot;3&quot; style=&quot;text-align:center;font-size:120%&quot; | &lt;b&gt; Critical Change &lt;br&gt; (unstandardized scores)&lt;/b&gt;
|-
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; A&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; B&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; C&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 95%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; 90%&lt;/b&gt;
| style=&quot;text-align:center;font-size:110%&quot; |&lt;b&gt; SE&lt;sub&gt;difference&lt;/sub&gt;&lt;/b&gt;
|-
| rowspan=&quot;5&quot; style=&quot;text-align:center;&quot; |'''EDE-Q &lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Mond|first=J. M.|last2=Hay|first2=P. J.|last3=Rodgers|first3=B.|last4=Owen|first4=C.|last5=Beumont|first5=P. J. V.|date=May 2004|title=Validity of the Eating Disorder Examination Questionnaire (EDE-Q) in screening for eating disorders in community samples|url=https://www.ncbi.nlm.nih.gov/pubmed/15033501|journal=Behaviour Research and Therapy|volume=42|issue=5|pages=551–567|doi=10.1016/S0005-7967(03)00161-X|issn=0005-7967|pmid=15033501}}&lt;/ref&gt;'''
| style=&quot;text-align:right;&quot; |''Global''
|1.4
|3.2
|2.3
|.7
|.6
|.3
|-
| style=&quot;text-align:right;&quot; |''Restraint''
|(-.3)
|3.6
|1.8
|1.5
|1.2
|.8
|-
| style=&quot;text-align:right;&quot; |''Eating Concern''
|.1
|2.0
|1.2
|1.1
|.9
|.6
|-
| style=&quot;text-align:right;&quot; |''Weight Concern''
|1.5
|3.9
|2.6
|1.0
|.9
|.5
|-
| style=&quot;text-align:right;&quot; |''Shape Concern''
|2.1
|4.8
|3.2
|.9
|.7
|.4
|-
| rowspan=&quot;5&quot; style=&quot;text-align:center;&quot; |'''EDE &lt;ref name=&quot;:2&quot; /&gt;'''
| style=&quot;text-align:right;&quot; |''Global''
|1.7
|2.3
|1.9
|1.9
|1.6
|1.0
|-
| style=&quot;text-align:right;&quot; |''Restraint''
|.3
|3.3
|1.9
|1.8
|1.5
|.9
|-
| style=&quot;text-align:right;&quot; |''Eating Concern''
|(-.5)
|.9
|.5
|.8
|.7
|.4
|-
| style=&quot;text-align:right;&quot; |''Weight Concern''
|2.0
|2.8
|2.4
|1.3
|1.1
|.7
|-
| style=&quot;text-align:right;&quot; |''Shape Concern''
|2.0
|3.2
|2.6
|1.2
|1.0
|.6
|-
| rowspan=&quot;1&quot; style=&quot;text-align:center;&quot; |'''EAT-26 &lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Mintz|first=L. B.|last2=O'Halloran|first2=M. S.|date=2000-06-01|title=The Eating Attitudes Test: validation with DSM-IV eating disorder criteria|url=http://www.ncbi.nlm.nih.gov/pubmed/10900574|journal=Journal of Personality Assessment|volume=74|issue=3|pages=489–503|doi=10.1207/S15327752JPA7403_11|issn=0022-3891|pmid=10900574}}&lt;/ref&gt;'''
| style=&quot;text-align:right;&quot; |''Total''
|6.5
|19.6
|15.0
|7.9
|6.7
|4.0
|}

'''Note:''' “A” = Away from the clinical range, “B” = Back into the nonclinical range, “C” = Closer to the nonclinical than clinical mean.

'''Search terms:''' [Anorexia Nervosa OR eating disorder] AND [validity OR clinical significance] in Google Scholar

=== Treatment ===
{{collapse top| Click here for more information on treatment for AN}}
* Treatment of AN typically consists of restoring the individual to a healthy weight and addressing thoughts and behaviors which are related to the eating disorder. It may involve re-nutrition, psychotherapy, nutritional counseling, and medication. 
* Literature reviews of existing research indicate that evidence supporting AN treatment is lacking. A systematic review of AN treatment efficacy studies by Bulik&lt;ref&gt;{{Cite journal|title = Anorexia nervosa treatment: a systematic review of randomized controlled trials|url = http://www.ncbi.nlm.nih.gov/pubmed/17370290|journal = The International Journal of Eating Disorders|date = 2007-05-01|issn = 0276-3478|pmid = 17370290|pages = 310–320|volume = 40|issue = 4|doi = 10.1002/eat.20367|first = Cynthia M.|last = Bulik|first2 = Nancy D.|last2 = Berkman|first3 = Kimberly A.|last3 = Brownley|first4 = Jan A.|last4 = Sedway|first5 = Kathleen N.|last5 = Lohr}}&lt;/ref&gt; found that evidence supporting medications, medications and behavioral interventions, and behavioral interventions alone in adults is weak. 
* There is moderately strong evidence suggesting that behavioral interventions may be helpful for adolescents. In particular, adolescents may benefit from family therapy. 
* Clinical trials investigating AN treatment suffer from high rates of attrition, as key features of AN (e.g., denial, fear of weight gain) may contribute to low motivation for remaining in treatment.
* More severe cases of AN may be treated in inpatient settings, which are equipped to manage the re-nutrition process and provide medical monitoring. 
* Partial hospitalization and intensive outpatient programs may provide intermediate levels of treatment intensity to assist individuals in the transition from intensive care to outpatient care after weight restoration.
{{collapse bottom}}
* Please refer to the page on [https://en.wikipedia.org/wiki/Anorexia_nervosa anorexia nervosa] for more information on available treatment or go to [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/eating-body-image-problems/ the Effective Child Therapy page for Eating &amp; Body Image Problems] for a curated resource on effective treatments for anorexia nervosa.

== '''External resources''' ==
# [http://apps.who.int/classifications/icd10/browse/2010/en#/F50.0 ICD-10 diagnostic criteria]
# [https://en.wikiversity.org/w/index.php?title=Helping_Give_Away_Psychological_Science/Resources/Annotated_List_of_Where_and_How_to_Find_a_Therapist&amp;wteswitched=1#Other_low-cost_options Find-a-Therapist] 
##This is a curated list of find-a-therapist websites where you can find a provider
# NIMH: [https://www.nimh.nih.gov/health/publications/eating-disorders/index.shtml ''Eating Disorders--About More Than Food'']  and [https://www.nimh.nih.gov/health/topics/eating-disorders/index.shtml ''Eating Disorders'']
##These NIMH website posts provide more information on anorexia nervosa
#[https://www.hopkinsmedicine.org/psychiatry/specialty_areas/eating_disorders/ John's Hopkins Resource] (guide about anorexia nervosa, treatment, and more)
# OMIM (Online Mendelian Inheritance in Man)
##[https://www.omim.org/entry/606788?search=anorexia%20nervosa&amp;highlight=nervosa%20anorexia Anorexia nervosa]
#[http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/eating-body-image-problems/ Effective Child Therapy page for anorexia nervosa]
##Effective Child Therapy is website sponsored by Division 53 of the American Psychological Association (APA), or The [https://sccap53.org Society of Clinical Child and Adolescent Psychology] (SCCAP), in collaboration with the Association for Behavioral and Cognitive Therapies (ABCT). Use for information on symptoms and available treatments.

== '''References''' ==
{{collapse top|Click here for references}}
{{Reflist|30em}}

[[Category:Psychological disorder portfolios|{{SUBPAGENAME}}]]
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&lt;!-- Template:Welcome --&gt;
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      <comment>New resource with &quot;{{RoundBoxTop|theme=3}} === [[w:Buckminster Fuller|Fuller, R. Buckminster (1895 - 1983)]] === &lt;hr /&gt; '''Notable Accomplishments''' * Synergetic * Tensgresity * Geodesic Dome * Ephemeralization * Dymaxion Car &amp; House &lt;br /&gt;&lt;hr /&gt; '''Publications''' {{User:Jtwsaddress42/Bibliography/Fuller, R. Buckminster}} {{RoundBoxBottom}} &lt;hr /&gt;&quot;</comment>
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=== [[w:Buckminster Fuller|Fuller, R. Buckminster (1895 - 1983)]] ===
&lt;hr /&gt;
'''Notable Accomplishments'''
* Synergetic
* Tensgresity
* Geodesic Dome
* Ephemeralization
* Dymaxion Car &amp; House
&lt;br /&gt;&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Fuller, R. Buckminster}}
{{RoundBoxBottom}}
&lt;hr /&gt;</text>
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=== [[w:Buckminster Fuller|Fuller, R. Buckminster (1895 - 1983)]] ===
&lt;hr /&gt;
[[File:BuckminsterFuller cropped.jpg|thumb|Buckminster Fuller (1895 - 1983)]]
[[File:Biosphère Montréal.jpg|thumb|Biosphère Montréal]]
[[File:Dynamaxion 1933.jpg|thumb|Dynamaxion 1933]]
[[File:Preserved R Buckminster Fuller and Anne Hewlitt Dome Home.jpg|thumb|Preserved R Buckminster Fuller and Anne Hewlitt Dome Home]]
'''Notable Accomplishments'''
* Synergetic
* Tensgresity
* Geodesic Dome
* Ephemeralization
* Dymaxion Car &amp; House
&lt;br /&gt;&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Fuller, R. Buckminster}}
{{RoundBoxBottom}}
&lt;hr /&gt;</text>
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=== [[w:Buckminster Fuller|Fuller, R. Buckminster (1895 - 1983)]] ===
&lt;hr /&gt;
[[File:BuckminsterFuller cropped.jpg|thumb|Buckminster Fuller (1895 - 1983)]]
'''Notable Accomplishments'''
* Synergetic
* Tensgresity
* Geodesic Dome
* Ephemeralization
* Dymaxion Car &amp; House
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|'''The Inventions of Buckminister Fuller'''
{{User:Jtwsaddress42/Gallery/Buckminster Fuller}}
&lt;br /&gt;
|}&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Fuller, R. Buckminster}}
{{RoundBoxBottom}}
&lt;hr /&gt;</text>
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=== [[w:Buckminster Fuller|Fuller, R. Buckminster (1895 - 1983)]] ===
&lt;hr /&gt;
[[File:BuckminsterFuller cropped.jpg|thumb|Buckminster Fuller (1895 - 1983)]]
[[File:BMCRP VM Bx92 SnelsonK Fuller 4cc--Sum1948-WmJoseph-EdeKooning-RBurkhardt (6506084311).jpg|thumb|BMCRP VM Bx92 SnelsonK Fuller 4cc--Sum1948-WmJoseph-EdeKooning-RBurkhardt (6506084311)]]
[[File:BMCRP VM Bx89 Nakagawa Portraits 11ee--Sum1949 (6506073429).jpg|thumb|BMCRP VM Bx89 Nakagawa Portraits 11ee--Sum1949 (6506073429)]]
[[File:R. Buckminster Fuller 1972.jpg|thumb|Spartan Daily (San Jose State University) Wednesday, January 12, 1972]]
[[File:Bucky TRIMTAB.jpg|thumb|Bucky TRIMTAB]]
'''Notable Accomplishments'''
* Synergetics
* Tensgresity
* Ephemeralization
* Dymaxion Car &amp; House
* Geodesic Dome
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|{{User:Jtwsaddress42/Gallery/Buckminster Fuller}}
|}&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Fuller, R. Buckminster}}
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|[[File:Buckminsterfullerene animated.gif|640px|center|Buckminsterfullerene animated]]
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=== [[w:Buckminster Fuller|Fuller, R. Buckminster (1895 - 1983)]] ===
&lt;hr /&gt;
[[File:BuckminsterFuller cropped.jpg|thumb|Buckminster Fuller (1895 - 1983)]]
[[File:BMCRP VM Bx92 SnelsonK Fuller 4cc--Sum1948-WmJoseph-EdeKooning-RBurkhardt (6506084311).jpg|thumb|BMCRP VM Bx92 SnelsonK Fuller 4cc--Sum1948-WmJoseph-EdeKooning-RBurkhardt (6506084311)]]
[[File:BMCRP VM Bx89 Nakagawa Portraits 11ee--Sum1949 (6506073429).jpg|thumb|BMCRP VM Bx89 Nakagawa Portraits 11ee--Sum1949 (6506073429)]]
[[File:Buckminster Fuller teaching at the College of Design.jpg|thumb|Buckminster_Fuller_teaching_at_the_College_of_Design]]
[[File:R. Buckminster Fuller 1972.jpg|thumb|Spartan Daily (San Jose State University) Wednesday, January 12, 1972]]
[[File:Bucky TRIMTAB.jpg|thumb|Bucky TRIMTAB]]
'''Notable Accomplishments'''
* Synergetics
* Tensgresity
* Ephemeralization
* Dymaxion Car &amp; House
* Geodesic Dome
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|{{User:Jtwsaddress42/Gallery/Buckminster Fuller}}
|}&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Fuller, R. Buckminster}}
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|[[File:Buckminsterfullerene animated.gif|480px|center|Buckminsterfullerene animated]]
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{{RoundBoxBottom}}
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=== [[w:Buckminster Fuller|Fuller, R. Buckminster (1895 - 1983)]] ===
&lt;hr /&gt;
[[File:BuckminsterFuller cropped.jpg|thumb|Buckminster Fuller (1895 - 1983)]]
[[File:BMCRP VM Bx92 SnelsonK Fuller 4cc--Sum1948-WmJoseph-EdeKooning-RBurkhardt (6506084311).jpg|thumb|BMCRP VM Bx92 SnelsonK Fuller 4cc--Sum1948-WmJoseph-EdeKooning-RBurkhardt (6506084311)]]
[[File:BMCRP VM Bx89 Nakagawa Portraits 11ee--Sum1949 (6506073429).jpg|thumb|BMCRP VM Bx89 Nakagawa Portraits 11ee--Sum1949 (6506073429)]]
[[File:Buckminster Fuller teaching at the College of Design.jpg|thumb|Buckminster_Fuller_teaching_at_the_College_of_Design]]
[[File:R. Buckminster Fuller 1972.jpg|thumb|Spartan Daily (San Jose State University) Wednesday, January 12, 1972]]
[[File:Bucky TRIMTAB.jpg|thumb|Bucky TRIMTAB]]
'''Notable Accomplishments'''
* Synergetics
* Tensgresity
* Ephemeralization
* Dymaxion Car &amp; House
* Geodesic Dome
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|{User:Jtwsaddress42/Gallery/Buckminster Fuller - The Dymaxion Car}}
|}&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Fuller, R. Buckminster}}
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
{{User:Jtwsaddress42/Gallery/Buckminster Fuller}}
|}&lt;hr /&gt;
{{RoundBoxBottom}}
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=== [[w:Buckminster Fuller|Fuller, R. Buckminster (1895 - 1983)]] ===
&lt;hr /&gt;
[[File:BuckminsterFuller cropped.jpg|thumb|Buckminster Fuller (1895 - 1983)]]
[[File:BMCRP VM Bx92 SnelsonK Fuller 4cc--Sum1948-WmJoseph-EdeKooning-RBurkhardt (6506084311).jpg|thumb|BMCRP VM Bx92 SnelsonK Fuller 4cc--Sum1948-WmJoseph-EdeKooning-RBurkhardt (6506084311)]]
[[File:BMCRP VM Bx89 Nakagawa Portraits 11ee--Sum1949 (6506073429).jpg|thumb|BMCRP VM Bx89 Nakagawa Portraits 11ee--Sum1949 (6506073429)]]
[[File:Buckminster Fuller teaching at the College of Design.jpg|thumb|Buckminster_Fuller_teaching_at_the_College_of_Design]]
[[File:R. Buckminster Fuller 1972.jpg|thumb|Spartan Daily (San Jose State University) Wednesday, January 12, 1972]]
[[File:Bucky TRIMTAB.jpg|thumb|Bucky TRIMTAB]]
'''Notable Accomplishments'''
* Synergetics
* Tensgresity
* Ephemeralization
* Dymaxion Car &amp; House
* Geodesic Dome
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|{{User:Jtwsaddress42/Gallery/Buckminster Fuller - The Dymaxion Car}}
|}&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Fuller, R. Buckminster}}
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
{{User:Jtwsaddress42/Gallery/Buckminster Fuller}}
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{{RoundBoxBottom}}
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=== [[w:Buckminster Fuller|Fuller, R. Buckminster (1895 - 1983)]] ===
&lt;hr /&gt;
[[File:BuckminsterFuller cropped.jpg|thumb|Buckminster Fuller (1895 - 1983)]]
[[File:BMCRP VM Bx92 SnelsonK Fuller 4cc--Sum1948-WmJoseph-EdeKooning-RBurkhardt (6506084311).jpg|thumb|BMCRP VM Bx92 SnelsonK Fuller 4cc--Sum1948-WmJoseph-EdeKooning-RBurkhardt (6506084311)]]
[[File:BMCRP VM Bx89 Nakagawa Portraits 11ee--Sum1949 (6506073429).jpg|thumb|BMCRP VM Bx89 Nakagawa Portraits 11ee--Sum1949 (6506073429)]]
[[File:Buckminster Fuller teaching at the College of Design.jpg|thumb|Buckminster_Fuller_teaching_at_the_College_of_Design]]
[[File:R. Buckminster Fuller 1972.jpg|thumb|Spartan Daily (San Jose State University) Wednesday, January 12, 1972]]
[[File:Bucky TRIMTAB.jpg|thumb|Bucky TRIMTAB]]
'''Notable Accomplishments'''
* Synergetics
* Tensgresity
* Ephemeralization
* Dymaxion Car &amp; House
* Geodesic Dome
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|{{User:Jtwsaddress42/Gallery/Buckminster Fuller - The Dymaxion Car}}
|}&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Fuller, R. Buckminster}}
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|{{User:Jtwsaddress42/Gallery/Buckminster Fuller - The Dymaxion House}}
&lt;br /&gt;&lt;hr /&gt;
{{User:Jtwsaddress42/Gallery/Buckminster Fuller}}
|}&lt;hr /&gt;
{{RoundBoxBottom}}
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=== [[w:Buckminster Fuller|Fuller, R. Buckminster (1895 - 1983)]] ===
&lt;hr /&gt;
[[File:BuckminsterFuller cropped.jpg|thumb|Buckminster Fuller (1895 - 1983)]]
[[File:BMCRP VM Bx92 SnelsonK Fuller 4cc--Sum1948-WmJoseph-EdeKooning-RBurkhardt (6506084311).jpg|thumb|BMCRP VM Bx92 SnelsonK Fuller 4cc--Sum1948-WmJoseph-EdeKooning-RBurkhardt (6506084311)]]
[[File:BMCRP VM Bx89 Nakagawa Portraits 11ee--Sum1949 (6506073429).jpg|thumb|BMCRP VM Bx89 Nakagawa Portraits 11ee--Sum1949 (6506073429)]]
[[File:Buckminster Fuller teaching at the College of Design.jpg|thumb|Buckminster_Fuller_teaching_at_the_College_of_Design]]
[[File:R. Buckminster Fuller 1972.jpg|thumb|Spartan Daily (San Jose State University) Wednesday, January 12, 1972]]
[[File:Bucky TRIMTAB.jpg|thumb|Bucky TRIMTAB]]
'''Notable Accomplishments'''
* Synergetics
* Tensgresity
* Ephemeralization
* Dymaxion Car &amp; House
* Geodesic Dome
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|{{User:Jtwsaddress42/Gallery/Buckminster Fuller - The Dymaxion Car}}
|}&lt;hr /&gt;
'''Publications'''
{{User:Jtwsaddress42/Bibliography/Fuller, R. Buckminster}}
&lt;br /&gt;&lt;hr /&gt;
{| align= center | width= 640px
|{{User:Jtwsaddress42/Gallery/Buckminster Fuller - The Dymaxion House}}
&lt;br /&gt;&lt;hr /&gt;
{{User:Jtwsaddress42/Gallery/Buckminster Fuller - The Biosphère Montréal}}
&lt;br /&gt;&lt;hr /&gt;
{{User:Jtwsaddress42/Gallery/Buckminster Fuller}}
|}&lt;hr /&gt;
{{RoundBoxBottom}}
&lt;hr /&gt;</text>
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 | File:Biosphère Montréal.jpg|alt1= Biosphère Montréal|Biosphère Montréal
 | File:Dynamaxion 1933.jpg|alt2= Dynamaxion 1933|Dynamaxion 1933
 | File:Preserved R Buckminster Fuller and Anne Hewlitt Dome Home.jpg|alt3= Preserved R Buckminster Fuller and Anne Hewlitt Dome Home|Preserved R Buckminster Fuller and Anne Hewlitt Dome Home
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 | title =  ''The Inventions of Buckminster Fuller''
 | width = 100
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 | File:Biosphère Montréal.jpg|alt1= Biosphère Montréal|Biosphère Montréal
 | File:Dymaxion car photo.jpg|alt2= The Dymaxion car, c.1933|The Dymaxion car, c.1933{{efn|Artist Diego Rivera shown entering the car, carrying coat}}
 | File:Dynamaxion 1933.jpg|alt3= A 1933 Dymaxion prototype.|A 1933 Dymaxion prototype.
 | File:Dymaxion House exterior.jpg|alt4= A Dymaxion house at The Henry Ford|A Dymaxion house at The Henry Ford
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|{{Gallery
 | title =  ''The Inventions of Buckminster Fuller''
 | width = 100
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 | align= center
 | File:Biosphère Montréal.jpg|alt1= Biosphère Montréal|Biosphère Montréal
 | File:Dymaxion car photo.jpg|alt2= The Dymaxion car, c.1933|The Dymaxion car, c.1933{{efn|Artist Diego Rivera shown entering the car, carrying coat}}
 | File:Dynamaxion 1933.jpg|alt3= A 1933 Dymaxion prototype.|A 1933 Dymaxion prototype.
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 | File:Dymaxion House exterior.jpg|alt5= A Dymaxion house at The Henry Ford|A Dymaxion house at The Henry Ford
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|{{Gallery
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 | File:Dymaxion car photo.jpg|alt2= The Dymaxion car, c.1933|The Dymaxion car, c.1933{{efn|Artist Diego Rivera shown entering the car, carrying coat}}
 | File:Dynamaxion 1933.jpg|alt3= A 1933 Dymaxion prototype.|A 1933 Dymaxion prototype.
 | File:Dynamixion car by Buckminster Fuller 1933 (side views).jpg|alt4= Dynamixion car (side view)|Dynamixion car (side view)
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 | File:Dynamaxion 1933.jpg|alt2= 1933 Dymaxion prototype|1933 Dymaxion prototype
 | File:Dynamixion car by Buckminster Fuller 1933 (side views).jpg|alt3= 1933 Dynamixion (side view)|1933 Dymaxion (side view)
 | File:1934 Dymaxion (1418436123).jpg|alt4= 1934 Dymaxion|1934 Dymaxion
 | File:Dymaxion House. Buckminster Fuller.jpg|alt5= Dymaxion House|Dymaxion House
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 | File:Preserved R Buckminster Fuller and Anne Hewlitt Dome Home.jpg|alt7= Fuller's home in Carbondale|Fuller's home in Carbondale
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      <comment>New resource with &quot;{| align= center | style=&quot;font-size:85%&quot; |{{Gallery  | title =  ''The Dymaxion Car''  | width = 100  | height = 100  | align= center  | File:Dymaxion car photo.jpg|alt1= The Dymaxion car, c.1933|The Dymaxion car, c.1933{{efn|Artist Diego Rivera shown entering the car, carrying coat}}  | File:Dynamaxion 1933.jpg|alt2= 1933 Dymaxion prototype|1933 Dymaxion prototype  | File:Dynamixion car by Buckminster Fuller 1933 (side views).jpg|alt3= 1933 Dynamixion (side view)|1933 Dy...&quot;</comment>
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 | File:Dynamaxion 1933.jpg|alt2= 1933 Dymaxion prototype|1933 Dymaxion prototype
 | File:Dynamixion car by Buckminster Fuller 1933 (side views).jpg|alt3= 1933 Dynamixion (side view)|1933 Dymaxion (side view)
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      <comment>New resource with &quot;{| align= center | style=&quot;font-size:85%&quot; |{{Gallery  | title =  ''The Dymaxion Car''  | width = 100  | height = 100  | align= center  | File:Dymaxion car photo.jpg|alt1= The Dymaxion car, c.1933|The Dymaxion car, c.1933{{efn|Artist Diego Rivera shown entering the car, carrying coat}}  | File:Dynamaxion 1933.jpg|alt2= 1933 Dymaxion prototype|1933 Dymaxion prototype  | File:Dynamixion car by Buckminster Fuller 1933 (side views).jpg|alt3= 1933 Dynamixion (side view)|1933 Dy...&quot;</comment>
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 | File:Dynamaxion 1933.jpg|alt2= 1933 Dymaxion prototype|1933 Dymaxion prototype
 | File:Dynamixion car by Buckminster Fuller 1933 (side views).jpg|alt3= 1933 Dynamixion (side view)|1933 Dymaxion (side view)
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 | File:Dymaxion car photo.jpg|alt2= The Dymaxion car, c.1933|The Dymaxion car, c.1933{{efn|Artist Diego Rivera shown entering the car, carrying coat}}
 | File:Dynamaxion 1933.jpg|alt3= 1933 Dymaxion prototype|1933 Dymaxion prototype
 | File:1933 Dynamaxion rear.jpg|alt4= 1933 Dymaxion (rear)|1933 Dymaxion (rear)
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 | File:Dynamaxion 1933.jpg|alt3= 1933 Dymaxion prototype|1933 Dymaxion prototype
 | File:1933 Dynamaxion rear.jpg|alt4= 1933 Dymaxion (rear)|1933 Dymaxion (rear)
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 | File:1934 Dymaxion (1418438271).jpg|alt8= 1934 Dymaxion|1934 Dymaxion
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    <title>User:Jtwsaddress42/Gallery/Buckminster Fuller - The Dymaxion House</title>
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      <comment>New resource with &quot;{| align= center | style=&quot;font-size:85%&quot; |{{Gallery  | title =  ''The Dymaxion House''  | width = 100  | height = 100  | align= center  | File:Dymaxion House. Buckminster Fuller.jpg|alt1= Dymaxion House|A Dymaxion House  | File:Dymaxion House exterior.jpg|alt2= A Dymaxion house at The Henry Ford|Dymaxion house at The Henry Ford  | File:Dymaxion structure.jpg|alt3= Dymaxion House Interior|Dymaxion House (interior)  | File:Pat2220482fig2.GIF|alt4= U.S. Patent 2,220,482|U.S...&quot;</comment>
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 | File:Dymaxion House exterior.jpg|alt2= A Dymaxion house at The Henry Ford|Dymaxion house at The Henry Ford
 | File:Dymaxion structure.jpg|alt3= Dymaxion House Interior|Dymaxion House (interior)
 | File:Pat2220482fig2.GIF|alt4= U.S. Patent 2,220,482|U.S. Patent 2,220,482, Prefabricated bathroom, by Richard Buckminster Fuller, issued 1940
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 | File:Dymaxion House. Buckminster Fuller.jpg|alt1= Dymaxion House|A Dymaxion House
 | File:Dymaxion House exterior.jpg|alt2= A Dymaxion house at The Henry Ford|Dymaxion house at The Henry Ford
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      <comment>New resource with &quot;{| align= center | style=&quot;font-size:85%&quot; |{{Gallery  | title =  ''The Biosphère Montréal''  | width = 100  | height = 100  | align= center  | File:Biosphère Montréal.jpg|alt1= Biosphère Montréal|Biosphère Montréal  | File:Mtl. Biosphere in Sept. 2004.jpg|alt2= Biosphere|Biosphere{{efn|September 2004}}  | File:Biosphere - Panorama.jpg|alt3= Biosphere - Panorama|Biosphere - Panorama }} |}&quot;</comment>
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 | File:Biosphère Montréal.jpg|alt1= Biosphère Montréal|Biosphère Montréal
 | File:Mtl. Biosphere in Sept. 2004.jpg|alt2= Biosphere|Biosphere{{efn|September 2004}}
 | File:Biosphere - Panorama.jpg|alt3= Biosphere - Panorama|Biosphere - Panorama
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    <title>History of Topics in Special Relativity/Lorentz transformation (introduction)</title>
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      <comment>New resource with &quot;{{../Lorentz transformation (header)}}  The '''history of Lorentz transformations''' comprises the development of [[w:linear map|w:linear transformations]] forming the [[w:Lorentz group]] or [[w:Poincaré group]] preserving the [[w:Lorentz interval]] &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}&lt;/math&gt; and the [[w:Minkowski inner product]] &lt;math&gt;-x_{0}y_{0}+\cdots+x_{n}y_{n}&lt;/math&gt;.  In [[w:mathematics]], transformations equivalent to what was later known as Lorentz transformations...&quot;</comment>
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      <text bytes="1788" xml:space="preserve">{{../Lorentz transformation (header)}}

The '''history of Lorentz transformations''' comprises the development of [[w:linear map|w:linear transformations]] forming the [[w:Lorentz group]] or [[w:Poincaré group]] preserving the [[w:Lorentz interval]] &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}&lt;/math&gt; and the [[w:Minkowski inner product]] &lt;math&gt;-x_{0}y_{0}+\cdots+x_{n}y_{n}&lt;/math&gt;.

In [[w:mathematics]], transformations equivalent to what was later known as Lorentz transformations in various dimensions were discussed in the 19th century in relation to the theory of [[w:quadratic form]]s, [[w:hyperbolic geometry]], [[w:Möbius geometry]], and [[w:Spherical wave transformation|w:sphere geometry]], which is connected to the fact that the group of [[w:hyperbolic motion|motions in hyperbolic space]], the [[w:Möbius group]] or [[w:projective special linear group]], and the [[w:Spherical wave transformation#Transformation by reciprocal directions|w:Laguerre group]] are [[w:Group isomorphism|w:isomorphic]] to the [[w:Lorentz group]].

In [[w:physics]], Lorentz transformations became known at the beginning of the 20th century, when it was discovered that they exhibit the symmetry of [[w:Maxwell's equations]]. Subsequently, they became fundamental to all of physics, because they formed the basis of [[w:special relativity]] in which they exhibit the symmetry of [[w:Minkowski spacetime]], making the [[w:speed of light]] invariant between different inertial frames. They relate the spacetime coordinates of two arbitrary [[w:inertial frame of reference|w:inertial frames of reference]] with constant relative speed ''v''. In one frame, the position of an event is given by ''x,y,z'' and time ''t'', while in the other frame the same event has coordinates ''x′,y′,z′'' and ''t′''.</text>
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      <text bytes="1853" xml:space="preserve">{{../Lorentz transformation (header)}}

The '''history of Lorentz transformations''' comprises the development of [[w:linear map|w:linear transformations]] forming the [[w:Lorentz group]] or [[w:Poincaré group]] preserving the [[w:Lorentz interval]] &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}&lt;/math&gt; and the [[w:Minkowski inner product]] &lt;math&gt;-x_{0}y_{0}+\cdots+x_{n}y_{n}&lt;/math&gt;.

In [[w:mathematics]], transformations equivalent to what was later known as Lorentz transformations in various dimensions were discussed in the 19th century in relation to the theory of [[w:quadratic form]]s, [[w:hyperbolic geometry]], [[w:Möbius geometry]], and [[w:Spherical wave transformation|w:sphere geometry]], which is connected to the fact that the group of [[w:hyperbolic motion|motions in hyperbolic space]], the [[w:Möbius group]] or [[w:projective special linear group]], and the [[w:Spherical wave transformation#Transformation by reciprocal directions|w:Laguerre group]] are [[w:Group isomorphism|w:isomorphic]] to the [[w:Lorentz group]].

In [[w:physics]], Lorentz transformations became known at the beginning of the 20th century, when it was discovered that they exhibit the symmetry of [[w:Maxwell's equations]]. Subsequently, they became fundamental to all of physics, because they formed the basis of [[w:special relativity]] in which they exhibit the symmetry of [[w:Minkowski spacetime]], making the [[w:speed of light]] invariant between different inertial frames. They relate the spacetime coordinates of two arbitrary [[w:inertial frame of reference|w:inertial frames of reference]] with constant relative speed ''v''. In one frame, the position of an event is given by ''x,y,z'' and time ''t'', while in the other frame the same event has coordinates ''x′,y′,z′'' and ''t′''.

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
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      <text bytes="2151" xml:space="preserve">:&lt;p style=&quot;background-color:Beige;border:1px solid black&quot;&gt;'''Note''': This and the following pages are based on the Wikipedia article [[w:History of Lorentz transformations]]. This extended Wikiversity version presents a long list of historical authors and their formulas in original notation.&lt;/p&gt;
{{../Lorentz transformation (header)}}

The '''history of Lorentz transformations''' comprises the development of [[w:linear map|w:linear transformations]] forming the [[w:Lorentz group]] or [[w:Poincaré group]] preserving the [[w:Lorentz interval]] &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}&lt;/math&gt; and the [[w:Minkowski inner product]] &lt;math&gt;-x_{0}y_{0}+\cdots+x_{n}y_{n}&lt;/math&gt;.

In [[w:mathematics]], transformations equivalent to what was later known as Lorentz transformations in various dimensions were discussed in the 19th century in relation to the theory of [[w:quadratic form]]s, [[w:hyperbolic geometry]], [[w:Möbius geometry]], and [[w:Spherical wave transformation|w:sphere geometry]], which is connected to the fact that the group of [[w:hyperbolic motion|motions in hyperbolic space]], the [[w:Möbius group]] or [[w:projective special linear group]], and the [[w:Spherical wave transformation#Transformation by reciprocal directions|w:Laguerre group]] are [[w:Group isomorphism|w:isomorphic]] to the [[w:Lorentz group]].

In [[w:physics]], Lorentz transformations became known at the beginning of the 20th century, when it was discovered that they exhibit the symmetry of [[w:Maxwell's equations]]. Subsequently, they became fundamental to all of physics, because they formed the basis of [[w:special relativity]] in which they exhibit the symmetry of [[w:Minkowski spacetime]], making the [[w:speed of light]] invariant between different inertial frames. They relate the spacetime coordinates of two arbitrary [[w:inertial frame of reference|w:inertial frames of reference]] with constant relative speed ''v''. In one frame, the position of an event is given by ''x,y,z'' and time ''t'', while in the other frame the same event has coordinates ''x′,y′,z′'' and ''t′''.

[[Category:Special Relativity]]
[[Category:History of Physics]]</text>
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