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# Prime number

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A prime number, or prime for short, is a natural number larger than 1 that has as its only positive divisor?s (factors?) 1 and itself. The first 10 prime numbers are

  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...

If a number greater than one is not a prime number then it is called a composite number.

### Representing natural numbers as products of primes

An important result is the [fundamental theorem of arithmetic]?, which states that every natural number can be written as a product of primes, and in essentially only one way. Primes are thus the "basic building blocks" of the natural numbers. For example, we can write

  23244 = 22 3 13 149.

### How many prime numbers are there?

There are infinitely many prime numbers: the proof of this was dates back to the Greeks, the oldest version of this proof that we know of was by Euclid. It is one of the oldest known proofs by contradiction.

Assume that there is only a finite number of primes. If you multiply all the primes together, and add one, the resulting number, when divided by any of the primes, has a remainder of one. Therefore it cannot be divided by any of what were supposedly all the primes, and it must be another prime, or divisible by a prime that we have omitted from our list of all the primes. We arrive at a contradiction, so our original assumption (that there is a finite number of primes) must be false. So there are an infinite number of primes.

Even though the total number of primes is infinite, one could still ask "how many primes are there below 10,000, or below 100,000?" Questions like this are answered by the prime number theorem.

### Finding prime numbers

An efficient way to compute a list of all the prime numbers up to a given limit is the algorithm called the "Sieve of Eratosthenes?". To find all prime numbers between 1 and n:

1. Write down the numbers 2, 3, 4, ... n. We will eliminate composites by marking them. Initially all numbers are unmarked.
2. Find the smallest number in the list that has not yet been identified as composite or prime. (The first number so found is 2.) Call it m and mark it as prime.
3. Mark all multiples of m (2m, 3m, 4m, ...) as composite.
4. Go back to step 2 and repeat with the next smallest number. Stop if every number in the list has been marked composite or prime.

For a random large number (say, up to a few thousand digits), you can test for primality with Fermats little theorem or the [Miller-Rabin test]?.

[Tests for primality]? would make an article in itself and would also bring in the notion of pseudoprime?s and witnesses?.

### The largest known prime

In August 2001, the largest known prime was 26972593-1. This is a Mersenne prime and was found by a collaborative effort of internet volunteers.

### Open Questions

There are many open questions about prime numbers. For example:

- Can every positive even integer greater than 2 be written as a sum of two primes? (see [Goldbach conjecture]?)
- A twin prime is a pair of primes with difference 2, e.g. 11 and 13. Are there infinitely many twin primes? (see [twin prime conjecture]?)

### Generalizations

The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. In number theory itself, one talks of pseudo primes which are numbers that, by virtue of having passed a certain test, are considered to be probably prime. To model some of the behavior of prime numbers, one defines prime and irreducible polynomials?. More generally, one can define prime and irreducible elements in every [integral domain]?. [Prime ideal]?s are an important tool and object of study in commutative algebra and algebraic geometry.

### See also

- [The prime pages]
- [MacTutor history of prime numbers]

/Talk

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