Prime number

A prime number (or simply prime) is a positive integer $p>1$ whose only positive divisors are 1 and itself. Note that $1$ is usually defined as being neither prime nor composite because it is its only factor among the natural numbers.

Identifying primes

The Sieve of Eratosthenes is a relatively quick method for generating a list of the prime numbers.

Importance of Primes

According to the Fundamental Theorem of Arithmetic, there is exactly one unique way to factor a positive integer into a product of primes ( permutations not withstanding). This unique prime factorization plays an important role in solving many kinds of number theory problems.

Famous Primes

Fermat Primes

A Fermat prime is a prime of the form $2^n+1$ . It can easily be shown that for such a number to be prime, n must not have any odd divisor larger than 1 and so must be a power of 2. Therefore all Fermat primes have the form $2^{2^n}+1$ . Fermat checked the cases n=0,1,2,3,4 and conjectured that all such numbers were prime. However, $2^{2^5}+1=641\cdot 6700417$ . In fact, no other Fermat primes have been found.

There is an easy proof of the infinitude of primes based on Fermat numbers (numbers of the form $2^{2^n} + 1$ ). One simply shows that any two Fermat numbers are relatively prime.

Mersenne Primes

A Mersenne prime is a prime of the form $2^n-1$ . For such a number to be prime, n must itself be prime. Compared to other numbers of comparable sizes, Mersenne numbers are easy to check for primality because of the Lucas-Lehmer test.

Twin Primes

Two primes that differ by exactly 2 are known as twin primes. The following are the smallest examples:
3, 5
5, 7
11, 13
17, 19
29, 31
41, 43

It is not known whether or not there are infinitely many pairs of twin primes. This is known as Goldbach's conjecture.

Advanced Definition

When the need arises to include negative divisors, a prime is defined as an integer p whose only divisors are 1, -1, p, and -p. More generally, if R is a unique factorization domain, then an element p of R is a prime if whenever we write $p=ab$ with $a,b\in R$ , then exactly one of a and b is a unit.

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