A **prime number** is a number which has precisely two positive factors: one and itself. Prime numbers are generally restricted to the domain of natural numbers. Note that the set of prime numbers excludes one, as one only has one positive factor. In any other algebraic structure with a multiplication operation, and a concept of primes, particularly rings with unity, the multiplicative identity is excluded as prime.

One very important theorem relating to prime numbers is that there are infinitely many primes. This can be proven by contradiction:

- If we assume that there are only a finite number of primes, then we can list them, $ p_1, p_2,...,p_k $. If we multiply all primes together and add one, we get $ P = p_1 * p_2 * ... * p_k + 1 $. Because 1 is only divisible by itself, this new number P is not divisible by any of the primes and is therefore itself a prime number. Therefore, the assumption is false and there must be an infinite number of primes.

## First 40 prime numbers

- 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173