A Prime Number is a positive whole number that is divisible only by itself and 1. (and their negatives (of course)).
You have to be a bit careful here. The usual definitions are a bit loose. There are two definitions that are much tighter.
  • A prime is a positive number $p$ such that if $p$ divides a product $ab,$ then $p$ divides $a,$ or $p$ divides $b,$ or both.
  • A prime is a positive number $p$ such that it has exactly two positive divisors, itself and 1.
The first of these would mean that 1 is a prime. The second would mean that 1 is not a prime. On every other positive number they agree, and that's why the question of whether or not 1 is a prime is regarded as a matter of convention, or ignored altogether.

For example 2, 3, 5, 7, 11, ...

Euclid in around 300BC proved that there are infinitely many primes. One proof of this (there is more than one) is on the page about Proof By Contradiction.

There are still many things unknown about primes. For example, look at Prime Pairs or Goldbach's Conjecture.

One way of thinking about primes is like this ...

One of the uses of prime numbers is in cryptography, especially the RSA Cryptosystem.
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