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.
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.
- A prime is a positive number $p$ such that it has exactly two positive divisors, itself and 1.
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 prime
s is like this ...
- Suppose you have 35 marbles. You can arrange them in a row, but you can also arrange them in a rectangle, 7 by 5.
- Suppose you have 28 marbles. You can arrange them in a rectangle, 7 by 4, or 14 by 2.
- Suppose now you have 29 marbles. You can't arrange them in a (non-trivial) rectangle, no matter how hard you try. That's because you can't find two positive whole numbers bigger than 1 that multiply to give 29.
One of the uses of prime numbers
is in cryptography
, especially the RSA Cryptosystem.