# Numbers, Infinitely Many Primes

## Infinitely Many Primes

Suppose there is a finite list of primes.
Multiply them together and add 1, giving n.
Now n is not prime -
it is larger than all the primes on our list,
which is suppose to be complete.
So n is composite.
Let p be a prime in the unique factorization of n.
Since p is on the list, it divides n-1, as well as n.
Hence it divides 1, which is impossible.
There are an infinite number of prime numbers.

The gaps between primes can be arbitrarily large.
Recall that n!, or n factorial,
is the product of the first n integers.
Now n!+2 is divisible by 2,
n!+3 is divisible by 3,
and so on through n!+n,
giving n-1 composite numbers in a row.

Primes that are only 2 apart are called twin primes.
Examples include 11 and 13, and 71 and 73.
We believe, but have not yet proved, that there are infinitely many twin primes.