# Proof: Primes are 6n +- 1

All primes (except 2 and 3) can be written in the form 6n plus or minus one

**Proof**

Let p be a prime number > 3

Let n be an appropriate integer.

**Case 1**If p = 6n, then p would have a factor of 6 and therefore p could not be prime.

**Case 2**

If p = 6n + 2, then p would have a factor of 2 and therefore p could not be prime.

**Case 3**

If p = 6n + 3, then p would have a factor of 3 and therefore p could not be prime.

**Case 4 **If p = 6n – 2, then p would have a factor of 2 and therefore p could not be prime.

This only leaves p being able to be 6n + 1 or 6n – 1

It’s a proof by exhaustion as there are no other possible cases.

( NB. Converse is not true. 65 = 6 x 11 – 1 and 65 is not prime.)

This proof is taken from the Mathematical Association’s book “Problem Pages 14-19”.

[I’m now considering joining the MA.]