Three prime numbers p, q and r, all greater than three, form an arithmetic progression. In other words, q = p + d, and r = q + d = p + 2d, where d is some integer.

Prove that d is divisible by 6.

Extensions:

What happens when I have four primes in arithmetic progression?

What about five?

Can you find five primes in arithmetic progression?

Fun fact: there exist arbitraily long sequences of primes in arithmetic progression! (We don't recommend trying to prove this, it's quite hard. See here for more info: http://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem.)