The prime numbers 5,11,17,23,29 form an arithmetic progression of length five, and a back-of-an-envelope calculation will confirm that the 22 numbers 11410337850553 + 4609098694200k, k = 0,...,21, are also all primes. Terry Tao and I have proved that in fact one can find arbitrarily long arithmetic progressions of primes.

In the talk I will describe some famous open problems concerning primes, and some famous open problems concerning arithmetic progressions, either of which would imply this statement. Then I will try and explain how we obtained our result without making the slightest dent in any of these conjectures.