Linear equations in primes

It has been known since 1937 that every large odd number is the sum of three primes, and since 1939 that there are infinitely many three-term arithmetic progressions of primes. Thus the equations p_1 + p_2 + p_3 = N and p_1 + p_3 = 2p_2 are relatively well-understood. I will describe recent work with T. Tao in which we have been able to handle systems of more than one linear equation. For example, we are able to count the number of four-term progressions of primes (which satisfy the simultaneous equations p_1 + p_3 = 2p_2, p_2 + p_4 = 2p_3).
The talk will be accessible to a general audience.