5, 11, 17, 23, 29 is an arithmetic progression of prime numbers. It is conjectured that there are such progressions with arbitrarily many terms (there cannot be infinite sequences because a, a + d, a + 2d,… sooner or later contains a + ad). The longest such known to the authors has first term 4943 and common difference 60 060 with 13 prime terms, but longer ones will have been found by now. In general, additive problems about the multiplicatively defined prime numbers are inevitably difficult (e.g. the Goldbach conjecture and twin primes) and may even be candidates for undecidability. We have been trying to prove a much more general conjecture of Erdos, spurred on by his offer of $3000 to the first proof or disproof of it.