Elemente der Mathematik

Remarks on the differences between consecutive primes In problem 654, Journal of Recreational Mathematics, Harry Nelson asks : "What is the most likely difference between consecutive primes?" Here a difference is`most likely' for primes < n if it occurs at least as often as any other difference. For a discussion see J. Rec. Math. 11, 231. We first show that a well-known conjecture of Hardy and Littlewood implies that the most likely difference tends to infinity with n, so that there is no most probable difference independent of n .