Gaps in Integer Sequences

Abstract

Let 6,: a, < a2 < ... be an infinite sequence of positive integers, and let dn = an ll an denote the n th gap. For our present purpose we regard any sequence d as interesting if its elements are so irregularly distributed that the gaps do not conform to any apparent pattern. Thus an arithmetic progression (a + bn: n = 0, 1,2,.. . } is uninteresting in our sense because all its gaps are equal to the common difference b, and so too is the sequence of squares {n2: n = 0, 1, 2,... } uninteresting because here the gaps form the arithmetic progression (1 + 2 n: n =0, 1,2,...}. Indeed, no sequence is interesting whose n th term an is given precisely by some algebraic formula, for the same is then true of the gaps dn. Given an interesting sequence (, there are various questions that one may ask about the fluctuation in size of its gaps, and the purpose of this article is to consider some of these questions in the context of three well-known sequences: (i) the primes, (ii) integers that are sums of two squares, and (iii) the squarefree numbers (integers that are products of distinct primes). I shall restrict myself to results that can be proved by elementary and simple arguments, and I shall only mention some of the deeper known results. You will get some impression of the difficulty of the subject if I add that even the deepest results now available fall far short of the likely ultimate truth.