Atkin's Theorem on Pseudo-squares

Abstract

We give an elementary proof of a theorem of A.O.L. Atkin on psuedo-squares. As pointed out by Atkin, from this theorem it immediately follows that there exists an infinite sequence of positive integers, whose $j$~th term $s(j)$ satisfies $s(j)=j^2 + O(\log(j))$, such that the set of integers representable as a sum of two distinct terms of this sequence is of positive asymptotic density.