Product of Integers in an Interval, Modulo Squares

Abstract

We prove a conjecture of Irving Kaplansky which asserts that between any pair of consecutive positive squares there is a set of distinct integers whose product is twice a square. Along similar lines, our main theorem asserts that if prime $p$ divides some integer in $[z,z+3\sqrt{z/2}+1)$ (with $z\geq 11$) then there is a set of integers in the interval whose product is $p$ times a square. This is probably best possible, because it seems likely that there are arbitrarily large counterexamples if we shorten the interval to $[z,z+3\sqrt{z/2})$.