Product of Integers in an Interval, Modulo Squares

Andrew Granville Department of Mathematics, University of Georgia Athens, GA 30602-7403 and J.L. Selfridge Department of Mathematics, Northern Illinois University De Kalb, IL 60115 Submitted: September 5, 2000; Accepted: January 15, 2001 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 + 3pz=2 + 1) (with z 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 + 3pz=2). AMS Subject Classi cation: 05E05