Product of integers in an interval, modulo squares (Q1593851)

For various algorithms for factoring large numbers and some related problems one gradually constructs a set of integers and tries to find efficiently a nonempty subset whose product is a square, before the set gets too large, see [\textit{C. Pomerance}, in Chatterji, S.D. (ed.), Proceedings of the international congress of mathematicians, ICM '94, vol. I, Basel, Birkhäuser, 411--422 (1995; Zbl 0854.11047)]. In the paper under review the authors study variants of this problem. The main result of the paper is the affirmative answer to the following conjecture of Irwin Kaplansky: For every integer \(u\geq 2\), there is a set of integers in the closed interval \([(u-1)^2,u^2]\) whose product is twice a square. More generally, the authors establish that for any real \(z\geq 10.22\) and any prime \(p\) dividing some integer in the interval \(J=[z,z+3\sqrt{z/2}+1)\), the product of some subset of the integers in \(J\) is equal to \(p\) times a square. The paper contains also some other interesting results and discussions on this kind of problems.