A problem of Erdős–Graham–Granville–Selfridge on integral points on hyperelliptic curves

Abstract

Abstract Erdős, Graham and Selfridge considered, for each positive integer n, the least value of $t_n$ so that the integers $n+1, n+2, \dots, n+t_n $ contain a subset the product of whose members with n is a square. An open problem posed by Granville concerns the size of $t_n$ , under the assumption of the ABC conjecture. We establish some results on the distribution of $t_n$ , and in the process solve Granville’s problem unconditionally.