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

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.

25 pages. Comments welcome