A problem of Erd\H{o}s-Graham-Granville-Selfridge on integral points on hyperelliptic curves

TL;DR

This paper investigates a problem related to the distribution of integers and squares in sequences, providing unconditional results that address an open question posed by Granville, with implications for understanding integral points on hyperelliptic curves.

Contribution

The authors solve Granville's open problem unconditionally and analyze the distribution of the sequence t_n without relying on the ABC Conjecture.

Findings

Unconditional bounds on the size of t_n.

Distribution results for the sequence t_n.

Resolution of Granville's problem without assuming the ABC Conjecture.

Abstract

Erd\H{o}s, 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.