Common factors among pairs of consecutive integers

Abstract

In this paper, we study products of consecutive integers and prove an optimal bound on their greatest common factors. In contrast, we obtain a modest upper bound for the greatest common factor in the perfect square situation. Using this and an upper bound on the size of solutions to hyperelliptic curves, we prove a gap principle when [Formula: see text] divides [Formula: see text] with some additional restrictions. We also obtain a stronger gap principle under the abc conjecture.