Common factors among pairs of consecutive integers
Tsz Ho Chan · International Journal of Number Theory · 2017
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.