Variants of the Diophantine equation <var>n</var>! + 1 = <var>y</var>²

Abstract

In this note we study variants of the Brocard–Ramanujan Diophantine equation n! + 1 = y2. For example, Berend and Harmse [1] proved that the equation n! = yr(y + 1) has only finitely many positive integer solutions (n,y) when r ≥ 4 is a fixed integer. Here we find all the integer solutions of this equation when r = 2, 3 under the additional assumption that y + 1 is square-free or cube-free, respectively.