Variants of the Diophantine equation $n!+1=y^2$

Abstract

In this note we study variants of the Brocard–Ramanujan Diophantine equation $n!+1=y^2$. For example, Berend and Harmse [1] proved that the equation $n!=y^r(y+1)$ has only finitely many positive integer solutions $(n,y)$ when $r\ge 4$ 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.