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

### Omar Kihel

Brock University, St. Catharines, Canada### Florian Luca

UNAM, Campus Morelia, Michoacán, Mexico### Alain Togbé

Purdue University North Central, Westville, USA

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## 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.

## Cite this article

Omar Kihel, Florian Luca, Alain Togbé, Variants of the Diophantine equation <var>n</var>! + 1 = <var>y</var><sup>2</sup> . Port. Math. 67 (2010), no. 1, pp. 1–11

DOI 10.4171/PM/1855