On the equation $ab(ab-1)-na=\Delta^2$

Abstract

Let $n$ be a positive integer. We study the diophantine equation $ab(ab-1)-na=\Delta^2$, where $a,b$ are positive integers. We also show that if a system of two congruences is soluble, then an equation which is a translation of Erd\H{o}s-Straus conjecture is soluble.