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

TL;DR

This paper investigates the diophantine equation involving positive integers and squares, establishing connections to the Erdős–Straus conjecture and providing conditions for solvability of related systems.

Contribution

It introduces new results on the solvability conditions of a specific diophantine equation and links these to the Erdős–Straus conjecture through congruence systems.

Findings

Conditions for the solvability of the diophantine equation.

A link between congruence systems and the Erdős–Straus conjecture.

New insights into the structure of solutions for the equation.

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.