Note on a paper "An Extension of a Theorem of Euler" by Hirata-Kohno et al

TL;DR

This paper extends a theorem related to a specific Diophantine equation involving products of linear terms and perfect squares, providing new insights into the solutions under certain coprimality conditions.

Contribution

It generalizes a previous result by Hirata-Kohno et al. on the Diophantine equation with additional parameters and conditions.

Findings

Extended the theorem to broader cases of the Diophantine equation.

Identified new solution constraints under coprimality conditions.

Enhanced understanding of the structure of solutions for the equation.

Abstract

In this paper we extend a result of Hirata-Kohno, Laishram, Shorey and Tijdeman on the Diophantine equation $n(n+d)...(n+(k-1)d)=by^2,$ where $n,d,k\geq 2$ and $y$ are positive integers such that $\gcd(n,d)=1.$