A Note On the Exponential Diophantine Equation (a^n-1)(b^n-1)=x^2

TL;DR

This paper investigates the solutions of the exponential Diophantine equation (a^n-1)(b^n-1)=x^2, providing new results on solution existence, including specific pairs and parity conditions, extending prior classifications.

Contribution

It extends previous work by proving non-existence of solutions under certain parity conditions and solving the equation for specific pairs of (a,b).

Findings

No solutions when a,b have opposite parity and n>4 with n even.

Solved the equation for specific pairs like (2,50) and (4,49).

Proved no solutions for a related equation when b is even.

Abstract

In 2002, F. Luca and G. Walsh solved the Diophantine equation in the title for all pairs (a,b) such that 1<a<b<101 with some exceptions. There are sixty nine exceptions. In this paper, we give some new results concerning the equation in the title. It is proved that the equation (a^n-1)(b^n-1)=x^2 has no solutions if a,b have opposite parity and n>4 with 2|n. Also, we solved (a^n-1)(b^n-1)=x^2 for the pairs (a,b)=(2,50),(4,49),(12,45),(13,76),(20,77),(28,49), and (45,100). Lastly, we show that when b is even, the equation (a^n-1)(b^(2n)a^n-1)=x^2 has no solutions n,x.