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

TL;DR

The article investigates the exponential Diophantine equation $(a^n-1)(b^n-1)=x^2$ for fixed $a,b>1$, delivering two main no-solution results. It uses a Pell-equation framework with $a^n-1=Dy^2$ and $b^n-1=Dz^2$, combined with gcd analysis and primitive-divisor theory to derive stringent parity and divisibility constraints. Specifically, it proves no solutions when $a$ is even and $b$ is a prime with $b\equiv3\pmod{8}$, and, under several congruence conditions on $a$ and $b$, shows that the only possible $n$ is $2$, correcting a previous incomplete argument related to Szalay's work. The results extend Szalay’s findings, clarify the role of Pell-type recurrences in this problem, and illustrate how primitive divisors can rule out larger exponents in exponential Diophantine equations.

Abstract

Let $a$ and $b$ be two distinct fixed positive integers such that $\min \{a,b\}>1.$ First, we correct an oversight from \cite{X-Z}. Then, we show that the equation in the title with $b \equiv 3 \pmod 8$, $b$ prime and $a$ even has no solution in positive integers $n, x$. This generalizes a result of Szalay \cite{L}.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2
• Lemma 3
• Lemma 4
• Lemma 5