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

Abstract

Let $a$ and $b$ be two distinct fixed positive integers such that $\min \{a,b\}> 1.$ First, we correct an oversight from [12]. 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 [10].