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 \cite{X-Z}. Then, we show that the equation in the title with $b\equiv 3(\mathrm{mod}8)$, $b$ prime and $a$ even has no solution in positive integers $n,x$. This generalizes a result of Szalay \cite{L}.

5 pages