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arXiv · 2504.14929

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

Armand Noubissie·Alain Togbe·Zhongfeng Zhang
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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}.

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Armand Noubissie, Alain Togbe, Zhongfeng Zhang. 2025-04-21. On the Exponential Diophantine Equation $(a^n-1)(b^n-1)=x^2$. https://arxiv.org/abs/2504.14929

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