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

Armand Noubissie; Alain Togbe; Zhongfeng Zhang

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

Number Theory 2025-04-22 v1

Authors: Armand Noubissie , Alain Togbe , Zhongfeng Zhang

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}.

Keywords

Diophantine equation exponential equation Pell equation number theory

Cite

@article{arxiv.2504.14929,
  title  = {On the Exponential Diophantine Equation $(a^n-1)(b^n-1)=x^2$},
  author = {Armand Noubissie and Alain Togbe and Zhongfeng Zhang},
  journal= {arXiv preprint arXiv:2504.14929},
  year   = {2025}
}

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5 pages

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