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arXiv · 2606.31223

On the exponential Diophantine equation $(a^n+1)(b^n+1)=x^2$

Paulius Virbalas
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Abstract

We study the Diophantine equation $(a^n+1)(b^n+1)=x^2$, which belongs to the family of equations originating from the work of Szalay in 2000. If $a>1$, it is shown that the equation of the title has only one solution in positive integers, when $a$ and $b$ are distinct powers of the same integer $t>1$. Also, a complete description of the solutions is obtained under the assumptions that $a$ and $b$ are coprime and $n$ is even. Several other special cases of the equation are considered, and two conjectures are proposed.

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Paulius Virbalas. 2026-06-30. On the exponential Diophantine equation $(a^n+1)(b^n+1)=x^2$. https://arxiv.org/abs/2606.31223

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