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

Paulius Virbalas

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

Number Theory 2026-06-30 v1

Authors: Paulius Virbalas

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.

Keywords

exponential Diophantine equation Szalay equation integer solutions coprime powers of integers

Cite

@article{arxiv.2606.31223,
  title  = {On the exponential Diophantine equation $(a^n+1)(b^n+1)=x^2$},
  author = {Paulius Virbalas},
  journal= {arXiv preprint arXiv:2606.31223},
  year   = {2026}
}