ON THE DIOPHANTINE EQUATION $p^x+(p-2)^y=z^2$
Year: 2026
Type: article
Abstract: In this paper, we investigate non-negative integer solutions $(x, y, z)$ of the Diophantine equation $p^x+(p-2)^y=z^2$, when $p$ is a prime number. Using elementary methods, we prove that if $p \equiv 5(\bmod 8)$, then the solution of the equation is $(x, y, z)=(0,1, \sqrt{p-1})$, where $z=\sqrt{p-1}$ is a positive integer. Moreover, if $p \equiv 7(\bmod 8)$, $p \equiv 11(\bmod 12)$ or $p \equiv 7(\bmod 20)$, then the equation has no solution.
Author: Suton Tadee
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W7166327970