On the exponential Diophantine equation ${\displaystyle p\cdot 3^{x}+p^{y}=z^2}$ with $p$ a prime number

TL;DR

This paper characterizes all non-negative integer solutions to the exponential Diophantine equation p·3^x + p^y = z^2 for prime p, revealing unique solutions for certain p and infinite solutions for p=3.

Contribution

It provides a complete solution classification for the equation depending on the prime p, including unique, infinite, and no solutions cases, using modular arithmetic techniques.

Findings

Unique solution for p ≡ 2 mod 3, p ≠ 2

Infinite solutions when p=3

No solutions for certain p ≡ 1 mod 3

Abstract

In this paper we find non-negative integer solutions for exponential Diophantine equations of the type $p \cdot 3^x+ p^y=z^2,$ where $p$ is a prime number. We prove that such equation has a unique solution $\displaystyle{(x,y,z)=\left(\log_3(p-2), 0, p-1\right)}$ if $2 \neq p \equiv 2 \pmod 3$ and $(x,y,z)=(0,1,2)$ if $p=2$. We also display the infinite solution set of that equation in the case $p=3$. Finally, a brief discussion of the case $p \equiv 1 \pmod 3$ is made, where we display an equation that does not have a non-negative integer solution and leave some open questions. The proofs are based on the use of the properties of the modular arithmetic.