On the exponential Diophantine equation $x^2+p^mq^n=2y^p$

TL;DR

This paper investigates the solvability of a specific exponential Diophantine equation involving prime powers and Lehmer sequences, combining number theory techniques to determine when solutions exist.

Contribution

It introduces a novel approach using primitive divisors of Lehmer sequences to analyze the solvability of the equation involving prime powers.

Findings

Conditions for the existence of solutions are established.

Certain cases are proven to have no solutions.

New connections between Lehmer sequences and exponential Diophantine equations are identified.

Abstract

We study the exponential Diophantine equation $x^2+p^mq^n=2y^p$ in positive integers $x,y,m,n$, and odd primes $p$ and $q$ using primitive divisors of Lehmer sequences in combination with elementary number theory. We discuss the solvability of this equation.