On the Diophantine equation $dx^2+p^{2a}q^{2b}=4y^p$

TL;DR

This paper thoroughly analyzes the solutions of a specific Diophantine equation involving squares and prime powers, using elementary methods and properties of Lehmer numbers to classify all solutions and explore related variants.

Contribution

It provides a complete characterization of solutions to the equation and applies these results to related equations, employing elementary techniques based on primitive divisors of Lehmer numbers.

Findings

All solutions to the main Diophantine equation are explicitly described.

The methods rely on elementary number theory and properties of Lehmer numbers.

Results are extended to solutions of related equations.

Abstract

We investigate the solvability of the Diophantine equation in the title, where $d>1$ is a square-free integer, $p, q$ are distinct odd primes and $x,y,a,b$ are unknown positive integers with $\gcd(x,y)=1$. We describe all the integer solutions of this equation, and then use the main finding to deduce some results concerning the integers solutions of some of its variants. The methods adopted here are elementary in nature and are primarily based on the existence of the primitive divisors of certain Lehmer numbers.