On the Diophantine equation $cx^2+p^{2m}=4y^n$

TL;DR

This paper classifies all integer solutions to a specific exponential Diophantine equation involving square-free integers, primes, and class numbers, using primitive divisor results in Lehmer sequences.

Contribution

It provides a complete description of solutions to the equation $cx^2+p^{2m}=4y^n$ under certain conditions, applying advanced number theory techniques.

Findings

Explicit solutions characterized for given parameters

Conditions on $c$, $p$, and $n$ for solutions to exist

Application of primitive divisor theorems to Diophantine equations

Abstract

Let $c$ be a square-free positive integer and $p$ a prime satisfying $p\nmid c$. Let $h(-c)$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-c})$. In this paper, we consider the Diophantine equation $$cx^2+p^{2m}=4y^n,~~x,y\geq 1, m\geq 0, n\geq 3, \gcd(x,y)=1, \gcd(n,2h(-c))=1,$$ and we describe all its integer solutions. Our main tool here is the prominent result of Bilu, Hanrot and Voutier on existence of primitive divisors in Lehmer sequences.