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

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 $$c{x}^2+p^{2m}=4y^n,x,y\ge 1,m\ge 0,n\ge 3,\gcd (x,y)=1,\gcd (n,2h(-c))=1,$$ 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.

12 pages. To appear in `Results in Mathematics'