On $px^2 + q^{2n}= y^p$ and related Diophantine equations

TL;DR

This paper investigates a class of Diophantine equations involving prime powers, proving non-existence of solutions for certain primes and analyzing specific cases like p=5 using advanced number theory techniques.

Contribution

It establishes new non-existence results for solutions when p ≡ 3 mod 8 and provides detailed analysis for p=5 using linear forms in logarithms and hypergeometric series.

Findings

No solutions when p ≡ 3 mod 8

Solutions analyzed for p=5

Results obtained via advanced Diophantine methods

Abstract

The title equation, where $p>3$ is a prime number $\not\equiv 7 \pmod 8$, $q$ is an odd prime number and $x,y,n$ are positive integers with $x,y$ relatively prime, is studied. When $p\equiv 3\pmod 8$, we prove (Theorem 2.3) that there are no solutions. For $p\not\equiv 3\pmod 8$ the treatment of the equation turns out to be a difficult task. We focus our attention to $p=5$, by reason of an article by F. Abu Muriefah, published in this journal, vol. 128 (2008), 1670-1675. Our main result concerning this special equation is Theorem 1.1, whose proof is based on results around the Diophantine equation $5x^2-4=y^n$ (integer solutions), interesting in themselves, which are exposed in Sections 3 and 4. These last results are obtained by using tools such as Linear Forms in Two Logarithms and Hypergeometric Series.