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

Abstract

The title equation, where $p>3$ is a prime number $\not\equiv 7(\mathrm{mod}8)$, $q$ is an odd prime number and $x,y,n$ are positive integers with $x,y$ relatively prime, is studied. When $p\equiv 3(\mathrm{mod}8)$, we prove (Theorem 2.3) that there are no solutions. For $p\not\equiv 3(\mathrm{mod}8)$ 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.

22 pages. In this version, Theorem 3.2 is corrected and relevant remarks are added. A few misprints are also corrected

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