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

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

2010-02-04

arxiv.org/abs/1002.1041v2

Journal of Number Theory 131 (2011), 1575-1596

Mathematics

Number Theory

23 pages, second version with minor revisions

Scientific paper

10.1016/j.jnt.2011.02.007

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.

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