On px2 + q2n = yp and related Diophantine equations

JOURNAL OF NUMBER THEORY

卷:131

Article

Laradji, A.²  Mignotte, M.³  Tzanakis, N.¹ 

[1] Univ Crete, Dept Math, Iraklion, Greece

[2] KFUPM, Dept Math & Stat, Dhahran, Saudi Arabia

[3] Univ Strasbourg, UFR Math, Strasbourg, France

关键词: Exponential Diophantine equation;    Linear forms in two logarithms;    Hypergeometric series;    Lehmer pair;    Cyclotomic polynomial;   

DOI  :  10.1016/j.jnt.2011.02.007

来源: Elsevier

【 摘 要 】

The title equation, where p > 3 is a prime number not equivalent to 7 (mod 8), q is an odd prime number and x, y, n are positive integers with x, y relatively prime, is studied. When p equivalent to 3 (mod 8), we prove (Theorem 2.3) that there are no solutions. For p not equivalent to 3 (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 J. Number Theory 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. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free