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A NOTE ON THE DIOPHANTINE EQUATION ${x}^{2} + {q}^{m} = {c}^{n} $

Part of: Diophantine equations

Published online by Cambridge University Press:  26 November 2013

NOBUHIRO TERAI
NOBUHIRO TERAI*
Affiliation:
Division of Information System Design, Ashikaga Institute of Technology, 268-1 Omae, Ashikaga, Tochigi 326-8558, Japan email terai@ashitech.ac.jp
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Abstract

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Let $q$ be an odd prime such that ${q}^{t} + 1= 2{c}^{s} $, where $c, t$ are positive integers and $s= 1, 2$. We show that the Diophantine equation ${x}^{2} + {q}^{m} = {c}^{n} $ has only the positive integer solution $(x, m, n)= ({c}^{s} - 1, t, 2s)$ under some conditions. The proof is based on elementary methods and a result concerning the Diophantine equation $({x}^{n} - 1)/ (x- 1)= {y}^{2} $ due to Ljunggren. We also verify that when $2\leq c\leq 30$ with $c\not = 12, 24$, the Diophantine equation ${x}^{2} + \mathop{(2c- 1)}\nolimits ^{m} = {c}^{n} $ has only the positive integer solution $(x, m, n)= (c- 1, 1, 2). $

Keywords

Diophantine equationinteger solution

MSC classification

Secondary: 11D61: Exponential equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 1 , August 2014 , pp. 20 - 27
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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