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

Published online by Cambridge University Press:  26 November 2013

## 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 $(x, m, n)= (c- 1, 1, 2).$

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Research Article
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