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February 2015 A note on the Diophantine equation $x^{2} + q^{m} = c^{2n}$
Mou-Jie Deng
Proc. Japan Acad. Ser. A Math. Sci. 91(2): 15-18 (February 2015). DOI: 10.3792/pjaa.91.15

Abstract

Let $q$ be an odd prime. Let $c>1$ and $t$ be positive integers such that $q^{t}+1=2c^{2}$. Using elementary method and a result due to Ljunggren concerning the Diophantine equation $\frac{x^{n}-1}{x-1}= y^{2}$, we show that the Diophantine equation $x^{2}+q^{m}=c^{2n}$ has the only positive integer solution $(x, m, n)=(c^{2}-1, t, 2)$. As applications of this result some new results on the Diophantine equation $x^{2}+q^{m} = c^{n}$ and the Diophantine equation $x^{2}+(2c-1)^{m} = c^{n}$ are obtained. In particular, we prove that Terai’s conjecture is true for $c=12,24$. Combining this result with Terai’s results we conclude that Terai’s conjecture is true for $2 \leq c \leq 30$.

Citation

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Mou-Jie Deng. "A note on the Diophantine equation $x^{2} + q^{m} = c^{2n}$." Proc. Japan Acad. Ser. A Math. Sci. 91 (2) 15 - 18, February 2015. https://doi.org/10.3792/pjaa.91.15

Information

Published: February 2015
First available in Project Euclid: 2 February 2015

zbMATH: 06441203
MathSciNet: MR3310965
Digital Object Identifier: 10.3792/pjaa.91.15

Subjects:
Primary: 11D61

Keywords: Diophantine equations , integer solution , Terai’s conjecture

Rights: Copyright © 2015 The Japan Academy

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Vol.91 • No. 2 • February 2015
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