Proceedings of the Japan Academy, Series A, Mathematical Sciences

A note on the Diophantine equation $x^{2} + q^{m} = c^{2n}$

Mou-Jie Deng

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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 $2 \leq c \leq 30$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 91, Number 2 (2015), 15-18.

Dates
First available in Project Euclid: 2 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.pja/1422885263

Digital Object Identifier
doi:10.3792/pjaa.91.15

Mathematical Reviews number (MathSciNet)
MR3310965

Zentralblatt MATH identifier
06441203

Subjects
Primary: 11D61: Exponential equations

Keywords
Diophantine equations integer solution Terai’s conjecture

Citation

Deng, Mou-Jie. A note on the Diophantine equation $x^{2} + q^{m} = c^{2n}$. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 2, 15--18. doi:10.3792/pjaa.91.15. https://projecteuclid.org/euclid.pja/1422885263


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