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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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

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

First available in Project Euclid: 2 February 2015

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Zentralblatt MATH identifier

Primary: 11D61: Exponential equations

Diophantine equations integer solution Terai’s conjecture


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.

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  • S. A. Arif and F. S. Abu Muriefah, On the Diophantine equation $x^{2}+q^{2k+1}=y^{n}$, J. Number Theory 95 (2002), no. 1, 95–100.
  • M. A. Bennett and C. M. Skinner, Ternary Diophantine equations via Galois representations and modular forms, Canad. J. Math. 56 (2004), no. 1, 23–54.
  • Z. Cao, Diophantine equation and its applications. (in Chinese), Shanghai Jio Tong Univ. Press, Shanghai, 2000.
  • C. Heuberger and M. Le, On the generalized Ramanujan-Nagell equation $x^{2}+D=p^{z}$, J. Number Theory 78 (1999), no. 2, 312–331.
  • W. Ljunggren, Some theorems on indeterminate equations of the form $x^{n}-1/x-1=y^{q}$, Norsk Mat. Tidsskr. 25 (1943), 17–20.
  • W. Ljunggren, Eine elementare Auflösung der diophantischen Gleichung $x^{3}+1=2y^{2}$, Acta Math. Acad. Sci. Hungar. 3 (1952), 99–101.
  • N. Terai, The Diophantine equation $x^{2}+q^{m}=p^{n}$, Acta Arith. 63 (1993), no. 4, 351–358.
  • N. Terai, A note on the Diophantine equation $x^{2}+q^{m}=c^{n}$, Bull. Aust. Math. Soc. 90 (2014), no. 1, 20–27.
  • H. L. Zhu, A note on the Diophantine equation $x^{2}+q^{m}=y^{3}$, Acta Arith. 146 (2011), no. 2, 195–202.