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A NOTE ON THE DIOPHANTINE EQUATION $x^{2}+(2c-1)^{m}=c^{n}$

  • MOU-JIE DENG (a1), JIN GUO (a2) and AI-JUAN XU (a3)

Abstract

Let $c\geq 2$ be a positive integer. Terai [β€˜A note on the Diophantine equation $x^{2}+q^{m}=c^{n}$ ’, Bull. Aust. Math. Soc.90 (2014), 20–27] conjectured that the exponential Diophantine equation $x^{2}+(2c-1)^{m}=c^{n}$ has only the positive integer solution $(x,m,n)=(c-1,1,2)$ . He proved his conjecture under various conditions on $c$ and $2c-1$ . In this paper, we prove Terai’s conjecture under a wider range of conditions on $c$ and $2c-1$ . In particular, we show that the conjecture is true if $c\equiv 3\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$ and $3\leq c\leq 499$ .

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This research was supported by the National Natural Science Foundation of China (grant no. 11601108) and the Natural Science Foundation of Hainan Province (grant no. 20161002).

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[1] Arif, S. A. and Abu Muriefah, F. S., β€˜On the Diophantine equation x 2 + q 2k+1 = y n ’, J. Number Theory 95 (2002), 95–100.
[2] Bennett, M. A. and Skinner, C. M., β€˜Ternary Diophantine equations via Galois representations and modular forms’, Canad. J. Math. 56 (2004), 23–54.
[3] Carmichael, R. D., β€˜On the numerical factor of the arithmetic forms 𝛼 n ±𝛽 n ’, Ann. Math. 15 (1913), 30–70.
[4] Deng, M.-J., β€˜A note on the Diophantine equation x 2 + q m = c 2n ’, Proc. Japan Acad. 91 (2015), 15–18.
[5] Ljunggren, W., β€˜Some theorems on indeterminate equations of the form (x n - 1/x - 1) = y q ’, Norsk Mat. Tidsskr. 25 (1943), 17–20; (in Norwegian).
[6] Terai, N., β€˜A note on the Diophantine equation x 2 + q m = c n ’, Bull. Aust. Math. Soc. 90 (2014), 20–27.
[7] Zhu, H. L., β€˜A note on the Diophantine equation x 2 + q m = y 3 ’, Acta Arith. 146 (2011), 195–202.
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A NOTE ON THE DIOPHANTINE EQUATION $x^{2}+(2c-1)^{m}=c^{n}$

  • MOU-JIE DENG (a1), JIN GUO (a2) and AI-JUAN XU (a3)

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