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On the Diophantine equation \(\varvec{x^2+b^m=c^n}\) with \(\varvec{a^2+b^4=c^2}\)

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Abstract

 Let a, b, c be pairwise relatively prime positive integers such that \(a^2 + b^4=c^2\) and b is odd. Then we show that the equation of the title has only one positive integer solution \((x, m, n)=(a, 4, 2)\) under some conditions.

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Acknowledgements

The author is supported by JSPS KAKENHI (No. 18K03247).

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Authors and Affiliations

  1. Division of Mathematical Sciences, Department of Integrated Science and Technology, Faculty of Science and Technology, Oita University, 700 Dannoharu, Oita, 870-1192, Japan

    Nobuhiro Terai

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  1. Nobuhiro Terai
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Correspondence to Nobuhiro Terai.

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Terai, N. On the Diophantine equation \(\varvec{x^2+b^m=c^n}\) with \(\varvec{a^2+b^4=c^2}\). Indian J Pure Appl Math 53, 162–169 (2022). https://doi.org/10.1007/s13226-021-00162-0

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  • DOI: https://doi.org/10.1007/s13226-021-00162-0

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