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On the Diophantine equation \(X^{2N}+2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5\)

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Abstract

We prove that for each prime p, positive integer \(\alpha \), and non-negative integers \(\beta \) and \(\gamma \), the Diophantine equation \(X^{2N} + 2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5\) has no solution with N, X, \(Z\in \mathbb {Z}^+\), \(N > 1\), and \(\gcd (X,Z) = 1\).

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Acknowledgements

The authors would like to thank Gary Walsh for noting a serious (and now corrected) error introduced in earlier editing.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Lebanon Valley College, Annville, PA, USA

    Eva G. Goedhart

  2. Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA, 19010, USA

    Helen G. Grundman

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  1. Eva G. Goedhart
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Correspondence to Eva G. Goedhart.

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Goedhart, E.G., Grundman, H.G. On the Diophantine equation \(X^{2N}+2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5\) . Period Math Hung 75, 196–200 (2017). https://doi.org/10.1007/s10998-017-0185-0

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  • DOI: https://doi.org/10.1007/s10998-017-0185-0

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