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