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On the Exponential Diophantine Equation \((m^2+m+1)^x+m^y=(m+1)^z \)

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Abstract

Let \(m \ge 1\) be a positive integer. We show that the exponential Diophantine equation \( (m^2+m+1)^x+m^y=(m+1)^z \) has no positive integer solutions other than \((x,y,z)=(1,1,2)\) when \(m \not \in \{1, 2, 3 \}\).

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Acknowledgements

I would like to thank the referees for their careful reading and valuable remarks.

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  1. Mathematics Department, Faculty of Arts and Sciences, Yildiz Technical University, Davutpasa Campus, Esenler, 34210, Istanbul, Turkey

    Murat Alan

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  1. Murat Alan
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Correspondence to Murat Alan.

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Alan, M. On the Exponential Diophantine Equation \((m^2+m+1)^x+m^y=(m+1)^z \). Mediterr. J. Math. 17, 189 (2020). https://doi.org/10.1007/s00009-020-01613-4

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  • DOI: https://doi.org/10.1007/s00009-020-01613-4

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