# 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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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