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arXiv · 2003.12749

On the exponential Diophantine equation $(n-1)^{x}+(n+2)^{y}=n^{z}$

Hairong Bai·Elif Kızıldere·Gökhan Soydan·Pingzhi Yuan
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Abstract

Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1)^{x}+(n+2)^{y}=n^{z},\ n\geq 2,\ xyz\neq 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2), (3,1,2,3)$. The main tools on the proofs are Baker's theory and Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers.

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Hairong Bai, Elif Kızıldere, Gökhan Soydan, Pingzhi Yuan. 2020-03-28. On the exponential Diophantine equation $(n-1)^{x}+(n+2)^{y}=n^{z}$. https://doi.org/10.4064/cm7668-6-2019

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