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

preprint2020arXivOpen access

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

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

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