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

Autor: Bai, Hairong, Kızıldere, Elif, Soydan, Gökhan, Yuan, Pingzhi
Rok vydání: 2020
Předmět:
Druh dokumentu: Working Paper
DOI: 10.4064/cm7668-6-2019
Popis: 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.
Comment: 12 pages, to appear, Colloquium Mathematicum (2020)
Databáze: arXiv