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finite solutions of F_{n,k}(x,y)=x!+n^k+n+1-y^k=0

Source: 2020 1st Memorial Mathematical Contest "Aleksandar Blazhevski-Cane" p3

April 30, 2021
number theory

Problem Statement

For given integers n>0n>0 and k>1k> 1, let Fn,k(x,y)=x!+nk+n+1−ykF_{n,k}(x,y)=x!+n^k+n+1-y^k. Prove that there are only finite couples (a,b)(a,b) of positive integers such that Fn,k(a,b)=0F_{n,k}(a,b)=0
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