A note on the number $a^n+ b^n – dc^n$

Abstract

We say that a positive integer $d$ is special number of degree $n$ if for every integer $m$, there exist nonzero integers $a,b,c$ such that $m=a^n+b^n-dc^n$. In this paper, we investigate some necessary conditions on $n$ for existing a special number of degree $n$.