Pairs where $2^ a-2^ b$ divides $n^ a-n^ b$ for all $n$

Abstract:

Recently Selfridge asked for what $a$ and $b$ does ${2^a} - {2^b}$ divide ${n^a} - {n^b}$ for all $n$? In this paper, the authors prove that there exist only fourteen such pairs $(a,b)$ when $0 \leq b < a$.