The divisibility of a^n-b^n by powers of n

TL;DR

This paper characterizes the set of integers n for which a^n - b^n is divisible by powers of n, identifying infinite cases, explicit exceptions, and conjecturing finiteness for higher powers using number theory tools.

Contribution

It explicitly determines the divisibility sets for a^n - b^n by n^j, including exceptional cases, and relates these to divisibility of a^n + b^n, advancing understanding of divisibility properties in number theory.

Findings

For j=1,2, the divisibility set is usually infinite.

Explicit exceptional cases are identified where the set is finite.

For j≥3 and gcd(a,b)=1, the set is probably always finite.

Abstract

For given integers a,b, and j at least 1 we determine the set of integers n for which a^n-b^n is divisible by n^j. For j=1,2, this set is usually infinite; we find explicitly the exceptional cases for which a,b the set is finite. For j=2, we use Zsigmondy's Theorem for this. For j at least 3 and gcd(a,b)=1, the set is probably always finite; this seems difficult to prove, however. We also show that determination of the set of integers n for which a^n+b^n is divisible by n^j can be reduced to that of the above set.