# The divisibility of a^n-b^n by powers of n - Mathematics > Number Theory

The divisibility of a^n-b^n by powers of n - Mathematics > Number Theory - Download this document for free, or read online. Document in PDF available to download.

Abstract: For given integers a,b, and j at least 1 we determine the set of integers nfor 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. Forj=2, we use Zsigmondy-s Theorem for this. For j at least 3 and gcda,b=1, theset 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 isdivisible by n^j can be reduced to that of the above set.

Author: ** Chris Smyth**

Source: https://arxiv.org/