On the divisibility of $a^n\pm b^n$ by powers of $n$

Abstract

We determine all triples $(a,b,n)$ of positive integers such that $a$ and $b$ are relatively prime and $n^k$ divides $a^n+b^n$ (respectively, $a^n-b^n$), when $k$ is the maximum of $a$ and $b$ (in fact, we answer a slightly more general question). As a by-product, it is found that, for $m,n\in {\mathbb{N}}^{+}$ with $n\ge 2$, $n^m$ divides $m^n+1$ if and only if $(m,n)=(2,3)$ or $(1,2)$, which generalizes problems from the 1990 and 1999 editions of the International Mathematical Olympiad. The results are related to a conjecture by K. Győry and C. Smyth on the finiteness of the sets $R_k^{\pm }(a,b):=\{n\in {\mathbb{N}}^{+}:n^k\mid a^n\pm b^n\}$, when $a,b,k$ are fixed integers with $k\ge 3$, $\gcd (a,b)=1$ and $|ab|\ge 2$.

arXiv commentChanged the title, fixed minor details and further improved readability (6 pages, no figures, to appear in Integers)