On the congruences a n ± b n ≡ 0

We determine all triples (a, b, n) of integers with gcd(a, b) = 1 and n ≥ 1 such that n divides a + b for k = max(|a|, |b|). In particular, for positive integers m,n we show that n | m+1 if and only if either (m,n) = (2, 3), (m,n) = (1, 2), or n = 1 and m is arbitrary; this generalizes a couple of problems from the 1990 and 1999 editions of the International Mathematical Olympiad. Then we solve the same question with a − b in place of a + b. The results are related to a conjecture by K. Győry and C. Smyth on the finiteness of {n ∈ N : n | a ± b} when a, b, k are fixed integers with k ≥ 3, gcd(a, b) = 1, and |a|, |b| not simultaneously equal to 1.