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We determine all triples (a, b, n) of positive integers such that a and b are relatively prime and nk divides an+bn (respectively, an bn), when k is the maximum of a and b (in fact, we answer a slightly more general question). As a by-product, we see that, for m,n 2 N+ with n 2, nm divides mn + 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 (a, b) := {n 2 N+ : nk | (an ± bn)}, where a, b, k are fixed integers with k 3, gcd(a, b) = 1 and |ab| 2; in particular, we find that the conjecture is true for k max(|a|, |b|).

ON THE DIVISIBILITY OF an ± bn BY POWERS OF n

ON THE DIVISIBILITY OF aⁿ ± bⁿ BY POWERS OF n