{"paper":{"title":"On the divisibility of $a^n \\pm b^n$ by powers of $n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Salvatore Tringali","submitted_at":"2013-01-01T20:16:27Z","abstract_excerpt":"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. Gy\\H{o}ry and C. Smyth on the finiteness of the sets $R"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.0131","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}