{"paper":{"title":"On the congruence $\\sum_{j=1}^{n-1} j^{k(n-1)} \\equiv -1 \\pmod n $. k-strong Giuga and k-Carmichael numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Antonio M. Oller-Marc\\'en, Jos\\'e Mar\\'ia Grau","submitted_at":"2013-11-14T14:52:57Z","abstract_excerpt":"In this work we consider the congruence $\\sum_{j=1}^{n-1} j^{k(n-1)} \\equiv -1 \\pmod n$ for each $k \\in \\mathbb{N}$, thus extending Giuga's ideas for $k=1$. In particular, it is proved that a pair $(n,k)\\in \\mathbb{N}^2$ satisfies this congruence if and only if $n$ is prime or a Giuga Number and $\\lambda(n) \\mid k(n-1)$. In passing, we establish new characterizations of Giuga numbers and we study some properties of the numbers $n$ satisfying $\\lambda(n) \\mid k(n-1)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.3522","kind":"arxiv","version":1},"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"}