{"paper":{"title":"On the Products $(1^\\ell+1)(2^\\ell+1)\\cdots (n^\\ell +1)$, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ming-Liang Gong, Yong-Gao Chen","submitted_at":"2013-11-22T03:32:59Z","abstract_excerpt":"In this paper, the following results are proved: (i) For any odd integer $\\ell$ with at most two distinct prime factors and any positive integer $n$, the product $(1^\\ell+1)(2^\\ell+1)\\cdots (n^\\ell +1)$ is not a powerful number; (ii) For any integer $r\\ge 1$, there exists a positive integer $T_r$ such that, if $\\ell$ is a positive odd integer with at most $r$ distinct prime factors and $n$ is an integer with $n\\ge T_r$, then $(1^\\ell+1)(2^\\ell+1)\\cdots (n^\\ell +1)$ is not a powerful number."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.5646","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"}