{"paper":{"title":"Powerful numbers in $(1^{\\ell}+q^{\\ell})(2^{\\ell}+q^{\\ell})\\cdots (n^{\\ell}+q^{\\ell})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Qing-Qing Zhao, Quan-Hui Yang","submitted_at":"2017-06-11T13:19:21Z","abstract_excerpt":"Let $q$ be a positive integer. Recently, Niu and Liu proved that if $n\\ge \\max\\{q,1198-q\\}$, then the product $(1^3+q^3)(2^3+q^3)\\cdots (n^3+q^3)$ is not a powerful number. In this note, we prove that (i) for any odd prime power $\\ell$ and $n\\ge \\max\\{q,11-q\\}$, the product $(1^{\\ell}+q^{\\ell})(2^{\\ell}+q^{\\ell})\\cdots (n^{\\ell}+q^{\\ell})$ is not a powerful number; (2) for any positive odd integer $\\ell$, there exists an integer $N_{q,\\ell}$ such that for any positive integer $n\\ge N_{q,\\ell}$, the product $(1^{\\ell}+q^{\\ell})(2^{\\ell}+q^{\\ell})\\cdots (n^{\\ell}+q^{\\ell})$ is not a powerful num"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03350","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"}