{"paper":{"title":"Note on vanishing power sums of roots of unity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"K. Senthil Kumar, Neeraj Kumar","submitted_at":"2015-03-25T05:11:12Z","abstract_excerpt":"For a given positive integers $m$ and $\\ell$, we give a complete list of positive integers $n$ for which their exist $m$th roots of unity $x_1,\\dots,x_n \\in \\mathbb{C}$ such that $x_1^{\\ell} + \\cdots + x_n^{\\ell}=0$. This extends the earlier result of Lam and Leung on vanishing sums of roots of unity. Furthermore, we prove that for which integers $n$ with $2 \\leq n \\leq m$, there are distinct $m$th roots of unity $x_1,\\dots,x_n \\in \\mathbb{C}$ such that $x_1^{\\ell} + \\cdots + x_n^{\\ell}=0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.07281","kind":"arxiv","version":2},"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"}