{"paper":{"title":"A multiplicative analogue of Schnirelmann's theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Aled Walker","submitted_at":"2015-05-13T11:01:24Z","abstract_excerpt":"The classical theorem of Schnirelmann states that the primes are an additive basis for the integers. In this paper we consider the analogous multiplicative setting of the cyclic group $\\left(\\mathbb{Z}/ q\\mathbb{Z}\\right)^{\\times}$, and prove a similar result. For all suitably large primes $q$ we define $P_\\eta$ to be the set of primes less than $\\eta q$, viewed naturally as a subset of $\\left(\\mathbb{Z}/ q\\mathbb{Z}\\right)^{\\times}$. Considering the $k$-fold product set $P_\\eta^{(k)}=\\{p_1p_2\\cdots p_k:p_i\\in P_\\eta \\}$, we show that for $\\eta \\gg q^{-\\frac{1}{4}+\\epsilon}$ there exists a con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03328","kind":"arxiv","version":4},"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"}