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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1505.03328","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-13T11:01:24Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0944427307a04adf5fb3731e4f629357a038b5c5526cf9138c7f74683eb9be28","abstract_canon_sha256":"7a7d6ed5f42e6435dff4e404d46b0b38c00c92922ba8b7a0406bccee0ed54683"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:52:24.886913Z","signature_b64":"N1CKcUD7c6cTSz2w9mOXW5RQPcIbJKzrqWUGu6+lhJDk+PA4tVRZRjh6wu/3AIen7c+AL4aqQuirqQ0XI8DjAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1c120b3b0783705cb2442d39021bc1f1798286c8628a455c0ea632e643f42756","last_reissued_at":"2026-05-17T23:52:24.886170Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:52:24.886170Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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