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Does there exist at least one prime $q$ such that $q$ divides $\\prod_{i \\in I} u_i - \\varepsilon(I)$ for some $I \\in \\mathcal D$, but it does not divide $u_1 \\cdots u_n$? We answer this question in the positive when the $u_i$ are prime powers and $\\varepsilon$ and $\\mathcal D$ are subjected to certain restrictions. We use the result to prove that, if $\\var"},"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":"1212.0802","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-12-04T17:33:01Z","cross_cats_sorted":[],"title_canon_sha256":"ef182c12ef3cf98ad66c276cad4b8502caafb7036891fbfd691cd255e09ec1da","abstract_canon_sha256":"df158704cd5a9b876546d26dbb4c3606bb72df4e2bfa5ab8557e7ecb03a06a71"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:21.142793Z","signature_b64":"guieeT+QOo9XpZXvJsYPSXga2w9j2QQHRNNuJMMKDE+l00or5KGwwQH/UEQgs7sQdomye/YUJ+XMOKbCID1GBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6d039b168d2627458e5a177eb5d8c54041eedf2e4d6c4077dd55655590d0fc72","last_reissued_at":"2026-05-18T02:28:21.142241Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:21.142241Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a system of equations with primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Paolo Leonetti, Salvatore Tringali","submitted_at":"2012-12-04T17:33:01Z","abstract_excerpt":"Given an integer $n \\ge 3$, let $u_1, \\ldots, u_n$ be pairwise coprime integers $\\ge 2$, $\\mathcal D$ a family of nonempty proper subsets of $\\{1, \\ldots, n\\}$ with \"enough\" elements, and $\\varepsilon$ a function $ \\mathcal D \\to \\{\\pm 1\\}$. Does there exist at least one prime $q$ such that $q$ divides $\\prod_{i \\in I} u_i - \\varepsilon(I)$ for some $I \\in \\mathcal D$, but it does not divide $u_1 \\cdots u_n$? We answer this question in the positive when the $u_i$ are prime powers and $\\varepsilon$ and $\\mathcal D$ are subjected to certain restrictions. 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