{"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. We use the result to prove that, if $\\var"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.0802","kind":"arxiv","version":5},"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"}