{"paper":{"title":"Numerical Semigroups generated by Primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Anton Rechenauer, Michael Hellus, Rolf Waldi","submitted_at":"2019-08-26T05:54:26Z","abstract_excerpt":"Let $p_1=2, p_2=3, p_3=5, \\ldots$ be the consecutive prime numbers, $S_n$ the numerical semigroup generated by the primes not less than $p_n$ and $u_n$ the largest irredundant generator of $S_n$. We will show, that\n  $\\bullet$ $u_n\\sim3p_n$.\n  Similarly, for the largest integer $f_n$ not contained in $S_n$, by computational evidence we suspect that\n  $\\bullet$ $f_n$ is an odd number for $n\\geq5$ and\n  $\\bullet$ $f_n\\sim3p_n$; further\n  $\\bullet$ $4p_n>f_{n+1}$ for $n\\geq1$.\n  If $f_n$ is odd for large $n$, then $f_n\\sim3p_n$. In case $f_n\\sim3p_n$ every large even integer $x$ is the sum of two"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09483","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.09483/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}