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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"},"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":"1908.09483","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-26T05:54:26Z","cross_cats_sorted":[],"title_canon_sha256":"709f66c441bcaccb1c94f96434b1f642748638f7236fd6011044cb49ad651b1c","abstract_canon_sha256":"e6dc03950a039c17f3d28730bc9a25da75970fe417355e03067ea2834675f266"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:08:24.528595Z","signature_b64":"XEVlIQhviV7Ah9R5XVdP9QKxXL0YYkplFL4bkT0ygF4Egy2zA6MyKI9ufPMW+zqNX786d8VDZ8JJDf5BVVM4Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f26cbcee949bec96f136b6fc9932e752a1954c78f98afdb0ffe85d818abf0f7","last_reissued_at":"2026-07-05T01:08:24.528139Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:08:24.528139Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1908.09483","created_at":"2026-07-05T01:08:24.528198+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.09483v3","created_at":"2026-07-05T01:08:24.528198+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.09483","created_at":"2026-07-05T01:08:24.528198+00:00"},{"alias_kind":"pith_short_12","alias_value":"F4TMXTXJJG7M","created_at":"2026-07-05T01:08:24.528198+00:00"},{"alias_kind":"pith_short_16","alias_value":"F4TMXTXJJG7MS3YT","created_at":"2026-07-05T01:08:24.528198+00:00"},{"alias_kind":"pith_short_8","alias_value":"F4TMXTXJ","created_at":"2026-07-05T01:08:24.528198+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU","json":"https://pith.science/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU.json","graph_json":"https://pith.science/api/pith-number/F4TMXTXJJG7MS3YTNNX4TEZOOU/graph.json","events_json":"https://pith.science/api/pith-number/F4TMXTXJJG7MS3YTNNX4TEZOOU/events.json","paper":"https://pith.science/paper/F4TMXTXJ"},"agent_actions":{"view_html":"https://pith.science/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU","download_json":"https://pith.science/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU.json","view_paper":"https://pith.science/paper/F4TMXTXJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.09483&json=true","fetch_graph":"https://pith.science/api/pith-number/F4TMXTXJJG7MS3YTNNX4TEZOOU/graph.json","fetch_events":"https://pith.science/api/pith-number/F4TMXTXJJG7MS3YTNNX4TEZOOU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU/action/storage_attestation","attest_author":"https://pith.science/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU/action/author_attestation","sign_citation":"https://pith.science/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU/action/citation_signature","submit_replication":"https://pith.science/pith/F4TMXTXJJG7MS3YTNNX4TEZOOU/action/replication_record"}},"created_at":"2026-07-05T01:08:24.528198+00:00","updated_at":"2026-07-05T01:08:24.528198+00:00"}