{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:SW2SR3YUSY2AIEMUUMSDCOHBXV","short_pith_number":"pith:SW2SR3YU","schema_version":"1.0","canonical_sha256":"95b528ef149634041194a3243138e1bd4777b5391c5b63912be5ee4b6771b8fc","source":{"kind":"arxiv","id":"2309.09149","version":2},"attestation_state":"computed","paper":{"title":"Generalized Frobenius Number of Three Variables","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kittipong Subwattanachai","submitted_at":"2023-09-17T04:14:41Z","abstract_excerpt":"For $ k \\geq 2 $, we let $ A = (a_{1}, a_{2}, \\ldots, a_{k}) $ be a $k$-tuple of positive integers with $\\gcd(a_{1}, a_2, \\ldots, a_k) =1$ and, for a non-negative integer $s$, the generalized Frobenius number of $A$, $g(A;s) = g(a_1, a_2, \\ldots, a_k;s)$, the largest integer that has at most $s$ representations in terms of $a_1, a_2, \\ldots, a_k$ with non-negative integer coefficients. In this article, we give a formula for the generalized Frobenius number of three positive integers $(a_1,a_2,a_3)$ with certain conditions."},"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":"2309.09149","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2023-09-17T04:14:41Z","cross_cats_sorted":[],"title_canon_sha256":"49f857a748587739d1fe6136f9cf6e5123f1299bdeed9c60ff302b672e64ab52","abstract_canon_sha256":"5b6ff528d8a486c6fdc049a1d01190b6f9e3ea764cf81c73f9cd1fc6fa15dc5a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:50:44.243268Z","signature_b64":"XjnWJhmVH20+clP3C6qJPWWY7A8I3ZDSMLJzYIBE5UUZNYbaxDvsWNJU7ts1s4Bzo9Xj36knWhfy4KaHc8pOCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95b528ef149634041194a3243138e1bd4777b5391c5b63912be5ee4b6771b8fc","last_reissued_at":"2026-07-05T09:50:44.242673Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:50:44.242673Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generalized Frobenius Number of Three Variables","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kittipong Subwattanachai","submitted_at":"2023-09-17T04:14:41Z","abstract_excerpt":"For $ k \\geq 2 $, we let $ A = (a_{1}, a_{2}, \\ldots, a_{k}) $ be a $k$-tuple of positive integers with $\\gcd(a_{1}, a_2, \\ldots, a_k) =1$ and, for a non-negative integer $s$, the generalized Frobenius number of $A$, $g(A;s) = g(a_1, a_2, \\ldots, a_k;s)$, the largest integer that has at most $s$ representations in terms of $a_1, a_2, \\ldots, a_k$ with non-negative integer coefficients. In this article, we give a formula for the generalized Frobenius number of three positive integers $(a_1,a_2,a_3)$ with certain conditions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.09149","kind":"arxiv","version":2},"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/2309.09149/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":"2309.09149","created_at":"2026-07-05T09:50:44.242745+00:00"},{"alias_kind":"arxiv_version","alias_value":"2309.09149v2","created_at":"2026-07-05T09:50:44.242745+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2309.09149","created_at":"2026-07-05T09:50:44.242745+00:00"},{"alias_kind":"pith_short_12","alias_value":"SW2SR3YUSY2A","created_at":"2026-07-05T09:50:44.242745+00:00"},{"alias_kind":"pith_short_16","alias_value":"SW2SR3YUSY2AIEMU","created_at":"2026-07-05T09:50:44.242745+00:00"},{"alias_kind":"pith_short_8","alias_value":"SW2SR3YU","created_at":"2026-07-05T09:50:44.242745+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.12052","citing_title":"Criteria and Curvatures for Singularities of Finite Multiplicities of Curves in $\\boldsymbol{R}^N$","ref_index":16,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SW2SR3YUSY2AIEMUUMSDCOHBXV","json":"https://pith.science/pith/SW2SR3YUSY2AIEMUUMSDCOHBXV.json","graph_json":"https://pith.science/api/pith-number/SW2SR3YUSY2AIEMUUMSDCOHBXV/graph.json","events_json":"https://pith.science/api/pith-number/SW2SR3YUSY2AIEMUUMSDCOHBXV/events.json","paper":"https://pith.science/paper/SW2SR3YU"},"agent_actions":{"view_html":"https://pith.science/pith/SW2SR3YUSY2AIEMUUMSDCOHBXV","download_json":"https://pith.science/pith/SW2SR3YUSY2AIEMUUMSDCOHBXV.json","view_paper":"https://pith.science/paper/SW2SR3YU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2309.09149&json=true","fetch_graph":"https://pith.science/api/pith-number/SW2SR3YUSY2AIEMUUMSDCOHBXV/graph.json","fetch_events":"https://pith.science/api/pith-number/SW2SR3YUSY2AIEMUUMSDCOHBXV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SW2SR3YUSY2AIEMUUMSDCOHBXV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SW2SR3YUSY2AIEMUUMSDCOHBXV/action/storage_attestation","attest_author":"https://pith.science/pith/SW2SR3YUSY2AIEMUUMSDCOHBXV/action/author_attestation","sign_citation":"https://pith.science/pith/SW2SR3YUSY2AIEMUUMSDCOHBXV/action/citation_signature","submit_replication":"https://pith.science/pith/SW2SR3YUSY2AIEMUUMSDCOHBXV/action/replication_record"}},"created_at":"2026-07-05T09:50:44.242745+00:00","updated_at":"2026-07-05T09:50:44.242745+00:00"}