{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:AWHC6ZPJ3TQ4KD5QJAUM73D3C6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"52662b3b14ecedb8ad51b3ed84b4462404d3ffdc7915466b725321a67359ef9a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-12-27T04:09:27Z","title_canon_sha256":"e7aec4e62478565e87453ac8f481f9ff90637059193d049c5890226a7fd666e6"},"schema_version":"1.0","source":{"id":"1312.7185","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.7185","created_at":"2026-05-18T02:19:06Z"},{"alias_kind":"arxiv_version","alias_value":"1312.7185v3","created_at":"2026-05-18T02:19:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.7185","created_at":"2026-05-18T02:19:06Z"},{"alias_kind":"pith_short_12","alias_value":"AWHC6ZPJ3TQ4","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"AWHC6ZPJ3TQ4KD5Q","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"AWHC6ZPJ","created_at":"2026-05-18T12:27:38Z"}],"graph_snapshots":[{"event_id":"sha256:9adfd44cbc4e38e36b28f0d6a4dad98ed83e64595c62d267a6443ff9a9670e24","target":"graph","created_at":"2026-05-18T02:19:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Given three pairwise coprime positive integers $a_1,a_2,a_3 \\in \\mathbb{Z}^+$ we show the existence of a relation between the sets of the first elements of the three quotients $\\frac{\\langle a_i,a_j \\rangle}{a_k}$ that can be made for every $\\{i.j,k\\}=\\{1,2,3\\}$. Then we use this result to give an improved version of Johnson's semi-explicit formula for the Frobenius number $g(a_1,a_2,a_3)$ without restriction on the choice of $a_1,a_2,a_3$ and to give an explicit formula for a particular class of numerical semigroups.","authors_text":"Alessio Moscariello","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-12-27T04:09:27Z","title":"The first elements of the quotient of a numerical semigroup by a positive integer"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.7185","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a489ecb0e0048f12f134829df273c48a1a617da2028905292c6b93861c2bd11b","target":"record","created_at":"2026-05-18T02:19:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"52662b3b14ecedb8ad51b3ed84b4462404d3ffdc7915466b725321a67359ef9a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-12-27T04:09:27Z","title_canon_sha256":"e7aec4e62478565e87453ac8f481f9ff90637059193d049c5890226a7fd666e6"},"schema_version":"1.0","source":{"id":"1312.7185","kind":"arxiv","version":3}},"canonical_sha256":"058e2f65e9dce1c50fb04828cfec7b17b2ac0b976311b592e1a64e86f50fffea","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"058e2f65e9dce1c50fb04828cfec7b17b2ac0b976311b592e1a64e86f50fffea","first_computed_at":"2026-05-18T02:19:06.189806Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:19:06.189806Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YiEzy0vi5hBZSZLiL0hhbwmZd1qsIKsjeYm1OrADNhHXTl/rxQxFVzqn++h3Xd1DgcRl3VlRVLGud/SdT5tDCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:19:06.190465Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.7185","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a489ecb0e0048f12f134829df273c48a1a617da2028905292c6b93861c2bd11b","sha256:9adfd44cbc4e38e36b28f0d6a4dad98ed83e64595c62d267a6443ff9a9670e24"],"state_sha256":"6df341e5c921e59148eb183ed43cb346a4f39fcf84d4eee8b58c652bd41d3dd5"}