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We show that when $k$ is greater than $r$, $kr - r - 1 \\le S_{\\mathfrak{z}}(k,r) \\le kr - 1$, and when $r$ is an odd prime, $S_{\\mathfrak{z}}(k,r)$ is in fact equal to $kr - r$."},"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":"1808.03851","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-11T19:12:08Z","cross_cats_sorted":[],"title_canon_sha256":"e84025f7207e0254088f912fad6346a361d6d9ea0b58e00539a6bdca1d614a5c","abstract_canon_sha256":"e9b09d0d8491c401d91d8b496d97fd65becdeabbad6d7e6fbcf547dab39146dc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:19.527391Z","signature_b64":"uow/LEwsl7A6dUpa4UdaNyzP4ONAcpqfEbRHmKbajSAGdCJjRm/CPHyacCyczcnjW4DNiPfWg1e1h1AJTiqHAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b62e34b69e27195c3f9d0dc37f605208873e3539ea214702362606435694df9","last_reissued_at":"2026-05-18T00:08:19.526896Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:19.526896Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Upper and Lower Bounds on Zero-Sum Generalized Schur Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Erik Metz","submitted_at":"2018-08-11T19:12:08Z","abstract_excerpt":"Let $S_{\\mathfrak{z}}(k,r)$ be the least positive integer such that for any $r$-coloring $\\chi : \\{1,2,\\dots,S_{\\mathfrak{z}}(k,r)\\} \\longrightarrow \\{1, 2, \\dots, r\\}$, there is a sequence $x_1, x_2, \\dots, x_k$ such that $\\sum_{i=1}^{k-1} x_i = x_k$, and $\\sum_{i=1}^{k} \\chi(x_i) \\equiv 0 \\pmod{r}$. We show that when $k$ is greater than $r$, $kr - r - 1 \\le S_{\\mathfrak{z}}(k,r) \\le kr - 1$, and when $r$ is an odd prime, $S_{\\mathfrak{z}}(k,r)$ is in fact equal to $kr - r$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.03851","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1808.03851","created_at":"2026-05-18T00:08:19.526969+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.03851v1","created_at":"2026-05-18T00:08:19.526969+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.03851","created_at":"2026-05-18T00:08:19.526969+00:00"},{"alias_kind":"pith_short_12","alias_value":"BNROGS3J4JYZ","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_16","alias_value":"BNROGS3J4JYZLQ7Z","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_8","alias_value":"BNROGS3J","created_at":"2026-05-18T12:32:16.446611+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/BNROGS3J4JYZLQ7Z2DODP5QFEC","json":"https://pith.science/pith/BNROGS3J4JYZLQ7Z2DODP5QFEC.json","graph_json":"https://pith.science/api/pith-number/BNROGS3J4JYZLQ7Z2DODP5QFEC/graph.json","events_json":"https://pith.science/api/pith-number/BNROGS3J4JYZLQ7Z2DODP5QFEC/events.json","paper":"https://pith.science/paper/BNROGS3J"},"agent_actions":{"view_html":"https://pith.science/pith/BNROGS3J4JYZLQ7Z2DODP5QFEC","download_json":"https://pith.science/pith/BNROGS3J4JYZLQ7Z2DODP5QFEC.json","view_paper":"https://pith.science/paper/BNROGS3J","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.03851&json=true","fetch_graph":"https://pith.science/api/pith-number/BNROGS3J4JYZLQ7Z2DODP5QFEC/graph.json","fetch_events":"https://pith.science/api/pith-number/BNROGS3J4JYZLQ7Z2DODP5QFEC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BNROGS3J4JYZLQ7Z2DODP5QFEC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BNROGS3J4JYZLQ7Z2DODP5QFEC/action/storage_attestation","attest_author":"https://pith.science/pith/BNROGS3J4JYZLQ7Z2DODP5QFEC/action/author_attestation","sign_citation":"https://pith.science/pith/BNROGS3J4JYZLQ7Z2DODP5QFEC/action/citation_signature","submit_replication":"https://pith.science/pith/BNROGS3J4JYZLQ7Z2DODP5QFEC/action/replication_record"}},"created_at":"2026-05-18T00:08:19.526969+00:00","updated_at":"2026-05-18T00:08:19.526969+00:00"}