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For any positive integer $k$, the $k$-th generalized Erd\\H{o}s-Ginzburg-Ziv constant $\\mathsf s_{km}(G)$ is defined as the smallest positive integer $t$ such that every sequence $S$ in $G$ of length at least $t$ has a zero-sum subsequence of length $km$. It is easy to see that $\\mathsf s_{kn}(C_n^r)\\ge(k+r)n-r$ where $n,r\\in\\mathbb N$. Kubertin conjectured that the equality holds for any $k\\ge r$. In this paper, we mainly prove the following results:\n  (1) For every positive integer $k\\ge 6$, we have $$\\mathsf s_{kn}(C_n^3)"},"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":"1809.06548","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-18T06:33:25Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"306025c82a7de98f22166bef9fc3abd0c778262b6a307c21fcd29eaeab6ef4f7","abstract_canon_sha256":"92346cea813849d557ba12513a7b6d1b84d38a8c9a43d5f6f929f2bb9929bb19"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:01.849627Z","signature_b64":"HG+QSSbbtxr9r8YDaaAWLcNhb6/8hKHJzZDNL9+JPZw+agGAhMbsY3MloPKd4vf2lcUHOtwBCqyMroKzNNvIDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2368ca784dc5bcbf68b9ff63e1a8b314223030fd6a743fd08b7c7ea8db444364","last_reissued_at":"2026-05-18T00:00:01.849094Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:01.849094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On generalized Erd\\H{o}s-Ginzburg-Ziv constants of $C_n^r$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Dongchun Han, Hanbin Zhang","submitted_at":"2018-09-18T06:33:25Z","abstract_excerpt":"Let $G$ be an additive finite abelian group with exponent $\\exp(G)=m$. For any positive integer $k$, the $k$-th generalized Erd\\H{o}s-Ginzburg-Ziv constant $\\mathsf s_{km}(G)$ is defined as the smallest positive integer $t$ such that every sequence $S$ in $G$ of length at least $t$ has a zero-sum subsequence of length $km$. It is easy to see that $\\mathsf s_{kn}(C_n^r)\\ge(k+r)n-r$ where $n,r\\in\\mathbb N$. Kubertin conjectured that the equality holds for any $k\\ge r$. In this paper, we mainly prove the following results:\n  (1) For every positive integer $k\\ge 6$, we have $$\\mathsf s_{kn}(C_n^3)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.06548","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":""},"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":"1809.06548","created_at":"2026-05-18T00:00:01.849184+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.06548v2","created_at":"2026-05-18T00:00:01.849184+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.06548","created_at":"2026-05-18T00:00:01.849184+00:00"},{"alias_kind":"pith_short_12","alias_value":"ENUMU6CNYW6L","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_16","alias_value":"ENUMU6CNYW6L62FZ","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_8","alias_value":"ENUMU6CN","created_at":"2026-05-18T12:32:22.470017+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/ENUMU6CNYW6L62FZ75R6DKFTCQ","json":"https://pith.science/pith/ENUMU6CNYW6L62FZ75R6DKFTCQ.json","graph_json":"https://pith.science/api/pith-number/ENUMU6CNYW6L62FZ75R6DKFTCQ/graph.json","events_json":"https://pith.science/api/pith-number/ENUMU6CNYW6L62FZ75R6DKFTCQ/events.json","paper":"https://pith.science/paper/ENUMU6CN"},"agent_actions":{"view_html":"https://pith.science/pith/ENUMU6CNYW6L62FZ75R6DKFTCQ","download_json":"https://pith.science/pith/ENUMU6CNYW6L62FZ75R6DKFTCQ.json","view_paper":"https://pith.science/paper/ENUMU6CN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.06548&json=true","fetch_graph":"https://pith.science/api/pith-number/ENUMU6CNYW6L62FZ75R6DKFTCQ/graph.json","fetch_events":"https://pith.science/api/pith-number/ENUMU6CNYW6L62FZ75R6DKFTCQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ENUMU6CNYW6L62FZ75R6DKFTCQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ENUMU6CNYW6L62FZ75R6DKFTCQ/action/storage_attestation","attest_author":"https://pith.science/pith/ENUMU6CNYW6L62FZ75R6DKFTCQ/action/author_attestation","sign_citation":"https://pith.science/pith/ENUMU6CNYW6L62FZ75R6DKFTCQ/action/citation_signature","submit_replication":"https://pith.science/pith/ENUMU6CNYW6L62FZ75R6DKFTCQ/action/replication_record"}},"created_at":"2026-05-18T00:00:01.849184+00:00","updated_at":"2026-05-18T00:00:01.849184+00:00"}