{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:EIO7TC6HDRVO5YHSPUVJDJMDME","short_pith_number":"pith:EIO7TC6H","schema_version":"1.0","canonical_sha256":"221df98bc71c6aeee0f27d2a91a5836116a1e373285a7d237350c88463a17f94","source":{"kind":"arxiv","id":"1303.1682","version":1},"attestation_state":"computed","paper":{"title":"Minimal zero-sum sequences of length four over finite cyclic groups II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jiangtao Peng, Yuanlin Li","submitted_at":"2013-03-07T13:30:25Z","abstract_excerpt":"Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\\cdot\\ldots\\cdot(n_lg)$ where $g\\in G$ and $n_1, \\ldots, n_l\\in[1, \\ord(g)]$, and the index $\\ind(S)$ of $S$ is defined to be the minimum of $(n_1+\\cdots+n_l)/\\ord(g)$ over all possible $g\\in G$ such that $\\langle g \\rangle =G$. An open problem on the index of length four sequences asks whether or not every minimal zero-sum sequence of length 4 over a finite cyclic group $G$ with $\\gcd(|G|, 6)=1$ has index 1. In this paper, we show that if $G=\\langle g\\rangle$ is a cyclic group with order of a pro"},"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":"1303.1682","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-07T13:30:25Z","cross_cats_sorted":[],"title_canon_sha256":"47e4cd4beb45ba73e70e885a6e9da758865d028a9237256ffa74cc0892052bd2","abstract_canon_sha256":"2aa4dde2ccb0199392caed109c9b05e922c530d05fb5bac2073a6234dea615f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:31:33.432099Z","signature_b64":"3NZfrVAP6DuvhGHHIRu/hLXlWXCCESESoiQXNaDXyfShWacACdIR9SP0gYB84sd/lHp/0JCOK2cEZjwtzAJhBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"221df98bc71c6aeee0f27d2a91a5836116a1e373285a7d237350c88463a17f94","last_reissued_at":"2026-05-18T03:31:33.431611Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:31:33.431611Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimal zero-sum sequences of length four over finite cyclic groups II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jiangtao Peng, Yuanlin Li","submitted_at":"2013-03-07T13:30:25Z","abstract_excerpt":"Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\\cdot\\ldots\\cdot(n_lg)$ where $g\\in G$ and $n_1, \\ldots, n_l\\in[1, \\ord(g)]$, and the index $\\ind(S)$ of $S$ is defined to be the minimum of $(n_1+\\cdots+n_l)/\\ord(g)$ over all possible $g\\in G$ such that $\\langle g \\rangle =G$. An open problem on the index of length four sequences asks whether or not every minimal zero-sum sequence of length 4 over a finite cyclic group $G$ with $\\gcd(|G|, 6)=1$ has index 1. In this paper, we show that if $G=\\langle g\\rangle$ is a cyclic group with order of a pro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.1682","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":"1303.1682","created_at":"2026-05-18T03:31:33.431682+00:00"},{"alias_kind":"arxiv_version","alias_value":"1303.1682v1","created_at":"2026-05-18T03:31:33.431682+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.1682","created_at":"2026-05-18T03:31:33.431682+00:00"},{"alias_kind":"pith_short_12","alias_value":"EIO7TC6HDRVO","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_16","alias_value":"EIO7TC6HDRVO5YHS","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_8","alias_value":"EIO7TC6H","created_at":"2026-05-18T12:27:43.054852+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/EIO7TC6HDRVO5YHSPUVJDJMDME","json":"https://pith.science/pith/EIO7TC6HDRVO5YHSPUVJDJMDME.json","graph_json":"https://pith.science/api/pith-number/EIO7TC6HDRVO5YHSPUVJDJMDME/graph.json","events_json":"https://pith.science/api/pith-number/EIO7TC6HDRVO5YHSPUVJDJMDME/events.json","paper":"https://pith.science/paper/EIO7TC6H"},"agent_actions":{"view_html":"https://pith.science/pith/EIO7TC6HDRVO5YHSPUVJDJMDME","download_json":"https://pith.science/pith/EIO7TC6HDRVO5YHSPUVJDJMDME.json","view_paper":"https://pith.science/paper/EIO7TC6H","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1303.1682&json=true","fetch_graph":"https://pith.science/api/pith-number/EIO7TC6HDRVO5YHSPUVJDJMDME/graph.json","fetch_events":"https://pith.science/api/pith-number/EIO7TC6HDRVO5YHSPUVJDJMDME/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EIO7TC6HDRVO5YHSPUVJDJMDME/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EIO7TC6HDRVO5YHSPUVJDJMDME/action/storage_attestation","attest_author":"https://pith.science/pith/EIO7TC6HDRVO5YHSPUVJDJMDME/action/author_attestation","sign_citation":"https://pith.science/pith/EIO7TC6HDRVO5YHSPUVJDJMDME/action/citation_signature","submit_replication":"https://pith.science/pith/EIO7TC6HDRVO5YHSPUVJDJMDME/action/replication_record"}},"created_at":"2026-05-18T03:31:33.431682+00:00","updated_at":"2026-05-18T03:31:33.431682+00:00"}