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We consider the following conjecture of Alspach: For any cyclic group $\\Z_n$ and any subset $A \\subseteq \\Z_n \\setminus \\{0\\}$ with $s_k \\neq 0$, it is possible to find an ordering of the elements of $A$ such that no two of its partial sums $s_i$ and $s_j$ are equal for $0 \\leq i < j \\leq k$. We show that Alspach's Conjecture holds for prime $n$ when $k \\g"},"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":"1809.02684","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-07T21:35:28Z","cross_cats_sorted":[],"title_canon_sha256":"ed29642e463b1e681f8bba3088f1268674b57184591f302201978034ac049b85","abstract_canon_sha256":"6816d6a63ab3ed5fff931dc5cf13fa413518ed0f65958a71adfb02ae3a0a0b80"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:15.619372Z","signature_b64":"2sSHKMPMmCcXypiWiJer26BaYvPs6VsB2bDEvHVO4xPGSsclPT3/FxQR92XxqC74m9LRo9VN+lQepIfu/dJ+CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"697172ec1f5a64bf493754a5768abb301a1c3047190203144d8637becd17d34d","last_reissued_at":"2026-05-18T00:06:15.618795Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:15.618795Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Distinct Partial Sums in Cyclic Groups: Polynomial Method and Constructive Approaches","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jacob Hicks, John. R. Schmitt, M. A. Ollis","submitted_at":"2018-09-07T21:35:28Z","abstract_excerpt":"Let $(G,+)$ be an abelian group and consider a subset $A \\subseteq G$ with $|A|=k$. Given an ordering $(a_1, \\ldots, a_k)$ of the elements of $A$, define its {\\em partial sums} by $s_0 = 0$ and $s_j = \\sum_{i=1}^j a_i$ for $1 \\leq j \\leq k$. We consider the following conjecture of Alspach: For any cyclic group $\\Z_n$ and any subset $A \\subseteq \\Z_n \\setminus \\{0\\}$ with $s_k \\neq 0$, it is possible to find an ordering of the elements of $A$ such that no two of its partial sums $s_i$ and $s_j$ are equal for $0 \\leq i < j \\leq k$. We show that Alspach's Conjecture holds for prime $n$ when $k \\g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.02684","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":"1809.02684","created_at":"2026-05-18T00:06:15.618878+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.02684v1","created_at":"2026-05-18T00:06:15.618878+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.02684","created_at":"2026-05-18T00:06:15.618878+00:00"},{"alias_kind":"pith_short_12","alias_value":"NFYXF3A7LJSL","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_16","alias_value":"NFYXF3A7LJSL6SJX","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_8","alias_value":"NFYXF3A7","created_at":"2026-05-18T12:32:40.477152+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/NFYXF3A7LJSL6SJXKSSXNCV3GA","json":"https://pith.science/pith/NFYXF3A7LJSL6SJXKSSXNCV3GA.json","graph_json":"https://pith.science/api/pith-number/NFYXF3A7LJSL6SJXKSSXNCV3GA/graph.json","events_json":"https://pith.science/api/pith-number/NFYXF3A7LJSL6SJXKSSXNCV3GA/events.json","paper":"https://pith.science/paper/NFYXF3A7"},"agent_actions":{"view_html":"https://pith.science/pith/NFYXF3A7LJSL6SJXKSSXNCV3GA","download_json":"https://pith.science/pith/NFYXF3A7LJSL6SJXKSSXNCV3GA.json","view_paper":"https://pith.science/paper/NFYXF3A7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.02684&json=true","fetch_graph":"https://pith.science/api/pith-number/NFYXF3A7LJSL6SJXKSSXNCV3GA/graph.json","fetch_events":"https://pith.science/api/pith-number/NFYXF3A7LJSL6SJXKSSXNCV3GA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NFYXF3A7LJSL6SJXKSSXNCV3GA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NFYXF3A7LJSL6SJXKSSXNCV3GA/action/storage_attestation","attest_author":"https://pith.science/pith/NFYXF3A7LJSL6SJXKSSXNCV3GA/action/author_attestation","sign_citation":"https://pith.science/pith/NFYXF3A7LJSL6SJXKSSXNCV3GA/action/citation_signature","submit_replication":"https://pith.science/pith/NFYXF3A7LJSL6SJXKSSXNCV3GA/action/replication_record"}},"created_at":"2026-05-18T00:06:15.618878+00:00","updated_at":"2026-05-18T00:06:15.618878+00:00"}