{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:UCGDQJK7MK7A3BR2IP5P7ZNBN7","short_pith_number":"pith:UCGDQJK7","schema_version":"1.0","canonical_sha256":"a08c38255f62be0d863a43faffe5a16fe8a935c7c7eac86f8c19f50c5aaeca98","source":{"kind":"arxiv","id":"1501.00573","version":1},"attestation_state":"computed","paper":{"title":"Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Ji-Cai Liu, Victor J. W. Guo","submitted_at":"2015-01-03T15:54:44Z","abstract_excerpt":"The numbers $R_n$ and $W_n$ are defined as \\begin{align*} R_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k}\\frac{1}{2k-1},\\ \\text{and}\\ W_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k}\\frac{3}{2k-3}. \\end{align*} We prove that, for any positive integer $n$ and odd prime $p$, there hold \\begin{align*} \\sum_{k=0}^{n-1}(2k+1)R_k^2 &\\equiv 0 \\pmod{n}, \\\\ \\sum_{k=0}^{p-1}(2k+1)R_k^2 &\\equiv 4p(-1)^{\\frac{p-1}{2}} -p^2 \\pmod{p^3}, \\\\ 9\\sum_{k=0}^{n-1}(2k+1)W_k^2 &\\equiv 0 \\pmod{n}, \\\\ \\sum_{k=0}^{p-1}(2k+1)W_k^2 &\\equiv 12p(-1)^{\\frac{p-1}{2}}-17p^2 \\pmod{p^3}, \\quad\\text{if $p>3$.} \\end{align*} The firs"},"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":"1501.00573","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-01-03T15:54:44Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b2cdea41e556fcfb9ba956745e12e5d721bf7bf3a9a58e75bf06d2cf7d8fbcd5","abstract_canon_sha256":"2bdcda7acc289a88853bd0e394b2362f019d48fc0a8daf86484c22c1f524be65"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:07.211780Z","signature_b64":"bc0kOelAidIaH3r+c31/B2IWosHp3AWrXE9VUb5rgpxPXZATAQOENoQY4fUtKSge76EjnO9C0ehoYHzYenLaAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a08c38255f62be0d863a43faffe5a16fe8a935c7c7eac86f8c19f50c5aaeca98","last_reissued_at":"2026-05-18T02:30:07.211045Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:07.211045Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Ji-Cai Liu, Victor J. W. Guo","submitted_at":"2015-01-03T15:54:44Z","abstract_excerpt":"The numbers $R_n$ and $W_n$ are defined as \\begin{align*} R_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k}\\frac{1}{2k-1},\\ \\text{and}\\ W_n=\\sum_{k=0}^{n}{n+k\\choose 2k}{2k\\choose k}\\frac{3}{2k-3}. \\end{align*} We prove that, for any positive integer $n$ and odd prime $p$, there hold \\begin{align*} \\sum_{k=0}^{n-1}(2k+1)R_k^2 &\\equiv 0 \\pmod{n}, \\\\ \\sum_{k=0}^{p-1}(2k+1)R_k^2 &\\equiv 4p(-1)^{\\frac{p-1}{2}} -p^2 \\pmod{p^3}, \\\\ 9\\sum_{k=0}^{n-1}(2k+1)W_k^2 &\\equiv 0 \\pmod{n}, \\\\ \\sum_{k=0}^{p-1}(2k+1)W_k^2 &\\equiv 12p(-1)^{\\frac{p-1}{2}}-17p^2 \\pmod{p^3}, \\quad\\text{if $p>3$.} \\end{align*} The firs"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.00573","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":"1501.00573","created_at":"2026-05-18T02:30:07.211164+00:00"},{"alias_kind":"arxiv_version","alias_value":"1501.00573v1","created_at":"2026-05-18T02:30:07.211164+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.00573","created_at":"2026-05-18T02:30:07.211164+00:00"},{"alias_kind":"pith_short_12","alias_value":"UCGDQJK7MK7A","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"UCGDQJK7MK7A3BR2","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"UCGDQJK7","created_at":"2026-05-18T12:29:44.643036+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/UCGDQJK7MK7A3BR2IP5P7ZNBN7","json":"https://pith.science/pith/UCGDQJK7MK7A3BR2IP5P7ZNBN7.json","graph_json":"https://pith.science/api/pith-number/UCGDQJK7MK7A3BR2IP5P7ZNBN7/graph.json","events_json":"https://pith.science/api/pith-number/UCGDQJK7MK7A3BR2IP5P7ZNBN7/events.json","paper":"https://pith.science/paper/UCGDQJK7"},"agent_actions":{"view_html":"https://pith.science/pith/UCGDQJK7MK7A3BR2IP5P7ZNBN7","download_json":"https://pith.science/pith/UCGDQJK7MK7A3BR2IP5P7ZNBN7.json","view_paper":"https://pith.science/paper/UCGDQJK7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1501.00573&json=true","fetch_graph":"https://pith.science/api/pith-number/UCGDQJK7MK7A3BR2IP5P7ZNBN7/graph.json","fetch_events":"https://pith.science/api/pith-number/UCGDQJK7MK7A3BR2IP5P7ZNBN7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UCGDQJK7MK7A3BR2IP5P7ZNBN7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UCGDQJK7MK7A3BR2IP5P7ZNBN7/action/storage_attestation","attest_author":"https://pith.science/pith/UCGDQJK7MK7A3BR2IP5P7ZNBN7/action/author_attestation","sign_citation":"https://pith.science/pith/UCGDQJK7MK7A3BR2IP5P7ZNBN7/action/citation_signature","submit_replication":"https://pith.science/pith/UCGDQJK7MK7A3BR2IP5P7ZNBN7/action/replication_record"}},"created_at":"2026-05-18T02:30:07.211164+00:00","updated_at":"2026-05-18T02:30:07.211164+00:00"}