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For example, for any prime $p>5$ we have $$\\sum_{k=1}^{p-1}\\frac{g_k(-1)}{k}\\equiv 0\\pmod{p^2}\\quad{and}\\quad\\sum_{k=1}^{p-1}\\frac{g_k(-1)}{k^2}\\equiv 0\\pmod p.$$ This is similar to Wolstenholme's classical congruences $$\\sum_{k=1}^{p-1}\\frac1k\\equiv0\\pmod{p^2}\\quad{and}\\quad\\sum_{k=1}^{p-1}\\frac{1}{k^2}\\equiv0\\pmod p$$ for any prime $p>3$."},"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":"1407.0967","kind":"arxiv","version":8},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-03T15:58:17Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"28e074b30b42740b06160e243e18ffb3e5824a1d88ab6c94e0c8f40f1a1f69dc","abstract_canon_sha256":"bbf530715e4676b3cec852634bf7802792c919e5c3f4a147156e80893183f71d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:54.669201Z","signature_b64":"u35VH/IV7t1RtcetSp3j0K//e3vr/V4K6dRlvBWfYcGGZ+tuVGVz4hlpH8lLO8eNwdTSMIFAf57La4Hm+RoxCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e55bf8b3c6a35a08407ee9c5fbdc4950c9a71ee5362583a213e1ef31eb201f6d","last_reissued_at":"2026-05-18T01:10:54.668604Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:54.668604Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Congruences involving $g_n(x)=\\sum_{k=0}^n\\binom nk^2\\binom{2k}kx^k$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2014-07-03T15:58:17Z","abstract_excerpt":"Define $g_n(x)=\\sum_{k=0}^n\\binom nk^2\\binom{2k}kx^k$ for $n=0,1,2,...$. Those numbers $g_n=g_n(1)$ are closely related to Ap\\'ery numbers and Franel numbers. In this paper we establish some fundamental congruences involving $g_n(x)$. For example, for any prime $p>5$ we have $$\\sum_{k=1}^{p-1}\\frac{g_k(-1)}{k}\\equiv 0\\pmod{p^2}\\quad{and}\\quad\\sum_{k=1}^{p-1}\\frac{g_k(-1)}{k^2}\\equiv 0\\pmod p.$$ This is similar to Wolstenholme's classical congruences $$\\sum_{k=1}^{p-1}\\frac1k\\equiv0\\pmod{p^2}\\quad{and}\\quad\\sum_{k=1}^{p-1}\\frac{1}{k^2}\\equiv0\\pmod p$$ for any prime $p>3$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.0967","kind":"arxiv","version":8},"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":"1407.0967","created_at":"2026-05-18T01:10:54.668698+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.0967v8","created_at":"2026-05-18T01:10:54.668698+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.0967","created_at":"2026-05-18T01:10:54.668698+00:00"},{"alias_kind":"pith_short_12","alias_value":"4VN7RM6GUNNA","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_16","alias_value":"4VN7RM6GUNNAQQD6","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_8","alias_value":"4VN7RM6G","created_at":"2026-05-18T12:28:14.216126+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/4VN7RM6GUNNAQQD65HC7XXCJKD","json":"https://pith.science/pith/4VN7RM6GUNNAQQD65HC7XXCJKD.json","graph_json":"https://pith.science/api/pith-number/4VN7RM6GUNNAQQD65HC7XXCJKD/graph.json","events_json":"https://pith.science/api/pith-number/4VN7RM6GUNNAQQD65HC7XXCJKD/events.json","paper":"https://pith.science/paper/4VN7RM6G"},"agent_actions":{"view_html":"https://pith.science/pith/4VN7RM6GUNNAQQD65HC7XXCJKD","download_json":"https://pith.science/pith/4VN7RM6GUNNAQQD65HC7XXCJKD.json","view_paper":"https://pith.science/paper/4VN7RM6G","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.0967&json=true","fetch_graph":"https://pith.science/api/pith-number/4VN7RM6GUNNAQQD65HC7XXCJKD/graph.json","fetch_events":"https://pith.science/api/pith-number/4VN7RM6GUNNAQQD65HC7XXCJKD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4VN7RM6GUNNAQQD65HC7XXCJKD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4VN7RM6GUNNAQQD65HC7XXCJKD/action/storage_attestation","attest_author":"https://pith.science/pith/4VN7RM6GUNNAQQD65HC7XXCJKD/action/author_attestation","sign_citation":"https://pith.science/pith/4VN7RM6GUNNAQQD65HC7XXCJKD/action/citation_signature","submit_replication":"https://pith.science/pith/4VN7RM6GUNNAQQD65HC7XXCJKD/action/replication_record"}},"created_at":"2026-05-18T01:10:54.668698+00:00","updated_at":"2026-05-18T01:10:54.668698+00:00"}