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We mainly establish for any prime $p>3$ the following congruences:\n  \\begin{align*}\\sum_{k=0}^{p-1}(-1)^kf_k&\\equiv\\left(\\frac p3\\right)\\ \\ (\\mbox{mod}\\ p^2), \\\\ \\sum_{k=0}^{p-1}(-1)^k\\,kf_k&\\equiv-\\frac 23\\left(\\frac p3\\right)\\ \\ (\\mbox{mod}\\ p^2), \\\\ \\sum_{k=1}^{p-1}\\frac{(-1)^k}kf_k &\\equiv0\\ \\ (\\mbox{mod}\\ p^2), \\\\ \\sum_{k=1}^{p-1}\\frac{(-1)^k}{k"},"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":"1112.1034","kind":"arxiv","version":11},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-12-05T19:38:56Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"351a5ce4a9674a36889fab453382895869fc5fcca46ff7841789205878743ff0","abstract_canon_sha256":"7a58b18a5821657e15df2b5069b8274ba76533d12aecd5e9c768eec710a5e660"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:21:28.003951Z","signature_b64":"ZZXup9JH4azMS25ThkwqbwAF/rqYsET7pDJS3J0MJav4V2dvuQaz4rkxwEoeF83b8pTfaMTgPoFAkHamN2nCBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc2e1d2902ee37a7b803febd4449a3051f7fbff4ba34ec7fed7f0236229b5d6e","last_reissued_at":"2026-05-18T02:21:28.003218Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:21:28.003218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Congruences for Franel numbers","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":"2011-12-05T19:38:56Z","abstract_excerpt":"The Franel numbers given by $f_n=\\sum_{k=0}^n\\binom{n}{k}^3$ ($n=0,1,2,\\ldots$) play important roles in both combinatorics and number theory. In this paper we initiate the systematic investigation of fundamental congruences for the Franel numbers. We mainly establish for any prime $p>3$ the following congruences:\n  \\begin{align*}\\sum_{k=0}^{p-1}(-1)^kf_k&\\equiv\\left(\\frac p3\\right)\\ \\ (\\mbox{mod}\\ p^2), \\\\ \\sum_{k=0}^{p-1}(-1)^k\\,kf_k&\\equiv-\\frac 23\\left(\\frac p3\\right)\\ \\ (\\mbox{mod}\\ p^2), \\\\ \\sum_{k=1}^{p-1}\\frac{(-1)^k}kf_k &\\equiv0\\ \\ (\\mbox{mod}\\ p^2), \\\\ \\sum_{k=1}^{p-1}\\frac{(-1)^k}{k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.1034","kind":"arxiv","version":11},"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":"1112.1034","created_at":"2026-05-18T02:21:28.003367+00:00"},{"alias_kind":"arxiv_version","alias_value":"1112.1034v11","created_at":"2026-05-18T02:21:28.003367+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.1034","created_at":"2026-05-18T02:21:28.003367+00:00"},{"alias_kind":"pith_short_12","alias_value":"3QXB2KIC5Y32","created_at":"2026-05-18T12:26:18.847500+00:00"},{"alias_kind":"pith_short_16","alias_value":"3QXB2KIC5Y32POAD","created_at":"2026-05-18T12:26:18.847500+00:00"},{"alias_kind":"pith_short_8","alias_value":"3QXB2KIC","created_at":"2026-05-18T12:26:18.847500+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/3QXB2KIC5Y32POAD726UISNDAU","json":"https://pith.science/pith/3QXB2KIC5Y32POAD726UISNDAU.json","graph_json":"https://pith.science/api/pith-number/3QXB2KIC5Y32POAD726UISNDAU/graph.json","events_json":"https://pith.science/api/pith-number/3QXB2KIC5Y32POAD726UISNDAU/events.json","paper":"https://pith.science/paper/3QXB2KIC"},"agent_actions":{"view_html":"https://pith.science/pith/3QXB2KIC5Y32POAD726UISNDAU","download_json":"https://pith.science/pith/3QXB2KIC5Y32POAD726UISNDAU.json","view_paper":"https://pith.science/paper/3QXB2KIC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1112.1034&json=true","fetch_graph":"https://pith.science/api/pith-number/3QXB2KIC5Y32POAD726UISNDAU/graph.json","fetch_events":"https://pith.science/api/pith-number/3QXB2KIC5Y32POAD726UISNDAU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3QXB2KIC5Y32POAD726UISNDAU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3QXB2KIC5Y32POAD726UISNDAU/action/storage_attestation","attest_author":"https://pith.science/pith/3QXB2KIC5Y32POAD726UISNDAU/action/author_attestation","sign_citation":"https://pith.science/pith/3QXB2KIC5Y32POAD726UISNDAU/action/citation_signature","submit_replication":"https://pith.science/pith/3QXB2KIC5Y32POAD726UISNDAU/action/replication_record"}},"created_at":"2026-05-18T02:21:28.003367+00:00","updated_at":"2026-05-18T02:21:28.003367+00:00"}