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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","authors_text":"Zhi-Wei Sun","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-12-05T19:38:56Z","title":"Congruences for Franel numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.1034","kind":"arxiv","version":11},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:4084bd44daedfc1469564f0a4a79570740e12945ea35297706d293a0502ad604","target":"record","created_at":"2026-05-18T02:21:28Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7a58b18a5821657e15df2b5069b8274ba76533d12aecd5e9c768eec710a5e660","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-12-05T19:38:56Z","title_canon_sha256":"351a5ce4a9674a36889fab453382895869fc5fcca46ff7841789205878743ff0"},"schema_version":"1.0","source":{"id":"1112.1034","kind":"arxiv","version":11}},"canonical_sha256":"dc2e1d2902ee37a7b803febd4449a3051f7fbff4ba34ec7fed7f0236229b5d6e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dc2e1d2902ee37a7b803febd4449a3051f7fbff4ba34ec7fed7f0236229b5d6e","first_computed_at":"2026-05-18T02:21:28.003218Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:21:28.003218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZZXup9JH4azMS25ThkwqbwAF/rqYsET7pDJS3J0MJav4V2dvuQaz4rkxwEoeF83b8pTfaMTgPoFAkHamN2nCBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:21:28.003951Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.1034","source_kind":"arxiv","source_version":11}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4084bd44daedfc1469564f0a4a79570740e12945ea35297706d293a0502ad604","sha256:442ba1d6179c2f25e215af753de9c8a035887e1454f6208ce5ef6a4c1dfa401d"],"state_sha256":"0365351495efa444a747604f14fa571df87d8a48236591d272ddee9ff22cc983"}