{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:3QXB2KIC5Y32POAD726UISNDAU","short_pith_number":"pith:3QXB2KIC","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"},"canonical_sha256":"dc2e1d2902ee37a7b803febd4449a3051f7fbff4ba34ec7fed7f0236229b5d6e","source":{"kind":"arxiv","id":"1112.1034","version":11},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.1034","created_at":"2026-05-18T02:21:28Z"},{"alias_kind":"arxiv_version","alias_value":"1112.1034v11","created_at":"2026-05-18T02:21:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.1034","created_at":"2026-05-18T02:21:28Z"},{"alias_kind":"pith_short_12","alias_value":"3QXB2KIC5Y32","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"3QXB2KIC5Y32POAD","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"3QXB2KIC","created_at":"2026-05-18T12:26:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:3QXB2KIC5Y32POAD726UISNDAU","target":"record","payload":{"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"},"canonical_sha256":"dc2e1d2902ee37a7b803febd4449a3051f7fbff4ba34ec7fed7f0236229b5d6e","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"},"source_kind":"arxiv","source_id":"1112.1034","source_version":11,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:21:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h6iwaJ8nAJekYHeCewugqK4/+l1ZilCVKn+KWfR+ztI/GdzQ3221SKBwQ4k3b2tHP6nD+6HrSxqI0r4pL7PnAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T22:14:59.709122Z"},"content_sha256":"4084bd44daedfc1469564f0a4a79570740e12945ea35297706d293a0502ad604","schema_version":"1.0","event_id":"sha256:4084bd44daedfc1469564f0a4a79570740e12945ea35297706d293a0502ad604"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:3QXB2KIC5Y32POAD726UISNDAU","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:21:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FORDcjiH1L7VzHxSHUidNlN/prtTEum9NBv9vHoB1i5y/KJwMiK4UX/JI7lbNP4N0oDoerHFQWfY8WAYjhViCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T22:14:59.709518Z"},"content_sha256":"442ba1d6179c2f25e215af753de9c8a035887e1454f6208ce5ef6a4c1dfa401d","schema_version":"1.0","event_id":"sha256:442ba1d6179c2f25e215af753de9c8a035887e1454f6208ce5ef6a4c1dfa401d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3QXB2KIC5Y32POAD726UISNDAU/bundle.json","state_url":"https://pith.science/pith/3QXB2KIC5Y32POAD726UISNDAU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3QXB2KIC5Y32POAD726UISNDAU/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-23T22:14:59Z","links":{"resolver":"https://pith.science/pith/3QXB2KIC5Y32POAD726UISNDAU","bundle":"https://pith.science/pith/3QXB2KIC5Y32POAD726UISNDAU/bundle.json","state":"https://pith.science/pith/3QXB2KIC5Y32POAD726UISNDAU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3QXB2KIC5Y32POAD726UISNDAU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:3QXB2KIC5Y32POAD726UISNDAU","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"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}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.1034","created_at":"2026-05-18T02:21:28Z"},{"alias_kind":"arxiv_version","alias_value":"1112.1034v11","created_at":"2026-05-18T02:21:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.1034","created_at":"2026-05-18T02:21:28Z"},{"alias_kind":"pith_short_12","alias_value":"3QXB2KIC5Y32","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"3QXB2KIC5Y32POAD","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"3QXB2KIC","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:442ba1d6179c2f25e215af753de9c8a035887e1454f6208ce5ef6a4c1dfa401d","target":"graph","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":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","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"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7lFeLsuWedKv6wadKqc1eS3i1q636+qAGF7uoDw/py8ooPkZFJCrrvNKhYzNVfeh4VIBpteWnV2xTftBBsaPCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T22:14:59.712843Z","bundle_sha256":"9f219d8a70cf09cc930a75b1f0204310f64875c648abe56d6bdc5a5cb22c6ea0"}}