{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:SEUYWIJ2F2LGIOU2DLXD27MMVO","short_pith_number":"pith:SEUYWIJ2","canonical_record":{"source":{"id":"0911.3060","kind":"arxiv","version":9},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-11-16T19:06:17Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"70ca84b6fb016a356eac06b684b9bc8f647c8892b118f72f62298a85e87c2ed3","abstract_canon_sha256":"b869e56e7c495db5fa756bce2f4e7d7fe5576997c8e20ef1b25fafa7e82d933e"},"schema_version":"1.0"},"canonical_sha256":"91298b213a2e96643a9a1aee3d7d8cab9bcad0141b8df9d61ddd90e84c932aad","source":{"kind":"arxiv","id":"0911.3060","version":9},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0911.3060","created_at":"2026-05-18T03:08:25Z"},{"alias_kind":"arxiv_version","alias_value":"0911.3060v9","created_at":"2026-05-18T03:08:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.3060","created_at":"2026-05-18T03:08:25Z"},{"alias_kind":"pith_short_12","alias_value":"SEUYWIJ2F2LG","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"SEUYWIJ2F2LGIOU2","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"SEUYWIJ2","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:SEUYWIJ2F2LGIOU2DLXD27MMVO","target":"record","payload":{"canonical_record":{"source":{"id":"0911.3060","kind":"arxiv","version":9},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-11-16T19:06:17Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"70ca84b6fb016a356eac06b684b9bc8f647c8892b118f72f62298a85e87c2ed3","abstract_canon_sha256":"b869e56e7c495db5fa756bce2f4e7d7fe5576997c8e20ef1b25fafa7e82d933e"},"schema_version":"1.0"},"canonical_sha256":"91298b213a2e96643a9a1aee3d7d8cab9bcad0141b8df9d61ddd90e84c932aad","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:08:25.442964Z","signature_b64":"5qqfxNgzR3T7g0MBeucM86XiMmAWikySnKiRKv/R4xrO3SLK23tO5IpuAnvggNhuNagDh3KC4MnI6vVnFBr2DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91298b213a2e96643a9a1aee3d7d8cab9bcad0141b8df9d61ddd90e84c932aad","last_reissued_at":"2026-05-18T03:08:25.442165Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:08:25.442165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0911.3060","source_version":9,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:08:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"62vWn/Z9hFp6k0uSzD0lbhTMntzaam5S/XO/5ndudBNcq1hNRZvRKCrwq7Iz3AZlzyGo3gArWqRMYJxx446oAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T21:23:21.207885Z"},"content_sha256":"1da381cc8d22c29ee933c18c81c983ee17c2b6758172bd0577b97233bf1d8dc0","schema_version":"1.0","event_id":"sha256:1da381cc8d22c29ee933c18c81c983ee17c2b6758172bd0577b97233bf1d8dc0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:SEUYWIJ2F2LGIOU2DLXD27MMVO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fibonacci numbers modulo cubes of primes","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":"2009-11-16T19:06:17Z","abstract_excerpt":"Let $p$ be an odd prime. It is well known that $F_{p-(\\frac p5)}\\equiv 0\\pmod{p}$, where $\\{F_n\\}_{n\\ge0}$ is the Fibonacci sequence and $(-)$ is the Jacobi symbol. In this paper we show that if $p\\not=5$ then we may determine $F_{p-(\\frac p5)}$ mod $p^3$ in the following way: $$\\sum_{k=0}^{(p-1)/2}\\frac{\\binom{2k}k}{(-16)^k}\\equiv\\left(\\frac{p}5\\right)\\left(1+\\frac{F_{p-(\\frac {p}5)}}2\\right)\\pmod{p^3}.$$ We also use Lucas quotients to determine $\\sum_{k=0}^{(p-1)/2}\\binom{2k}k/m^k$ modulo $p^2$ for any integer $m\\not\\equiv0\\pmod{p}$; in particular, we obtain $$\\sum_{k=0}^{(p-1)/2}\\frac{\\bino"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.3060","kind":"arxiv","version":9},"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":"2026-05-18T03:08:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a5nDzu3s0G3QEsLU9sT6uyTrqQsvkUBRthsUBgzIAy0JHqzaz4TkldB9Xq7Y1U5g9UhwZD5y5koEjNXqE16iDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T21:23:21.208599Z"},"content_sha256":"59cec4d841502d7408c6b651840b528e5fb6c1c796806a093496eee1725fd9de","schema_version":"1.0","event_id":"sha256:59cec4d841502d7408c6b651840b528e5fb6c1c796806a093496eee1725fd9de"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SEUYWIJ2F2LGIOU2DLXD27MMVO/bundle.json","state_url":"https://pith.science/pith/SEUYWIJ2F2LGIOU2DLXD27MMVO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SEUYWIJ2F2LGIOU2DLXD27MMVO/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":"2026-05-22T21:23:21Z","links":{"resolver":"https://pith.science/pith/SEUYWIJ2F2LGIOU2DLXD27MMVO","bundle":"https://pith.science/pith/SEUYWIJ2F2LGIOU2DLXD27MMVO/bundle.json","state":"https://pith.science/pith/SEUYWIJ2F2LGIOU2DLXD27MMVO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SEUYWIJ2F2LGIOU2DLXD27MMVO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:SEUYWIJ2F2LGIOU2DLXD27MMVO","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":"b869e56e7c495db5fa756bce2f4e7d7fe5576997c8e20ef1b25fafa7e82d933e","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-11-16T19:06:17Z","title_canon_sha256":"70ca84b6fb016a356eac06b684b9bc8f647c8892b118f72f62298a85e87c2ed3"},"schema_version":"1.0","source":{"id":"0911.3060","kind":"arxiv","version":9}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0911.3060","created_at":"2026-05-18T03:08:25Z"},{"alias_kind":"arxiv_version","alias_value":"0911.3060v9","created_at":"2026-05-18T03:08:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.3060","created_at":"2026-05-18T03:08:25Z"},{"alias_kind":"pith_short_12","alias_value":"SEUYWIJ2F2LG","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"SEUYWIJ2F2LGIOU2","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"SEUYWIJ2","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:59cec4d841502d7408c6b651840b528e5fb6c1c796806a093496eee1725fd9de","target":"graph","created_at":"2026-05-18T03:08:25Z","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":"Let $p$ be an odd prime. It is well known that $F_{p-(\\frac p5)}\\equiv 0\\pmod{p}$, where $\\{F_n\\}_{n\\ge0}$ is the Fibonacci sequence and $(-)$ is the Jacobi symbol. In this paper we show that if $p\\not=5$ then we may determine $F_{p-(\\frac p5)}$ mod $p^3$ in the following way: $$\\sum_{k=0}^{(p-1)/2}\\frac{\\binom{2k}k}{(-16)^k}\\equiv\\left(\\frac{p}5\\right)\\left(1+\\frac{F_{p-(\\frac {p}5)}}2\\right)\\pmod{p^3}.$$ We also use Lucas quotients to determine $\\sum_{k=0}^{(p-1)/2}\\binom{2k}k/m^k$ modulo $p^2$ for any integer $m\\not\\equiv0\\pmod{p}$; in particular, we obtain $$\\sum_{k=0}^{(p-1)/2}\\frac{\\bino","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":"2009-11-16T19:06:17Z","title":"Fibonacci numbers modulo cubes of primes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.3060","kind":"arxiv","version":9},"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":"sha256:1da381cc8d22c29ee933c18c81c983ee17c2b6758172bd0577b97233bf1d8dc0","target":"record","created_at":"2026-05-18T03:08:25Z","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":"b869e56e7c495db5fa756bce2f4e7d7fe5576997c8e20ef1b25fafa7e82d933e","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-11-16T19:06:17Z","title_canon_sha256":"70ca84b6fb016a356eac06b684b9bc8f647c8892b118f72f62298a85e87c2ed3"},"schema_version":"1.0","source":{"id":"0911.3060","kind":"arxiv","version":9}},"canonical_sha256":"91298b213a2e96643a9a1aee3d7d8cab9bcad0141b8df9d61ddd90e84c932aad","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"91298b213a2e96643a9a1aee3d7d8cab9bcad0141b8df9d61ddd90e84c932aad","first_computed_at":"2026-05-18T03:08:25.442165Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:08:25.442165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5qqfxNgzR3T7g0MBeucM86XiMmAWikySnKiRKv/R4xrO3SLK23tO5IpuAnvggNhuNagDh3KC4MnI6vVnFBr2DA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:08:25.442964Z","signed_message":"canonical_sha256_bytes"},"source_id":"0911.3060","source_kind":"arxiv","source_version":9}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1da381cc8d22c29ee933c18c81c983ee17c2b6758172bd0577b97233bf1d8dc0","sha256:59cec4d841502d7408c6b651840b528e5fb6c1c796806a093496eee1725fd9de"],"state_sha256":"41d7aef180f042933da66134ff108758b8ec9af48c1ade08392eb04396623e36"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wC/llEnLYEoaj6a3VIY2KflPgvDaWzHVDZD9l7P3piswKbh2Qs0rq3fXXC9RQR6FTAD9Dh0T65Gg5htVgsAHBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-22T21:23:21.212246Z","bundle_sha256":"5c77f6b3ed130c8caaa88c1271be677827088811b5edbbe44f8e235170de1b34"}}