{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:SEUYWIJ2F2LGIOU2DLXD27MMVO","short_pith_number":"pith:SEUYWIJ2","schema_version":"1.0","canonical_sha256":"91298b213a2e96643a9a1aee3d7d8cab9bcad0141b8df9d61ddd90e84c932aad","source":{"kind":"arxiv","id":"0911.3060","version":9},"attestation_state":"computed","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"},"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":"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0911.3060","created_at":"2026-05-18T03:08:25.442292+00:00"},{"alias_kind":"arxiv_version","alias_value":"0911.3060v9","created_at":"2026-05-18T03:08:25.442292+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.3060","created_at":"2026-05-18T03:08:25.442292+00:00"},{"alias_kind":"pith_short_12","alias_value":"SEUYWIJ2F2LG","created_at":"2026-05-18T12:26:01.383474+00:00"},{"alias_kind":"pith_short_16","alias_value":"SEUYWIJ2F2LGIOU2","created_at":"2026-05-18T12:26:01.383474+00:00"},{"alias_kind":"pith_short_8","alias_value":"SEUYWIJ2","created_at":"2026-05-18T12:26:01.383474+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/SEUYWIJ2F2LGIOU2DLXD27MMVO","json":"https://pith.science/pith/SEUYWIJ2F2LGIOU2DLXD27MMVO.json","graph_json":"https://pith.science/api/pith-number/SEUYWIJ2F2LGIOU2DLXD27MMVO/graph.json","events_json":"https://pith.science/api/pith-number/SEUYWIJ2F2LGIOU2DLXD27MMVO/events.json","paper":"https://pith.science/paper/SEUYWIJ2"},"agent_actions":{"view_html":"https://pith.science/pith/SEUYWIJ2F2LGIOU2DLXD27MMVO","download_json":"https://pith.science/pith/SEUYWIJ2F2LGIOU2DLXD27MMVO.json","view_paper":"https://pith.science/paper/SEUYWIJ2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0911.3060&json=true","fetch_graph":"https://pith.science/api/pith-number/SEUYWIJ2F2LGIOU2DLXD27MMVO/graph.json","fetch_events":"https://pith.science/api/pith-number/SEUYWIJ2F2LGIOU2DLXD27MMVO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SEUYWIJ2F2LGIOU2DLXD27MMVO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SEUYWIJ2F2LGIOU2DLXD27MMVO/action/storage_attestation","attest_author":"https://pith.science/pith/SEUYWIJ2F2LGIOU2DLXD27MMVO/action/author_attestation","sign_citation":"https://pith.science/pith/SEUYWIJ2F2LGIOU2DLXD27MMVO/action/citation_signature","submit_replication":"https://pith.science/pith/SEUYWIJ2F2LGIOU2DLXD27MMVO/action/replication_record"}},"created_at":"2026-05-18T03:08:25.442292+00:00","updated_at":"2026-05-18T03:08:25.442292+00:00"}