{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:UTN2IRYH7JDPXEKCRAHYY6FL5H","short_pith_number":"pith:UTN2IRYH","schema_version":"1.0","canonical_sha256":"a4dba44707fa46fb9142880f8c78abe9eb2f2def7fd27668e53aa617fa435399","source":{"kind":"arxiv","id":"1302.3678","version":1},"attestation_state":"computed","paper":{"title":"Morley's other miracle: $\\displaystyle 4^{p-1}\\equiv\\pm (\\smallmatrix p-1 \\frac{p-1}{2} \\endsmallmatrix) \\pmod {p^3} $","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christian Aebi, Grant Cairns","submitted_at":"2013-02-15T05:19:59Z","abstract_excerpt":"Frank Morley is famous for his theorem concerning the angle trisectors of a triangle. This note gives an elementary proof of another result of Morley's, which relates the middle binomial coefficient to a certain power of two. The striking thing about Morley's congruence is that it is valid modulo the third power of the prime being considered."},"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":"1302.3678","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-02-15T05:19:59Z","cross_cats_sorted":[],"title_canon_sha256":"03a9775891b217d7feb45c1e4a470aaeabe2d67054027911404b7c09b7ff714d","abstract_canon_sha256":"554427c3abd17b96d52b3198d59328a1d9e4f26a7817bec062925767d3689e91"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:33:36.151614Z","signature_b64":"P+umEYclYwccdwOr2D677pEgU1EyLhRRspwPiGwLgcp1lPNj2mIfuP/LTiLA0xjfkrMGBmwNTFFnROX1W6vHDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a4dba44707fa46fb9142880f8c78abe9eb2f2def7fd27668e53aa617fa435399","last_reissued_at":"2026-05-18T03:33:36.150718Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:33:36.150718Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Morley's other miracle: $\\displaystyle 4^{p-1}\\equiv\\pm (\\smallmatrix p-1 \\frac{p-1}{2} \\endsmallmatrix) \\pmod {p^3} $","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christian Aebi, Grant Cairns","submitted_at":"2013-02-15T05:19:59Z","abstract_excerpt":"Frank Morley is famous for his theorem concerning the angle trisectors of a triangle. This note gives an elementary proof of another result of Morley's, which relates the middle binomial coefficient to a certain power of two. The striking thing about Morley's congruence is that it is valid modulo the third power of the prime being considered."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.3678","kind":"arxiv","version":1},"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":"1302.3678","created_at":"2026-05-18T03:33:36.150862+00:00"},{"alias_kind":"arxiv_version","alias_value":"1302.3678v1","created_at":"2026-05-18T03:33:36.150862+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.3678","created_at":"2026-05-18T03:33:36.150862+00:00"},{"alias_kind":"pith_short_12","alias_value":"UTN2IRYH7JDP","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_16","alias_value":"UTN2IRYH7JDPXEKC","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_8","alias_value":"UTN2IRYH","created_at":"2026-05-18T12:28:02.375192+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/UTN2IRYH7JDPXEKCRAHYY6FL5H","json":"https://pith.science/pith/UTN2IRYH7JDPXEKCRAHYY6FL5H.json","graph_json":"https://pith.science/api/pith-number/UTN2IRYH7JDPXEKCRAHYY6FL5H/graph.json","events_json":"https://pith.science/api/pith-number/UTN2IRYH7JDPXEKCRAHYY6FL5H/events.json","paper":"https://pith.science/paper/UTN2IRYH"},"agent_actions":{"view_html":"https://pith.science/pith/UTN2IRYH7JDPXEKCRAHYY6FL5H","download_json":"https://pith.science/pith/UTN2IRYH7JDPXEKCRAHYY6FL5H.json","view_paper":"https://pith.science/paper/UTN2IRYH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1302.3678&json=true","fetch_graph":"https://pith.science/api/pith-number/UTN2IRYH7JDPXEKCRAHYY6FL5H/graph.json","fetch_events":"https://pith.science/api/pith-number/UTN2IRYH7JDPXEKCRAHYY6FL5H/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UTN2IRYH7JDPXEKCRAHYY6FL5H/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UTN2IRYH7JDPXEKCRAHYY6FL5H/action/storage_attestation","attest_author":"https://pith.science/pith/UTN2IRYH7JDPXEKCRAHYY6FL5H/action/author_attestation","sign_citation":"https://pith.science/pith/UTN2IRYH7JDPXEKCRAHYY6FL5H/action/citation_signature","submit_replication":"https://pith.science/pith/UTN2IRYH7JDPXEKCRAHYY6FL5H/action/replication_record"}},"created_at":"2026-05-18T03:33:36.150862+00:00","updated_at":"2026-05-18T03:33:36.150862+00:00"}