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As applications we extend Lucas' law of repetition and characterize the square prime factors of $a^n+1$ or $S_n$, where $\\{S_n\\}$ is given by $S_1=P^2+2$ and $S_{k+1}=S_k^2-2\\ (k\\ge 1)$."},"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":"1312.3511","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-12-12T15:01:02Z","cross_cats_sorted":[],"title_canon_sha256":"ff316182988e37cc075702fb56964fdda2218bdcf49bf48457afeb00ec17e90c","abstract_canon_sha256":"17ad34473b5128cd95ee562b0ed1909ff6fde1ebcf705a4c61b636834b55eeb8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:50.703010Z","signature_b64":"E9h8AK3WYTGrkfxUjkuF6weLyyuTwt7fDOHyhfvUSd+gQ14PIrVXi+1dJhOZ4yNow/sttHWPhKTiYt6L61SeAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95b702ae7e9e8777bc8d92c1bd8c10a5a47de65e243bd6c1ad54ff7928fd79fb","last_reissued_at":"2026-05-18T03:04:50.702363Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:50.702363Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Congruences concerning Lucas' law of repetition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2013-12-12T15:01:02Z","abstract_excerpt":"Let $P,Q\\in\\Bbb Z$, $U_0=0,\\ U_1=1$ and $U_{n+1}=PU_n-QU_{n+1}$. 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As applications we extend Lucas' law of repetition and characterize the square prime factors of $a^n+1$ or $S_n$, where $\\{S_n\\}$ is given by $S_1=P^2+2$ and $S_{k+1}=S_k^2-2\\ (k\\ge 1)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3511","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":"1312.3511","created_at":"2026-05-18T03:04:50.702456+00:00"},{"alias_kind":"arxiv_version","alias_value":"1312.3511v1","created_at":"2026-05-18T03:04:50.702456+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.3511","created_at":"2026-05-18T03:04:50.702456+00:00"},{"alias_kind":"pith_short_12","alias_value":"SW3QFLT6T2DX","created_at":"2026-05-18T12:27:59.945178+00:00"},{"alias_kind":"pith_short_16","alias_value":"SW3QFLT6T2DXPPEN","created_at":"2026-05-18T12:27:59.945178+00:00"},{"alias_kind":"pith_short_8","alias_value":"SW3QFLT6","created_at":"2026-05-18T12:27:59.945178+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/SW3QFLT6T2DXPPENSLA33DAQUW","json":"https://pith.science/pith/SW3QFLT6T2DXPPENSLA33DAQUW.json","graph_json":"https://pith.science/api/pith-number/SW3QFLT6T2DXPPENSLA33DAQUW/graph.json","events_json":"https://pith.science/api/pith-number/SW3QFLT6T2DXPPENSLA33DAQUW/events.json","paper":"https://pith.science/paper/SW3QFLT6"},"agent_actions":{"view_html":"https://pith.science/pith/SW3QFLT6T2DXPPENSLA33DAQUW","download_json":"https://pith.science/pith/SW3QFLT6T2DXPPENSLA33DAQUW.json","view_paper":"https://pith.science/paper/SW3QFLT6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1312.3511&json=true","fetch_graph":"https://pith.science/api/pith-number/SW3QFLT6T2DXPPENSLA33DAQUW/graph.json","fetch_events":"https://pith.science/api/pith-number/SW3QFLT6T2DXPPENSLA33DAQUW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SW3QFLT6T2DXPPENSLA33DAQUW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SW3QFLT6T2DXPPENSLA33DAQUW/action/storage_attestation","attest_author":"https://pith.science/pith/SW3QFLT6T2DXPPENSLA33DAQUW/action/author_attestation","sign_citation":"https://pith.science/pith/SW3QFLT6T2DXPPENSLA33DAQUW/action/citation_signature","submit_replication":"https://pith.science/pith/SW3QFLT6T2DXPPENSLA33DAQUW/action/replication_record"}},"created_at":"2026-05-18T03:04:50.702456+00:00","updated_at":"2026-05-18T03:04:50.702456+00:00"}