{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:DVMBTDXFLVEMEATEMX2AILPHMG","short_pith_number":"pith:DVMBTDXF","schema_version":"1.0","canonical_sha256":"1d58198ee55d48c2026465f4042de761bfc58bbc2d7992cde21a1ffbe037ed9e","source":{"kind":"arxiv","id":"1505.00668","version":2},"attestation_state":"computed","paper":{"title":"Congruences for Catalan-Larcombe-French numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Xiao-Juan Ji, Zhi-Hong Sun","submitted_at":"2015-05-04T15:02:27Z","abstract_excerpt":"Let $\\{P_n\\}$ be the Catalan-Larcombe-French numbers given by $P_0=1,\\ P_1=8$ and $n^2P_n=8(3n^2-3n+1)P_{n-1}-128(n-1)^2P_{n-2}$ $(n\\ge 2)$, and let $S_n=P_n/2^n$. In this paper we deduce congruences for $S_{mp^r}\\pmod{p^{r+2}}$, $S_{mp^r-1}\\pmod{p^r}$ and $S_{mp^r+1}\\pmod{p^{2r}}$, where $p$ is an odd prime and $m,r$ are positive integers. We also prove that $S_{(p^2-1)/2}\\equiv 0\\pmod {p^2}$ for any prime $p\\equiv 5,7\\pmod 8$, and show that $\\{S_m\\}$ is log-convex."},"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":"1505.00668","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-04T15:02:27Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ea640ad5ef02ae6351f899573f5a47cc0646dc86a76025b412e32d4712ed3913","abstract_canon_sha256":"d041a9f35bc28943b1ceaf9bd787fe7214616ba93a134d60f95402897233751d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:53.668088Z","signature_b64":"SEO6CJP0YjWl+gq+v3S6wJ5936kCK1/P6xWxImWbQNRzfbuYdYlte4Wub8jijEW/2D3CUHCgq18ORzCVvrdPBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d58198ee55d48c2026465f4042de761bfc58bbc2d7992cde21a1ffbe037ed9e","last_reissued_at":"2026-05-18T01:36:53.667553Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:53.667553Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Congruences for Catalan-Larcombe-French numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Xiao-Juan Ji, Zhi-Hong Sun","submitted_at":"2015-05-04T15:02:27Z","abstract_excerpt":"Let $\\{P_n\\}$ be the Catalan-Larcombe-French numbers given by $P_0=1,\\ P_1=8$ and $n^2P_n=8(3n^2-3n+1)P_{n-1}-128(n-1)^2P_{n-2}$ $(n\\ge 2)$, and let $S_n=P_n/2^n$. In this paper we deduce congruences for $S_{mp^r}\\pmod{p^{r+2}}$, $S_{mp^r-1}\\pmod{p^r}$ and $S_{mp^r+1}\\pmod{p^{2r}}$, where $p$ is an odd prime and $m,r$ are positive integers. We also prove that $S_{(p^2-1)/2}\\equiv 0\\pmod {p^2}$ for any prime $p\\equiv 5,7\\pmod 8$, and show that $\\{S_m\\}$ is log-convex."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.00668","kind":"arxiv","version":2},"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":"1505.00668","created_at":"2026-05-18T01:36:53.667638+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.00668v2","created_at":"2026-05-18T01:36:53.667638+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.00668","created_at":"2026-05-18T01:36:53.667638+00:00"},{"alias_kind":"pith_short_12","alias_value":"DVMBTDXFLVEM","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_16","alias_value":"DVMBTDXFLVEMEATE","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_8","alias_value":"DVMBTDXF","created_at":"2026-05-18T12:29:17.054201+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/DVMBTDXFLVEMEATEMX2AILPHMG","json":"https://pith.science/pith/DVMBTDXFLVEMEATEMX2AILPHMG.json","graph_json":"https://pith.science/api/pith-number/DVMBTDXFLVEMEATEMX2AILPHMG/graph.json","events_json":"https://pith.science/api/pith-number/DVMBTDXFLVEMEATEMX2AILPHMG/events.json","paper":"https://pith.science/paper/DVMBTDXF"},"agent_actions":{"view_html":"https://pith.science/pith/DVMBTDXFLVEMEATEMX2AILPHMG","download_json":"https://pith.science/pith/DVMBTDXFLVEMEATEMX2AILPHMG.json","view_paper":"https://pith.science/paper/DVMBTDXF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.00668&json=true","fetch_graph":"https://pith.science/api/pith-number/DVMBTDXFLVEMEATEMX2AILPHMG/graph.json","fetch_events":"https://pith.science/api/pith-number/DVMBTDXFLVEMEATEMX2AILPHMG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DVMBTDXFLVEMEATEMX2AILPHMG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DVMBTDXFLVEMEATEMX2AILPHMG/action/storage_attestation","attest_author":"https://pith.science/pith/DVMBTDXFLVEMEATEMX2AILPHMG/action/author_attestation","sign_citation":"https://pith.science/pith/DVMBTDXFLVEMEATEMX2AILPHMG/action/citation_signature","submit_replication":"https://pith.science/pith/DVMBTDXFLVEMEATEMX2AILPHMG/action/replication_record"}},"created_at":"2026-05-18T01:36:53.667638+00:00","updated_at":"2026-05-18T01:36:53.667638+00:00"}