{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:LN2JOK4LFJ5E5GX4OKL5XMHFWR","short_pith_number":"pith:LN2JOK4L","schema_version":"1.0","canonical_sha256":"5b74972b8b2a7a4e9afc7297dbb0e5b44e691cdc0ef2be8c3a4b23ee7ad25b62","source":{"kind":"arxiv","id":"1201.0617","version":3},"attestation_state":"computed","paper":{"title":"Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Victor J. W. Guo","submitted_at":"2012-01-03T12:40:50Z","abstract_excerpt":"For all nonnegative integers n, the Franel numbers are defined as $$ f_n=\\sum_{k=0}^n {n\\choose k}^3.$$ We confirm two conjectures of Z.-W. Sun on congruences for Franel numbers: \\sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\\equiv 0 \\pmod{2n^2}, \\sum_{k=0}^{p-1}(3k+2)(-1)^k f_k &\\equiv 2p^2 (2^p-1)^2 \\pmod{p^5}, where n is a positive integer and p>3 is a prime."},"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":"1201.0617","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-01-03T12:40:50Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d60a98edc7259fa9bd7afbcbff13e9e5eb7c38a6b203c6b6e0b4d3cee32e4f69","abstract_canon_sha256":"efebd10db83944ad125d08c849ee1a71cf07b120c56e68d389e7fa3be11fb3a0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:25.554581Z","signature_b64":"L3N8VBw5N0lyQZ+nmPkWAhi/MP/7nvgvLt+VWshnCRSyXTqKyrvCF2Ni+rk1djYF8CQ1PwoKWrNwtaC67jikAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5b74972b8b2a7a4e9afc7297dbb0e5b44e691cdc0ef2be8c3a4b23ee7ad25b62","last_reissued_at":"2026-05-18T03:50:25.553715Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:25.553715Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Victor J. W. Guo","submitted_at":"2012-01-03T12:40:50Z","abstract_excerpt":"For all nonnegative integers n, the Franel numbers are defined as $$ f_n=\\sum_{k=0}^n {n\\choose k}^3.$$ We confirm two conjectures of Z.-W. Sun on congruences for Franel numbers: \\sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\\equiv 0 \\pmod{2n^2}, \\sum_{k=0}^{p-1}(3k+2)(-1)^k f_k &\\equiv 2p^2 (2^p-1)^2 \\pmod{p^5}, where n is a positive integer and p>3 is a prime."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.0617","kind":"arxiv","version":3},"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":"1201.0617","created_at":"2026-05-18T03:50:25.553843+00:00"},{"alias_kind":"arxiv_version","alias_value":"1201.0617v3","created_at":"2026-05-18T03:50:25.553843+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.0617","created_at":"2026-05-18T03:50:25.553843+00:00"},{"alias_kind":"pith_short_12","alias_value":"LN2JOK4LFJ5E","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_16","alias_value":"LN2JOK4LFJ5E5GX4","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_8","alias_value":"LN2JOK4L","created_at":"2026-05-18T12:27:14.488303+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/LN2JOK4LFJ5E5GX4OKL5XMHFWR","json":"https://pith.science/pith/LN2JOK4LFJ5E5GX4OKL5XMHFWR.json","graph_json":"https://pith.science/api/pith-number/LN2JOK4LFJ5E5GX4OKL5XMHFWR/graph.json","events_json":"https://pith.science/api/pith-number/LN2JOK4LFJ5E5GX4OKL5XMHFWR/events.json","paper":"https://pith.science/paper/LN2JOK4L"},"agent_actions":{"view_html":"https://pith.science/pith/LN2JOK4LFJ5E5GX4OKL5XMHFWR","download_json":"https://pith.science/pith/LN2JOK4LFJ5E5GX4OKL5XMHFWR.json","view_paper":"https://pith.science/paper/LN2JOK4L","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1201.0617&json=true","fetch_graph":"https://pith.science/api/pith-number/LN2JOK4LFJ5E5GX4OKL5XMHFWR/graph.json","fetch_events":"https://pith.science/api/pith-number/LN2JOK4LFJ5E5GX4OKL5XMHFWR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LN2JOK4LFJ5E5GX4OKL5XMHFWR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LN2JOK4LFJ5E5GX4OKL5XMHFWR/action/storage_attestation","attest_author":"https://pith.science/pith/LN2JOK4LFJ5E5GX4OKL5XMHFWR/action/author_attestation","sign_citation":"https://pith.science/pith/LN2JOK4LFJ5E5GX4OKL5XMHFWR/action/citation_signature","submit_replication":"https://pith.science/pith/LN2JOK4LFJ5E5GX4OKL5XMHFWR/action/replication_record"}},"created_at":"2026-05-18T03:50:25.553843+00:00","updated_at":"2026-05-18T03:50:25.553843+00:00"}