{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:XVHIYIHQNOGOBAJFO224NAX4HM","short_pith_number":"pith:XVHIYIHQ","schema_version":"1.0","canonical_sha256":"bd4e8c20f06b8ce0812576b5c682fc3b2110cf7f79dd9b5c4ada044e274c45e8","source":{"kind":"arxiv","id":"1407.5436","version":2},"attestation_state":"computed","paper":{"title":"New Congruences of Partitions With Odd Parts Distinct","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Liuquan Wang","submitted_at":"2014-07-21T09:44:22Z","abstract_excerpt":"Let $\\mathrm{pod}(n)$ denote the number of partitions of $n$ with odd parts distinct, and ${{r}_{k}}(n)$ be the number of representations of $n$ as sum of $k$ squares. We find the following two arithmetic relations: for any integer $n\\ge 0$, \\[\\mathrm{pod}(3n+2)\\equiv 2{{(-1)}^{n+1}}{{r}_{5}}(8n+5) \\pmod{9}, \\] and \\[\\mathrm{pod}(5n+2)\\equiv 2{{(-1)}^{n}}{{r}_{3}}(8n+3) \\pmod{5}.\\] From which we deduce many interesting congruences including the following two infinite families of Ramanujan-type congruences: for $a \\in \\{11, 19\\}$ and any integers $\\alpha \\ge 1$ and $n \\ge 0$, we have \\[\\mathrm{"},"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":"1407.5436","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-21T09:44:22Z","cross_cats_sorted":[],"title_canon_sha256":"a705bc4f522dd5defb870631f11a5cdb65c9907035e7081449c653815c01d18b","abstract_canon_sha256":"c3e66eae6512ec21e0f7ff2657acfe3a9a0c49036053c960306bfff40ceb6ec6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:59.257377Z","signature_b64":"Wt0mR1hQPUFRbWHqWKYcxCpWURdLYw1k+Mcc4g19VQgby5/y9gT60lP5I5QoiKQq+AiBZTVnwy+ajGvYBf4WBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bd4e8c20f06b8ce0812576b5c682fc3b2110cf7f79dd9b5c4ada044e274c45e8","last_reissued_at":"2026-05-18T02:38:59.256943Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:59.256943Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New Congruences of Partitions With Odd Parts Distinct","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Liuquan Wang","submitted_at":"2014-07-21T09:44:22Z","abstract_excerpt":"Let $\\mathrm{pod}(n)$ denote the number of partitions of $n$ with odd parts distinct, and ${{r}_{k}}(n)$ be the number of representations of $n$ as sum of $k$ squares. We find the following two arithmetic relations: for any integer $n\\ge 0$, \\[\\mathrm{pod}(3n+2)\\equiv 2{{(-1)}^{n+1}}{{r}_{5}}(8n+5) \\pmod{9}, \\] and \\[\\mathrm{pod}(5n+2)\\equiv 2{{(-1)}^{n}}{{r}_{3}}(8n+3) \\pmod{5}.\\] From which we deduce many interesting congruences including the following two infinite families of Ramanujan-type congruences: for $a \\in \\{11, 19\\}$ and any integers $\\alpha \\ge 1$ and $n \\ge 0$, we have \\[\\mathrm{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.5436","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":"1407.5436","created_at":"2026-05-18T02:38:59.257002+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.5436v2","created_at":"2026-05-18T02:38:59.257002+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.5436","created_at":"2026-05-18T02:38:59.257002+00:00"},{"alias_kind":"pith_short_12","alias_value":"XVHIYIHQNOGO","created_at":"2026-05-18T12:28:57.508820+00:00"},{"alias_kind":"pith_short_16","alias_value":"XVHIYIHQNOGOBAJF","created_at":"2026-05-18T12:28:57.508820+00:00"},{"alias_kind":"pith_short_8","alias_value":"XVHIYIHQ","created_at":"2026-05-18T12:28:57.508820+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/XVHIYIHQNOGOBAJFO224NAX4HM","json":"https://pith.science/pith/XVHIYIHQNOGOBAJFO224NAX4HM.json","graph_json":"https://pith.science/api/pith-number/XVHIYIHQNOGOBAJFO224NAX4HM/graph.json","events_json":"https://pith.science/api/pith-number/XVHIYIHQNOGOBAJFO224NAX4HM/events.json","paper":"https://pith.science/paper/XVHIYIHQ"},"agent_actions":{"view_html":"https://pith.science/pith/XVHIYIHQNOGOBAJFO224NAX4HM","download_json":"https://pith.science/pith/XVHIYIHQNOGOBAJFO224NAX4HM.json","view_paper":"https://pith.science/paper/XVHIYIHQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.5436&json=true","fetch_graph":"https://pith.science/api/pith-number/XVHIYIHQNOGOBAJFO224NAX4HM/graph.json","fetch_events":"https://pith.science/api/pith-number/XVHIYIHQNOGOBAJFO224NAX4HM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XVHIYIHQNOGOBAJFO224NAX4HM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XVHIYIHQNOGOBAJFO224NAX4HM/action/storage_attestation","attest_author":"https://pith.science/pith/XVHIYIHQNOGOBAJFO224NAX4HM/action/author_attestation","sign_citation":"https://pith.science/pith/XVHIYIHQNOGOBAJFO224NAX4HM/action/citation_signature","submit_replication":"https://pith.science/pith/XVHIYIHQNOGOBAJFO224NAX4HM/action/replication_record"}},"created_at":"2026-05-18T02:38:59.257002+00:00","updated_at":"2026-05-18T02:38:59.257002+00:00"}