{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:ASHOG3H33YX6LEBJE7INPQ5NC2","short_pith_number":"pith:ASHOG3H3","canonical_record":{"source":{"id":"1410.5628","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-21T12:05:22Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0d7f74cfffbc64bb29e599951d62886d477509464af0789620a71afc382d86d9","abstract_canon_sha256":"8560682ec666f84b2447ae232909ead8cacbaea1da0bed09ff23d6f44ac0bed5"},"schema_version":"1.0"},"canonical_sha256":"048ee36cfbde2fe5902927d0d7c3ad16844b05ed68283410edeb25aae81d5e0b","source":{"kind":"arxiv","id":"1410.5628","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.5628","created_at":"2026-05-18T01:28:05Z"},{"alias_kind":"arxiv_version","alias_value":"1410.5628v2","created_at":"2026-05-18T01:28:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.5628","created_at":"2026-05-18T01:28:05Z"},{"alias_kind":"pith_short_12","alias_value":"ASHOG3H33YX6","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_16","alias_value":"ASHOG3H33YX6LEBJ","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_8","alias_value":"ASHOG3H3","created_at":"2026-05-18T12:28:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:ASHOG3H33YX6LEBJE7INPQ5NC2","target":"record","payload":{"canonical_record":{"source":{"id":"1410.5628","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-21T12:05:22Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0d7f74cfffbc64bb29e599951d62886d477509464af0789620a71afc382d86d9","abstract_canon_sha256":"8560682ec666f84b2447ae232909ead8cacbaea1da0bed09ff23d6f44ac0bed5"},"schema_version":"1.0"},"canonical_sha256":"048ee36cfbde2fe5902927d0d7c3ad16844b05ed68283410edeb25aae81d5e0b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:28:05.799303Z","signature_b64":"Q00hQw3LtIvwlwZQoLNReLKv9rbP5YPeae3YYBlFUYf0rC5BB4hFqRY3PAiVcB1WOwZwMHN/oYszWlj67ulYDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"048ee36cfbde2fe5902927d0d7c3ad16844b05ed68283410edeb25aae81d5e0b","last_reissued_at":"2026-05-18T01:28:05.798715Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:28:05.798715Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.5628","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:28:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KaNH51aT8ncyGnti/E8Lk/OeQGSKOl9NASDuT9koeXa4gFuXLY5j7g6Vyfr9Gn2PN3rHP9Jv4LNJ+2TljJSfCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T06:12:57.811091Z"},"content_sha256":"aaf4e4e52da2eb37846ddc921ab6e9555619b2199718e08c211307de34eddf7f","schema_version":"1.0","event_id":"sha256:aaf4e4e52da2eb37846ddc921ab6e9555619b2199718e08c211307de34eddf7f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:ASHOG3H33YX6LEBJE7INPQ5NC2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Arithmetic Properties of Partition Quadruples With Odd Parts Distinct","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Liuquan Wang","submitted_at":"2014-10-21T12:05:22Z","abstract_excerpt":"Let $\\mathrm{pod}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\\mathrm{pod}_{-4}(n)$ involving the following infinite family of congruences: for any integers $\\alpha \\ge 1$ and $n \\ge 0$,\n  \\[\\mathrm{pod}_{-4}\\Big({{3}^{\\alpha +1}}n+\\frac{5\\cdot {{3}^{\\alpha }}+1}{2}\\Big)\\equiv 0 \\pmod{9}.\\] We also establish some internal congruences and some congruences modulo 2, 5 and 8 satisfied by $\\mathrm{pod}_{-4}(n)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5628","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:28:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/N60uYna9RfENvt4ScB/fH/g6o6YQkZOQe5IqHPTAMJ+9ja+bcgjw9RNFs8nihbcLtRm9T6X0NqMaQVEHDAfBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T06:12:57.811453Z"},"content_sha256":"37bf59efd2dc11347b29d389081c1fbecde0b81d9327347d4af5f0f7b184d882","schema_version":"1.0","event_id":"sha256:37bf59efd2dc11347b29d389081c1fbecde0b81d9327347d4af5f0f7b184d882"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ASHOG3H33YX6LEBJE7INPQ5NC2/bundle.json","state_url":"https://pith.science/pith/ASHOG3H33YX6LEBJE7INPQ5NC2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ASHOG3H33YX6LEBJE7INPQ5NC2/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-03T06:12:57Z","links":{"resolver":"https://pith.science/pith/ASHOG3H33YX6LEBJE7INPQ5NC2","bundle":"https://pith.science/pith/ASHOG3H33YX6LEBJE7INPQ5NC2/bundle.json","state":"https://pith.science/pith/ASHOG3H33YX6LEBJE7INPQ5NC2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ASHOG3H33YX6LEBJE7INPQ5NC2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:ASHOG3H33YX6LEBJE7INPQ5NC2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8560682ec666f84b2447ae232909ead8cacbaea1da0bed09ff23d6f44ac0bed5","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-21T12:05:22Z","title_canon_sha256":"0d7f74cfffbc64bb29e599951d62886d477509464af0789620a71afc382d86d9"},"schema_version":"1.0","source":{"id":"1410.5628","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.5628","created_at":"2026-05-18T01:28:05Z"},{"alias_kind":"arxiv_version","alias_value":"1410.5628v2","created_at":"2026-05-18T01:28:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.5628","created_at":"2026-05-18T01:28:05Z"},{"alias_kind":"pith_short_12","alias_value":"ASHOG3H33YX6","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_16","alias_value":"ASHOG3H33YX6LEBJ","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_8","alias_value":"ASHOG3H3","created_at":"2026-05-18T12:28:19Z"}],"graph_snapshots":[{"event_id":"sha256:37bf59efd2dc11347b29d389081c1fbecde0b81d9327347d4af5f0f7b184d882","target":"graph","created_at":"2026-05-18T01:28:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $\\mathrm{pod}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\\mathrm{pod}_{-4}(n)$ involving the following infinite family of congruences: for any integers $\\alpha \\ge 1$ and $n \\ge 0$,\n  \\[\\mathrm{pod}_{-4}\\Big({{3}^{\\alpha +1}}n+\\frac{5\\cdot {{3}^{\\alpha }}+1}{2}\\Big)\\equiv 0 \\pmod{9}.\\] We also establish some internal congruences and some congruences modulo 2, 5 and 8 satisfied by $\\mathrm{pod}_{-4}(n)$.","authors_text":"Liuquan Wang","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-21T12:05:22Z","title":"Arithmetic Properties of Partition Quadruples With Odd Parts Distinct"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5628","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:aaf4e4e52da2eb37846ddc921ab6e9555619b2199718e08c211307de34eddf7f","target":"record","created_at":"2026-05-18T01:28:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8560682ec666f84b2447ae232909ead8cacbaea1da0bed09ff23d6f44ac0bed5","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-21T12:05:22Z","title_canon_sha256":"0d7f74cfffbc64bb29e599951d62886d477509464af0789620a71afc382d86d9"},"schema_version":"1.0","source":{"id":"1410.5628","kind":"arxiv","version":2}},"canonical_sha256":"048ee36cfbde2fe5902927d0d7c3ad16844b05ed68283410edeb25aae81d5e0b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"048ee36cfbde2fe5902927d0d7c3ad16844b05ed68283410edeb25aae81d5e0b","first_computed_at":"2026-05-18T01:28:05.798715Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:28:05.798715Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Q00hQw3LtIvwlwZQoLNReLKv9rbP5YPeae3YYBlFUYf0rC5BB4hFqRY3PAiVcB1WOwZwMHN/oYszWlj67ulYDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:28:05.799303Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.5628","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aaf4e4e52da2eb37846ddc921ab6e9555619b2199718e08c211307de34eddf7f","sha256:37bf59efd2dc11347b29d389081c1fbecde0b81d9327347d4af5f0f7b184d882"],"state_sha256":"857ecf8b3bb96768e1b0dbe22424acfafbad8415835fc4588aff7fc489d4d6b1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HPOwD7FGFKiqnd87XD1P+mWpSSPF8jkwIP4Qrr/h6EntFmgDZENsHp4dHtWTgoFNR4Bxb3oJQnQxedYCkIepCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T06:12:57.813326Z","bundle_sha256":"e2d8fd074f9436fd99096e2f53dc97dd5953ee6d0a27253dd805ccb5d23a9a9f"}}