{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:JLRQWHAMDQBLCSHYTUOI4REG6O","short_pith_number":"pith:JLRQWHAM","canonical_record":{"source":{"id":"1410.7898","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-29T08:27:49Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5d6713cf2fa9c79381906bd01323c0da2861bf02239840c692a83c3b73b289f8","abstract_canon_sha256":"7856249dc749963a97fba00207a9ed13b95069317f393e55de858bde52b55bca"},"schema_version":"1.0"},"canonical_sha256":"4ae30b1c0c1c02b148f89d1c8e4486f3827077a8778e5021581dd1d53403e34a","source":{"kind":"arxiv","id":"1410.7898","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.7898","created_at":"2026-05-18T02:14:17Z"},{"alias_kind":"arxiv_version","alias_value":"1410.7898v2","created_at":"2026-05-18T02:14:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.7898","created_at":"2026-05-18T02:14:17Z"},{"alias_kind":"pith_short_12","alias_value":"JLRQWHAMDQBL","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"JLRQWHAMDQBLCSHY","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"JLRQWHAM","created_at":"2026-05-18T12:28:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:JLRQWHAMDQBLCSHYTUOI4REG6O","target":"record","payload":{"canonical_record":{"source":{"id":"1410.7898","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-29T08:27:49Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5d6713cf2fa9c79381906bd01323c0da2861bf02239840c692a83c3b73b289f8","abstract_canon_sha256":"7856249dc749963a97fba00207a9ed13b95069317f393e55de858bde52b55bca"},"schema_version":"1.0"},"canonical_sha256":"4ae30b1c0c1c02b148f89d1c8e4486f3827077a8778e5021581dd1d53403e34a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:14:17.489694Z","signature_b64":"bkwqLleA6SOSX86k6c75NSQ2Cdx9FYVw8eNVgF685ROUepBORHXZVgQXxoPEukz2i3jExnZUIHeLZrlnpU0kCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ae30b1c0c1c02b148f89d1c8e4486f3827077a8778e5021581dd1d53403e34a","last_reissued_at":"2026-05-18T02:14:17.488920Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:14:17.488920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.7898","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":"2026-05-18T02:14:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SPBSfmW8wMKlRJZSbMhuAxhirkEW74Mjqe4NhTm112pwP1w9lX5Bq/SUbidG7GDfg+5TD4Jp3tLo44R2H4ZPCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T13:27:37.329401Z"},"content_sha256":"43d19e947fd1b4671b38849e8f023636edabb75997d7559392c72d05d347fe8b","schema_version":"1.0","event_id":"sha256:43d19e947fd1b4671b38849e8f023636edabb75997d7559392c72d05d347fe8b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:JLRQWHAMDQBLCSHYTUOI4REG6O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Arithmetic Properties of Overpartition Triples","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Liuquan Wang","submitted_at":"2014-10-29T08:27:49Z","abstract_excerpt":"Let ${{\\overline{p}}_{3}}(n)$ be the number of overpartition triples of $n$. By elementary series manipulations, we establish some congruences for ${\\overline{p}}_{3}(n)$ modulo small powers of 2, such as\n  \\[{{\\overline{p}}_{3}}(16n+14)\\equiv 0 \\pmod{32}, \\quad {{\\overline{p}}_{3}}(8n+7)\\equiv 0 \\pmod{64}.\\] We also find many arithmetic properties for ${{\\overline{p}}_{3}}(n)$ modulo 7, 9 and 11, involving the following infinite families of Ramanujan-type congruences: for any integers $\\alpha \\ge 1$ and $n \\ge 0$, we have ${{\\overline{p}}_{3}}\\big({{3}^{2\\alpha +1}}(3n+2)\\big)\\equiv 0$ (mod $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7898","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":"2026-05-18T02:14:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TaVDoeikvUUEMavlga0Ajzy8oa+TexFeKScgn3dL62j3duOF5Qft5xZlTPrKBE0AYnveUGfP4JwhSsQjRM/KAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T13:27:37.329744Z"},"content_sha256":"918314d1d1d7997f9574fb89a652b8d015b4e621ea91ec0a1fd84bc59a552efb","schema_version":"1.0","event_id":"sha256:918314d1d1d7997f9574fb89a652b8d015b4e621ea91ec0a1fd84bc59a552efb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JLRQWHAMDQBLCSHYTUOI4REG6O/bundle.json","state_url":"https://pith.science/pith/JLRQWHAMDQBLCSHYTUOI4REG6O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JLRQWHAMDQBLCSHYTUOI4REG6O/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":"2026-07-03T13:27:37Z","links":{"resolver":"https://pith.science/pith/JLRQWHAMDQBLCSHYTUOI4REG6O","bundle":"https://pith.science/pith/JLRQWHAMDQBLCSHYTUOI4REG6O/bundle.json","state":"https://pith.science/pith/JLRQWHAMDQBLCSHYTUOI4REG6O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JLRQWHAMDQBLCSHYTUOI4REG6O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:JLRQWHAMDQBLCSHYTUOI4REG6O","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":"7856249dc749963a97fba00207a9ed13b95069317f393e55de858bde52b55bca","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-29T08:27:49Z","title_canon_sha256":"5d6713cf2fa9c79381906bd01323c0da2861bf02239840c692a83c3b73b289f8"},"schema_version":"1.0","source":{"id":"1410.7898","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.7898","created_at":"2026-05-18T02:14:17Z"},{"alias_kind":"arxiv_version","alias_value":"1410.7898v2","created_at":"2026-05-18T02:14:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.7898","created_at":"2026-05-18T02:14:17Z"},{"alias_kind":"pith_short_12","alias_value":"JLRQWHAMDQBL","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"JLRQWHAMDQBLCSHY","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"JLRQWHAM","created_at":"2026-05-18T12:28:33Z"}],"graph_snapshots":[{"event_id":"sha256:918314d1d1d7997f9574fb89a652b8d015b4e621ea91ec0a1fd84bc59a552efb","target":"graph","created_at":"2026-05-18T02:14:17Z","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 ${{\\overline{p}}_{3}}(n)$ be the number of overpartition triples of $n$. By elementary series manipulations, we establish some congruences for ${\\overline{p}}_{3}(n)$ modulo small powers of 2, such as\n  \\[{{\\overline{p}}_{3}}(16n+14)\\equiv 0 \\pmod{32}, \\quad {{\\overline{p}}_{3}}(8n+7)\\equiv 0 \\pmod{64}.\\] We also find many arithmetic properties for ${{\\overline{p}}_{3}}(n)$ modulo 7, 9 and 11, involving the following infinite families of Ramanujan-type congruences: for any integers $\\alpha \\ge 1$ and $n \\ge 0$, we have ${{\\overline{p}}_{3}}\\big({{3}^{2\\alpha +1}}(3n+2)\\big)\\equiv 0$ (mod $","authors_text":"Liuquan Wang","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-29T08:27:49Z","title":"Arithmetic Properties of Overpartition Triples"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7898","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":"sha256:43d19e947fd1b4671b38849e8f023636edabb75997d7559392c72d05d347fe8b","target":"record","created_at":"2026-05-18T02:14:17Z","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":"7856249dc749963a97fba00207a9ed13b95069317f393e55de858bde52b55bca","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-29T08:27:49Z","title_canon_sha256":"5d6713cf2fa9c79381906bd01323c0da2861bf02239840c692a83c3b73b289f8"},"schema_version":"1.0","source":{"id":"1410.7898","kind":"arxiv","version":2}},"canonical_sha256":"4ae30b1c0c1c02b148f89d1c8e4486f3827077a8778e5021581dd1d53403e34a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4ae30b1c0c1c02b148f89d1c8e4486f3827077a8778e5021581dd1d53403e34a","first_computed_at":"2026-05-18T02:14:17.488920Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:14:17.488920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bkwqLleA6SOSX86k6c75NSQ2Cdx9FYVw8eNVgF685ROUepBORHXZVgQXxoPEukz2i3jExnZUIHeLZrlnpU0kCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:14:17.489694Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.7898","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:43d19e947fd1b4671b38849e8f023636edabb75997d7559392c72d05d347fe8b","sha256:918314d1d1d7997f9574fb89a652b8d015b4e621ea91ec0a1fd84bc59a552efb"],"state_sha256":"55b61fa80f583a4dc40f9329ec34e9649856c757b53bf47c2f564b2e7c620b99"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bDslFByjBzV+DljWc1vZxY8B+qYAzI+5AGkkjdsERxz/xVyDYcAH9XohX9oco+J7x3eX2QYJcQQQaJgp1Y/wDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T13:27:37.331644Z","bundle_sha256":"9629a64f7b2024404341aa3d29ad4d1677413c1c0182be1566223f2e773fbb85"}}