{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:WDNIYHVDD6HXCWMJLI5NTYJ5AY","short_pith_number":"pith:WDNIYHVD","schema_version":"1.0","canonical_sha256":"b0da8c1ea31f8f7159895a3ad9e13d06176d09f885060fcb3780637d6c136ed2","source":{"kind":"arxiv","id":"2201.04291","version":2},"attestation_state":"computed","paper":{"title":"Congruences for odd class numbers of quadratic fields with odd discriminant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jigu Kim, Yoshinori Mizuno","submitted_at":"2022-01-12T03:55:23Z","abstract_excerpt":"For any distinct two primes $p_1\\equiv p_2\\equiv 3$ $(\\text{mod }4)$, let $h(-p_1)$, $h(-p_2)$ and $h(p_1p_2)$ be the class numbers of the quadratic fields $\\mathbb{Q}(\\sqrt{-p_1})$, $\\mathbb{Q}(\\sqrt{-p_2})$ and $\\mathbb{Q}(\\sqrt{p_1p_2})$, respectively. Let $\\omega_{p_1p_2}:=(1+\\sqrt{p_1p_2})/2$ and let $\\Psi(\\omega_{p_1p_2})$ be the Hirzebruch sum of $\\omega_{p_1p_2}$. We show that $h(-p_1)h(-p_2)\\equiv h(p_1p_2)\\Psi(\\omega_{p_1p_2})/n$ $(\\text{mod }8)$, where $n=6$ (respectively, $n=2$) if $\\min\\{p_1,p_2\\}>3$ (respectively, otherwise). We also consider the real quadratic order with conduct"},"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":"2201.04291","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-01-12T03:55:23Z","cross_cats_sorted":[],"title_canon_sha256":"27dca6cceca7a4fa382a80f595bfe18262a75ecf425fc924b2ad8a66bcdf87b8","abstract_canon_sha256":"5c703d37b65f52cbbd313397bfc509af3ce003fa1494a974d0ade42a4d6d48a5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:52:16.045307Z","signature_b64":"yi/jI/xNZ76r1lryeUOw2iXo7MVywBGYDP6eHMStFVlmTjyJhGfRIj3N1O/Xj4Znx75W8yDJJnH/pgT6HC1QCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0da8c1ea31f8f7159895a3ad9e13d06176d09f885060fcb3780637d6c136ed2","last_reissued_at":"2026-07-05T03:52:16.044808Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:52:16.044808Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Congruences for odd class numbers of quadratic fields with odd discriminant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jigu Kim, Yoshinori Mizuno","submitted_at":"2022-01-12T03:55:23Z","abstract_excerpt":"For any distinct two primes $p_1\\equiv p_2\\equiv 3$ $(\\text{mod }4)$, let $h(-p_1)$, $h(-p_2)$ and $h(p_1p_2)$ be the class numbers of the quadratic fields $\\mathbb{Q}(\\sqrt{-p_1})$, $\\mathbb{Q}(\\sqrt{-p_2})$ and $\\mathbb{Q}(\\sqrt{p_1p_2})$, respectively. Let $\\omega_{p_1p_2}:=(1+\\sqrt{p_1p_2})/2$ and let $\\Psi(\\omega_{p_1p_2})$ be the Hirzebruch sum of $\\omega_{p_1p_2}$. We show that $h(-p_1)h(-p_2)\\equiv h(p_1p_2)\\Psi(\\omega_{p_1p_2})/n$ $(\\text{mod }8)$, where $n=6$ (respectively, $n=2$) if $\\min\\{p_1,p_2\\}>3$ (respectively, otherwise). We also consider the real quadratic order with conduct"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.04291","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2201.04291/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2201.04291","created_at":"2026-07-05T03:52:16.044865+00:00"},{"alias_kind":"arxiv_version","alias_value":"2201.04291v2","created_at":"2026-07-05T03:52:16.044865+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.04291","created_at":"2026-07-05T03:52:16.044865+00:00"},{"alias_kind":"pith_short_12","alias_value":"WDNIYHVDD6HX","created_at":"2026-07-05T03:52:16.044865+00:00"},{"alias_kind":"pith_short_16","alias_value":"WDNIYHVDD6HXCWMJ","created_at":"2026-07-05T03:52:16.044865+00:00"},{"alias_kind":"pith_short_8","alias_value":"WDNIYHVD","created_at":"2026-07-05T03:52:16.044865+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/WDNIYHVDD6HXCWMJLI5NTYJ5AY","json":"https://pith.science/pith/WDNIYHVDD6HXCWMJLI5NTYJ5AY.json","graph_json":"https://pith.science/api/pith-number/WDNIYHVDD6HXCWMJLI5NTYJ5AY/graph.json","events_json":"https://pith.science/api/pith-number/WDNIYHVDD6HXCWMJLI5NTYJ5AY/events.json","paper":"https://pith.science/paper/WDNIYHVD"},"agent_actions":{"view_html":"https://pith.science/pith/WDNIYHVDD6HXCWMJLI5NTYJ5AY","download_json":"https://pith.science/pith/WDNIYHVDD6HXCWMJLI5NTYJ5AY.json","view_paper":"https://pith.science/paper/WDNIYHVD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2201.04291&json=true","fetch_graph":"https://pith.science/api/pith-number/WDNIYHVDD6HXCWMJLI5NTYJ5AY/graph.json","fetch_events":"https://pith.science/api/pith-number/WDNIYHVDD6HXCWMJLI5NTYJ5AY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WDNIYHVDD6HXCWMJLI5NTYJ5AY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WDNIYHVDD6HXCWMJLI5NTYJ5AY/action/storage_attestation","attest_author":"https://pith.science/pith/WDNIYHVDD6HXCWMJLI5NTYJ5AY/action/author_attestation","sign_citation":"https://pith.science/pith/WDNIYHVDD6HXCWMJLI5NTYJ5AY/action/citation_signature","submit_replication":"https://pith.science/pith/WDNIYHVDD6HXCWMJLI5NTYJ5AY/action/replication_record"}},"created_at":"2026-07-05T03:52:16.044865+00:00","updated_at":"2026-07-05T03:52:16.044865+00:00"}