{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:3IVGBEIYZZQLEYEBJYA74Q2L5A","short_pith_number":"pith:3IVGBEIY","schema_version":"1.0","canonical_sha256":"da2a609118ce60b260814e01fe434be8105f4e42f0805320a3be9a505996f075","source":{"kind":"arxiv","id":"2210.02668","version":1},"attestation_state":"computed","paper":{"title":"Congruences on the class numbers of $\\mathbb{Q}(\\sqrt{\\pm 2p})$ for $p\\equiv3$ $(\\text{mod }4)$ a prime","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jigu Kim, Yoshinori Mizuno","submitted_at":"2022-10-06T04:08:58Z","abstract_excerpt":"For a prime $p\\equiv 3$ $(\\text{mod }4)$, let $h(-8p)$ and $h(8p)$ be the class numbers of $\\mathbb{Q}(\\sqrt{-2p})$ and $\\mathbb{Q}(\\sqrt{2p})$, respectively. Let $\\Psi(\\xi)$ be the Hirzebruch sum of a quadratic irrational $\\xi$. We show that $h(-8p)\\equiv h(8p)\\Big(\\Psi(2\\sqrt{2p})/3-\\Psi\\big((1+\\sqrt{2p})/2\\big)/3\\Big)$ $(\\text{mod }16)$. Also, we show that $h(-8p)\\equiv 2h(8p)\\Psi(2\\sqrt{2p})/3$ $(\\text{mod }8)$ if $p\\equiv 3$ $(\\text{mod }8)$, and $h(-8p)\\equiv \\big(2h(8p)\\Psi(2\\sqrt{2p})/3\\big)+4$ $(\\text{mod }8)$ if $p\\equiv 7$ $(\\text{mod }8)$."},"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":"2210.02668","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-10-06T04:08:58Z","cross_cats_sorted":[],"title_canon_sha256":"2b6cb8fc0a9f83057db811ab28573ba5e077a55094af575cf90d77b57e86e1c9","abstract_canon_sha256":"71172ea41408d562fa35425a4000bb1c96aef4f0e87bd7957a9248d00d0b5c0e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:03:55.250145Z","signature_b64":"jTmuXddcMYo3B0qloaLRV4khhrpQDnQWYBI9ovqf7K78yol/ewk2a2EC6efGSAoO+ViacyZur/v7jJnP4D7tCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"da2a609118ce60b260814e01fe434be8105f4e42f0805320a3be9a505996f075","last_reissued_at":"2026-07-05T05:03:55.249672Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:03:55.249672Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Congruences on the class numbers of $\\mathbb{Q}(\\sqrt{\\pm 2p})$ for $p\\equiv3$ $(\\text{mod }4)$ a prime","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jigu Kim, Yoshinori Mizuno","submitted_at":"2022-10-06T04:08:58Z","abstract_excerpt":"For a prime $p\\equiv 3$ $(\\text{mod }4)$, let $h(-8p)$ and $h(8p)$ be the class numbers of $\\mathbb{Q}(\\sqrt{-2p})$ and $\\mathbb{Q}(\\sqrt{2p})$, respectively. Let $\\Psi(\\xi)$ be the Hirzebruch sum of a quadratic irrational $\\xi$. We show that $h(-8p)\\equiv h(8p)\\Big(\\Psi(2\\sqrt{2p})/3-\\Psi\\big((1+\\sqrt{2p})/2\\big)/3\\Big)$ $(\\text{mod }16)$. Also, we show that $h(-8p)\\equiv 2h(8p)\\Psi(2\\sqrt{2p})/3$ $(\\text{mod }8)$ if $p\\equiv 3$ $(\\text{mod }8)$, and $h(-8p)\\equiv \\big(2h(8p)\\Psi(2\\sqrt{2p})/3\\big)+4$ $(\\text{mod }8)$ if $p\\equiv 7$ $(\\text{mod }8)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.02668","kind":"arxiv","version":1},"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/2210.02668/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":"2210.02668","created_at":"2026-07-05T05:03:55.249731+00:00"},{"alias_kind":"arxiv_version","alias_value":"2210.02668v1","created_at":"2026-07-05T05:03:55.249731+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2210.02668","created_at":"2026-07-05T05:03:55.249731+00:00"},{"alias_kind":"pith_short_12","alias_value":"3IVGBEIYZZQL","created_at":"2026-07-05T05:03:55.249731+00:00"},{"alias_kind":"pith_short_16","alias_value":"3IVGBEIYZZQLEYEB","created_at":"2026-07-05T05:03:55.249731+00:00"},{"alias_kind":"pith_short_8","alias_value":"3IVGBEIY","created_at":"2026-07-05T05:03:55.249731+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/3IVGBEIYZZQLEYEBJYA74Q2L5A","json":"https://pith.science/pith/3IVGBEIYZZQLEYEBJYA74Q2L5A.json","graph_json":"https://pith.science/api/pith-number/3IVGBEIYZZQLEYEBJYA74Q2L5A/graph.json","events_json":"https://pith.science/api/pith-number/3IVGBEIYZZQLEYEBJYA74Q2L5A/events.json","paper":"https://pith.science/paper/3IVGBEIY"},"agent_actions":{"view_html":"https://pith.science/pith/3IVGBEIYZZQLEYEBJYA74Q2L5A","download_json":"https://pith.science/pith/3IVGBEIYZZQLEYEBJYA74Q2L5A.json","view_paper":"https://pith.science/paper/3IVGBEIY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2210.02668&json=true","fetch_graph":"https://pith.science/api/pith-number/3IVGBEIYZZQLEYEBJYA74Q2L5A/graph.json","fetch_events":"https://pith.science/api/pith-number/3IVGBEIYZZQLEYEBJYA74Q2L5A/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3IVGBEIYZZQLEYEBJYA74Q2L5A/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3IVGBEIYZZQLEYEBJYA74Q2L5A/action/storage_attestation","attest_author":"https://pith.science/pith/3IVGBEIYZZQLEYEBJYA74Q2L5A/action/author_attestation","sign_citation":"https://pith.science/pith/3IVGBEIYZZQLEYEBJYA74Q2L5A/action/citation_signature","submit_replication":"https://pith.science/pith/3IVGBEIYZZQLEYEBJYA74Q2L5A/action/replication_record"}},"created_at":"2026-07-05T05:03:55.249731+00:00","updated_at":"2026-07-05T05:03:55.249731+00:00"}