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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 $p\\equiv 7$ $(\\text{mod }8)$.","authors_text":"Jigu Kim, Yoshinori Mizuno","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-10-06T04:08:58Z","title":"Congruences on the class numbers of $\\mathbb{Q}(\\sqrt{\\pm 2p})$ for $p\\equiv3$ $(\\text{mod }4)$ a prime"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.02668","kind":"arxiv","version":1},"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:f8a6b86a9bd34e5e490d8fdcf585638b6ce1123bfa925cee2f34b5ce066e2def","target":"record","created_at":"2026-07-05T05:03:55Z","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":"71172ea41408d562fa35425a4000bb1c96aef4f0e87bd7957a9248d00d0b5c0e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-10-06T04:08:58Z","title_canon_sha256":"2b6cb8fc0a9f83057db811ab28573ba5e077a55094af575cf90d77b57e86e1c9"},"schema_version":"1.0","source":{"id":"2210.02668","kind":"arxiv","version":1}},"canonical_sha256":"da2a609118ce60b260814e01fe434be8105f4e42f0805320a3be9a505996f075","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"da2a609118ce60b260814e01fe434be8105f4e42f0805320a3be9a505996f075","first_computed_at":"2026-07-05T05:03:55.249672Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:03:55.249672Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jTmuXddcMYo3B0qloaLRV4khhrpQDnQWYBI9ovqf7K78yol/ewk2a2EC6efGSAoO+ViacyZur/v7jJnP4D7tCg==","signature_status":"signed_v1","signed_at":"2026-07-05T05:03:55.250145Z","signed_message":"canonical_sha256_bytes"},"source_id":"2210.02668","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f8a6b86a9bd34e5e490d8fdcf585638b6ce1123bfa925cee2f34b5ce066e2def","sha256:ec47ca5d3a1b43a08e28d2aab1abfdbfa990a46382342d79d3a2c8005eb954bc"],"state_sha256":"19eca82d26dfdf5d7e882b45957fe7dc6c4856c8683a627b50b03f95b44f6dc7"}