{"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"}