{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:RR2MMKB3NPTEYTBDKPIOMTT7XJ","short_pith_number":"pith:RR2MMKB3","schema_version":"1.0","canonical_sha256":"8c74c6283b6be64c4c2353d0e64e7fba5060b3ba8aa7ff4562e791abddfe309e","source":{"kind":"arxiv","id":"1611.10337","version":3},"attestation_state":"computed","paper":{"title":"Spins of prime ideals and the negative Pell equation $x^2 - 2py^2 = -1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Djordjo Milovic, Peter Koymans","submitted_at":"2016-11-30T20:06:45Z","abstract_excerpt":"Let $p\\equiv 1\\bmod 4$ be a prime number. We use a number field variant of Vinogradov's method to prove density results about the following four arithmetic invariants: (i) $16$-rank of the class group $\\mathrm{Cl}(-4p)$ of the imaginary quadratic number field $\\mathbb{Q}(\\sqrt{-4p})$; (ii) $8$-rank of the ordinary class group $\\mathrm{Cl}(8p)$ of the real quadratic field $\\mathbb{Q}(\\sqrt{8p})$; (iii) the solvability of the negative Pell equation $x^2 - 2py^2 = -1$ over the integers; (iv) $2$-part of the Tate-\\v{S}afarevi\\v{c} group of the congruent number elliptic curve $E_p: y^2 = x^3-p^2x$."},"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":"1611.10337","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-30T20:06:45Z","cross_cats_sorted":[],"title_canon_sha256":"718ce4b6470a8dfc400dfba63b7c99b35616e7c042a77b198524ac4fa54d9516","abstract_canon_sha256":"e1198c55e04b1588edea0e465ef618211921cb99ccb322ec0da6c1be433b4a1e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:34.071066Z","signature_b64":"knON0WYOCEm706bq/pH1G5O/ITwJ3dxUHqCetlZTrhAWnH4rLIf4WB/hWDREh925gqkzj9UXSUBzseH6p+vUDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c74c6283b6be64c4c2353d0e64e7fba5060b3ba8aa7ff4562e791abddfe309e","last_reissued_at":"2026-05-17T23:53:34.070412Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:34.070412Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spins of prime ideals and the negative Pell equation $x^2 - 2py^2 = -1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Djordjo Milovic, Peter Koymans","submitted_at":"2016-11-30T20:06:45Z","abstract_excerpt":"Let $p\\equiv 1\\bmod 4$ be a prime number. We use a number field variant of Vinogradov's method to prove density results about the following four arithmetic invariants: (i) $16$-rank of the class group $\\mathrm{Cl}(-4p)$ of the imaginary quadratic number field $\\mathbb{Q}(\\sqrt{-4p})$; (ii) $8$-rank of the ordinary class group $\\mathrm{Cl}(8p)$ of the real quadratic field $\\mathbb{Q}(\\sqrt{8p})$; (iii) the solvability of the negative Pell equation $x^2 - 2py^2 = -1$ over the integers; (iv) $2$-part of the Tate-\\v{S}afarevi\\v{c} group of the congruent number elliptic curve $E_p: y^2 = x^3-p^2x$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.10337","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1611.10337","created_at":"2026-05-17T23:53:34.070513+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.10337v3","created_at":"2026-05-17T23:53:34.070513+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.10337","created_at":"2026-05-17T23:53:34.070513+00:00"},{"alias_kind":"pith_short_12","alias_value":"RR2MMKB3NPTE","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_16","alias_value":"RR2MMKB3NPTEYTBD","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_8","alias_value":"RR2MMKB3","created_at":"2026-05-18T12:30:41.710351+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/RR2MMKB3NPTEYTBDKPIOMTT7XJ","json":"https://pith.science/pith/RR2MMKB3NPTEYTBDKPIOMTT7XJ.json","graph_json":"https://pith.science/api/pith-number/RR2MMKB3NPTEYTBDKPIOMTT7XJ/graph.json","events_json":"https://pith.science/api/pith-number/RR2MMKB3NPTEYTBDKPIOMTT7XJ/events.json","paper":"https://pith.science/paper/RR2MMKB3"},"agent_actions":{"view_html":"https://pith.science/pith/RR2MMKB3NPTEYTBDKPIOMTT7XJ","download_json":"https://pith.science/pith/RR2MMKB3NPTEYTBDKPIOMTT7XJ.json","view_paper":"https://pith.science/paper/RR2MMKB3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.10337&json=true","fetch_graph":"https://pith.science/api/pith-number/RR2MMKB3NPTEYTBDKPIOMTT7XJ/graph.json","fetch_events":"https://pith.science/api/pith-number/RR2MMKB3NPTEYTBDKPIOMTT7XJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RR2MMKB3NPTEYTBDKPIOMTT7XJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RR2MMKB3NPTEYTBDKPIOMTT7XJ/action/storage_attestation","attest_author":"https://pith.science/pith/RR2MMKB3NPTEYTBDKPIOMTT7XJ/action/author_attestation","sign_citation":"https://pith.science/pith/RR2MMKB3NPTEYTBDKPIOMTT7XJ/action/citation_signature","submit_replication":"https://pith.science/pith/RR2MMKB3NPTEYTBDKPIOMTT7XJ/action/replication_record"}},"created_at":"2026-05-17T23:53:34.070513+00:00","updated_at":"2026-05-17T23:53:34.070513+00:00"}