{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:37C3DSBUHN6MEKPAUHYIQAAM57","short_pith_number":"pith:37C3DSBU","schema_version":"1.0","canonical_sha256":"dfc5b1c8343b7cc229e0a1f088000cefe7b48f73f2bc41b9ecb0b1bf3a4ddb75","source":{"kind":"arxiv","id":"1605.00088","version":1},"attestation_state":"computed","paper":{"title":"Density of solutions to quadratic congruences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Neha Prabhu","submitted_at":"2016-04-30T10:13:38Z","abstract_excerpt":"A classical result in number theory is Dirichlet's theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly k prime factors for k>1. Building upon a proof by E.M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree n not greater than x with k prime factors such that a fixed quadratic equation has exactly 2^k solutions modulo n."},"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":"1605.00088","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-04-30T10:13:38Z","cross_cats_sorted":[],"title_canon_sha256":"1c4e323c8e00317b56ceee3f2ecd409b5757597cfac63ac33787ef15a671da7d","abstract_canon_sha256":"42686693376b5b6b2ad9f65cb2329f3f6a8f9325f493d03961056012fe202bec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:56.896373Z","signature_b64":"CzDIUTnmY68zo8GwQKZoLIxwYAL0hdI7dwrht5GECdSZJrXy5Cj419iVmr6KGRMSTfDi7ORj9RvtKzZU1LHACQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dfc5b1c8343b7cc229e0a1f088000cefe7b48f73f2bc41b9ecb0b1bf3a4ddb75","last_reissued_at":"2026-05-18T01:15:56.895852Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:56.895852Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Density of solutions to quadratic congruences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Neha Prabhu","submitted_at":"2016-04-30T10:13:38Z","abstract_excerpt":"A classical result in number theory is Dirichlet's theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly k prime factors for k>1. Building upon a proof by E.M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree n not greater than x with k prime factors such that a fixed quadratic equation has exactly 2^k solutions modulo n."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.00088","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":""},"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":"1605.00088","created_at":"2026-05-18T01:15:56.895917+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.00088v1","created_at":"2026-05-18T01:15:56.895917+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.00088","created_at":"2026-05-18T01:15:56.895917+00:00"},{"alias_kind":"pith_short_12","alias_value":"37C3DSBUHN6M","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"37C3DSBUHN6MEKPA","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"37C3DSBU","created_at":"2026-05-18T12:29:55.572404+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/37C3DSBUHN6MEKPAUHYIQAAM57","json":"https://pith.science/pith/37C3DSBUHN6MEKPAUHYIQAAM57.json","graph_json":"https://pith.science/api/pith-number/37C3DSBUHN6MEKPAUHYIQAAM57/graph.json","events_json":"https://pith.science/api/pith-number/37C3DSBUHN6MEKPAUHYIQAAM57/events.json","paper":"https://pith.science/paper/37C3DSBU"},"agent_actions":{"view_html":"https://pith.science/pith/37C3DSBUHN6MEKPAUHYIQAAM57","download_json":"https://pith.science/pith/37C3DSBUHN6MEKPAUHYIQAAM57.json","view_paper":"https://pith.science/paper/37C3DSBU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.00088&json=true","fetch_graph":"https://pith.science/api/pith-number/37C3DSBUHN6MEKPAUHYIQAAM57/graph.json","fetch_events":"https://pith.science/api/pith-number/37C3DSBUHN6MEKPAUHYIQAAM57/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/37C3DSBUHN6MEKPAUHYIQAAM57/action/timestamp_anchor","attest_storage":"https://pith.science/pith/37C3DSBUHN6MEKPAUHYIQAAM57/action/storage_attestation","attest_author":"https://pith.science/pith/37C3DSBUHN6MEKPAUHYIQAAM57/action/author_attestation","sign_citation":"https://pith.science/pith/37C3DSBUHN6MEKPAUHYIQAAM57/action/citation_signature","submit_replication":"https://pith.science/pith/37C3DSBUHN6MEKPAUHYIQAAM57/action/replication_record"}},"created_at":"2026-05-18T01:15:56.895917+00:00","updated_at":"2026-05-18T01:15:56.895917+00:00"}