{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:HZ3NLV6KIJAMOSTHCTQRJP6ZIJ","short_pith_number":"pith:HZ3NLV6K","schema_version":"1.0","canonical_sha256":"3e76d5d7ca4240c74a6714e114bfd9425e1c5db90a40784ea8e655279d790d97","source":{"kind":"arxiv","id":"1509.07406","version":1},"attestation_state":"computed","paper":{"title":"Shortest Distance in Modular Hyperbola and Least Quadratic Nonresidue","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tsz Ho Chan","submitted_at":"2015-09-24T15:25:43Z","abstract_excerpt":"In this paper, we study how small a box contains at least two points from a modular hyperbola $x y \\equiv c \\pmod p$. There are two such points in a square of side length $p^{1/4 + \\epsilon}$. Furthermore, it turns out that either there are two such points in a square of side length $p^{1/6 + \\epsilon}$ or the least quadratic nonresidue is less than $p^{1/(6 \\sqrt{e}) + \\epsilon}$."},"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":"1509.07406","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2015-09-24T15:25:43Z","cross_cats_sorted":[],"title_canon_sha256":"1de377d4b1ea87eec57732434bbe0305d5745c138ef16f6932543cc7d7abd6f9","abstract_canon_sha256":"45bea42d39a694b6cc7d567bf979333b89fc4d0e0c5596b544d21a018e11b095"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:09.089492Z","signature_b64":"UuRbOj6SkndjdrWYU9UPkF+EcOX9huQ1K7ssqQefdlDeK017P9Ohtlp9hbYiE2bzDGWd2Tc1CNLf5rq8XY2eAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e76d5d7ca4240c74a6714e114bfd9425e1c5db90a40784ea8e655279d790d97","last_reissued_at":"2026-05-18T01:32:09.088718Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:09.088718Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Shortest Distance in Modular Hyperbola and Least Quadratic Nonresidue","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tsz Ho Chan","submitted_at":"2015-09-24T15:25:43Z","abstract_excerpt":"In this paper, we study how small a box contains at least two points from a modular hyperbola $x y \\equiv c \\pmod p$. There are two such points in a square of side length $p^{1/4 + \\epsilon}$. Furthermore, it turns out that either there are two such points in a square of side length $p^{1/6 + \\epsilon}$ or the least quadratic nonresidue is less than $p^{1/(6 \\sqrt{e}) + \\epsilon}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07406","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":"1509.07406","created_at":"2026-05-18T01:32:09.088852+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.07406v1","created_at":"2026-05-18T01:32:09.088852+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07406","created_at":"2026-05-18T01:32:09.088852+00:00"},{"alias_kind":"pith_short_12","alias_value":"HZ3NLV6KIJAM","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_16","alias_value":"HZ3NLV6KIJAMOSTH","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_8","alias_value":"HZ3NLV6K","created_at":"2026-05-18T12:29:25.134429+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/HZ3NLV6KIJAMOSTHCTQRJP6ZIJ","json":"https://pith.science/pith/HZ3NLV6KIJAMOSTHCTQRJP6ZIJ.json","graph_json":"https://pith.science/api/pith-number/HZ3NLV6KIJAMOSTHCTQRJP6ZIJ/graph.json","events_json":"https://pith.science/api/pith-number/HZ3NLV6KIJAMOSTHCTQRJP6ZIJ/events.json","paper":"https://pith.science/paper/HZ3NLV6K"},"agent_actions":{"view_html":"https://pith.science/pith/HZ3NLV6KIJAMOSTHCTQRJP6ZIJ","download_json":"https://pith.science/pith/HZ3NLV6KIJAMOSTHCTQRJP6ZIJ.json","view_paper":"https://pith.science/paper/HZ3NLV6K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.07406&json=true","fetch_graph":"https://pith.science/api/pith-number/HZ3NLV6KIJAMOSTHCTQRJP6ZIJ/graph.json","fetch_events":"https://pith.science/api/pith-number/HZ3NLV6KIJAMOSTHCTQRJP6ZIJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HZ3NLV6KIJAMOSTHCTQRJP6ZIJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HZ3NLV6KIJAMOSTHCTQRJP6ZIJ/action/storage_attestation","attest_author":"https://pith.science/pith/HZ3NLV6KIJAMOSTHCTQRJP6ZIJ/action/author_attestation","sign_citation":"https://pith.science/pith/HZ3NLV6KIJAMOSTHCTQRJP6ZIJ/action/citation_signature","submit_replication":"https://pith.science/pith/HZ3NLV6KIJAMOSTHCTQRJP6ZIJ/action/replication_record"}},"created_at":"2026-05-18T01:32:09.088852+00:00","updated_at":"2026-05-18T01:32:09.088852+00:00"}