{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:H53HMXM4AVPTMB4VPIFYIGX4AQ","short_pith_number":"pith:H53HMXM4","schema_version":"1.0","canonical_sha256":"3f76765d9c055f3607957a0b841afc0420631fac22cc9cc32a8f50ef09cbae21","source":{"kind":"arxiv","id":"1812.07275","version":1},"attestation_state":"computed","paper":{"title":"Experiments with the Markoff surface","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Matthew de Courcy-Ireland, Seungjae Lee","submitted_at":"2018-12-18T10:18:50Z","abstract_excerpt":"We confirm, for the primes up to 3000, the conjecture of Bourgain, Gamburd, and Sarnak on strong approximation for the Markoff surface $x^2+y^2+z^2 = 3xyz$ modulo primes. For primes congruent to 3 modulo 4, we find data suggesting that some natural graphs constructed from this equation are asymptotically Ramanujan. For primes congruent to 1 modulo 4, the data suggest a weaker spectral gap. In both cases, there is close agreement with the Kesten-McKay law for the density of states for random 3-regular graphs. We also study the connectedness of other level sets $x^2+y^2+z^2-3xyz = k$. In the deg"},"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":"1812.07275","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-18T10:18:50Z","cross_cats_sorted":[],"title_canon_sha256":"f6d14df3334ceca46c3ee67d6db996dfc0fbb1b08afb2670ad1242a2abd2375e","abstract_canon_sha256":"ebc55b20dbed2a716efabe865e0c4c7fb3eb67f3e83aa1e71254376ed11d2623"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:00.995073Z","signature_b64":"PwQ5IabzI5adtSwK1RmvpX1NMeLvcOGucDflu7NYRGOJkK1ngxMv6CxIaSFr7imcSF8ifg5AyaUVLhmCYhyzCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f76765d9c055f3607957a0b841afc0420631fac22cc9cc32a8f50ef09cbae21","last_reissued_at":"2026-05-17T23:58:00.994377Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:00.994377Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Experiments with the Markoff surface","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Matthew de Courcy-Ireland, Seungjae Lee","submitted_at":"2018-12-18T10:18:50Z","abstract_excerpt":"We confirm, for the primes up to 3000, the conjecture of Bourgain, Gamburd, and Sarnak on strong approximation for the Markoff surface $x^2+y^2+z^2 = 3xyz$ modulo primes. For primes congruent to 3 modulo 4, we find data suggesting that some natural graphs constructed from this equation are asymptotically Ramanujan. For primes congruent to 1 modulo 4, the data suggest a weaker spectral gap. In both cases, there is close agreement with the Kesten-McKay law for the density of states for random 3-regular graphs. We also study the connectedness of other level sets $x^2+y^2+z^2-3xyz = k$. In the deg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.07275","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":"1812.07275","created_at":"2026-05-17T23:58:00.994478+00:00"},{"alias_kind":"arxiv_version","alias_value":"1812.07275v1","created_at":"2026-05-17T23:58:00.994478+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.07275","created_at":"2026-05-17T23:58:00.994478+00:00"},{"alias_kind":"pith_short_12","alias_value":"H53HMXM4AVPT","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_16","alias_value":"H53HMXM4AVPTMB4V","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_8","alias_value":"H53HMXM4","created_at":"2026-05-18T12:32:28.185984+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/H53HMXM4AVPTMB4VPIFYIGX4AQ","json":"https://pith.science/pith/H53HMXM4AVPTMB4VPIFYIGX4AQ.json","graph_json":"https://pith.science/api/pith-number/H53HMXM4AVPTMB4VPIFYIGX4AQ/graph.json","events_json":"https://pith.science/api/pith-number/H53HMXM4AVPTMB4VPIFYIGX4AQ/events.json","paper":"https://pith.science/paper/H53HMXM4"},"agent_actions":{"view_html":"https://pith.science/pith/H53HMXM4AVPTMB4VPIFYIGX4AQ","download_json":"https://pith.science/pith/H53HMXM4AVPTMB4VPIFYIGX4AQ.json","view_paper":"https://pith.science/paper/H53HMXM4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1812.07275&json=true","fetch_graph":"https://pith.science/api/pith-number/H53HMXM4AVPTMB4VPIFYIGX4AQ/graph.json","fetch_events":"https://pith.science/api/pith-number/H53HMXM4AVPTMB4VPIFYIGX4AQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/H53HMXM4AVPTMB4VPIFYIGX4AQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/H53HMXM4AVPTMB4VPIFYIGX4AQ/action/storage_attestation","attest_author":"https://pith.science/pith/H53HMXM4AVPTMB4VPIFYIGX4AQ/action/author_attestation","sign_citation":"https://pith.science/pith/H53HMXM4AVPTMB4VPIFYIGX4AQ/action/citation_signature","submit_replication":"https://pith.science/pith/H53HMXM4AVPTMB4VPIFYIGX4AQ/action/replication_record"}},"created_at":"2026-05-17T23:58:00.994478+00:00","updated_at":"2026-05-17T23:58:00.994478+00:00"}