{"id":"palomar/PALOMAR-2026-08-27-000009","url":"https://github.com/yuma-mizuno/markoff-modp/tree/aea702e1116c5688a0c570127ebcd3f194924439","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-08-27-000009-v1.json","repo":"https://github.com/yuma-mizuno/markoff-modp","tags":["math.NT","msc:11D25","number-theory","lean","strong-approximation","markoff-surface"],"score":0,"title":"yuma-mizuno/markoff-modp","trust":"high","domain":"math.NT","source":"palomar","authors":["Yuma Mizuno"],"reviews":0,"subject":"math.NT","summary":"Lean 4 formalization of strong approximation for the Markoff surface modulo sufficiently large primes. The primary source is Bourgain-Gamburd-Sarnak.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-08-27","reactions":{},"formalized":true,"formalizes":[{"title":"Strong Approximation and Diophantine Properties of Markoff Triples","identifier":"https://doi.org/10.1090/jams/1061","relationship":"formalizes"},{"title":"Greatest common divisors of u - 1, v - 1 in positive characteristic and rational points on curves over finite fields","identifier":"https://doi.org/10.4171/JEMS/409","relationship":"formalizes"},{"title":"Some cases of Vojta's Conjecture on integral points over function fields","identifier":"https://doi.org/10.1090/S1056-3911-07-00489-4","relationship":"formalizes"},{"title":"A new proof of Chen's theorem for Markoff graphs","identifier":"https://doi.org/10.1007/s00222-025-01346-9","relationship":"formalizes"},{"title":"Connectivity of Markoff mod-p graphs and maximal divisors","identifier":"https://doi.org/10.1112/plms.70027","relationship":"adapts"}]}