{"title":"Formalizing Mason-Stothers Theorem and its Corollaries in Lean 4","authors":"Jineon Baek, Seewoo Lee","doi":"arxiv-2408.15180","DOIUrl":null,"url":null,"abstract":"The ABC conjecture implies many conjectures and theorems in number theory,\nincluding the celebrated Fermat's Last Theorem. Mason-Stothers Theorem is a\nfunction field analogue of the ABC conjecture that admits a much more\nelementary proof with many interesting consequences, including a polynomial\nversion of Fermat's Last Theorem. While years of dedicated effort are expected\nfor a full formalization of Fermat's Last Theorem, the simple proof of\nMason-Stothers Theorem and its corollaries calls for an immediate\nformalization. We formalize an elementary proof of by Snyder in Lean 4, and also formalize\nmany consequences of Mason-Stothers, including (i) non-solvability of\nFermat-Cartan equations in polynomials, (ii) non-parametrizability of a certain\nelliptic curve, and (iii) Davenport's Theorem. We compare our work to existing\nformalizations of Mason-Stothers by Eberl in Isabelle and Wagemaker in Lean 3\nrespectively. Our formalization is based on the mathlib4 library of Lean 4, and\nis currently being ported back to mathlib4.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"49 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.15180","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The ABC conjecture implies many conjectures and theorems in number theory,
including the celebrated Fermat's Last Theorem. Mason-Stothers Theorem is a
function field analogue of the ABC conjecture that admits a much more
elementary proof with many interesting consequences, including a polynomial
version of Fermat's Last Theorem. While years of dedicated effort are expected
for a full formalization of Fermat's Last Theorem, the simple proof of
Mason-Stothers Theorem and its corollaries calls for an immediate
formalization. We formalize an elementary proof of by Snyder in Lean 4, and also formalize
many consequences of Mason-Stothers, including (i) non-solvability of
Fermat-Cartan equations in polynomials, (ii) non-parametrizability of a certain
elliptic curve, and (iii) Davenport's Theorem. We compare our work to existing
formalizations of Mason-Stothers by Eberl in Isabelle and Wagemaker in Lean 3
respectively. Our formalization is based on the mathlib4 library of Lean 4, and
is currently being ported back to mathlib4.