Formalizing Mason-Stothers Theorem and its Corollaries in Lean 4
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914056806334464 |
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| author | Baek, Jineon Lee, Seewoo |
| author_facet | Baek, Jineon Lee, Seewoo |
| contents | 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 by Snyder in Lean 4, and also formalize many consequences of Mason-Stothers, including nonsolvability of Fermat-Cartan equations in polynomials, nonparametrizability of a certain elliptic curve, and Davenport's Theorem. We compare our work to existing formalizations of the Mason-Stothers by Eberl in Isabelle and Wagemaker in Lean 3 respectively. Our formalization has been integrated into the mathlib library of Lean 4. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15180 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Formalizing Mason-Stothers Theorem and its Corollaries in Lean 4 Baek, Jineon Lee, Seewoo Logic in Computer Science Rings and Algebras 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 by Snyder in Lean 4, and also formalize many consequences of Mason-Stothers, including nonsolvability of Fermat-Cartan equations in polynomials, nonparametrizability of a certain elliptic curve, and Davenport's Theorem. We compare our work to existing formalizations of the Mason-Stothers by Eberl in Isabelle and Wagemaker in Lean 3 respectively. Our formalization has been integrated into the mathlib library of Lean 4. |
| title | Formalizing Mason-Stothers Theorem and its Corollaries in Lean 4 |
| topic | Logic in Computer Science Rings and Algebras |
| url | https://arxiv.org/abs/2408.15180 |