[Merged by Bors] - feat(FermatLastTheorem): Add a relaxed variant of FLT allowing nonzero unit solutions #16060
[Merged by Bors] - feat(FermatLastTheorem): Add a relaxed variant of FLT allowing nonzero unit solutions #16060
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PR summary 73cadf8e10Import changes for modified filesNo significant changes to the import graph Import changes for all files
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Ruben-VandeVelde
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Some small comments:
Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>
Ruben-VandeVelde
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No objection from me.
maintainer merge
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🚀 Pull request has been placed on the maintainer queue by Ruben-VandeVelde. |
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Yeah this is all fine by me. To be quite honest I do think this work is interesting -- it makes formal the statement "Mason-Stothers is FLT for polynomial rings". The naive generalization is false (loads of constant solutions). bors merge |
…o unit solutions (#16060) Adds a variant `FermatLastTheoremWith'` of FLT for general commutative semirings, allowing nonzero unit solutions and their multiples. Co-authored-by: Jineon Baek <34874130+jcpaik@users.noreply.github.com>
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Pull request successfully merged into master. Build succeeded: |
…o unit solutions (#16060) Adds a variant `FermatLastTheoremWith'` of FLT for general commutative semirings, allowing nonzero unit solutions and their multiples. Co-authored-by: Jineon Baek <34874130+jcpaik@users.noreply.github.com>
Adds a variant
FermatLastTheoremWith'of FLT for general commutative semirings, allowing nonzero unit solutions and their multiples.Supersedes #14270. I find this version less intrusive to the original statement of FLT.
The relaxed 'unit' variant$a^n+b^n=c^n$ should be units $(a, b, c)$ or their common multiples. The stronger original $(a, b, c)$ from the beginning.
FermatLastTheoremWith' αstates that any nonzero solution to Fermat equationFermatLastTheoremWith αstates that there is no nonzero solutionFor$k$ , but
α = ℕorℤthe two statements are equivalent to the original FLT. A purpose of this is to include a polynomial version of FLT which is easy to prove (relevant Zulip discussion here).FermatLastTheoremWith k[X]is false for general fieldFermatLastTheoremWith' k[X]is.I speculate that for some integral ring$\mathbb{Z}[w]$ where $w = (-1 + \sqrt{3} i ) / 2$ is the 3rd root of unity satisfying $w^3 = 1$ . The original statement for $n = 6k + 1$ is false as $w^n + (w^2)^n = (-1)^n$ , but note that the solutions only consists of units.
αoverℤ, the original statement of FLT is false while the suggested unit variant is true. For example, take the integral closure of