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[Merged by Bors] - feat(FermatLastTheorem): Add a relaxed variant of FLT allowing nonzero unit solutions #16060

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flt-unit-variant
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[Merged by Bors] - feat(FermatLastTheorem): Add a relaxed variant of FLT allowing nonzero unit solutions #16060
jcpaik wants to merge 6 commits intomasterfrom
flt-unit-variant

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@jcpaik jcpaik commented Aug 22, 2024

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 FermatLastTheoremWith' α states that any nonzero solution to Fermat equation $a^n+b^n=c^n$ should be units $(a, b, c)$ or their common multiples. The stronger original FermatLastTheoremWith α states that there is no nonzero solution $(a, b, c)$ from the beginning.

For α = ℕ or the 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 field $k$, but FermatLastTheoremWith' k[X] is.

I speculate that for some integral ring α over , the original statement of FLT is false while the suggested unit variant is true. For example, take the integral closure of $\mathbb{Z}[w]$ 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.

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@github-actions github-actions bot added the new-contributor This PR was made by a contributor with at most 5 merged PRs. Welcome to the community! label Aug 22, 2024
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github-actions bot commented Aug 22, 2024

PR summary 73cadf8e10

Import changes for modified files

No significant changes to the import graph

Import changes for all files
Files Import difference

Declarations diff

+ FermatLastTheoremWith'
+ FermatLastTheoremWith'.fermatLastTheoremWith
+ FermatLastTheoremWith.fermatLastTheoremWith'
+ fermatLastTheoremWith'_iff_fermatLastTheoremWith
+ fermatLastTheoremWith'_nat_int_tfae
+ fermatLastTheoremWith'_of_field

You can run this locally as follows
## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

@jcpaik jcpaik added the t-number-theory Number theory (also use t-algebra or t-analysis to specialize) label Aug 22, 2024
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@Ruben-VandeVelde Ruben-VandeVelde left a comment

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Some small comments:

@Ruben-VandeVelde Ruben-VandeVelde added the awaiting-author A reviewer has asked the author a question or requested changes. label Aug 27, 2024
jcpaik and others added 2 commits August 30, 2024 20:57
Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>
@jcpaik jcpaik removed the awaiting-author A reviewer has asked the author a question or requested changes. label Aug 30, 2024
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No objection from me.

maintainer merge

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github-actions bot commented Sep 9, 2024

🚀 Pull request has been placed on the maintainer queue by Ruben-VandeVelde.

@github-actions github-actions bot added the maintainer-merge A reviewer has approved the changed; awaiting maintainer approval. label Sep 9, 2024
@jcommelin jcommelin requested a review from kbuzzard September 10, 2024 06:37
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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

@github-actions github-actions bot added the ready-to-merge This PR has been sent to bors. label Sep 10, 2024
mathlib-bors bot pushed a commit that referenced this pull request Sep 10, 2024
…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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mathlib-bors bot commented Sep 10, 2024

Pull request successfully merged into master.

Build succeeded:

@mathlib-bors mathlib-bors bot changed the title feat(FermatLastTheorem): Add a relaxed variant of FLT allowing nonzero unit solutions [Merged by Bors] - feat(FermatLastTheorem): Add a relaxed variant of FLT allowing nonzero unit solutions Sep 10, 2024
@mathlib-bors mathlib-bors bot closed this Sep 10, 2024
@mathlib-bors mathlib-bors bot deleted the flt-unit-variant branch September 10, 2024 12:03
bjoernkjoshanssen pushed a commit that referenced this pull request Sep 11, 2024
…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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