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wlyaninWhat's the shortest proof of Fermat's Last Theorem currently available?
What's the shortest proof of Fermat's Last Theorem currently available, starting from "first principles", measured in terms of say page numbers?
Since Wiles/Taylor-Wiles 1994,1995 work, much progress has been made in the field and new routes to FLT have been established. See https://mathoverflow.net/questions/361714/recent-developments-in-the-proof-of-fermats-last-theorem particular the overview given by Ken Ribet at the 2020 JMM in Denver, CO, "A 2020 View of Fermat's Last Theorem": youtube.com/watch?v=mq9BS6S2E2k https://www.youtube.com/watch?v=mq9BS6S2E2k
However, even looking at the major papers referenced, we do not get an accurate picture for exactly how much math goes into the full proof. See Kevin Buzzard's comments in https://mathoverflow.net/questions/97820/a-recommended-roadmap-to-fermats-last-theorem?rq=1:
>Let me make the following "philosophical" point. If you read these books like Cornell-Silverman-Stevens, then probably you'll be happy, but in some sense all you will have learnt is how to deduce FLT from stuff that was regarded as standard in the early 90s. So, for example, you will have to take on trust the modularity of E\3\] (proved by Langlands and Tunnell using a lot of very complicated analysis, e.tic continuation of Eisenstein series, cyclic base change for GL(2), non-Galois cubic base change...).
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>As another example you'll have to believe in the Neron model of the Jacobian of a curve over a p-adic field, the relationship between the reduction of the curve and the reduction of the model. You'll have to believe in local-global for modular forms, a hard theorem of Carayol involving some very delicate vanishing cycles calculations. You'll have to believe in SGA7. You'll have to believe in the reduction of Shimura curves at primes dividing the discriminant (to follow Ribet's work) and this is very technical...
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>and you'll have to believe in Fontaine's work on p-divisible groups in order to follow Ramakrishna's thesis, which is crucial. Those are just a few things that spring to mind. In books like Cornell-Silverman-Stevens a lot of these things are regarded as "standard" (because they *were*!) and references are given. On the other hand R=T theorems are now regarded as "standard"! And FLT follows "via a standard argument" from such theorems! So in some sense it's hard to see where to logically draw the line :-)
with the following addendum by Felipe Voloch
>To continue on Kevin's riff, you'll also have to believe Faltings's proof of the Tate conjecture, the Hecke-Weil theorem relating modular forms and L-series with functional equations and a bunch of other stuff.
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Even the Wikipedia says that the most commonly recommended book is not complete [https://en.wikipedia.org/wiki/Wiles%27s\_proof\_of\_Fermat%27s\_Last\_Theorem#Overviews\_available\_in\_the\_literature https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem#Overviews_available_in_the_literature:
>For those in search of a commercially available book to guide them, he recommended that those familiar with abstract algebra read Hellegouarch, then read the Cornell book,\[9\] https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem#cite_note-CornellBook-hich is claimed to be accessible to "a graduate student in number theory". The Cornell book does not cover the entirety of the Wiles proof.\[12\] https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem#cite_note-AMS-review-12
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In fact some people don't even believe that there's a single person on Earth who knows the complete proof (from first principles) in their head https://xenaproject.wordpress.com/2019/09/27/does-anyone-know-a-proof-of-fermats-last-theorem/.
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So again, **Questions:**
* what are all the routes to FLT currently known,
* how long is each one (in terms of page numbers, if one were to write a complete book say starting from first principles --- I'll allow the baseline of standard undergraduate level analysis and algebra, like basic complex analysis (of which modular forms are not a part of lol), and basic algebra (basic ring theory for instance)),
* and which one is the shortest? 9 ga.
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