Відео

@@the_eternal_student Ribenboim's "Fermat's Last Theorem for Amateurs"

@@noahtaul sorry, I'm not following this. What is w?

The third root of unity!

@@noahtaul Got it. I will try to write down the details later!

What does this sentence fragment mean: "But the only cubes are 0 and 1 mod 2 (I.e. not w or w+1)?"

@@Entropize1We’re working mod 2, and Z[w]/2Z[w] is F4, the field of 4 elements. The cubes in this field are only 0 and 1, the other two elements are not cubes

@@MichaelGuerry yep! This is fixed in my notes.

Next one tomorrow!

Not really. The theorem is false in Q_p, destroying any hopes of a congruence based proof, which is what most people think of when they think of "simple proof." There are simplifications to the method I presented, but the main narrative remains the same. There is also the Khare-Wintenberger proof, but I am not knowledgeable of how that works, and it is still quite complicated.

@@tahsinabrar-sl7yq running an FLT course this fall with notes that utterly dwarf these videos in quality. The course notes along with background reading notes will be publicly available.

@@aronhegedus it would take far longer than a month to get through all this, even without understanding. Years!

Surprisingly, kind of

...and consequently, as was indeed the case with Wile's discovery of his proof, it is a far better approach to be completely secretive until a point of certainty is reached that at least some merit would be earned for novelty or effort, even in the event of still being completely in error. A difficulty then immediately arises that any simple proof of FLT will be so underwhelming and deflating that the greater majority of high schooled mathematicians with a passing interest in a simple proof will overwhelm and moderate any public narrative with their incredulity such that a (largely) correct simple proof will never reach the attention of the infinitesimal minority of those able to sufficiently understand Wile's proof to make an authoritative determination as to its veracity. The latter, being mostly already convinced anyway that there is no such simple proof, will never take notice of yet another simple proof of FLT to be bothered with it, lest they put their professional reputation and career at risk. This is the current state of any purported simple proof of FLT, and it is extremely unlikely to ever change, despite Wile's proof's failure to explain "Why is it so?" (ie. axiom of choice) apart from the apparently patently obvious observation that "A circle is a elliptic curve in disguise". Thus any simple proof of FLT, and especially one that can also show "Why it is so." will be lauded only in the Hall of Ridicule and thereafter quickly lost to science, if not utterly banished from, the journals of western academic literature.

The likelihood of there being a quick proof of FLT is incredibly low, since it is false in Q_p. Moreover, the interesting part of FLT is the Modularity Theorem, which a quick proof of FLT would not provide.

Next one up tomorrow!

Hi there. Admittedly, by this far into the playlist, the intended audience is advanced graduate students in Number Theory. I recommend doing a quick UA-cam search along the lines "what is an elliptic curve?" and checking out any one of the excellent videos that pop up.

@@Entropize1 Yup, one already came up - haven't watched it yet. I remember how excited I was in high school to see (in my own library research) how a full-strength elliptic integral solves the general pendulum problem. I did browse some of your other vids, particularly the one with the list of hundreds of books. It goes without saying that 95% of that math did not exist in the time of Fermat, no? I can't imagine that even Euler or Gauss had it available either(?) Either Fermat didn't really have a proof (was mistaken), or there seems (naively) something awry in the structure of the universe --- thank you for your answer!

Elliptic curves are related to ellipses by a very long historical line, with the name "elliptic" being passed down until the object no longer had anything to do with ellipses. First, we had elliptic integrals. These were originally developed to find the arc length of an ellipse, but grew to be much more general. Then, we had elliptic functions, which arose when you took the inverse of elliptic integrals and extended that to the complex plane. The Jacobi elliptic function is still closely related to ellipses, but it is the last link you'll find. Of course, we finally have elliptic curves, which come from the Weierstrass elliptic function, and that comes from a certain elliptic integral which has nothing to do with ellipses. :)

@@zswu31416 I wouldn't say they have nothing to do with ellipses since they show up in the formula for the arc length of an ellipse. There's also a somewhat recent theorem relating elliptic curves to ellipses, but I can't recall it. I'll have to go digging.

@@Entropize1 The elliptic integral that comes from the arc length of an ellipse and the elliptic integral that leads to elliptic curves are quite different integrals, and only look similar. Correct me if I'm wrong

Well, almost everything. I'm still learning myself! Check out the linked video below for what you really need to get started.

Idk Honestly that guy got into math late in his life cause he was a lawyer. So you’d have to do math 24/7 to study all this

He didn't, because half of this math didn't exist when Fermat "proved" his theorem. We don't actually have his original proof, he just said it as a fact, added a footnote that he can't fit the proof into the book because it's too long and promptly died before passing it forward. It took about 3 centuries after that for the theorem to be confirmed, so unless Fermat was simply that ahead of his time and just didn't tell anyone, he probably made a mistake somewhere

​@@asd-wd5bj ...or there is a different way to prove it...

@@callejohansson5732 We can't rule out that possibility, but ever since Fermat's death mathematicans, some of the greatest ones among them (Euler, Lebesque, HIlbert to name a few) have tried to crack that proof without success (aside from proving few specific cases) Do not underestimate the amount of work that went into solving this problem, if there was another way achievable by 17th century math alone, I'm 100% sure somebody would've found it. Fermat was still human, it's way more reasonable to assume that he simply made an error than that he was a better mathematician that virtually all number theorists (and more) after him combined

@asd-wd5bj Prejudice and preconceptions often make us miss the obvious. It may be better to stay open-minded if we want to learn the truth.

Yea, but Google Shoenfield's Absoluteness Theorem to see why it doesn't matter.

@@Entropize1 Ok, thank you! I'll check it out.

@@Entropize1 I just checked it out. Math is so cool! Thank you very much

At least!

@@Entropize1 I already made a good way, ie a Ph.D. in Arithmetic Geometry. So yes, the rest of my life still might be too short.

I have a couple of videos on my channel about this exact topic. Do some digging and let me know if you still have any questions.

Watch my pinned video below before you do that :)

I think it's pretty exciting to have a chance for someone to give such a comprehensive and concise set of options for learning a broad subset of mathematical ideas! Overwhelming eventually gives way to engaging if you get bored enough :)

Depressive??? On the contrary. This is one of the most exciting and uplifting videos I have seen in a long time. If anyone is even remotely interested in math, this video is a goldmine. Everyone knows about Fermat's Last Theorem and it comes up in discussions now and then. It is an extremely simple problem and often people wonder why it took so long to solve. This video is phenomenal and very helpful to see the math required to prove FLT. It is the first time I have seen it listed in a structured way like this. Thank you for making this video! I will recommend it to my friends.

Not my forte or my cup of tea, unfortunately.

That's the spirit!

Fermat didn't know this stuff.

@@Viewpoint314 He did, but didn't have space to write it down.

​@@Viewpoint314 Of course! I was joking. But there is still the possibility that Fermat found an alternative proof involving some simple procedure: Wouldn't that be wonderful?!

​@@r.w.emersonii3501He did. It's buried in a secret forest, under a special tree. Also, there's a probably a goblin, or something.

@@r.w.emersonii3501 It would! but the boring answer is that he probably just made an error somewhere, still his intuition is incredible to just "feel" the right answer in this case

I haven't met one, and I've talked to many experts. I've even heard several experts say no.

@@Entropize1 Well, i'm welcoming myself to the No Club lol. I didnt finish my Maths degree and now being in my mid 50s and moving to Europe soon, from Australia, wondered whether to go finish it for curiosity sake as i love maths yet there is so much online re lectures etc, i dont know if i could handle stringent academia at my age plus i like debating things as am very unconventional. Back in the days of torrents, i once downloaded a file with maths textbooks only to find there were literally like 5000 of them. I went, oops lol That is a lot of textbooks :-)

@@Entropize1 Then how does anyone know it's true?

@@johnbyrne1022 because multiple experts have come together, each of whom know all the details of various parts of the proof, and have collaboratively exchanged results that allowed the result to be proven. This happens all the time in mathematics. Even though no one person knows all the details, all of the details are written down somewhere in a published journal or book.

10 years, you think? No way. You must be a genius. 😆