Full podcast episode: kzfaq.info/get/bejne/hdmYY5B7mLqWno0.html Lex Fridman podcast channel: kzfaq.info Guest bio: Edward Frenkel is a mathematician at UC Berkeley working on the interface of mathematics and quantum physics. He is the author of Love and Math: The Heart of Hidden Reality.

Edward is such a great communicator, is obviously brilliant and yet projects no ego which seeks to diminish the average man. Rare qualities. Bravo!

I'm pretty sure working on Fermat's Last Theorem was considered professional suicide which was another reason why Andrew Wiles worked on it in secret. So many mathematicians had tried to solve it and failed over the centuries it had a stigma of being a problem you could waste your career on.

I have a truly marvelous reaction to this video that this comment section is too narrow to contain.

Loved his explanation of the Riemann Hypothesis on Numberphile.

An episode of Star Trek TNG that aired in 1989 had Captain Picard discussing the “unsolved” Fermat’s Last Theorem. This is an awesome goof because although the story takes place in the distant future, it was created five years before the proof was published.

I tried proving this theorem and quickly learned the difference between a math student and a mathematician!

Amazing. Please more people like him

The premise for Fermat's Last Theorem is not difficult to understand. Solve it, though, took around 300 years. There's a book by Simon Singh called Fermat's Last Theorem for non-mathematicians. It talks about the story and the history behind the problem.

Fermat proved the theorem for fourth powers (in fact, he proved a stronger statement for fourth powers). Euler (almost) proved the theorem for cubes, but his proof had a gap that was later filled in.

The most charming mathematician I've ever heard !

It really is like an adventure story. Max Tegmark tells a similar adventure-like story regarding decoherence in his book Our Mathematical Universe, if anyone's interested.

This guy is amazing, pls bring him again

Fermat proved the case n = 4; Euler did the case of n = 3 (well he has credit for it; its a bit complicated) and other people have credit for specific exponents up to n = 11.

Please bring more mathematicians.....

Great guest! I've watched his videos in the past and they were always inspirational. I almost didn't recognize him until I heard his voice. His accent is so mathematician that can make anyone who listen to him long enough to pursue math for his career ;)

He’s so proud of his tweet he said it out loud.

One way I found to look at Fermat's Last Theorem, was to see it as a delta not as a sum. In other words, it's obvious that the difference between any two consecutive squares, is the set of all odd numbers. Some odd numbers are also squares. So far so good. If you could look at the entire set of all the differences between any two cubes, and demonstrate that for some reason that no cubes could be included in that set, you might then generalize that to any whole-number exponent.

In order for the multiplication operator to exist, both its elements must exist. Russell's Paradox: 1^2 1 # = 2 = 1+1 (first order) Then #^2 = (1 + 1)^2 = [1^2 + 1^2] + [2(1)(1)] = 4(1^2) (second order - via Binomial Expansion) where the first term is existence and the second is interaction (multiiplication, entanglement, entropy) Note that existence and interaction are not 4D (1,1,1,1) which diagonal is 4 elements without multiplication. Every number is prime relative to its own base. n = n(n/n) = n(1_n) Goldbach's Theorem: every even number is the sum of two primes: n + n = 2n n is odd. Godel's characterization of wff's in his meta-language only uses odd numbers (products of primes). Therefore, the sums of odd numbers (even numbers) cannot be represented by his wff's. In cluding products of sums (a + b)^2 in second order. So it is just Goedel's meta-language that is incomplete, not positive real numbers. Together with Fermat's Last Theorem (applied to multinomials of arbitray powers), the arithmetic system is complete and consistent for positive real numbers. There are no negative numbers: -c = a - b, b > a b - c = a, a + 0 = a, a - a = 0.. If there are no negative numbers, there are no square roots of negative numbers. Proof of Fermat's Theorem for Village Idiots (n>2) c = a + b c^n = a^n + b^n +f(a,b,n) (Binomial Expansion) c^n = a^n + b^n iff f(a,b,n) = 0 f(a,b,n)0 c^n a^n + b^n QED Also valid for n > 1 c^2 = [a^2 + b^2] + [2ab]] 2ab < >0 c^2 a^2 + b^2 QED (Pythagoras was wrong; use your imagination) Check out my pdfs in physicsdiscussionforum "dot" org.

There's an old BBC doc on this.

Well, an analogue to FLT could be there do not exist a,b,c,d positive integers, e,f positive integers and n>=3 positive integer such that: a^n+b^n+c^n+d^n=e^n+f^n. Somebody tries this conjecture.

They begin by explaining what Fermat's Last Theorem is, something anybody with basic math can understand. Tony Padilla in a Numberphile video mistakenly said that the Collatz Conjecture is one of the $1m Millennium Problems, and I realised why that was not true, and why, if it had not been proved before 2001, Fermat would also not be one of the Millennium Problems. If Fermat WAS added to the Millennium Problems, it would have been the only problem that this would be true of: that it can be understood by someone with elementary school math. All the other Millennium Problems are very deep, complex math that you have to be a postgrad to even understand what the problem is! Whereas Fermat is "Prove why the sum of two like integer powers higher than the second power is never the same power of an integer."

Also choice of space as it real numbers or etc dimentions

Fermat may have made a distinction between the simple identities which we encounter in algebra like (a+b)^2=a^2+2ab+b^2 , a^3_b^3=(a_b)(a^2+ab+b^2) and the derived identities like Euclid's identity: (m^2_n^2)^2+(2mn)^2=(m^2+n^2)^2 in the following sense: The first identities are simple , in the sense that they stand alone , they are immediately given. The others like Euclid's were derived and have the property of bridging the gap between the set of couples (E=3 , there is no such a connecting identity, an identity of Euclid's type. Numeration cannot apply to a,b,c if a^n+b^n=c^n , n>=3, therefore they do not exist.

Visually it is easy to demonstrate! For x2 + y2 = z2 is a square sharing the hypothesis (multiply opposite sides: side x times opposite side x + side y x opposite side y = shared z x shared z [itself] ) BTW, this is a 2D triangle and 2D square. However, for any other, such as a cube, or 4th power, etc. there is no shared z face for all 6 faces of a cube as this is no longer a 2D figure, but 3D, 4D, etc. For example, top and bottom faces will not share z face just the top and bottom line of z.

I think I’m going to fall in love with Mathematics after watching this. I’m 33.

I spoke with one disciple of Pythagoras; Yes they are still around since the 5th century b.c. I told him about the Fermat's conjecture; I could see the anger, the dismay in his eyes. He says to me That Fermat is guilty of a great sin in the eyes of the Pythagorician fraternity. That it was sinful and devilish to even suggest that the expression a^n+b^n=c^n where a,b,c are integers and n an integer >=3, was worthy of consideration for, he rejected one of our core beliefs, actually our main first principle. In our eyes the sphere , this perfect geometrical figure actually, the circle this perfect geometrical figure and unity are identical, which seem bizarre and paradoxycal to the neophyte. Our Master Pythagoras may he dwell in the realm of numbers left his theorem for posterity. The meaning of his great theorem, given that the one, the unit is identical to the sphere, the circle , this perfect geometrical shape is that the one, the unit can be written as the sum of the squares of two rationnal numbers. Fermat's great sin is to suggest otherwise! That the one , the unit could be written as the sum of two powers greater than 2 of rationnal numbers. Such a doubter is anathema to us. And he goes with a cruel smirk on his face, too bad he was not in the 5th century b.c. I tried to explain to him that beautiful mathematics came out of this consideration , the latest of which was Wiles beautiful work, which resulted in the proof of Fermat's conjecture as a corollary. He starred at me silently , contemptuously. I decided to cut short the discussion and split. I thought I understood why Fermat sinned greatly in their eyes. He suggested that a more perfect geometrical figure could exist , more perfect than the sphere , than the circle!

That picture of Wiles shows a board with a false statement because he should have said nontrivial because x=y=z=0 is an integer solution for all n>=3.

Isn’t it the case of Euclid 2dwith parallel lines axiom vs other geometry sphere hyperbola geometry or Poincaré 2d ?

thank you for this

The answer is 42

lol - I remember when Taniyama - Shimura was just a conjecture...

эх, придется видимо заказать его книжку

I haven't yet finished watching the whole - GREAT - discussion with Edward Frenkel, but I have serious doubt that Fermat had a proof of his last theorem. It took 350 years to find that proof and they did it indirectly by solving another - equivalent - problem; so Fermat, if you had a proof, she wasn't correct. 🤔

This fellow is amazing! Embarrassed I don't know his name... He's like Max Tegmark without the ticks.

I want to take his class!

It’s math teacher Jamie Lannister!

sorry lex clips guy, im pretty sure its just one mathematician not multiple mathematicians

Imagine if fermat knew this problem was impossibly difficult and just decided to troll us.

in one of my comments presented an elementary proof of wiles theorem(FLT.the proof is using a second factorization of the binomX^N +Y^N.using this second factorization of this bind find a second proof of wiles theorem(FLT.good luck.As you see i claim to discover tow elementary proofs of FLT.now i claim too to be able to prove collate conjecture.and i am only an amateur.

Wouldn't it be odd if the Modularity Theorem, the key to proving Fermat's Last Theorem, also turns out to be the key in proving Riemann's Hypothesis? Maybe Ken Ribet can make another connection?

Note that the equation of a circle is wrong: c= a + b c^2 = a^2 + b^2 + 2ab c^2 = a^2 + b^2 iff 2ab = 0 2ab 0 c^2 a^2 + b^2 (I edited this for the inequality; for some reason I had it equal originally which didn't make sense given the previous line. My bad, sorry .. :) Please work this out for a 5,4,3 right triangle, and note that 5:= 4 + 3i 55* = 16 + 9 = 25 BUT i = sqr(-1) i^2 = sqr(-1)sqr(-1)= sqr[(-1)(-1)] = sqr(1^2) = 1 -1 This has profound consequences for conventional physics (Relativity, Quantum Mechanics, QFT) Much more to this story, but I don't have the spacetime to write it here; write if you get work... :) (I have developed a lot of it in pdfs, which are available on request.)

1:38 you'll thank me.

Russell's Paradox "A barber in a village shaves all those and only those that don't shave themselves. Does the barber shave himself?? - Bertrand Russell Answer: The barber doesn't exist (a barber cant both shave and not shave himself) This is actually an expression of the relation 1^2 1 (a unit cannot both multiply and not multiply itself). not an relation in set theory. well, ok (1,1^2) are independent sets...... x dot x^2 = 0 x cross x^2 = 0 (polynomials f = 1 + x + x^2 + .... x^n = Tr|M | as bases for sets 1 dot x = 0)

Proof of Fermat's Last Theorem (6 Lines) Hypothesis c^n a^n + b^n for all a,b,c, n positive real numbers Proof Let c,a,b, n be positive real numbers, n > 1 (so n>2 is automatic) Define addition as : c = a + b c^n = (a + b)^n = [a^n + b^n] + f(a,b,n) (Binomial expansion on r.h.s.) c^n = [a^n + b^n] iff f(a,b,n) = 0 f(a,b,n) 0 c^n [a^n + b^n] Also true for multinomials of any order, so system is complete and consistent (see Godel Urban legend says this proof was discovered within three days after its appearance by a math "C" student, who was then hustled away by the men in black (or white) coats, never to be heard from again. OTH, you may have read it here first. Please tell Dr. Wiles...

The conjecture is called Taniyama-Shimura/Taniyama-Weil/Taniyama-Shimura-Weil conjecture, AKA the modularity theorem - please rename the section.

You can't start with c=a+b.

Bring Grigori Perelman!

You don’t want to dig a hole ...

My greatest intellectual achievement was walking around the park. You have to be bored enough to let your mind wander, and imagine, and fill in the empty space.

Lex and Andrew Wiles would be a great episode

So in love for the last time EVER EVER EVER ❤

Para mi, los matemáticos se cansaron y aceptaron un camino muy complicado y de 100 páginas por lo que solo un número mínimo de matemáticos entiende cual es la prueba. Aquí un enfoque diferente hacia el Último Teorema en una sola página: kzfaq.info/get/bejne/Y9CgdJCl15ycp6s.html

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10:50

I going outside to make mud mud pies now!

Fermat was probably lying/trolling but had the audacity to make the claim knowing that someone might use the claim as a clue that it could be done, inspiring others to work on it. Just like Frankel mentions here with Wiles.

Energy propagates always orthogonal to the direction of mass movement. Speed is the link.

Hello Professor Jay Daigle, I am looking forward to meeting you online 4/26/2024 about my presentation of Collatz Sequence. Taha M. Muhammad/ USA Kurd Iraq Owner of Collatz, Euler, and Fermat's both last Theories

A businessman once told me that it's hard to attract scientists to industry because they have very different motivation. They care about their ego, not money. They want to be a first author on the paper instead of their results being owned by a company. In most fields we do care, to a certain extent, about ownership of ideas. For example, on biology conferences you oftentimes can't make photos of slides. But unfortunately this egoism reaches its most disguising forms in math, where people never share their best ideas for the sake of accelerating the research. Interestingly enough, in physics they situation is quite different, as there scientists oftentimes "speculate" on certain things, mainly to initiate a discussion.

I can relate to the feeling of being the only person who possesses a piece of valuable knowledge. I'm a composer who explores and uses harmony in ways that I have never heard elsewhere. It's lonely not because I don't want to share it, but because I don't know anyone who is actually interested.

I understand why antisemitism exists. The world is full of ignorant people. But I'll never understand how is it possible for things like antisemitism to exist in such a place full of world-class intellectuals (Soviet Mathematicians)

why would it matter if they were negative if they were being squared

When it takes 4minutes and 31 seconds in to simply begin to explain fermat's last theorem after being asked you can begin to understand why math is flagging in america. I have a proof as to why but the ....

Proof of Goldbach's conjecture define 1_n := n/n 1_m = 1_n iff m = n Then n = n(1_n) for all n (All numbers are prime relative to their own base) The n + n = 2n QED Send beer and pizza

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It was a nerd joke. He'd made a nerd joke. Everybody knew it was just Pythagorean Theorem. I've been wondering if the guy whom solved it merely proved you can't have more than three-dimensional space. And am too unintellectual to care.

BINGO

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Define f(a)=a^n, for any n at all (integers fine). Now, f(a) + f(b) = f(c) is the new equation, which is always feasible for any continuous line. All n are possible, if all functions f(a) are lines (turn curves into lines, as example). Yay! Hooray! Thanks for listening.

As a former management consultant - can I observe - if you can't spell Pythagoras then no one is going to listen to everything else you have to say.

Why do people pronounce the letter Z as "zee"? It's not like that, the correct way is "zeta".

Pytgagoras. Fire your editor.

MMAT to the moon

Ex soviet jews ftw.

the subtitles are hilarious