Mathematicians explains Fermat's Last Theorem | Edward Frenkel and Lex Fridman
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- เผยแพร่เมื่อ 1 ก.ค. 2024
- Lex Fridman Podcast full episode: • Edward Frenkel: Realit...
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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.
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Full podcast episode: th-cam.com/video/Osh0-J3T2nY/w-d-xo.html
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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.
Fermat's Great Theorem 1637 - 2016 !
I proved on 09/14/2016 the ONLY POSSIBLE proof of the Fermat's Great! Theorem (Fermata!).
I can pronounce the formula for the proof of Fermat's Great Theorem:
1 - Fermat's Great Theorem NEVER! and nobody! NOT! HAS BEEN PROVEN !!!
2 - proven! THE ONLY POSSIBLE proof of Fermat's Great Theorem !
3 - Fermat's Great Theorem is proved universally-proven for all numbers !
4 - Fermat's Great Theorem is proven in the requirements of himself! Fermata 1637 y.
5 - Fermat's Great Theorem proved in 2 pages of a notebook !
6 - Fermat's Great Theorem is proved in the apparatus of Diophantus arithmetic !
7 - The proof of the great Fermat's Great Theorem, as well as the formulation,
is easy for a student of the 5th grade of the school to understand !!!
8 - Me! opened the GREAT! A GREAT Mystery! Fermat's Great Theorem !
(not a "simple" "mechanical" proof)
Explicit formula or recursion formula ? Infinite ♾️ or pie=3.? The patterns 1,4,7,10,13
Pt
Y9
I can pronounce the formula for the proof of Fermat's Great Theorem:
1 - Fermat's Great Theorem NEVER! and nobody! NOT! HAS BEEN PROVEN !!!
2 - proven! THE ONLY POSSIBLE proof of Fermat's Great Theorem !
3 - Fermat's Great Theorem is proved universally-proven for all numbers !
4 - Fermat's Great Theorem is proven in the requirements of himself! Fermata 1637 y.
5 - Fermat's Great Theorem proved in 2 pages of a notebook !
6 - Fermat's Great Theorem is proved in the apparatus of Diophantus arithmetic !
7 - The proof of the great Fermat's Great Theorem, as well as the formulation,
is easy for a student of the 5th grade of the school to understand !!!
8 - Me! opened the GREAT! A GREAT Mystery! Fermat's Great Theorem !
(not a "simple" "mechanical" proof
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.
Try and others may add to your effort, thus widening the comment section.
😅
Too narrow for your big head? 🥱
Yeah bro… keep telling yourself that.
@@jamauldrew it's a meta joke 😂😂😂
Loved his explanation of the Riemann Hypothesis on Numberphile.
Yes. Even for a non-math person the problem was explained very clearly
@@conforzo that’s the most frustrating thing about the Reimann Hypothesis. Simple to understand, yet extremely difficult to prove. He does a remarkable job of explaining it though.
Good old numberphile
If you fully understand tell me why 1+2+3+.... =-1/12 (in analytic). This remain a mistery to me
He sparked my love for math with that video
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.
That's not a goof, that's just time. Goofs refer to preventable mistakes, but there was no way the writers could have known it would be solved that quickly.
Or they can explain it away by saying that the theorem would remain unsolved for many more years in the future in that particular timeline? 😃
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.
I read it and it's a really beautiful book, even for people studying mathematics like me.
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.
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.
The most charming mathematician I've ever heard !
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 ;)
numberphile brought me here
He’s so proud of his tweet he said it out loud.
There's an old BBC doc on this.
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.
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.
Isn’t it the case of Euclid 2dwith parallel lines axiom vs other geometry sphere hyperbola geometry or Poincaré 2d ?
Also choice of space as it real numbers or etc dimentions
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.
эх, придется видимо заказать его книжку
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."
The P vs NP problem can be understood by anyone with only elementary math.
lol - I remember when Taniyama - Shimura was just a conjecture...
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.
@@macminty_ So, what will be the geometric shape of an object when n=3? a cube? I interpreted an imaginary shape in which the top and bottom faces will only intersect z line and not the z face. Because, would it not be an extension of the 2D triangle (or 2D square) when n=2.
@@macminty_ So basically, what you are stating is that when n is greater than 2, the geometrical object becomes an abstraction. Also, for z will not be able to face all the geometrical faces, like z does in the 2D triangle as in that situation z increases when the other sides increase in proportional ratio. I hope you can see what I am trying to get at.
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.
The equation y^2 = x^3 +1 also has no such rational parametrization, but it admits integer solutions (3,2), (-3,2), same with x^3 + y^3 + z^3 = 3, etc. Rational parametrizations help find solutions, but not prove that their aren't any.
@@theflaggeddragon9472
Yet the equation x^2+y^2=z^2 , has its true meaning in an identity. The Euclid's identity.
Your equation solvable by writing x^2_x+1=y and y=x+1, these two relations are the meaning of this equation when we require x and y to belong to Z.
I think I’m going to fall in love with Mathematics after watching this. I’m 33.
It's such a beautiful subject 😊
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.
thank you for this
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!
It’s math teacher Jamie Lannister!
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...
I need whatever you're smoking buddy
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. 🤔
sorry lex clips guy, im pretty sure its just one mathematician not multiple mathematicians
The answer is 42
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.)
why don't you publish it? I can't make sense of what you wrote there, maybe formulate what you are trying to prove more clearly?
i^2 != sqrt[(-1)(-1)], this is markedly incorrect. You cannot combine square roots like this for complex values.
@@marcyeo1 Why not? (Number lines are not vectors)
In fact, negative numbers do not exist (so neither do their square roots)..
-c = a-b, b>a b-c = a
a-a = 0
a=a
x+1=0 iff x=-1
-1+1 = 0
1=1
i^4 = (i^2)(i^2) =(-1)(-1) = 1 ???
1=sqr[(-1)(-1)] = sqr[(i^2)(i^2)] = sqr[i^4] = 1
Nice bait
@@BuleriaChkBecause sqrt(1) = 1 is only a convention. There are always two numbers that square to any complex number x (which only coincide for x = 0). That is why people have made an arbitrary choice that the square root is defined to be the non-negative root of a non-negative real number. As you see yourself your "proof" that -1 = i^2 = 1 is a contradiction, from which we conclude that one of your equations is incorrect. It is not always the mathematical community who needs to change, but sometimes it‘s yourself.
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!
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.
You don’t want to dig a hole ...
Imagine if fermat knew this problem was impossibly difficult and just decided to troll us.
You can't start with c=a+b.
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)
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?
How? Modularity gives an analytic continuation for L-functions of elliptic curves. The analytic continuation of the Riemann zeta function was well understood before modularity.
1:38 you'll thank me.
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.
The conjecture is called Taniyama-Shimura/Taniyama-Weil/Taniyama-Shimura-Weil conjecture, AKA the modularity theorem - please rename the section.
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: th-cam.com/video/-jpA-tr68ww/w-d-xo.html
0:00
Bring Grigori Perelman!
10:50
Lex and Andrew Wiles would be a great episode
I going outside to make mud mud pies now!
So in love for the last time EVER EVER EVER ❤
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
Energy propagates always orthogonal to the direction of mass movement. Speed is the link.
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.
Fermat later in his life proved the case for N=4 via infinite decent so it seems to be accepted thought he believed he had a proof and later realized he didn't but never intended his note in Arithmetica be read by anyone. To me, the mystery is why for hundreds of years people pursued the proof ignoring the timeline showing that he didn't (Why say a general proof then years later specifically prove case for N=4).
@@justinsutter3602 In regards to your last sentence, just because people sought _a_ proof doesn't mean that they specifically sought _the_ "claimed" proof in the margin. It's a somewhat interesting problem that's easy to "get into" which no one had yet solved. This is a recipe for a lot of people to work on it.
@@MuffinsAPlenty Yes I agree. I find the whole story of this problem and sought of proof fascinating.
I imagine he just had a flawed proof that he thought was correct
why would it matter if they were negative if they were being squared
For classifying numbers, Natural are all positive whole numbers. When you include negatives, that class is called Integers. Integers also contain 0, which would be cheating or give results like a2 = c2. So to easily define the rules, Natural numbers is the correct term.
this comment was not about terminology. I think the point was that when restricting Fermat's Theorem to even powers, you can also allow negative numbers, as the sign disappears. It's potentially different with odd powers, but you can reduce that case easily to the case of natural numbers too (if you exclude using 0)
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.
Mathematics is mostly collaborative, with professor's within departments working together on problems. Most landmark results are either the product of sequences of people each adding a bit to the eventual solution, or a collaboration by a large number of people. The idea that mathematicians hide their research from everyone else is completely incorrect
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.
Give me a shot
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
....
Well?...... what? You lost it?
Unbelievable......
*Scrambles to find the proof myself in secret*
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)
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
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.
If i cloud to sead on back stage further troupe Key note thé ascendant number WE take a choycess récolte between daily for remember Time at s.l c compté Samer by solitude and a coriaces aid in webley stadium if WE Can t give Samer or weeklly regroupe to filer Mike attitude formulate a carburating sell in march of palestine
❤❤❤
BINGO
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.
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.
What on earth did I just read? For large n, f(a) is not a line. "Turn curves into lines" makes no sense. Not even sure what point you are trying to make. This is just a mess of a comment.
@@estolee5485 Take the (n-1) derivative of even large n, is one way. “This comment makes no sense.” You are ready to insult someone but haven’t seen even slightly a reason why?
@The Jealous God I'm not insulting you, just trying to get to understand what you are claiming, which I still don't know. Knowing what you are trying to show would be a good start.
That being said, taking derivatives doesn't make much sense here. If you want to replace f(a) by its (n-1)th derivative, that doesn't say anything about the original equation. So you might be insinuating that we can take some kind of partial derivative of the whole equation with respect to each variable in a sort of "piecewise" fashion where each term gets replaced by its corresponding partial derivative, which obviously doesn't tell us anything about the original equation. All you've done is said a+b=c in that case.
@@estolee5485 Sure, let’s walk through it. The difficulty in the original a^n + b^n = c^n is in how unusual each power of n is to each other. I realized that motion along each curve of n is independent of it’s relationship to other curves. So each curve is like unrelated to others, and the entire set of all dimensions simplifies to a line. The use of information from the equation needs to be “re-elevated” back into relation with other curves, but the solutions are clearly shown to exist for all n.
It is a + b = c. So easy! But the basis is exactly the curve (for any power n) when implementing along it.
@The Jealous God "The entire set of all dimensions" doesn't mean anything here. You will have to explain what you mean by that. You will also have to explain why each term simplifies to a line. You will also need to explain that even if that is the case, how that implies anything about the original equation. "The basis is a curve"... I don't see any vector spaces here, nor do I see how a curve can be a basis, so you will have to explain that. And lastly, I would also appreciate if you explained what you are trying to prove. I still don't know
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.
There called bolshiviks .
The best Jews in the land!😬👌🏼
Lex must interview Grigori Perelman.
@@lenyabloko he is too busy picking mushrooms
@@lenyabloko💯
the subtitles are hilarious