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Everytime I listen to Wiles's work on Fermat's last theorem I get incredibly amazed by his passion and determination to complete the dream of his life. Truly a remarkable person

I don't think Andrew's proof would fit in the margin.

sucks that in the world we live in today people are hesitant to share their ideas due to it being stolen etc. Imagine how much further we could progress in science if it were the opposite

I felt that!.. when he broke into tears 💪

Having the motivation to solve an unsolved problem at the age of 10 is just mind boggling

Wow, that video was absolutely incredible! Watching Andrew Wiles persevere through all those challenges and finally solve the theorem was so inspiring. It really shows that with enough determination and hard work, you can overcome any obstacle. Truly amazing!

There are those with otherworldly intellect... and there are those with an otherworldly relentless work ethic... in the overlap of the two.. you will find Andrew Wiles.

Nice video. A woman was involved in the early attempts: Sophie Germain who had to disguised herself as Monsieur Leblanc to avoid discrimination. She was communicating to Euler and eventually had to admit that she was a woman when Euler planned to me M. Leblanc in Paris.... She had feared that Euler would be angry that she posed as a woman. He was not at all.

And here I am trying to solve the mystery of why my steak is always overcooked.

Fermat's Last Theorem: "I'm the World's Hardest Math Problem!" Riemann Hypothesis: "Hold my nontrivial beers..." (This is a joke. I am not denigrating Andrew Wiles' incredible achievement at all.)

I've watched that original documentary and its amazing how emotional Andrew gets. You can see the passion for the problem in how he barely is able to get out the words "Nothing I ever do again will...." as he almost bursts into tears. I would say 99% of people would just say "umm..its just a math problem!". To him though, this represented the culmination of an entire lifetime of dedication that has finally been realized. A quote given by Simon Singh from what Piet Hein said: "Problems worthy of attack prove their worth by fighting back". That is this entire journey in a nutshell.

Dear Ms Pom, thank you again for a fantastic exposition of a truly inspiring story. All your videos are amazing. I remember that I was doing my PhD at the time, watched the documentary on British tv, and read the book by Simon Singh.

I read somewhere that after Wiles completed his proof, only a handful of people in the world could understand it, let alone verify and cross examine it.

The reason that Wiles kept his work on Fermat’s Last Theorem secret wasn’t that he feared others might steal his thunder. The reason was that by the mid-20th century trying to prove the theorem was something that drew in cranks and crackpots. No serious mathematician worked on Fermat’s theorem if he valued his reputation in the world of mathematicians.

I remember reading about this when it happened. It made world news.

I wonder what Fermat’s idea for the solution was. We’ll never know, but it’s interesting to think about

I have learnt an incredible amount without understanding..........from this man.

This was an excellent talk.

Thank you, thank you, thank you for an excellent video. I've been interested for a long time in the Poincare Conjecture and both the drama and the personalities that accompanied its solution by Grigori (Grisha) Perelman in 2006 and onward. I didn't think that I'd have the chance to learn about another mathematicians of comparable heart, drive, and determination as Perelman, until this video about Andrew Wiles. Both Perelman and Wiles built upon the work of other great mathematicians to resolve hurdles and advance pure mathematics, which in the decades and centuries to come, will enable the creation of new technologies that we cannot yet know at this point in time.

Thanks for explaining. I could not get such holistic, simplified explanation elsewhere

Excellent summary of the Wiles timeline. You missed out Gerhard Frey's work which enabled Ken Ribet to tie the conjecture to Fermat.

I can listen to this woman all day❤

"Fermat's Enigma" is really beautifully written book. Joy to read.

this is a truly marvelous video!

Wiles and Forrest Gump have something in common: "I think I'll stop here". LOL.

It is my thought that what Fermat decided was a proof of the notion that there are no solutions when x > 2, was later determined by him to be incorrect and invalid. He chose not to publish anything about it until he could go back to the drawing board to work through it again.

Very good presentation! Unsure if Wiles initially sent his paper to Inventiones Mathematicae (A Springer Verlag publication), his eventual thesis was published by Annals of Mathematics. Also, like the fact that you mentioned Mazur. Perhaps should also mention Frey.

A stunning and astonishing achievement by Andrew. The real deal...

@4:00 !!! Wow, look at the stacks of paper on and around his desk! I'm almost more amazed he could sort through all this.

Idea - An iterated x^n + y^n = z^n curve where you iterate with a remainder term for each x y z can still find an integer solution. So Fermats last theorem is a remainder

With the mathematical knowledge of Fermat's day it was impossible to solve the last theorem

The real question is, did Fermat actually have a "marvelous proof" to the equation, as indicated in the margin?

Great video

Excellent video narrated by a beautifully perfect narrator

Wiles showed that the proof was indeed too large to write in the margin.

I've have the same obsession with the trisection theorem and despite there being a proof that it's unsolvable for arbitrary angles, I was never satisfied with it. I've spent 35 years looking for a solution. At this point I doubt I'll find my proof, but it's something I could do forever and never get bored with. It's like a rabbit hole of mathematics. Like calculating digits of Pi. You're essentially marrying a quadratic and a circular system together. You can divide by 2 easily in geometry, and a 3rd can be derived by scaling some standard angle like 90 degrees, where you can split it in 3, by using sine of 30 degrees... When solving geometrically, you end up with a pattern that emerges from all the results extrapolated from a division by 2, and that reference division by 3, and you're supposed to scale it to other possible angles, and that's where the systems diverge never to interact again, and it draws the craziest flower-like patterns if you attempt to bruteforce it with an algorithm. I wonder if AI will at some point be able to definitively solve it.

so what is the solution 🤔

I don't know where I read it but this thought marked on me indelibly, "Success is built upon adjacent innovation/possible." Wile's story is the most realistic example of this.

I very distinctly remember when this happened ... people in the math world were angry it was all done in secret.

We aren't sure that this is the hardest math proof until someone solves the Riemann hypothesis.

Man solves problem that was already solved 357 years ago Congrats

That is impressive. One thing though is he does seem to rely on major building blocks done by others though. You kind of wonder if there was a more direct algebra method than making proofs about surfaces.

I never understood maths especially when the letters were used. Then years later i saw a Futurama episode about these maths formulas and it all made sense 😊. Then later i forgot it again 😢.

There was just no space to write my proof on it.😂

0^3+0^3=0^3. There's another rule: x, y and z need to be natural numbers.

129 pages ? I made a shorter proof but youtube says its too long a comment. Your loss...

Good to hear the backstory including Prof Andrew Wiles' tireless determination to solve it. Kudos to him!Speaking of which, as I processed his proof strategy - and I paraphrase - Prove Taniyama-Shimura conjecture by proving the linkage between elliptic curves and modular forms which then automatically proves that Fermat’s last theorem is true Assuming I have gotten the causality correct, I am wondering whether our CPT(α,Φ) function proof of Riemann Hypothesis can also become an alternate proof for Fermat's last theorem (complementing Prof Andrew Wiles proof, without eclipsing it? Let me repeat -- an alternate proof without eclipsing it only! So, hear me out! For example When we think about the Modular forms, they are a special case of a much larger “periodical automorphic form functions behind the symmetrical Lie group” only This is where I see a linkage to our CPT(α,Φ) function proof of Riemann Hypothesis In other words, when we prove that Riemann Zeta function is indeed is the automorphic function (or action generator) for the generic Lie group, then it adds another linkage - this brings me to our firm’s CPT(α,Φ) function based meta proof strategy of solving all Clay Institute problems using one META-PROOF of Riemann hypothesis (aka CPT(α,Φ) function proof)! For example Our META-PROOF strategy has been developed by “ Gödel completing” Sir Michael Aityah’s Riemann hypothesis proof as CPT(α,Φ) function proof, by additionally framing FSC(α) as an Omniscient AoC chooser of Banach-Tarski paradox/Hidden Variable/Maxwell daemon of our TOE, with a symbiotic symmetrical fractal causality of Lie group! What do I mean by that? As explained in this article ((www.linkedin.com/pulse/summary-our-firms-10yrs-toe-work-wa-request-world-form-prabakar-k25sc/?trackingId=3oeFnfoaRT61kGCCQ7VNNQ%3D%3D), one of the best ways to explain my proof is using our TOE model of our universe (more specifically the QVF sourced quantum fields of universe), with the following 5 fundamental ingredients 1. Scale to measure unit growth distance -> 1 2. Scale to measure exponential symmetrical (infinite) unit growth -> e 3. Scale to break symmetry so that we can re-normalize/limit the exponential infinite growth as a finite asymmetrical exponential growth -> α (Note : our TOE is the only one that has framed this ingredient like this which is absolutely needed to explain why probability is α sourced phenomenon and not a statistics sourced phenomenon as framed by all other TOEs) 4. Scale to balance symmetrical scale and asymmetrical scale -> π 5. Scale to measure both the asymmetrical-symmetrical unit balanced geometry shapes (especially unit circle and square as they both are foundation of all other Lie group shapes) -> eiπ That said, yet another uniqueness of our TOE is that ours is the only approach that has integrated all these 5 ingredients (with this type of intuition) using one such “symmetry breaking CPT(α,Φ) function driven hidden variable/Maxwell daemon mechanism with 5 AITGE origins (Riemann Zeta function Action of Lie group, Inertia, Time. Gravity and Entropy by framing fine structure constant (FSC) as the hidden variable including its flip sided golden ratio) so that we can solve all problems using this meta proof strategy What do I mean by that? In other words, our integrated meta-proof strategy (with the following steps) not only solves all Clay institute problems but also it is the best way to answer all 10+ centuries old unanswered physics questions 1. Riemann Zeta function Generator function to model the sinusoidal QVF fields so that it can source the least action for Lie group (not done by any TOE) 2. Fourier transform operator to chunk QVF as 137 chunks using FSC( not done by any TOE) 3. Schrodinger's wave function to model the unit cycle/circle (we have kept it unchanged from current QM, however we have added zeta function on top of it) 2. Mobius transformation operator(including Banach-Tarski/Russel paradox) to project it into projected plane (not done by any TOE) 5. 10 meta dualities of math (including clay institute dualism) mapped to projected plane (not done by any TOE This brings us to the unique steps of our TOE engine operation 1. Our CPT(α,Φ) function starts rolling two-sided (α,Φ) Solomon’s dice( or Krishna’s leela),wherein, α ends up splitting the frequencies of QVF as 137 dipoles by matching spin to frequency using the analytic continuation property of Riemann Zeta function. This is what maintains causality even after the symmetry of Lie group is broken by α as our whole TOE is anchored on this analyticity property of Riemann hypothesis proof 2. Then CPT(α,Φ) function transforms each frequency as the 137 ratioed spin of unit circle using Fourier transformation in such a way that the electric field of each dipole gets rotated as nπ cycles (using Euler’s identity of eiπ) before getting rotated by their magnetic field (by 90 degrees) to create least action during each iteration 3. This way, all frequency cycles of e are mapped to one Schrodinger's wave function. 4. This is when only one iteration of e succeeds as least action (aka 1 out of 137) of Lie group. This is what gives raise to probability including classical reality of Lie group! 5. Then this least action (A) is twisted to flow along the eigen-valued nodes of Ramanujan graph(and all dualities of Lie group including elliptic curves), using the 2/3:1/3 rule of α (ratioed by its flip sided golden ratio Φ)! This is it! With these unique differentiating intuitions , how about we get to all Clay Institute problems including an alternate proof for both Piuncare conjecture and Fermat last theorem? We imagine our Universe as the Riemann Zeta function governed LMFDB universe (that is a motivic/metamorphic/Galois representation based SU 2/SU3/SU4 symmetrical engine. In other words, Z(1) is the fundamental frequency of this Universe’s TOE engine that is QVF/ZPE sourced, FSC(α)-Einstein-Bohr-HV-Maxwell Daemon governed frequency of Riemann's zeta function Ζ(S) with a singularity of S=1+0i, that is made up of his harmonic oscillating zeros(S=1/2+it stacked on his 1/2 critical line, before being transformed as a 137 frequency-spin momentum matched dipole, using our FSC(α)-GR-PLA+5 AITGE origin formulas(see exhibit) In other words, our TOE/SOE engine is the one that is transforming the Riemann's zeros into an artistic unit charge SU2 dipole(see visual), by contracting/expanding its electric flux as the center of mass (as r = αR), before rotating its magnetic flux by 90 in such a way that it can be extended into the left plane as a paired unit charge, using the "only possible analytical continuation of Zeta". Sure enough, this engine function is nothing but universe's wave function only, transforming itself from position/time space into frequency/momentum space, using the Fourier transform operator -- ψ(k) = ∫ ψ(t) e^-iwt dt - This brings us to our next point about CPT function This "one & only allowed analytically continued/functional equation allowed symmetrical dipole" is what limits/constrains the ∞ pole of Riemann sphere to a value of 137 cycles( per Laurent/Cauchy residue including the α=r/R,=fe/fp=we/wp logic of our CP function as explained in my post and attached one page exhibit for details lim t→ ∞ CP(1/2 + ti) = 1/α cycles of dipole In other words, this CPT function proof(lnkd.in/drGQ44Mt) for Riemann hypothesis is a polynomial in the convex region of the Riemann Sphere only (thanks to the "one and only allowed analytical continuation logic of dipole & its 137 cycle ratio logic"), limiting/constraining the ∞ pole of Riemann sphere to the convex region - which brings us to the need for META PROOF strategy to solve all CI problems as summarized below

When I was a senior in college, I, like many others, studied this question. I believe I partially solved this problem before I graduated. I'm not a math major.

edit: I just saw the equation and thought I had already solved it. 25 squared plus 60 squared equals 65 squared. I would play a card game with my mum and we would be assigned random colours. I would make stupid codes to indicate the players colours. One night both players were blue. So I made an eiffle 65 reference by writing 4225 (65 squared). Then I looked at it and was sure it was the sum of 2 squares which it is.

fermat needed that extra chunk of his notebook to take care of a roving dingeberry

Now we have to get busy and prove (or disprove!) Goldbach's conjecture: Every even number larger than 2 is the sum of exactly two primes.

So what numbers or groups of numbers solve that equation?

Andrew Wiles and his proof also was covered in Star Trek Deep Sapce Nine, possibly also in an episode of Star Trek The Next Generation. His proof went beyond the news media.

These people are incredibly smart, and their analytic skills are just applicable to literally every thing in our world. We need more people in STEM.

I would say, Poincare conjecture is way harder than this one.

129 page proof, using methods not known 357 years ago… are we sure that is what Fermat called a "truly marvelous proof"?

Yet, academia has profoundly failed to comprehend the simplicity of the construct of consciousness.

The proof is brilliant, has anyone connected it to what Fermat may have been thinking of? Not as a way of negating this person's work, but to better understand Fermat.

What's the answer?

Great video! However, I find myself wondering about that statement that Fermat had in his book about having proved it. No accusation of lying on Fermat’s part, but how could it be that Fermat only complained about needing a wider margin whereas the modern proof took hundreds of pages? Was Fermat fibbing? Or was his proof erroneous? Or, is there yet a much simpler way to prove it but we just have not found it?

Imagine how DEVASTATING that must have been for Andrew when one of the referees (his colleague, no less) found that discrepancy in his proof. Having to go, once again, back to the drawing board after so many years of work. Oof!

There's an Horizon documentary about him and this on BBC iPlayer that well worth a watch.

Where are you guys from

Please 🙏 make a video about origin of calculus invention and the development and history of changes in this field of math,

Taniyama- Shimura is the most beautiful theorem in all of mathematics😍

WOW, such perseverance. You are so intelligent, but this was so far over my head. I couldn't follow it at all.

If Fermat didn't lie, he couldn't have used the same tools. But I deem it probable that Fermat hat a flawed or ratehr unfinished prrof that he thought was complete and valid, without any peer review.

If riemann's hypothesis will not be solved in the next 200 years then it would beat the 357 years i guess

Seems Fermat had a completely different and even simpler proof which is suggested by his remark that it wouldn't fit in the margin. "The world wonders."

Although a very great achievement, I kind of have the idea that Fermat had a less elaborate proof, if he indeed had one.

How is this equation applied in physics, chemistry, engineering, economics, and informatics?

Mr Wiles worked alone pursuing the fame of “the one who solved it”… except that he got a companion in the end. i wonder what if mathematicians collaborated more? Maybe it wouldn’t have taken 350y of struggle.

5:45 Man, they couldn't have met literally anywhere other than on campus?

If anything, Wiles' proof also indicates Fermat most likely did not prove anything, as the instrumentarium Wiles used was completely inaccessible during the time of Fermat. He likely *thought* he had proved it, which is not the same as having proven it.

You can't also find cube root of x plus cube root of y equal cube root of z. From this fermat's theorem can be proved more easily.

If Napoleon Dynamite can do it - so can you.

“The world hardest math problem”. Apparently not.

Still waiting for a proof using the mathematics of Fermat's time.

Love of Math > Love of the woman 😅

Is your channel name a play on 1984?

7:10 "...this was subject matter that perhaps only half a dozen people in the world could fully grasp." John Lynch, Fermat's Last Theorem (1996) Unless you're a KZfaq commenter. Then you understand. Ha ha. /sarcasm

I wonder how this solved math problem will help humankind.

The guy who ended classical math :)

wth is a modular form? you didn't explain that at all and what the connection between that an an elliptical curve means

its a rather bold claim to say its the hardest, given it was solved... And remain so many unsolved

That's pretty much the problem with humanity in a nutshell. A guy's research might get scalped.

I suspect Fermat's proof was substantially more elegant. But I guess we'll never know.

Imagine a rectangle of size y by y^n-1 on top of a rectangle of size x by x^n-1 where x = y^n-1 and the sum of the two are equal to the size of some variation of z^m by z^k where k and m sum to n 😊

Even if Fermat's proof was not exactly correct, it would be interesting to see his "approximation" of a proof. With that as a start, perhaps a mathematician today would develop a complete, more concise proof.

Sounds like a very indirect proof based on a lot of "if this, then that" assumptions. Too bad he couldn't prove it more directly.

I feel hopeless to understand the proof!

This type of Nerdism reaches metaphysical levels.

Does anyone know what the consensus is on weather Fermat could have found this or if there maybe is still a different proof out there? Or do people just think he found something thats most likely just wrong?

The problem belongs to mass since 2D version is a circle. Since matter should not find any solution in another higher dimension n more than 2 does not work.

what has Wiles been doing since then?

Now remember that Pierre de Fermat proved that hundreds of years ago. I wonder if his proof was the same. Probably not...?

As much as i loved the Story of Andrew Wiles and Fermats last Thereom i dont think that Fermat would have had any idea of elliptical curves and so although Andrew might have solved the problem i dont think its what Fermat was alluding too , which opens up a whole can of worms about how Fermat came to his conclusion

I wonder if Fermat had a valid proof. If so, I suspect it would be much simpler than Wiles' proof.

This is what the school expected to give yet the school is just making us study things that do not illuminate our curiosity most of the time. If you want to escape this you should make a side hobby that is extra to your curriculum if not you will become a person of no interest!