# The Bridges to Fermat’s Last Theorem – Numberphile

KR: Well, the usual way that people, um, discuss

it is to talk about perfect squares first. So, if you have studied the Pythagorean

Theorem there’s a formula that the square of the hypotenuse is the sum of the two squares of the sides of a right triangle. So you might

ask if the, err, the sides of a right triangle are whole

numbers might it happen that the hypotenuse is also a whole number.

They’re called Pythagorean triples and the Greeks figured out all of the Pythagorean Triples. Okay,

so what happened next is I think it must have been very natural

to look at the equation A cubed plus B cubed equals C cubed and A to the fourth plus B fourth equals C to the fourth. And so on. So, this is now in the realm of

algebra and you can plug in values for ‘A’ and ‘B’

you’ll see that, err, these equations, infinite number of

equations, seem not to have solutions. Well, so the

story is that Fermat was reading the text of Diophantus, who was the Greek mathematician best known

for this type of problem. And in the margin of his copy of Diophantus, he wrote some marginal note

saying that I can prove it for higher exponents

bigger than two there are no positive integer solutions and then famously he said but my solution is too long to write in this margin. And in his copy of Diophantus which

was inherited by Fermat’s son Samuel de Fermat there are many, many marginal notes and

Samuel published a revised version, annotated version of Diophatus’s Book in which all his father’s writings

are reproduced. And the reason that Fermat’s

Last Theorem was called Fermat’s Last Theorem, is that all the previously on examined

marginal notes proved to be correct for they were, ermm prove to be correct or incorrect mostly

correct by mathematicians after the

seventeenth-century and there was this only one assertion that remained it was the last

one was hit list. And you know again, I’m not an expert in the history of mathematics but I think most

people are convinced that when Fermat was

a mature mathematician he realized he did have no proof of this thing that he wrote in the marginal

note and in fact later in his life he devoted a lot of

attention to special cases of that type of equation. It’s not just the cubes but it’s the cubes

the fourth powers and the fifth powers and so on. And Fermat himself later in his life took care the fourth powers. Well, the cubes

certainly must’ve bothered him because after all he knew

about squares and he knew about fourth powers and cubes, you know, right in the middle. BH: The theory is that

there is no solution. There’s no whole number solution. KR: That’s right. BH: Would it have been or wouldn’t it have been more exciting or more attractive to a mathematician to prove him wrong and to find these three

numbers that you could plug in that would do it? Would that have been more triumphant? KR: Well, it would have been- would have been very exciting and would

have made lots of news. And then the proof, so to speak, would

have been simply on a single counter-example that would

have fit, you know, very neatly in a newspaper

column, you known, there would have been some story, you know. ‘Mathematician in Australia finds the

first counter-example to Fermat’s Last Theorem’. Certainly was big news either

way. Ermm, the main thing about Fermat’s Last

Theorem is that over the centuries it gave rise to a tremendous amount of new, and fruitful mathematics. That people

really develop new techniques to try to solve that equation that proved to be, you know, infinitely versatile. BH: So a bit like going to

the moon and let[ting] them develop other lots of other technologies along the way. This was for going to the moon. KR: That’s a-that’s a fine

analogy. I mean, you, when you formulate a problem in

mathematics no one knows whether this problem will really be

helpful for mathematics as a general enterprise. And, you know, for whatever reason Fermat’s equation has led to tremendous amount of- of new mathematics KR: So, when I was a graduate student. We’re

talking, you know, roughly forty years ago or forty-five years ago. Fermat’s Last Theorem

was kinda some curiosity. Everybody knew about

the problem but no one really had any approach to it and there was no kinda fundamental reason why the problem

was going to have a positive solution. And when I was a student there was a

tremendous interest in what are called elliptic curves and it was realized around nineteen sixty-eight, nineteen sixty-nine, nineteen seventy by André Weil and Goro Shimura in

Princeton who built on work of Yutaka Taniyama who was his buddy in Tokyo in the nineteen-fifties.

Taniyama had passed away and there was this

gradual understanding that elliptic curves really were related modular forms and again that was some

completely are apparently inaccessible outstanding

conjecture analysts thought to be incredibly hard

conjecture. And then in the early nineteen-eighties a

mathematician came around, and I say that literally because you know he was

sitting here in this office his name is Gerhard Frey. At the time he

was at the University of Saarland in Saarbrücken on and he came- BH: Is that a famous one? I don’t know that one. KR: Well, um, I didn’t know it either and came around

and he had the scheme for relating Fermat’s Last Theorem and the modularity conjecture. Somehow he realized for the first time that, um, if- if he had a solution to Fermat’s equation this would give him an elliptic curve

which apparently could never be modular. I was a little

skeptical because I knew that some of the things

that he said that he needed in his argument, um, were

very hard. Well, for me a really important thing was in nineteen eighty-five Jean-Pierre Serre, who was one of my mathematical heroes in Paris

sat down and kind of tried to distill everything that had been said

about the Prey’s idea and he came out with a very clear letter and this letter basically said if

you can prove this tiny little result which he called epsilon because the number in calculus, epsilon is a small number. He

said if you can prove this tiny little epsilon then you can really show that on the

modularity conjecture, which he called Weil at the time, implies Fermat’s Last Theorem. So, epsilon became, for me, a holy grail because not only was it something fairly well

encapsulated and very clear, and this was the kind of thing that

I’d thought about before without success but nonetheless while

thinking about yet another problem, it occurred to me that I really

had some new information that I could possibly bring to bear on this epsilon. When I first did was say to myself: “What’s

the simplest possible case where I can think about this? What’s the simplest

possible prediction?” I tried to settle that. I said “Well, you

know, if you have to start somewhere. can I-” Well, I know about this simplest

case so I have this thing in my mind and I

thought about it, off and on, during the academic year- nineteen eighty-five and eighty-six. And during this year I was teaching calculus and I was busy with things, there are, you know, colleagues come in and they want you to be

on committees. And finally, um, my final grades were submitted in May nineteen eighty-six and I started thinking about this problem again, first when I was on the east

coast for a while, going to Europe on and I found myself

in Paris. And finally I found myself at an empty desk and on the Max Planck

Institute in Bonn where I can just sit and think and I

thought about this yet again – BH: Is that what you do? Do you just sit at a desk with your head and your hands, and think, think, think, I mean – what if I was a fly on the

wall when you were trying to crack this problem. What would I have seen? What was it looked like? Well, it’s probably not have probably had

a pad of paper in front of me and a pencil or a pen we didn’t really have computers

quite yet although they were starting to come, and when you visit some institute you

don’t have all your stuff with you don’t have lots of books you don’t have all your

paraphernalia. So I was really just kind of sitting there like that, and I started writing, you know.

And I- If you look in my office you’ll find

lots of old pads where I sit and write, and, you know, a page might start what do we

know what are we trying to do summarize the situation so far. “Let’s-

let’s try this”, and to my astonishment, I was finally able to- to cracked that problem – the simplest case of that epsilon. BH: It’s like inspiration

comes at preordained times.

Like you sit down and say, okay for the next two hours before I make my friend Bob a hot dog, I’m gonna have an inspiration and trying to solve the problem, and that’s when it happens. It doesn’t happen on time, so you’re lying on bed-

Oh it can, I mean, if you

if you think about something very intensively then you can just carry

around with you and you know lots of my colleagues say

that while they’re swimming had some insight or you know lots of

people say in the middle of the night they jumped up and they said my god you

know why don’t I try a minus sign instead of instead of a plus sign and this- this can indeed happen,

you know. BH: Not for you though. You are an at-the-desk kind of guy? I-I think in this particular situation I

was at the desk I knew for the first time that I could do something that I hadn’t done before that no one had done before. This is a

special case is this so-called epsilon, And I kinda wrote and rewrote it just to make sure I was really right. And finally, using a very first

Macintosh that I had access to I typed out a letter to Barry Mazur at Harvard, and I sent him this letter explaining what I had done. And then I spent weeks and weeks trying to do the general case. I couldn’t see what to do and then fortunately, there was an international congress of

mathematicians. They occur once every four years on the campus of UC Berkeley. So I was right here.

I came back in August for this International Congress and Barry Mazur was there. And on I spoke to him, and I said Barry I sent you this letter

explaining how I did a special case, and I really wonder, you know, what I

have to do now in the general case. And he looked at me completely quizzically.

He said: “Well, you’ve already done the general case.” and basically, he- when he read my letter to seemed obvious to him that there was this extra little thing you carry alone as a supplemental

object. And this extra thing which is do the bookkeeping. And, so we sat down at this Caffe Strada

which is at the corner of the Berkeley campus, and we had a Cappuccino and he kinda said well you just carry this and run with you and I was astonished you know it’s like this amazing moment were all the sudden I realized that in principle at least there was

no obstacle to proving this epsilon conjecture. I mean, it like, you

know, my god, you know, it’s some special thing, you know, an angel comes down

and the light shines from the heavens or something. So this was really quite a thing and on since or my colleagues were walking around the campus in this

big international congress I kind of told, you know, a number of them

that I have done this thing. You know of course

in beginning I was cautious I said well I think I’ve done this thing, and- but

people kept running up to me. They said was it really

true you’ve done epsilon? I lived in a world and think I still do

were my colleagues are very generous with

attributions and praise and there isn’t too much of this kinda race to be the

first to publish and people basically- I had little fear that

someone would try to scoop me on the other hand what was a little

worrisome was that I hadn’t written down all the

details, you know, so I kinda written down very closely the special case. But I hadn’t kind of worried about writing

thing more generally so what happened was that during the

year following that, nineteen 86-87 there was a special program on algebraic

geometry and arithmetic geometry at the Mathematical Sciences Research

Institute in Berkeley, the MSRI. So I was up there with my callings

and I gave lectures every week I gave

another lecture I was kinda giving like a min course on my proof and when I wasn’t lecturing I was furiously

typing into the computer to try to expand and

revise my manuscript in kinda more and more

details got put in. There were certainly people in the

audience who were appropriately skeptical. They asked me hard questions and I

didn’t always know the answer to our questions you know like “How do we really know that this

thing is in such-and-such way”, and I would say well

you know this was proved in the 1960s by Grothendieck but it isn’t quite written down in the place you

expect then I have find the argument. Well epsilon is small. It was clearly a misnomer. And let me just get up and pull off a reprint. Here’s my article. It was published

in 1990 which really makes are a lot of time between 1986 and 1990. BH: How many pages? It’s like a book. Well it’s a more than a pamphlet. If you

open up this paper on the very first or second page there’s Fermat’s equation and an elliptic curve but then they disappear. If you look at

the actual article it’s more about the

technicalities of the subject. It was a big deal in mathematics. It got write ups in the press,

including Science magazine if I’m not mistaken. You know, whenever there’s a serious

announcement, of a major theorem that gets a lot of

attention. And people stop what they’re doing to try to

see what the basis of proof is. BH: who called this Ribet Theorem? you know obviously it to make that name

that is above all other whenever addition both well the yeah that’s right so you certainly there

there’s a general aversion to naming things after yourself

very often people give lectures and I’ll say such-and-such

the room and princes the right name the person who on proves it and if it’s a fair

amount they prove themselves in typically right there the first

initial their last name or something like that for all say they

don’t on they have no attribution listeners

understand all this is the fear that the persons is providing in this lecture. – You’ve done

this thing which is important if you have a bridge

between two things which no other which also Granja. – Well that’s

right so the um what what I when I showed was that

this modularity conjecture which was thought to be was

was felt to be true but um wasn’t expected to be provable I show that that thing actually implies

there was less there I think the main takeaway from my

theorem was that since the very conjecture short on the OMA was likely to be true

that now you should think that for miles last year it was likely to be true for example it probably wouldn’t have

been a really good idea to spend a lot of computing

power looking for a counterexample farrell’s less there well as anyone else

who’s in the room well I was the whole thing so any Wells

you know coursing came around a few times I guess but

basically he was holed up in his attic and I didn’t know that and I think most

people didn’t know that so he was on I mean it’s his I’m it’s a statement that he was

certainly intrigued by Fermat’s last theorem as a child and then as soon as he heard

about this Ridge he decided to try to prove the

modularity theorem because of its relation to the Fermat’s last theorem. so was it is a sort of the situation that Andrew Wiles wanted to prove Fermat’s last theorem by any means possible he given him the means that’s what he says I mean if this is

surprising to me because modularity theorem by itself is a

tremendous price and you know it seems kind of strange

that you needed the motivation to Fermats to go and

try to prove it if you thought that there was some way side well anyway that’s that’s that’s

that’s what he says yet we were all together meaning kinda

lots of people in the subject in a conference in Cambridge England in June 1993 and usually go to a

conference and you see the list speakers and you know you speaker has an hour and the thing and then set up so that

Andrew Wiles had three hours I’m so here’s the organizers he said I

have something I want to discuss it will take me three hours and I like those

slots I said okay from and then in the first lecture you could see he

was proving He was trying to prove the modularity

of some class elliptic curves but the class elliptic curves at least in

the first lecture had no intersection with the class that you

needed for Fermat’s last theorem within the second lecture got a lot

closer and it became more and more parents have

people at the conference third lecture he was going to be able to

do enough to get fair was last year the crowd was

growing so love the mathematicians you know brought their colleagues who

happen to be in Cambridge in you know some other family members and I think there was a little doubt that he

would announce approval for was left there I’m so I can i sat in the front row center and

brought my camera and snap the picture of him I wasn’t another moment when you know

those cuts time stop and when I first realized what was going

to happen .. and then you know when it actually did

happen there was this some frends of activity I mean the director love of the Isaac Newton Institute where these lectures were held on had ordered a case a Napa Valley brute

champagne for the end the lecture is after the

lecture we came out into the you know common area only is all the

champagne flutes were produced in his bottle started popping you know this is kinda really an amazing

thing between that time when you- originally published a paper and this

lecture it was a cold or write you a nasty lol nah to work not a word you know it i no idea what he

was doing on you he whole who absolutely he was so he was a a friend and a

colleague I first met him when he was a graduate student in Cambridge and in fact when I first came to

berkeley the first year that I was here he was one of the first people that I

invited came to spend a week and gave a lecture to on do you that delight he kept a secret

that make a surprise I to this day you battle by I’m using a

little collegiate you’re telling me when you thought you

open your running around telling everyone NEC said well it’s very unusual behavior and the you know you can say in

retrospect that it was really um justified because he was afraid that

people would kind of jump all over it because %uh the important so the modularity conjecture and fair I was

last there on button he did not on tell me what he was doing he did from confide in to this calling to princeton

as he was giving is graduate course where he was working

out the ideas that on were instrumental in this paper

I think it was only a few days later that I got a copy of this manuscript and the manuscript was very thick nose like 300 pages from single space you know typeset by test mathematics typesetting program and on we were supposed to be the referee is on looking at different parts of the paper

and I was assigned to a par that be turn out to be problem-free arm and I’m the cats look at a park but turn out to have problems a problem was

there shattering for you a do you like that doesn’t make

you think this is good I like the reader a lot about it really

deflating up to I have champagne mean finding the problem well from want more of the aspects of the

whole thing is that I had gone out pretty for a limb from telling everybody that Andrew Wiles

improve this thing on in the way that started was that when the New York Times first heard

about his announcement may ask to speak to him and returns me and said and you know you speak to Gina Kolata so

I went and on went upstairs and spoke to her for a very long time

explaining the whole thing and then I’m somehow the media kept

calling me because it was understood from that

somehow I was is designated spokesperson so I nice spent the entire summer I’m lecturing about his theorem and I’m fielding phone calls from I’m the

press hello for was announcement cast oh I think it was like two weeks what

happens in mathematics is if if you have a proof that seems to really be on seems to have a lot or from internal logic to it and there’s something that’s wrong

with the proof your immediate reaction is to say well you know this is just some technical

problem that I need to fix and I think that are Andrew Wiles when

he found out about the gap arm expected

that you would be able to repair insured or so it really took a while before there

was some realization on his part that this was a

serious problem couldn’t be fixed on immediately just by

their tools improper if you love the person

who was taking all the calls from the press whatever reason did that mean you will

you are contacting and while get updates on the game

because my i don’t know what to tell them what we call him but those you want to say it

was a he gone right not especially Inc he kind

of went dark he was working on on mathematics and I wasn’t wasn’t getting a lot of

information out of him at that time you know it was all poison about right i

mean I’m wasn’t clear whether or not this

will get repaired or not and then you know wonderfully it turned

out that there was this so repair to the proof from release in completely new insight

actually by Andrew Wiles and Richard Taylor who work together over the summer and

this will be the summer 1994 and they had a or preprint out in in the fall 1994 really explain their

new method in which is now call Taylor Wiles method and that really I’m circumvented a lot or the

more complicated arguments in the original preprint and the two articles were

published together on the following summer which is a

summer 1995 in the annals of mathematics it was a relief but it was it was not

our kinda joyous relief you know and Richard

Taylor came to me one day again MSRI where I happen to be in fall far 1994 any sand we have some good news for you and then on the next day and make up in this

paper so that was kinda very good and you know I don’t remember there

being a huge celebration on the moment the releases that reprint been just

kinda gradually build people understand that the proof is complete and then there was a very large

instructional conference at Boston University in summer 1995 different people got up and explain

the different elements approve well it was a fabulous

pieces mathematics arm as impressed arm with by what what I did and presses I am by my

my own were coming I’m even much more impressed by what

they did on they had some really knew inside and this inside has I’m percolated through our whole subject for the last twenty

years arm one way to say what they did is they

showed that somehow modularity is contagious in the sense that if you have something

that hope has a little pieces that the modular you can parlay that into modularity over

forty first evolved Wiles was obviously intensely driven to do this he

was really motivated and he said you know

I’m gonna do this one way or another and me brought in all sorts a different techniques that hadn’t been

related on to the the problem before on what some what’s amazing for example

there’s a whole subject an algebraic geometry call deformation theory that was going on when I was a graduate

student that I didn’t learn I thought that was really far removed

from anything involving elliptic curves and on in the years before wells announcement

Harry Mazer had developed this new technique of

deformation theory in gallery presentations and it turned out to be

exactly what Wiles needed the language in the

techniques the median for his discussion on and so you know something that seems to be far

away technically turn out to be crucial well I think I’m you know just

informally everyone refer to it as the Fermat’s last theorem but in fact when you when you look at

the article just like my article was something internal to on series modular forms and get our

presentations and Wiles article in my article share the trait that for was a question covers

exactly once in the article in the beginning and then you get down to the real stuff

on very happy with the recognition that I’ve gotten and I very proud of what I did and be I when students come and my classes they

may know about my role for Fermat’s last theorem. A lot of them have seen the video that was broadcast on over and BBC and a I think you know if you do

something very very very very famous and you have an encore problem where people saying you know what great

thing you’re going to do next and I try to be on more or less a normal

guy and not you know walk around with angels on

your shoulder britt to rediscover what FAMAS proof

might it be is that they have widely-helds that he

had done it already disclaim lot was he telling the truth ? I think by the

time we get to 20th century it’s quite clear that this is an incredibly complex problem.

there is a solution for power 42… but the numbers are too big to fit the margin

The biggest surprise in this video is that Ken is 70 years old.

Wonderful! I saw the Nova program on this years ago, brought back many memories. Thank you!

A mathematical argument that only awaits refutation

(it can not be proved)

Green number + green number = number that is not green

A green number is a natural number that has a third root .

Refutation will appear if a green number is found to the left side of

the equation

Aetzbar

This guy can’t go a sentence without dropping names lol

Proved in the 60s by Grothendieck

Can I like this video twice? There is only one like button.

USE THE CODE TO SOLVE FERMAT Always be correct

(x^1/a+y^1/a)^na=(z^1/a+x^1//a+y^1/a – z^1/a)^na. Call d=x^1/a+y^1/a –

z^1/a =>(x^1/a+y^1/a)^na=(z^1/a+d)^na. They are composed of two

groups One group contains x^n,y^n and z^n and the other contains all

irrational numbers. z^n=x^n+y^n. Impossible!

Ken rebit is Fermat !

One of the best video

Numberphile knows NOTHING . I PROVED THIS IN 1978 WITHJA AM GTAPHS WITH ASYMTORES – I SUSPET FRMAT DID THJAT

draw the graph of a circle with radius 1

as the exponent increases, the limit of the graph reaches an asymtotic square

draw a line from any curve down to the x axis

By integration, we know the area under that curve

the remaining area as we reach the limit of 1

we have two irrational areas adding up to a rational number

Numberphile is an ameteur

Not a math geek here but I always found this story fascinating. I remember reading Simon Singh's book on it years ago so I don't remember if he covered this but I have a question. Can any one of you Numperphile's tell me what the connection is between elliptic curves and the pythagorean equation that Fermat was referencing?

What I'm getting at, is the general case equation of the x^n + y^n = z^n an elliptic curve itself? I looked up the equation for an elliptic curve and it doesn't look like that. Gerhard Frey's argument was that if Fermat was wrong and that general case equation was solvable at n > 2 then it would produce an elliptic curve that had odd properties. One of which is that it would not be modular. OK fine. But that says to me that the general case equation x^n + y^n = z^n is a subset, (or in the family), of elliptic curve equations. Is that right? Again, just trying to understand and follow the explanation. I have no higher math knowledge. Thanks!

No real explanations …

How about this equation:

a^3 + b^3 + c^3 = d^3Is there anything special about 6 in that 1*2*3=6, as well as summing to 6?

Ken Ribet: What a great human being. I love how humble he is and never speaks low of Wiles, even though he hid his stuff from him. Thanks for the interesting talk with this fine man.

"Andrew Wiles gently smiles,

Does his thing, and voila!

Q.E.D., we agree,

And we all shout hurrah!

As he confirms what Fermat

Jotted down in that margin,

Which could've used some enlargin' "

–Tom Lehrer

I never liked that line, Q.E.D., we "agree"

Before agreement comes understanding. How many people on the planet can honestly say that they truly understand what Ribet and Wiles did? Less than 10?

Those may be the least accurate and most hilarious subtitles I have ever seen.

It seem imposible Fermat had a correct proof to his theorem but… wouldn't it be amazing if in some decades somebody came up with a simple and elegant solution and it turned out that Fermat really had had a proof?

Too much history and not enough math. Or as you guys say maths.

Does this guy remind anyone of Richard Feynman?

I was on the virge of sleep when I thought: x^2+2x+1=y^2

(X+1)^2= (x+1)^2. Let x=5

5^+10+1

25+10+1=6^2: 36

Let x= 92

92^2+184+1=?

The close-captioning on this talk is nonsense that doesn't reflect what is being said. I would be very surprised if any deaf person could get the gist of what is being discussed. Can't you guys proofread the captioning after the computer is done mangling everybody's words into what it thinks you meant, and make corrections? Seems like a 2-3 hour job at most.

Proper clickbait:

"Mathematic prof. stumped by margin note!"

As a layman in mathematics, my question is whether there are other routes to solving FLT? Is this Occum's route?

Is this the origin thread of the Fermat Comment Meme

(N0)P=P00

Bathroom mathematics

Я получил короткое доказательство теоремы до 1985 года,но одну операцию я не мог осознать(только на уровне интуиции).Осознание пришло позже.У меня нет высшего образования,поэтому мне трудно это опубликовать. Когда это произойдет-все будут в шоке… .

Fascinating thank you

Really, a great video – thanks for recording and posting. Thumbs up!

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Close your eyes and listen as Kermit the frog explains Fermats last theorem

This is one of the most interesting videos I ever saw on YouTube. And I am not a mathematician myself 🙂

Binomial theorem is the basis of Fermat's proof.

in the mysterious mathematical world, I have an equation.This equation when squared up gives Fermat's results in one unique line. that equation is x^n/2+y^n/2+d=z^n/2.suppose x^n+y^n=z^n use new equation we see integer=irrational number

5:20 and a mathematician literally came around, and I say that because he was literally sitting here in this office 😂

8:37 is that a T-Rex??? 🦖

The easy solution to a^3 + b^3 = c^3 would be 1, 1, 1. 🙂

He turns his whole head when most people would just turn their eyes slightly. It's quite noticeable after a while.

I really like Ken Ribet!

Fermat's Last Theorem Proof Simplified

a^n +b^n =c^n

odd+even=odd

a^3+b^3=c^3

let c=2x+1

8x^3+12x^2+6x+1=c^3

let b=2x

b^3+12x^2+6x+1=c^3

a^3=12x^2+6x+1

Only valid solution to a is when x=0 for all

c=2x+1,2x+3,2x+5,…

n=3,4,5,…

No whole number solution for a,b,c when n>2

I wonder what he is looking at, at this left, where he is always pointing his head at.

I am fairly certain that this story was told to me by my college math professor. If it wasn't this story, it was very, very similar.

The video example for a^3 + b^3 = c^3 is wrong (if im correct but i tried it few times). It's 414 and a bit.

I loved it. What a nice document.

The cucks offscreen needs to stfu.

"People have an aversion to attributing their name to something…". Except one president I can name (but won't).

Congratulations Ken!

Nice vid, let me check out your Channel and ser what else you got.

Oh,

Nevermind

Actually not everything Fermat conjectured was correct. The n powers of n problem has a counterexample as pointed out by this channel

But isn't 0 a whole no.

0^x + 0^x = 0^x.

MY SOLUTION TO FERMAT'S LAST THEOREM

Ryuji Kanadani – São Paulo (SP), december 30th, 2018.

Brazil

.

1 – If x^n + y^n = z^n

then x^n = z^n – y^n

and there is a d = z – y

and a k = (z + y) / 2

.

2 – Also there is a K EQUATION

k = x * [x / (n * d)]^[1/(n-1)]

.

3 – If x^n =[z ]^n – [y ]^n

then x^n =[k+(d/2)]^n – [k-(d/2)]^n

.

4 – For "n" equal to 2, we can square [k+(d/2)]

and subtract [k-(d/2)] squared and we get

x^n = [4*k*(d/2)].

For "n" equal to 2 we have an equality

.

5 – For "n" equal to 3, we can cube [k+(d/2)]

and subtract [k-(d/2)] cubed.

The result is x^n = [6*(k^2)*(d/2)] + [2*(d/2)^n].

Since x^n = [6*(k^2)*(d/2)] is an equaltiy,

[2*(d/2)^n] is a difference.

For "n" equal to 3 there is allways a difference

.

6 – The same occurs for "n" greater than 3

.

7 – Numerical examples that confirm my proof:

7.1 – Find FERMAT'S EQUATION FOR "x"=3, "n"=2 AND "d"=1

.

7.1.1 – Find "k":

k = x * [x / (n * d)]^[1/(n-1)]

k = 3 * [3 / (2 * 1)]^[1/(2-1)]

k = 4.5

.

7.1.2 – Find "y":

y = k – (d / 2)

y = 4.5 – (1 / 2)

y = 4

.

7.1.3 – Find "z":

z = k + (d / 2)

z = 4.5 + (1 / 2)

z = 5

.

7.1.4 – Find the difference of x^n = [4*k*(d/2)].

This expression is an EQUALITY

.

7.1.5 – The FERMAT'S EQUATION IS AN EQUALITY

AND has "y" and "x" INTEGERS for "x"=3, "n"=2 AND "d"=1.

x^n + y^n = z^n

3^2 + 4^2 = 5^2

.

7.2 – Find FERMAT'S EQUATION FOR "x"=6, "n"=2 AND "d"=2

.

7.2.1 – Find "k":

k = x * [x / (n * d)]^[1/(n-1)]

k = 6 * [6 / (2 * 2)]^[1/(2-1)]

k = 9

.

7.2.2 – Find "y":

y = k – (d / 2)

y = 9 – (2 / 2)

y = 8

.

7.2.3 – Find "z":

z = k + (d / 2)

z = 8 + (2 / 2)

z = 10

.

7.2.4 – Find the difference of x^n = [4*k*(d/2)].

This expression is an EQUALITY.

.

7.2.5 – The FERMAT'S EQUATION IS AN EQUALITY

and has "y" and "x" INTEGERS for "x"=6, "n"=2 AND "d"=2.

x^n + y^n = z^n

6^2 + 8^2 = 10^2

.

7.3 – Find FERMAT'S EQUATION FOR "x"=27, "n"=3 AND "d"=1

.

7.3.1 – Find "k":

k = x * [ x / (n * d)]^[1/(n-1)]

k = 27 * [27 / (3 * 1)]^[1/(3-1)]

k = 81

.

7.3.2 – Find "y":

y = k – (d / 2)

y = 81 – (1 / 2)

y = 80.5

.

7.3.3 – Find "z":

z = k + (d / 2)

z = 81 + (1 / 2)

z = 81.5

.

7.3.4 – Find the difference of FERMAT'S EQUATION:

x^n = [6*(k^2)*(d/2)] + [2*(d/2)^n].

The expression x^n = [6*(k^2)*(d/2)] is an EQUALITY

AND has a difference of [2*(d/2)^n]:

[2*(d/2)^n]

[2*(1/2)^3]

[ 0.25]

.

7.3.5 – The FERMAT'S EQUATION has a DIFFERENCE of 0.25

AND has "y" and "x" NON INTEGERS for "x"=27, "n"=3 AND "d"=1.

x^n + y^n = z^n

27^2 + 80.5^2 = 81.5^2

.

7.4 – Find FERMAT'S EQUATION FOR "x"=24, "n"=3 AND "d"=2

.

7.4.1 – Find "k":

k = x * [ x / (n * d)]^[1/(n-1)]

k = 24 * [24 / (3 * 2)]^[1/(3-1)]

k = 48

.

7.4.2 – Find "y":

y = k – (d / 2)

y = 48 – (2 / 2)

y = 47

.

7.4.3 – Find "z":

z = k + (d / 2)

z = 48 + (2 / 2)

z = 49

.

7.4.4 – Find the difference of

x^n = [6*(k^2)*(d/2)] + [2*(d/2)^n].

This expression is an INEQUALITY

AND has a DIFFERENCE of [2*(d/2)^n]:

[2*(d/2)^n]

[2*(2/2)^3]

and the DIFFERENCE is 2

.

7.4.5 – The FERMAT'S EQUATION has a DIFFERENCE of 2

AND has "y" and "x" INTEGERS for "x"=24, "n"=3 AND "d"=2.

x^n + y^n = z^n

24^2 + 47^2 = 49^2 – 2

@3:14 Numberphile is inferring (perhaps stating) that a correct solution for a^3+b^3=c^3 is 255^3 + 414^3 =444^3

This is entirely incorrect .The difference between ( a^3 + b^3 ) and 444^3 is 10935 !!

Well did he prove that it is proved or not. Not much of a math student but still like to understand it.

some of which that are proved – FLT , poincaré conjecture.

Reimann hypothesis and other clay math problems yet to be proved.

Come on guys Godel's clock is ticking..

I don't get it

23:16 what is "poised in the ballar"?

Except no one has been to the moon.

8999^3 + 89970^3 = 89999.99999999996^3 … Its the best I could do

4:51 tiddies

How did I get here I was accidentally laying on my tablet and this started playing LOL

This is the best numberphile video. Extremely inspirational yet sad too

Prove that I have proved this solition.

I would love to see the proof of :

p/(p-1) = 1/p^0 + 1/p^1 + 1/p^2 + 1/p^3… for all p>1

A^2+B^2=C^2

A^3+B^3+C^3=D^3

A^4+B^4+C^4+D^4=E^4

.

.

.

George R. R. Martin looks like Pierre de Fermat.

This one needs a continuation with Andrew Wiles.

What a fascinating video!

Sound quality 🙁

Wow!!

Solid edit at 4:30

This is the most inspirational mathematical story

There is one oft-heard error he repeats here. Of all the proofs Fermat made statements about, there is not one case where he was proven wrong. Wrt his remarks about the Fermat numbers being primes he never claimed to have a proof: there is an extant letter to Pascal asking for help in finding a proof.

Thus it seems quite reasonable to believe that Fermat's short proof exists.

Somehow, I do not think that Fermat's Last Theorem ran to 300 pages.

Also, Fermat was not aware of the TS Conjecture.

z^n=(x+y)^n x^n+y^2≠z^n

The value of the expansion of the equation that is the multiplication of different prime numbers. It does not become √ of a natural number.

a^mn+b^mn=c^mn 💔? C^Mn=(aa +bb).c^(MN-2) >=a^mn +b^mn :a=<b =<C⚠!

Interesting to note that Euclidean metric is approximately the local metric of physical space and that it's also the maximum p-norm for which there are "Pythagorean triples"

Excellent video!

SIR Andrew Wiles now

Great interview that, it touches on the spirit and drive that pushes science forward. I've read a book on Fermat's Last Theorem, so heard about this before, and have seen the excellent BBC documentary for which Ken is referring to; but to hear another opinion about solving this great problem from another angle is fascinating.

Good questions as well Brady, you didn't always go for the easy ones, a sign of a great interviewer.

This guy is so pompous

C'mon it was so easy why u guys wait 100 years to solve my theorem ?

I'd like to see a playlist – "Fermat's Last Theorem Proof for Dummies"

Too bad he didn't talk about the math at all 🙁

This was so well explained. Well done to all involved.

Fail fuk

There are two really amazing things here: first that a few modern mathematicians could actually prove this in the first place, and second, that Fermat apparently could also even though his toolbox was so much sparser. Which of course leaves the suggestion that there may be a much more primitive solution still waiting in the wings.

These vids not havin an intro makes me feel like i should know more before it even starts

Haha his notes look a lot like my notes when I am doing math for fun

Wow, this guy teaches at my school

I asked Ken to explain it to me. He said here, read this, and shoved a little pamphlet into my hands.

This is a truly fantastic video. Ken is such an amazingly nice guy, he's saying it, but you have to listen really carefully to hear it.

just trust me on this one