The Heart of Fermat's Last Theorem - Numberphile

Modularity... Simon Pampena gets to the heart of proving Fermat's Last Theorem.
More links & stuff in full description below ↓↓↓
Audible: www.audible.com/numberphile
More Numberphile videos on Fermat: bit.ly/fermat_videos
Simon Pampena is Australia's Numeracy Ambassador --- / mathemaniac
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Videos by Brady Haran
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[@EugeneKhutoryansky]

Now that we have the heart of Fermat's last theorem, we just need its liver, kidneys, gallbladder, etc...

[@iugoeswest]

+TheAnubis022 waaah

[@iugoeswest]

+DekuStickGamer well said!

[@derekbaldwin8314]

one of my favorite channels on one of my other favorite channels!

[@DekuStickGamer]

***** He's just replying dude. Never does he mention "Please come to my channel" or any sort of begging. He's just making a funny comment, which in turn may result in noticement for him, BUT he does not "advertise". He's not spamming, or self-promoting excessively. If he wanted attention, he could go to much more popular education channels. I mostly just see him on Numberphile.

[@raymondfabi1140]

Physics Videos by Eugene Khutoryans

[@bentoth9555]

I love how excited Brady sounds when he says "it's like a slinky."

[@michaelbauers8800]

Slinkies are easy to like :)

[@PC_Simo]

@@michaelbauers8800 They are :).

[@pime32]

*whispers* My attempt at a circle. -Yea it's not bad. -Thanks man. i lol'd

[@McPlaySpot]

Same xD

[@randomjunkohyeah1]

+PIME32 "That was an oval! It has to be a circle!"

[@ABumInABox]

+PIME32 I lol'd too.

[@DekuStickGamer]

Simon

[@jerome1335]

"That's my attempt at a circle" - "its not that bad" ... "Thanks man" idk why I laughed so much at this ffs😂😂

[@xnax1993]

+Jerome Hart The way he said "Thanks, man" was definitely what did it for me :D

[@jerome1335]

+Lime Icing same OMG 😂😂

[@JorgetePanete]

Jerome it's*

[@Triantalex]

??

[@ArjunBahuguna]

One of my favourite Numberphile videos now. With visual intuition, it really makes modular elliptic curves accessible.

[@sams.1597]

I know there is a lot more to it, but this is the single best explanation of the Taniyama-Shimura conjecture I’ve ever seen! “modular forms with many symmetries are actually elliptic curves in disguise”. I never really could wrap my head around modular forms until now.

[@phyein4815]

This is exactly the kind of math vid I look for - I'm a long way from being able to understand advanced math but I love attempts to paint an intuitive analog (difficult as it may be on it's own as well) to extremely technical stuff that seems so alien. Nobody can just look at a wikipedia summary of modular forms and elliptic curves and even begin to get an idea like one that is conveyed here for example. This is the kind of thing I want to get out of learning more and more advanced math.

[@fSateQ]

fisrt numberphile video i understand nothing

[@roberttimmins415]

same

[@msclrhd]

+fSateQ A function f(x) that is infinitely tileable repeats every distance k. That is: f(x) = f(x + nk) for all integer n A circle x^2 + y^2 = r^2 has a similar property. Every time it repeats, it loops back to the start and then draws over itself for the next cycle. This is akin to f(x) getting back to the point where it starts to repeat (i.e. any value nk). Because of this property, you can write an equation that transforms f(x) to the circle, such that k is the circumference of that circle. Fermat's last theorem states that a^n + b^n = c^n for all n > 2 does not hold for integral values of a, b and c. Thus, the inverse of that you would use in a proof by contradiction is that you can find integral values for that equation at all n > 2. Where modularity would come in is that I assume that if it is used in the proof, that proof would then be alternatively expressed as a generalised modularity (mapping tileable functions to x^n + y^n = r^n) being valid (i.e. there exists a mapping that can get you exactly back to the start of the hypercircle for tileable functions). If you could prove this to be false, you could prove Fermat's last theorem.

[@thebigcapitalism9826]

+msclrhd oh wow that really clears it up thanks

[@HasnainHossain_h]

+fSateQ Really? You understand the higher dimension videos and have no difficulty visualizing those?

[@callummunro7380]

+Hasnain Hossain I think he was being sarcastic...

[@Joeobrown1]

this all goes way over my head (and probably most of our heads), but i think we can all agree that's a pretty great shirt

[@McPlaySpot]

*reading the comment* "oh ikr , yea yea ikr " *reaches the shirt part* oh

[@lights473]

+・ﾍﾘｵﾃﾞｨﾝ what are you talking about? He's talking about his shirt not mathematics

[@mighty8357]

+Joe Brown I laughed really hard at this comment :D

[@TheMrvidfreak]

Greit shiet, Dolan

[@DK-py2qx]

Oh! Ok. I get it now. All the explanations i saw before were kind of confusing.

[@colebutler4442]

"That's not bad" "Thanks man"

[@only4giga]

This video is just great. I'm tempted to say it's my favorite video of all numberphile videos so far (although the coin flip ones are a strong contestant for my favorite video as well). I've watched all videos on this channel and all your other channels Brady and ever since I saw your videos on Fermat's Last Theorem wondered how the conjecture works. Great stuff, keep up the good work!

[@prae197]

3:22-3:25 is my favorite part tbh

[@hayhayeht]

"That's not bad" "Thanks man" real bros there, I loved it too

[@RoflZack]

The infinite plane joke at 1:40 hahaha

[@owenlever]

This is a truly marvellous demonstration of Fermat's last theorem.

[@marsgal42]

+Owen Lever Pauca sed matura. Oops. Sorry. Wrong mathematician. Me bad.

[@jazzsoul69]

He’s so passionate that’s beautiful to see

[@modolief]

Thanks for making so many high quality videos!

[@lybeclt]

The best video I saw it on this channel so far. Thank you guys 😃 see you next year!

[@dominiclohry1782]

what an absolutely amazingly elegant basic explanation. Thank you!

[@anon8109]

Loved it! More videos about this please!

[@Goheeca]

Please do more about this topic, this is exactly the video where I've expected an extra content.

[@recklessroges]

The "infinite plane" animation: ICWYDT and laughed, (so thanks), but I wondered if that might confuse newer students that don't play as comfortably with math. My second thought was that I love Dr Pampena's calmed-mania style of presentation, and the irony of describing a helix with his fingers while we are looking at many lovely examples attached to his head.

[@general9064]

yay!! some concrete math appreciation! loved it

[@ThomasGiles]

That was pretty awesome! We actually glimpse some of the meat without being blasted away by the nitty gritty. Great work!

[@randyzeitman1354]

Severely well explained ... to get the core of the idea of modularity and whole numbers.

[@ItsGlizda]

Great film! I was watching it with a smile.

[@itchybrother21]

An incredibly brave topic to go for guys, keep it up and thanks a bunch!

[@Srsbzns_5150]

Why was it brave?

[@samueltaylor9935]

@@Srsbzns_5150 because this is ridiculously dumbed down for normal people to grasp

[@Kilroyan]

I've read love & math and I absolutely loved it! Even as a mathematically illiterate person I was able to grasp a huge amount of what Mr. Frenkel is talking about and his own story is absolutely fascinating. I definitely recommend checking it out!

[@ChristopherSLucas]

Very well done. Thank you.

[@TechXSoftware]

What about Fermat's first theorem?

[@cortster12]

+TechXSoftware Baby+food=shit.

[@ReuvenF957]

Well done! Simon actually accomplishes what he started out to do. Namely, to give a FEELING for what is involved in the proof of FLT. I'm not really a mathematician, but I can "sense" that there is a 'situation' which Simon's explanation is a metaphor of.

[@hakkbak]

MORE VIDEOS ON THIS STUFF: I LIKE IT LOTS. might be appropriate to put it in numberphile2 but still, MORE PLEASE.

[@coryrobertson6367]

+hakkihan tunbak who knew there was a numberphile2. Thanks.

[@hakkbak]

+Cory Robertson no prob, :D

[@PetraKann]

This presentation is like trying to give a person a taste of what "calculus" is about by introducing them to the addition of fractions. I personally dont see the point. It would have been better to post this video as a description of what modular forms were (and perhaps make a reference at the end to how it was useful in the early work carried out to develop a proof of Fermat's Last theorem)

[@phuonghovan8092]

+Peter Kan calculus? nope.

[@shepherdpiers]

Hi Brady Thanks for continuing to put out superb content. Computerphile and Numberphile are my most favourite channels on youtube. I can't get enough! I'm not sure if you're aware, or maybe it's just me, but all of you videos seem a lot quieter when compared to other videos on KZbin. This results in me having to crank my speakers up quite a bit. Looking forward to more. Piers

[@gilberttheisen9270]

Je n'ai jamais écrit ça !

[@Epoch11]

I'd love to see more on this or any of the Millennium Problems.

[@hankyg329]

Well, only one of them has actually been solved

[@KMusic_13]

+Mark G I'd love to learn more about Navier-Stokes! I'm sure any engineers watching would also be interested in seeing them discuss it.

[@juliusdictatorperpetuus2147]

The riemann hypothesis and Poincaré Theorem / Conjecture have been done on Numberphile. I think the issue with many of the other problems is that they're so abstract that it would take way too many videos to explain them.

[@shivtekoriginal]

just came home from a royal society lecture on ramanujan where his work on modular mathematics came up. Was reading Fermat's Last Theorem on the train and planned to search up about modular forms. Then this happens... love you guys

[@nathansmith3608]

amazing video. this is a great high level description mathematicians usig graphs & conceptual spaces to demonstrate things about numbers & priciples

[@IsoYear]

I am so glad I started watching this channel as it reignited my love of math. This particular video really blew my mind (and annoyed the family a tad haha)

[@gilberttheisen9270]

Je n'ai jamais écrit ça !

[@gilberttheisen9270]

Je n'ai jamais écrit ça !

[@singingblueberry]

An amazing explaination! I already knew some stuff about it because of the book from Simon Singh about fermats last theorem, but this was a great visualisation though.

[@michaelsheffield6852]

Nicely done.

[@bernardofitzpatrick5403]

Like Simon's way of teaching and he seems a really cool, chilled dude.

[@AnstonMusic]

I have always wanted to understand this one!

[@AnstonMusic]

Oh well, I guess it's a bit too complicated to fit on a 10 minute video. Also I would need to get a few courses of background info.

[@TheMothew]

+Anston [Music] I'd really recommend Simon Singh's book "Fermat's Last Theorem" from 1997. While it doesn't delve very deeply into the mathematics, it presents the story in an accessible and readable manner.

[@AnstonMusic]

Matthew Shepherd Ok then.

[@Airblader]

+Matthew Shepherd Agreed, that book is a great read!

[@MOHNAKHAN]

Great Explanation 👍👍👍 Thanks.

[@picknikbasket]

Enjoyed that - any extra footage ;-)

[@veggiet2009]

The volume in the video seems kinda low

[@EddSjo]

seems pretty normal on my end.

[@U014B]

WHAT?

[@Load_the_ducks]

+veggiet2009 then turn it up

[@Graemyr]

+veggiet2009 Some of its contents must have spilled between their end and yours. Be sure to mop it up, will you?

[@PetraKann]

+MichaelKingsfordGray professional software? must be right then

[@ehtikhet]

Yes Simon! The enthusiasm us palpable!

[@TheMrvidfreak]

What.

[@mechanicalsnail4703]

Same.

[@Kuba87687]

yeah...

[@charlienam42]

+TheMrvidfreak wat

[@aliqatishat3790]

Who

[@isaacc7]

So am I correct in thinking that proving the conjecture was actually a greater addition to mathematics than proving Fermat's last theorem? I know the last theorem was what motivated Wiles but it sounds like proving the conjecture will have a long lasting impact while Fermat's last theorem is more of a curiosity.

[@busTedOaS]

+isaacc7 The conjecture was a key part in the proof of Fermat's last theorem, so yeah, in a way the theorem is just a concrete application of this very general and surprising conjecture. Mathematicians had a harder time believing it than the theorem itself, it's almost like cheating yourself some symmetry out of the aether into your problem.

[@ollehellemaa4789]

+isaacc7 correct, the proof of fermats theorem only gained so much attention mainly because no one has solved it for over 350 years. Andrew wiles proved the theorem by proving the conjucture which was a relatively new discovery that will have a greater impact on mathematics.

[@bushyconn]

Absolutely Fabulous. I wish could understand any of it , but you are so convincing I almost want to study maths again.

[@SPLEclipse]

That coil shape is also used to describe wave phase transitions, except it's displayed from the "side" rather than from the "top".

[@CraaaabPeople]

one of the better numberphile videos

[@nikhilwardrobe]

I came here for answers but found none and found many questions

[@denisdaly1708]

X dark congratulations, you have reached the first stage of enlightenment.

[@ItumelengS]

Listen to Denis

[@Bwern]

This guy is totally the real guy from numb3rs

[@bemusedindian8571]

This is awesome.

[@scottmiller5331]

Brady, the slinky footage off the brown paper is really clever. Well played sir, well played.

[@EmaDaCuz]

This guy is unreally cool, my favourite in the great set of mathematicians on this channel.

[@bernardfinucane2061]

In chess you have light square bishops and dark square bishops. The bishops move along diagonals, so the color of the square the bishop is on never changes. In the end game, if you opponent only has a dark square bishop, you try to keep all your pieces on light squares, because you know that no combination of legal bishop moves will ever allow that dark square bishop to threaten the light squares. This is similar to the Galois Theory proof that you can't square a circle. Using construction, you can start with length one and create line segments of irrational length. For example the square root of two is pretty easy to construct. You can create an infinite number of different irrational lengths, like seven plus the square root of two divided by two. But you can't create ALL irrational lengths. Like the dark square bishop, you can get lots of places within the rules of the game, but you can't get everywhere. The proof shows that squaring the circle involves the wrong sort of irrational numbers -- the square root of pi I think, but that is just detail. The video seems to be suggesting that similar restrictions apply to curves and modular forms. If I understand correctly, the Fermat equations look like elliptic equations, but they represent places on the playing field that the rules of modular forms don't allow.

[@glufull]

Infinite plane FTW! And Simon, you're a great speaker.

[@arturia-leafgreen5127]

that's my attempt at a circle its not bad thanks man

[@robin5171]

I'd love to see a full numberphile video about the proof of Fermats last theorm. Even if it is like 2-3 hours long, I'd watch it.

[@clubberdang3955]

Fascinating

[@GaneshNayak]

very well explained

[@Egonkiller]

+Ganesh Nayak sarcasm?

[@8ytan]

+Jack Freeman I don't think so. It was well explained, it's just a difficult subject.

[@josephasghar]

I just love this guy. Why couldn’t I have a maths teacher like this at school.

[@swaggercr7]

When is dr James grimes going to come back?

[@suave319]

+SwaggerCR7 when he has a new numbah to show us

[@magicalpencil]

+SwaggerCR7 I'm pining for the G-man

[@NickCybert]

+SwaggerCR7 You can always go to his channel, singingbannana, if you're yearning for him.

[@janopawski183]

+SwaggerCR7 he has his own channel too (singingbanana)

[@swaggercr7]

NickCybert thanks

[@selimbaydar123]

Thanks so much for taking up modularity i kept coming across it (mainly in the proof of the abc conjecture ) once i saw ur video with the bridges (says a lot about my mathematical knowledge that bridge was the word i chose to refer to it), probably cause of the Baader-Meinhoff phenomenon but still my mind feels much more clarified. U make me want to become a mathematician BRADY, U R JUST FUCKIN AWESOME

[@GlutesEnjoyer]

4:15"Let's have fun"That looked so wrong in all the right ways. thanks Simon.

[@Speireata4]

Now I wish you would break down this video to a simpler explanation again, because I have been out of school for ten years now and my native language is not English so I only have a very vage idea on what elliptic curves might be. You explained the modular forms well enough that I understood that part.

[@orthographiechecker820]

This is magical stuff.

[@3VILmonkey]

That was explained really well. I just graduated architecture school and we spent a lot of time doing partis utilizing modulars and the variations they produce (think golden triangle), so maybe that's why the visuals you used explained it to me so well. Regardless, we'll done.

[@arpyzero]

Aw, you could have at least shown an example of an elliptic curve to extend the concept slightly.

[@briangreen3496]

A elliptic curve is a circle

[@jdhalligan]

+Brian Green it's not though, Google it

[@eeshan3955]

a circle is an elliptic curve*

[@ruroruro]

@@eeshan3955 it's not though, Google it Like seriously. A circle is an *ellipse*, not an elliptic *curve*. Those are completely different things.

[@hybmnzz2658]

@@briangreen3496 no

[@thomaskember4628]

I was once asked to leave the Mitchell library, the main public library in Sydney, when I was discussing Fermat's Last Theorem with a friend. We were talking too loudly so we were both showen the exit.

[@45640uberfreak]

Please a video with more explanation, this looks really interesting but this video is really hard to understand with so little information

[@StephenMortimer]

I sit here and try to imagine my feeling if Simon was my grandson... would be lovely to have a chat with his grandfather !!

[@Dazbog373]

I appreciate this almost as much as I do the hyacinthine curls :D

[@ScottEltringhamMusic]

The best guy at numberphile

[@duckymomo7935]

Not grimes or the other guy(he did largest prime video)? These 3 are the best imo

[@jonyb0b13]

I am doing my senior capstone on Fermat's Last Theorem. And this is basically what my capstone was about lol. This worked out perfectly.

[@lawrencecalablaster568]

This guy is brilliant :)

[@bluespartan3000]

He was having way too much fun playing with that circle.

[@Xanderqwerty123]

Hot damn thats neat. My calc 3 teacher metioned this when we graphed a coil, but never explained how it was applied. I now want to know more about modularity.

[@ARP2wefightforyou]

I love this guy.

[@TheNBKiller]

This is amazing, the transition of a seemingly finite object or number to an infinite wave which loops back onto itself on a different plane... am I missing something, or completely wrong altogether?

[@AlexanderBachmann]

There goes my hero. He's ordinaaaaryyyy!!!!! ;) Great video! Great guy.

[@LittlePeng9]

I don't think this was emphasised: circle is _not_ an elliptic curve.

[@phuonghovan8092]

+LittlePeng9 yeah, im confuse too

[@salmjak]

If I got the "heart" of the proof its that some curves can be represented by a modular form, even though the modular form exists in another plane/dimension.

[@jamieg2427]

Simon Pampena is one of my favorites.

[@In-N-Out333]

Can anyone link me to an image of an elliptical curve that becomes a modular form? I'd really like to see how that 2d curve or 3d doughnut becomes a highly symmetrical object.

[@EGarrett01]

You lost me at 3:22 because I went to watch freehand circle drawing vids.

[@metallsnubben]

+EGarrett01 You just made a (probably unintentional) far-fetched joke for people who play DotA 2, just so you know ;)

[@karamtraktor]

+metallsnubben I am dota 2 player and didn't understand the joke. Could you explain it please

[@metallsnubben]

I see You! Have you heard about 322? It's basically about how some half famous player lost a game on purpose because he bet against himself on betting sites, which earned him 322$ apparently but for understandable reasons got him banned from further competition. So "322" has become an in-joke in the DotA community that you use when someone makes a really bad mistake, sort of jokingly implying that the only way you could screw up that bad was by doing it on purpose. So that's why "you lost me at 3.22" could kinda be thought of as a dota joke :) It's not really something that you see a whole lot in-game, it's more common if one watches pro games or read forums such as the dota 2 subreddit and such

[@xenon9562]

+metallsnubben Lol Seriously?

[@karamtraktor]

metallsnubben Yeah I've heard about it but I didn't linked the comment with it. I guess I am getting rusty

[@Callinder]

Of course the really infuriating thing is that Fermat made a note in the margin of his book where he stated this theorem which said that he had a proof but it wouldn't fit in the margin. Even Wiles said that even though he proved the theorem he still wondered what Fermat had done.

[@denisdaly1708]

I got that the proof entailed merging, or at least connecting, two different branches of mathematics and how one branch can soft of be represented in another. Also the proof hinged on a contradiction. Finally, the proof is actually very complicated and probably is best done by following the argument as trying to visualise what is going on is too mindbending. Is that the feeling I was supposed to get? I liked this video.

[@dariuscygler7611]

i loved the infinite airplane :)

[@ceza237]

Can you make a video about Perrin Numbers?

[@master_blizzard]

I don't know why but I just love how he says "now".

[@Merrida100]

Brady, I like how excited you get: "Like a SLINKY!!" :) .......PS: The volume's been low on your videos lately.....

[@43labontepetty]

We have Ron Graham explaining Graham’s number. We have Neil Sloane explaining many of the great things he’s had a hand in. We need Andrew Wyles going through Fermat’s last theorem.

[@TheGamblermusic]

I almost understood what he said but that made me plenty happy

[@Hecatonicosachoron]

Reading the title I thought the video would be about the pedal curves of Lamé curves - so I got excited as Lamé curves are among the best things that exist - but alas no geometry for today. Still, interesting video.

[@alexwilson8034]

Ong this made sense for the first time. Every time he said modularity, think periodicity like periodic functions,ones that rePEat

[@thaneross]

@Brady: I know sometimes new discoveries in math allows for simpler / more elegant versions of old proofs. I'm curious if Simon thinks that will ever happen with Fermat's Last Theorem.

[@insainsin]

like when you take x to the root of x on a 3d complex graph. For x where it is negative like -10 you get a complex number. It spirals out to infinity in the closer it gets to 0(a 3d shape). However when it gets to a number that is positive it is a real number, so it looks completely 2d. Also you get a spring shape when you take i to the power of x. btw(for those who do not no what I am talking about) i is complex. i=sqrt(-1) || is absolute value. (the distance from 1) |-1|=1 |i|=1 |0.7071067812+0.7071067812j (a estimate of the sqirt(i)) |=1 |0.9238795325+0.3826834324j|=1 You can change the value of any of these number and they will still have an absolute value of 1 |-0.9238795325+0.3826834324j|=1 Their are an infinite amount of numbers that have a absolute value of 1.

[@pedroscoponi4905]

I have a blue screen in my head right now :( But I think I at least understand what all the surprise was about - using the very rules of the game to do something that no one thought could be done...

[@_bender4143]

4:40 This must look so ridiculous for someone who doesn't know what is he trying to say...

[@Graskian]

You made me laugh, thank you :D

[@chrissidiras]

This is awesome, but still a lot confusing. Please make more videos on Fermat's last theorem!!! What exactly is a modural???