***** 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.
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.
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 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.
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!
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.
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)
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!
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.
+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.
+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.
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.
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.
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
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.
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.
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 TH-cam. This results in me having to crank my speakers up quite a bit. Looking forward to more. Piers
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.
+Tom Mulligan Pastebin is awesome if your proof is any more tangible than popping a 3rd dimension onto a 2-dimensional problem. I suspect this is most mathematicians problems with the slinky explanation. Fermat's Last Theorem is specifically a 2-dimensional problem but the slinky solution bolts on a 3rd dimension. You go from aa + bb = cc to aa + bb + cc = dd; you've changed the equation to make it work at that point even if it's just following the same pattern.
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.
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.
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)
Is there an integer solution (a, b, c) for fractional powers of n? For example, a^n + b^n = c^n for n = 2/3 Answer: For n = 2/3, a=27, b=64, c=625 Theorem: Equation has a solution if and only if the numerator of the irreducible fraction is 1 or 2! This is a rather interesting result, in that it distinguishes rational numbers on the basis of their numerator! Let's define f( 1/n) = min( abs( c^n - a^n - b^n)) for all integers a, b, c. Then all of the zeros of this function are multiples of 1/2. Correct?
I used ds/dn to derive an approximation of the power n for which s(n) = 6^n + 8^n = 9^n. The derivative of s(n) is log(6) * 6^n + log(8) * 8^n, which evaluates to 1456 when n=3. s(3) = 9^3 - 1, so we need to add 1 to s(n) get equality. That gives us 3 + 1/1456 as the approximate value for n! How would we refine the approximation further? What would the series look like?
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.
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?
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.
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.
Thinking about it abstractly, a circle is "something that comes around at a constant rate". Anything with selfsame properties can in theory be expressed as a circle.
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
i took an attempt at fermats last theorem as an engineer and i started with a^2+b^2 = c^2 and i figured out that you can create a square made out of 8 triangles thats length b and width a and at the center there was a smaller square so the entire square is 7 * 7 . it turns out that the formula for c can be written out as c = sqrt( (2 * Volume of a*b) + (|a+b|)^2 ) e.g sqrt((2* 3 * 4) + 1) = 25 . anyway i thought that if in 2d space u can perceive a^2+b^2 = c^2 as a square than it may follow that in 3d space u can make a cube following the same rules . so i tried making a cube using 6 sides that follow the rule and it turns out that no matter how you do it the cube always ends up with 4 a^2+b^2 = c^2 sides and 2 sides that are oneven rectangles instead of squares . and so in conclusion i think you can prove fermats last theorem because the geometry in 3d space isnt even and this is because the legs of the triangles have to be adjacent to a different length for the cube to actually be geometrical
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.
0:30 But 1 ≡ 2 (mod 1)! So that is not trivial if you think in a modular way. Anyway, does this also mean the heart can be found in other periodic parametric equations? Such as x(t)=16 sin^3(t) y(t)=13 cos(t) - 5 cos(2 t) - 2 cos(3 t) - cos(4 t) That gives a real modular hearty feel to it, like a heartbeat ;)
But 2%1==0==1%1, which is the only thing 2 ≡ 1 (mod 1) could represent! (sorry for previous bad notation) Granted, it would be a trivial case when not allowing for fractional values, but the beautiful thing is that modular arithmetic can be extended to do these kind of things, so in a way 7/2 ≡ 1/2 (mod 1) and 735° ≡ 15° (mod 2π)!
Exactly 😊 I just found it ironical to use the example of unity not equaling duality to compare to the proof of Fermat's last theorem, which uses modularity in its proof, that, in return although trivially, could contradict the former statement. One should know that the trivial cases are thus very important in mathematics, for example in the standard mistake made by solving x^2=x.
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.
An explanation for the confused: (NOTE: Again as the video repeats, this is a simplified layman interpretation) Modularity is trying to take a formula/number, and cut it up into pieces that all look the same. So y = sin x, can be cut up into those strips, and all those strips look exactly the same! But what about a circle? If we try to split this up into strips, we can't get them all to be the same. And so we take advantage of an extra dimension (3D!). We make a helix/coil/slinky, that goes up from the circle. Looking down, it's still a "circle". We then look at the helix from the side, and what do we see? The exact same thing as the sin graph! So we can then split these up into strips, and we've demonstrated modularity on a circle. If you didn't get the last part, just imagine or look up the side of a stretched slinky. Then you can see we can cut it up into strips.
Okay, but... How does that relate to numbers? Does it work for other shapes than circles, like parabolas? For anything? And how does this tie in to Fermat???
Ryan Wilson The circle can be expressed as a formula. Fermat's last theorem is trying to prove there is no a^x + b^x = c^x for x > 2, and so modularity breaks this down into something we can use.
This was more to get a feel of what modularity was, with the concept of clever manipulation. So sadly not, nothing outside the vague concept applies. E.g if I explained that cars work by burning gas which then gives it energy.
There is a story (perhaps apocryphal) that Pythagoras came up with his famous theorem after looking at a pattern in a tile floor. I have always believed that the 'remarkable' proof that Fermat referenced was something like that; not pages and pages of formulae.
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...
So is the slinky a representation of the modularity theorem? And who decides when a proof is a proof and what sort of exclusions, substitutions, omissions, and limitations are permitted in developing one? Is there any connection between patterns observed in the natural world (in flora, fauna, patterns on the surface of water, etc.) and the modularity theorem?
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.
Here's one thing you may have wanted to point out: Fermat's theorem is a^n + b^n = c^n ; a circle's equation is x^n + y^n = r^n . I'm guessing there's a connection there.
simple explanation of Taniyama-Shimura-Weil conjecture by 3*n/2^2 - 2 = m^2 turn into 2^3 - (3*t)*2 = (2*m)^2 (-m have symmetry with m at x-axis) have elliptic form x^3 - ax = y^2 for x = 2, a = 3*t, n=2*t, n is positive nature number, have integer solution for m at 1,4,7,10...and 2,5,8,11...all of multiple of 3 to infinity for modular form, 3 and 2 could be any prime number, if it have solution for m at start(m=1 or 2 for 3(=1+2), m=2 or 5 for 7(=2+5),etc..)must be modular form, 2^d for d>2 is not elliptic curve any more, 2 could be any prime numbers.
With the exclusive combination between the two theories, now you need to curve the elliptical wave back around to touch itself. Point of origin to touch the point of termination and call it the 4th dimension: time.
Worth pointing out though that the solution provides by Wiles couldn't have been Fermat's solution as the information Wiles uses was only recently developed.
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.
Now that we have the heart of Fermat's last theorem, we just need its liver, kidneys, gallbladder, etc...
+TheAnubis022 waaah
+DekuStickGamer well said!
one of my favorite channels on one of my other favorite channels!
***** 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.
Physics Videos by Eugene Khutoryans
One of my favourite Numberphile videos now. With visual intuition, it really makes modular elliptic curves accessible.
"That's my attempt at a circle" - "its not that bad" ... "Thanks man" idk why I laughed so much at this ffs😂😂
+Jerome Hart The way he said "Thanks, man" was definitely what did it for me :D
+Lime Icing same OMG 😂😂
Jerome it's*
??
I love how excited Brady sounds when he says "it's like a slinky."
Slinkies are easy to like :)
@@michaelbauers8800 They are :).
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.
*whispers* My attempt at a circle.
-Yea it's not bad.
-Thanks man.
i lol'd
Same xD
+PIME32
"That was an oval! It has to be a circle!"
+PIME32 I lol'd too.
Simon
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.
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
*reading the comment*
"oh ikr , yea yea ikr "
*reaches the shirt part*
oh
+・ﾍﾘｵﾃﾞｨﾝ what are you talking about? He's talking about his shirt not mathematics
+Joe Brown I laughed really hard at this comment :D
Greit shiet, Dolan
Oh! Ok. I get it now. All the explanations i saw before were kind of confusing.
fisrt numberphile video i understand nothing
same
+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.
+msclrhd oh wow that really clears it up thanks
+fSateQ Really? You understand the higher dimension videos and have no difficulty visualizing those?
+Hasnain Hossain I think he was being sarcastic...
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!
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.
"That's not bad"
"Thanks man"
This is a truly marvellous demonstration of Fermat's last theorem.
+Owen Lever Pauca sed matura.
Oops. Sorry. Wrong mathematician. Me bad.
3:22-3:25 is my favorite part tbh
"That's not bad" "Thanks man" real bros there, I loved it too
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)
+Peter Kan calculus? nope.
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!
what an absolutely amazingly elegant basic explanation. Thank you!
The infinite plane joke at 1:40 hahaha
He’s so passionate that’s beautiful to see
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.
+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.
+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.
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.
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.
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
Like Simon's way of teaching and he seems a really cool, chilled dude.
Please do more about this topic, this is exactly the video where I've expected an extra content.
I don't think this was emphasised: circle is _not_ an elliptic curve.
+LittlePeng9 yeah, im confuse too
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.
Severely well explained ... to get the core of the idea of modularity and whole numbers.
MORE VIDEOS ON THIS STUFF: I LIKE IT LOTS.
might be appropriate to put it in numberphile2 but still, MORE PLEASE.
+hakkihan tunbak who knew there was a numberphile2. Thanks.
+Cory Robertson no prob, :D
4:40 This must look so ridiculous for someone who doesn't know what is he trying to say...
You made me laugh, thank you :D
That coil shape is also used to describe wave phase transitions, except it's displayed from the "side" rather than from the "top".
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.
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 TH-cam. This results in me having to crank my speakers up quite a bit.
Looking forward to more.
Piers
Je n'ai jamais écrit ça !
The volume in the video seems kinda low
seems pretty normal on my end.
WHAT?
+veggiet2009 then turn it up
+veggiet2009 Some of its contents must have spilled between their end and yours. Be sure to mop it up, will you?
+MichaelKingsfordGray professional software? must be right then
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.
Guys, guys! I have a truly wonderful proof which differs from Andrew Wiles but unfortunately this comment section is too small to contain it.
+Tom Mulligan cool
+Tom Mulligan
Pastebin is awesome if your proof is any more tangible than popping a 3rd dimension onto a 2-dimensional problem. I suspect this is most mathematicians problems with the slinky explanation. Fermat's Last Theorem is specifically a 2-dimensional problem but the slinky solution bolts on a 3rd dimension. You go from aa + bb = cc to aa + bb + cc = dd; you've changed the equation to make it work at that point even if it's just following the same pattern.
nice one :)
+Tom Mulligan
lol, I get it. ;)
+Alan Hunter Tom is making a reference to Fermat's original statement of the theorem...
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.
I'd love to see more on this or any of the Millennium Problems.
Well, only one of them has actually been solved
+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.
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.
When is dr James grimes going to come back?
+SwaggerCR7 when he has a new numbah to show us
+SwaggerCR7 I'm pining for the G-man
+SwaggerCR7 You can always go to his channel, singingbannana, if you're yearning for him.
+SwaggerCR7 he has his own channel too (singingbanana)
NickCybert thanks
The best video I saw it on this channel so far. Thank you guys 😃 see you next year!
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)
Je n'ai jamais écrit ça !
Je n'ai jamais écrit ça !
This guy is totally the real guy from numb3rs
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.
What about Fermat's first theorem?
+TechXSoftware Baby+food=shit.
What.
Same.
yeah...
+TheMrvidfreak wat
Who
An incredibly brave topic to go for guys, keep it up and thanks a bunch!
Why was it brave?
@@Srsbzns_5150 because this is ridiculously dumbed down for normal people to grasp
That was pretty awesome! We actually glimpse some of the meat without being blasted away by the nitty gritty. Great work!
This guy is unreally cool, my favourite in the great set of mathematicians on this channel.
Is there an integer solution (a, b, c) for fractional powers of n? For example, a^n + b^n = c^n for n = 2/3
Answer: For n = 2/3, a=27, b=64, c=625
Theorem: Equation has a solution if and only if the numerator of the irreducible fraction is 1 or 2!
This is a rather interesting result, in that it distinguishes rational numbers on the basis of their numerator!
Let's define f( 1/n) = min( abs( c^n - a^n - b^n)) for all integers a, b, c. Then all of the zeros of this function are multiples of 1/2. Correct?
I used ds/dn to derive an approximation of the power n for which s(n) = 6^n + 8^n = 9^n. The derivative of s(n) is log(6) * 6^n + log(8) * 8^n, which evaluates to 1456 when n=3. s(3) = 9^3 - 1, so we need to add 1 to s(n) get equality. That gives us 3 + 1/1456 as the approximate value for n! How would we refine the approximation further? What would the series look like?
4:15"Let's have fun"That looked so wrong in all the right ways. thanks Simon.
Brady, the slinky footage off the brown paper is really clever. Well played sir, well played.
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.
yay!! some concrete math appreciation! loved it
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?
Aw, you could have at least shown an example of an elliptic curve to extend the concept slightly.
A elliptic curve is a circle
+Brian Green it's not though, Google it
a circle is an elliptic curve*
@@eeshan3955 it's not though, Google it
Like seriously. A circle is an *ellipse*, not an elliptic *curve*. Those are completely different things.
@@briangreen3496 no
Thanks for making so many high quality videos!
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.
I came here for answers but found none and found many questions
X dark congratulations, you have reached the first stage of enlightenment.
Listen to Denis
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.
I have always wanted to understand this one!
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.
Matthew Shepherd Ok then.
+Matthew Shepherd Agreed, that book is a great read!
Thinking about it abstractly, a circle is "something that comes around at a constant rate".
Anything with selfsame properties can in theory be expressed as a circle.
amazing video. this is a great high level description mathematicians usig graphs & conceptual spaces to demonstrate things about numbers & priciples
Infinite plane FTW!
And Simon, you're a great speaker.
You lost me at 3:22 because I went to watch freehand circle drawing vids.
+EGarrett01 You just made a (probably unintentional) far-fetched joke for people who play DotA 2, just so you know ;)
+metallsnubben I am dota 2 player and didn't understand the joke. Could you explain it please
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
+metallsnubben Lol
Seriously?
metallsnubben Yeah I've heard about it but I didn't linked the comment with it. I guess I am getting rusty
Please a video with more explanation, this looks really interesting but this video is really hard to understand with so little information
i took an attempt at fermats last theorem as an engineer and i started with a^2+b^2 = c^2 and i figured out that you can create a square made out of 8 triangles thats length b and width a and at the center there was a smaller square so the entire square is 7 * 7 . it turns out that the formula for c can be written out as c = sqrt( (2 * Volume of a*b) + (|a+b|)^2 ) e.g sqrt((2* 3 * 4) + 1) = 25 . anyway i thought that if in 2d space u can perceive a^2+b^2 = c^2 as a square than it may follow that in 3d space u can make a cube following the same rules . so i tried making a cube using 6 sides that follow the rule and it turns out that no matter how you do it the cube always ends up with 4 a^2+b^2 = c^2 sides and 2 sides that are oneven rectangles instead of squares . and so in conclusion i think you can prove fermats last theorem because the geometry in 3d space isnt even and this is because the legs of the triangles have to be adjacent to a different length for the cube to actually be geometrical
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.
I don't know why but I just love how he says "now".
Brady, I like how excited you get: "Like a SLINKY!!" :) .......PS: The volume's been low on your videos lately.....
Great film! I was watching it with a smile.
0:30 But 1 ≡ 2 (mod 1)! So that is not trivial if you think in a modular way.
Anyway, does this also mean the heart can be found in other periodic parametric equations? Such as
x(t)=16 sin^3(t)
y(t)=13 cos(t) - 5 cos(2 t) - 2 cos(3 t) - cos(4 t)
That gives a real modular hearty feel to it, like a heartbeat ;)
But 2%1==0==1%1, which is the only thing 2 ≡ 1 (mod 1) could represent! (sorry for previous bad notation)
Granted, it would be a trivial case when not allowing for fractional values, but the beautiful thing is that modular arithmetic can be extended to do these kind of things, so in a way 7/2 ≡ 1/2 (mod 1) and 735° ≡ 15° (mod 2π)!
Exactly 😊
I just found it ironical to use the example of unity not equaling duality to compare to the proof of Fermat's last theorem, which uses modularity in its proof, that, in return although trivially, could contradict the former statement.
One should know that the trivial cases are thus very important in mathematics, for example in the standard mistake made by solving x^2=x.
one of the better numberphile videos
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.
An explanation for the confused:
(NOTE: Again as the video repeats, this is a simplified layman interpretation)
Modularity is trying to take a formula/number, and cut it up into pieces that all look the same.
So y = sin x, can be cut up into those strips, and all those strips look exactly the same!
But what about a circle? If we try to split this up into strips, we can't get them all to be the same. And so we take advantage of an extra dimension (3D!). We make a helix/coil/slinky, that goes up from the circle.
Looking down, it's still a "circle". We then look at the helix from the side, and what do we see? The exact same thing as the sin graph! So we can then split these up into strips, and we've demonstrated modularity on a circle.
If you didn't get the last part, just imagine or look up the side of a stretched slinky. Then you can see we can cut it up into strips.
Okay, but... How does that relate to numbers?
Does it work for other shapes than circles, like parabolas? For anything?
And how does this tie in to Fermat???
Ryan Wilson The circle can be expressed as a formula. Fermat's last theorem is trying to prove there is no a^x + b^x = c^x for x > 2, and so modularity breaks this down into something we can use.
Then is that to say we're expanding a^n + b^n = c^n upwards into a fourth axis (via modularity) and working from there?
This was more to get a feel of what modularity was, with the concept of clever manipulation.
So sadly not, nothing outside the vague concept applies. E.g if I explained that cars work by burning gas which then gives it energy.
There is a story (perhaps apocryphal) that Pythagoras came up with his famous theorem after looking at a pattern in a tile floor. I have always believed that the 'remarkable' proof that Fermat referenced was something like that; not pages and pages of formulae.
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...
Yes Simon! The enthusiasm us palpable!
He was having way too much fun playing with that circle.
I sit here and try to imagine my feeling if Simon was my grandson... would be lovely to have a chat with his grandfather !!
I just love this guy. Why couldn’t I have a maths teacher like this at school.
So is the slinky a representation of the modularity theorem?
And who decides when a proof is a proof and what sort of exclusions, substitutions, omissions, and limitations are permitted in developing one?
Is there any connection between patterns observed in the natural world (in flora, fauna, patterns on the surface of water, etc.) and the modularity theorem?
This is awesome, but still a lot confusing. Please make more videos on Fermat's last theorem!!! What exactly is a modural???
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.
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.
What if the width of the sections of the plane were the same as the diameter of the circle?
Loved it!
More videos about this please!
I think I know why a^n + b^n don't = c^n with integers because of you need 4 right triangles for cubes and I don't think it's can't be done
+kordell curl what?
Here's one thing you may have wanted to point out: Fermat's theorem is a^n + b^n = c^n ; a circle's equation is x^n + y^n = r^n . I'm guessing there's a connection there.
+Marconius Of course, because that's the same equation, only different variables.
+Marconius this is actually one of the most underrated comments here.
IsorokuPND Not really. They are similar because they are the same equation.
+Marconius Wait, to plot graph you just need x and y right? What is that r^n doing there?
AMOGHA JAYANTH MK r is the radius of the circle.
Ong this made sense for the first time. Every time he said modularity, think periodicity like periodic functions,ones that rePEat
simple explanation of Taniyama-Shimura-Weil conjecture by 3*n/2^2 - 2 = m^2 turn into 2^3 - (3*t)*2 = (2*m)^2 (-m have symmetry with m at x-axis) have elliptic form x^3 - ax = y^2 for x = 2, a = 3*t, n=2*t, n is positive nature number, have integer solution for m at 1,4,7,10...and 2,5,8,11...all of multiple of 3 to infinity for modular form, 3 and 2 could be any prime number, if it have solution for m at start(m=1 or 2 for 3(=1+2), m=2 or 5 for 7(=2+5),etc..)must be modular form, 2^d for d>2 is not elliptic curve any more, 2 could be any prime numbers.
i gotta show the proof to someone and then just say, "this is basically just saying 1 doesn't equal 2".
Simon Pampena is one of my favorites.
With the exclusive combination between the two theories, now you need to curve the elliptical wave back around to touch itself. Point of origin to touch the point of termination and call it the 4th dimension: time.
that's my attempt at a circle
its not bad
thanks man
1 ≠ 2
This is almost as simple. :)
I need a follow-up though! Is there more from Simon on the way?
Bottom line is spiral hides behind the circle.
Slinky is modular - repetitive.
Circle is whole number
very well explained
+Ganesh Nayak sarcasm?
+Jack Freeman
I don't think so. It was well explained, it's just a difficult subject.
Worth pointing out though that the solution provides by Wiles couldn't have been Fermat's solution as the information Wiles uses was only recently developed.
Absolutely Fabulous. I wish could understand any of it , but you are so convincing I almost want to study maths again.
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.
The best guy at numberphile
Not grimes or the other guy(he did largest prime video)?
These 3 are the best imo