The math behind Fermat's Last Theorem | Modular Forms

The fascinating piece of math hidden behind the proof of Fermat's last Theorem for most people.

===Chapters===
0:00 Introduction
1:14 Lattices
4:05 Modular Forms
7:26 The Modular Group
10:46 Fourier Series
13:14 How many are there
13:50 Outro

===Made with===

===Music===
Music by Vincent Rubinetti
Download the music on Bandcamp:
Stream the music on Spotify:

===Tags===
#modularforms #fermat #modularity

[TheDavidlloydjones]

"There are two ways of doing anything, the smart way and the dumb way. When you do it the smart way, that's mathematics." ~ Definition attributed to a kid in a fifth grade Lego class in Lexington, Massachusetts.

[se]

Made it halfway through before I got lost — but that was much further than I expected to get! Great video!

[Risuchan]

This is an appetizing introduction to the topic. Easily accessible overview to undergraduate students, with a vertiginous glimpse into the bottomless depths of modern analytic number theory.

[omargaber]

This is actually genius. I hope you explain the entire proof of FLT

[ЛевЛокуциевский]

Very nice, thank you! Simple explanation of very non trivial things. Would love to see a sequel about Fermat's theorem. Subscribed.

[Jooolse]

3:57 "The connection between elliptic functions and modular forms is the key part of the proof of Fermat's Last Theorem": there is a confusion here between elliptic CURVES and elliptic functions (they are totally different).

The modularity theorem which implies FLT is about the relationship between elliptic CURVES and modular forms.

[ytang]

Thank you so much. Great video. I tried to read Weil's introduction to his proof and was overwhelmed by how many things I couldn't understand. I still make my students read it, to see the amazing emotional journey, but I enjoy steadily plugging away at learning the bits of terminology. Your videos are wonderful!

[TalkingBook]

These are very excellent videos. I really have to concentrate and hit pause and rewind a lot and take time to make notes, There's no fluff. Enjoy!

[eamonnsiocain]

Thank you. This was concise and clear.

[cryptonative]

Great video! Would love a sequel on the relation of modular forms to FLT

[lordeji]

no words can describe what i'm feeling.
HOW IN THE WORLDS NOBODY EXPLAINED THE "why" OF THE DEFINITION !

I always wondered why we asked the function to change (accordingly to the weight) with SL2 matrices and now, with this approach, the need of matrices of determinant 1 come as evident.
Truly wonderful +1 :))

[aieousavren]

How beautiful! Well done, such a great presentation! I had never seen modular forms explained like this before. I am very grateful!

[Ava-LeeWillow]

I’ve just been watching Kimberly Brehm’s linear algebra series, and this is the first group theory video I’ve watched since then, and you have finally given me the necessary insight to have an intuition on what groups are.

[ShinCChin]

Thank you for this excellent video about Modular Forms which I learnt a lot just by viewing this excellent video.

[p_m_]

It is really nice to see a video on modularity:) I really enjoyed your video! Thanks!
Subscribed:)

[beamathematician]

Really awesome, and appreciate your work. From my childhood, I always dream, one day i will understand proof of FLT. Now I may hope, it is easy to complete my dream through your videos. Thank you 😊

[EldarUrmanov-qx]

12:04 Great attention to small design elements. If you pause a video in landscape mode at 12:04, the third circle matches exactly with the play/pause gray UI circle of the YouTube app (at least on the iPhone).

[harisserdarevic]

I'm curious what your sources were for the definition. In my studies, we define the modularity as f(gamma(z)) = (cz +d)^k f(z), where gamma is some SL(2, Z) matrix. You had a negative k in your definition.

[Strawberry-qw]

Great video <3 you are underrated genuinely

[BuleriaChk]

Proof of Fermat's Last Theorem for Village Idiots
(works for the case of n=2 as well)

To show: c^n <> a^n + b^n for all natural numbers, a, b, c, n, n >1
c = a + b
c^n = (a + b)^n = [a^n + b^n] + f(a, b, n) Binomial Expansion
c^n = [a^n + b^n] iff f(a, b, n) = 0
f(a, b, n) <> 0
c^n <> [a^n + b^n] QED

n=2
"rectangular coordinates"
c^2 = a^2 + b^2 + 2ab

Note that 2ab = 4[(1/2)ab] represents the areas of four right triangles)

"radial coordinates"
Lete p:= pi, n= 2
multiply by pi
pc^2 = pa^2 + pb^2 + p2ab
Note that pc^2, pa^2, and pb^2 represent areas of circles, wile p2ab = a(2pb) is the product of a radius (a) and a circumference (2pb).

This proof also works for multi-nomial functions.
Note: every number is prime relative to its own base: a = a(a/a) = a(1_a)
a + a = 2a (Godbach's Conjecture (now Theorem.... :)

(Wiles' proof) used modular functions defined on the upper half of the complex plane. Trying to equate the two models is trying to square the circle.


c = a + ib
c* - a - ib
cc* = a^2 + b^2 <> #^2
But #^2 = [cc*] +[2ab] = [a^2 + b^2] + [2ab] so complex numbers are irrelevant.
Note: there are no positive numbers: - c = a-b, b>a iff b-c = a, a + 0 = a, a-a=0, a+a =2a
Every number is prime relative to its own base: n = n(n/n), n + n = 2n (Goldbach)
1^2 <> 1 (Russell's Paradox)
In particular the group operation of multiplication requires the existence of both elements as a precondition, meaning there is no such multiplication as a group operation)
(Clifford Algebras are much ado about nothing)

Remember, you read it here first)

There is much more to this story, but I don't have the spacetime to write it here.