Why was this visual proof missed for 400 years? (Fermat's two square theorem)
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- Опубликовано: 23 май 2024
- Today's video is about a new really wonderfully simple and visual proof of Fermat's famous two square theorem: An odd prime can be written as the sum of two integer squares iff it is of the form 4k+1. This proof is a visual incarnation of Zagier's (in)famous one-sentence proof.
0:00 Intro
2:20 Chapter 1: Discovering a theorem
7:05 Chapter 2: 400 years worth of proofs
9:59 Chapter 3: Zagier's one-sentence proof
15:40 Chapter 4: The windmill trick
22:12 Chapter 5: Windmill maths interlude
25:08 Chapter 6: Uniqueness !!
33:08 Credits
The first ten minutes of the video are an introduction to the theorem and its history. The presentation of the new proof runs from 10:00 to 21:00. Later on I also present a proof that there is only one way to write 4k+1 primes as the sum of two squares of positive integers.
I learned about the new visual proof from someone who goes by the RUclips name TheOneThreeSeven. What TheOneThreeSeven pointed out to me was a summary of the windmill proof by Moritz Firsching in this mathoverflow discussion: mathoverflow.net/questions/31...
In turn Moritz Firsching mentions that he learned this proof from Günter Zieger and he links to a very nice survey of proofs of Fermat's theorem by Alexander Spivak that also contains the new proof (in Russian): Крылатые квадраты (Winged squares), Lecture notes for the mathematical circle at Moscow State University, 15th lecture 2007: mmmf.msu.ru/lect/spivak/summa_...
Here is a link to JSTOR where you can read Zagier's paper for free:
www.jstor.org/stable/2323918
Here are the Numberphile videos on Zagier's proof that I mention in my video:
• The Prime Problem with...
• The One Sentence Proof...
Finally here is a link to my summary of the different cases for the windmill pairing that need to be considered (don't read until you've given this a go yourself :)
www.qedcat.com/misc/windmill_s...
Today's t-shirt is one of my own: "To infinity and beyond"
Enjoy!
P.S.: Added a couple of hours after the video went live:
One of the things that I find really rewarding about making these videos is all the great feedback here in the comments. Here are a few of the most noteworthy observations so far:
- Based on feedback by one of you it looks like it was the Russian math teacher and math olympiad coach Alexander Spivak discovered the windmill interpretation of Zagier's proof; see also the link in the description of this video.
- Challenge 1 at the very end should (of course :) be: an integer can be written as a difference of two squares if and only if it is odd or a multiple of 4.
-one of you actually ran some primality testing to make sure that that 100 digit number is really a prime. Based on those tests it's looking good that this is indeed the case :)
- one of you actually found this !!! 6513516734600035718300327211250928237178281758494417357560086828416863929270451437126021949850746381 = 16120430216983125661219096041413890639183535175875^2 + 79080013051462081144097259373611263341866969255266^2
- a nice insight about the windmill proof for Pythagoras's theorem is that you can shift the two tilings with respect to each other and you get different dissection proofs this way. Particularly nice ones result when you place the vertices of the large square at the centres of the smaller squares :)
- proving that there is only one straight square cross: observe that the five pieces of the cross can be lined up into a long rectangle with short side is x. Since the area of the rectangle is the prime p, x has to be 1. Very pretty :)
- Mathologer videos covering the ticked beautiful proofs in the math beauty pageant:
e^i pi=-1 : • e to the pi i for dummies (there are actually a couple of videos in which I talk about this but this is the main one)
infinitely many primes: Mentioned a couple of times: This video has a really fun proof off the beaten track: • Euler’s Pi Prime Produ...
pi^2/6: Again mentioned a couple of times but this one here is the main video: • Euler's real identity ...
root 2 is irrational: one of the videos in which I present a proof: • Root 2 and the deadly ...
pi is transcendental: • The dark side of the M...
And actually there is one more on the list, Brouwer's fixed-point theorem that is a corollary of of what I do in this video: • NYT: Sperner's lemma d...
- When you start with the 11k windmill and then alternate swapping yz and the footprint construction, you'll start cycling through different windmill solutions and will eventually reach one of the solutions we are really interested in. Zagier et al talk about this in an article in the American Mathematical Monthly "New Looks at Old Number Theory" www.jstor.org/stable/10.4169/...
"4k+1, now can you see the patter on the left?"
"Yeah 😄, 4k-1!"
"4k+3!
"😑"
Is there a reason that it is notated as 4k+3 in stead of 4k-1?
@@McDaldo There is nothing wrong with using 4k-1 instead of 4k+3. An integer is 1 less than a multiple of 4 if and only if it is 3 more than a multiple of 4. So 4k-1 and 4k+3 describe the same sets of integers. The arguments/proofs in this video would work exactly as well using 4k-1 as it does using 4k+3.
So why does Mathologer use 4k+3? Because of modular arithmetic! In modular arithmetic, we work with the _remainders._ So if you were asked, "what is 7 modulo 4; in other words, what is the remainder when you divide 7 by 4?" you would probably answer with "3", not with "-1".
And Mathologer's next video (after this one), uses modular arithmetic, so feel free to check it out: watch?v=X63MWZIN3gM
@@McDaldo it's because when you do modulo the remainder of 7 / 4 is 3 not -1. Because of this is more standardized to use 4k 4k+1 4k+2 4k+3 and not things like 4k-1 or 4k+4
MuffinsAPlenty Thank you so much for answering this. This makes so much sense
@@MuffinsAPlenty Isn't it a trade-off of domain consistency for the consistency of modular arithmetic? For 4k+1 --> k>=1 but for 4k+3 --> k>=0.
"The proof is left as an exercise for the reader" -Fermat
Fermat on every ''theorem'' and conjecture
lol
Yeye
I have a proof but they go to another school and you wouldn't know them
To be fair, he had us pretty exercised over his theorem for many, many years...
This proof is so beautiful that I wrote an entire essay about numbers as the sum of two squares. When the essay was "finished" (I admit that it wasn't), I sent it to the main competition for this type of math essays in the Netherlands, and it got third place.
Also, because I heavily studied the subject in my spare time and Olympiad training, I got really good at this type of number theory.
When I participated at the IMO in Oslo this year (second time), I solved question 3 with full points, which was about this type of number theory. I got a perfect score on the first day, and scored 7+5+4=16 points on the second day, for a total of 37 points! GOLD! 19th place worldwide! Relative best for my country ever!
I really don't know if I would have gotten this score without this proof, so thank you so much for making this video. I hope that you are going to inspire lots of other people as well!
*_WOW!_* That’s really impressive 😮👏🏻! *_CONGRATULATIONS!_* 🥳😃👍🏻
You are a legend
This is great! Congratulations. And I hope that maybe you will be one inspiring people as well!
25/3/2024.
La preuve se démontre en 4 lignes. Niveau: classe de 4e en France !
Plus, revoir sa copie;
Bon courage.
This proof was discovered by Roger Heath-Brown in 1971, and was later condensed into the one sentence version by Don Zagier. It's one of two proofs of this theorem found in the wonderful book "Proofs from THE BOOK" 6th ed by Martin Aigner and Günter M. Ziegler in chapter 4.
Thanks for that. I bought the book when it came out (1ed.). Loved it then. Looks like I should have a look at the most recent edition. Who knows what other gems have found their way in there :)
I love Zagier's sentence, even without the windmills. It serves as a great exercise in reading proofs. If I ever teach one of those "intro to proofs" class, I would assign the task of deciphering it as some sort of class discussion for the day.
@mikemthify
Roger Heath-Brown was 19 in 1971. Could you post some sources?
@@Macieks300 page 21 of the book I mentioned.
As a source it cites: D. R. Heath-Brown: Fermat's two squares theorem, Invariant (1984), 2-5. latex version, with appendix on history, January 2008, at eprints.maths.ox.ac.uk/677/1/invariant.pdf
The URL is archived at: web.archive.org/web/20110606154228/eprints.maths.ox.ac.uk/677/1/invariant.pdf
@@mikemthify He said "My original
notes date from 1971." I don't know if that means he came up with the proof then but if he did he really would've been 19 and that just blows my mind.
Fermat: "Hey, here's this cool thing about numbers."
Mathematicians: "Amazing! Can you prove it?"
Fermat: "I already did."
Mathematicians: "Wow! Can we see it?"
Fermat: "Hmmm... nah."
Fermat: *dies*
Fermat : I'm sorry but I run out of space to write the stuff anyway bye
Everyone : ... you can just get another paper
Fermat's👻: Aaaahh, now let's just sit and enjoy their struggle !
perhaps fermat chose to let other people work on the problem than to just spoonfeed the proofs for them, so as to not spoil the pursuit of mathematics for people. since he knew he was able to prove it he can reasonably assume that anyone else could be able to as well
When it came to granting access to his proofs, he seems to have been slightly on the egg plant side of behaviors 🤔 On the other hand, he also was an extremely strict judge, so maybe he wanted people to demonstrate their ability to grok things on their own while watching with a frown 🙂
Videos like this make me marvel at the internet. Growing up I could never have access to content like this but now I can watch a brilliant mathematical mind explain fascinating concepts to me. this channel is an example that should give everyone faith in the future of humanity.
I never made it past geometry in public school, and yet I was able to follow most of this well, and appreciate how beautiful this proof really is. I chalk that up not only to your ability to explain things in various ways, but also to just how clean and professionally edited this video was. Well done. You have yourself a new fan. (Or... a new windmill.)
That's great :)
I don't know what I am doing on this video but that last bit of your comment is better than the proof
Thanks for the video! When I first heard about this proof, I asked Alexander Spivak who invented the visual version. And he said that there was no other source, it was his own idea. Because we don't know anybody who came up with this before 2007, it's almost certainly that he was the first. Unbelievable, but the Zagier's proof (and the previous proof by Heath-Brown) had appeared without any connection to geometry.
I actually had a link to a writeup by Spivak and I dug up an e-mail address. Sadly he never replied to my e-mail asking him whether he discovered the windmills (neither did Don Zagier) :(
@@Mathologer I have a few friends in common with him, and it was easier for me.
@@vsevolodvoronov7526 No harm in him replying now?
I like, that 3blue1brown is also a patron
Yes. I love that "3lue1brown" is a "patreon."
Shirt = To infinity and beyond?
why is > beyond?
@@dimitrispapadakis2122 Im thinking it refers to a number greater than infinity (>inf). In other words beyond infinity
@@dimitrispapadakis2122 because it's not >=
2 Infinity or greater than. And is after all the multiplicative function.
There is no "beyond" the boundless aka infinite...Buzz Lightyear was stoned on "star command," a powerful strain of marijuana...
Thank you, Mathologer for your wonderful videos! David Wells's survey sadly omits Cantor's diagonalization, which, in my opinion, belongs no lower than position 2 on his list of most beautiful proofs. Cantor's proof is also the granddaddy (through Goedel) of Turing's proof of the undecidability of the halting problem (which also sends chills down my spine whenever I read it), and which ushered in the field of computer science.
Speaking of omissions. What about Pythagoras's theorem ? :)
I kept hearing "a 4k+1 prime" and wondered how or if the primality mattered. It's amazing how late, and how crucially, it finally comes into play.
Where in the proof it mattered?
Can you give me timestamp?
I still don't understand why it has to be a 4k+1 prime.
@@programmer4047 20:07 primality comes in
So elegant. At 19:17, I understood where this proof is going, that is the happiest moment of your video when I understand where the proof is going 😃
Chirayu Jain Nice!
I agree! That kind of feeling is just amazing!
Hi Chairayu
Jo didn't realize it was 19 minutes of Math already at that moment
@@blackpenredpen you are too here!!!! 😮
20:55 I had to immediately upvote here. I love when a proof concludes and it all comes together and makes sense. I wish that visuals were more commonplace in math papers (and in maths in general), because I feel like less people would feel like math is something they'll never be able to understand. Great video, very easy to follow, very enlightening!
So far I'm utterly hopeless. Your eureka moment went right by me, I don't see how anything fits together. I was completely lost every inch of the way. I believe there are people who just CAN NOT understand math no matter how gifted the teacher. And I HATE that I am one of those people, because I think I'd really like math if I could just catch on.
@@johnnysparkleface3096 That's ok! Even though this video is aimed at being a simple proof, it is still somewhat advanced to be able to grasp. Don't beat yourself up, there is always plenty of math for you to enjoy that you'd be able to digest, not matter your skill level.
I love that eureka-moment, as well; and this proof and video certainly delivers. 👍🏻
I love the structure of this video. The moment when I understood how the visual proof would go (just before we moved to visual representations of it) is why I watch videos like this.
You guys really outdid yourselves with the presentation of this visual proof. Nice addition of the uniqueness proof. Spectacular job!
I really love the graphical intuition added onto that one sentence proof. It makes it a lot clearer WHY that function is an involution and has exactly 1 fixed point.
Also, you misspoke. At 28:54 you said "b squared" instead of "c squared." >.< Gotta be tough to get through that stuff without any mistakes. At least it's clear what you meant cause of the written equations.
A great reason for redundancy in information given!
It is good to see that mathloger is back online...
wooo!!! Yes! Re-subbed!
And to see all the comments restored aswell!
Yes! In the era of 2+2=4 is racist!
Why was he offline?
I am numerically challenged. I have a bachelor's degree in nursing and have never passed algebra...(please don't ask).
I am addicted to your channel and genuinely understand the pleasure that you exhibit from elegant solutions.
Thank you for this long undiscovered pleasure that you have introduced me to.
WOW! Definitiv eines der besten Mathe-Videos auf RUclips! Und auch sehr schön aufbereitet und präsentiert!
Euler's formula for polyhedra can easily reach #1 if you realise it's actually d0-d1+d2-d3+d4...dn=1 where di is the number of i-dimensional objects that form an n-dimensional polyhedron
Out of curiosity: are you sure about the right side? I am certainly no expert in this particular subject, but having an odd number there seems....
Well... Odd 😁😁
Jokes aside though, this is kinda new form of knowledge for me and I want to see where you got this from :)
@@csDiablo1 It checks out for the familiar 3D case - V-E+F-1=1 (the last 1 on the left is the body itself). In 2D, it can be rewritten as V=E (the shape and the constant 1 on the right cancel).
I noticed that pattern in high school when playing with polytopes. Never tried to prove it though.
I think I also noticed that the n-1 dimensional surface of an n-dimensional sphere is the derivative of its hyper-volume. I think that might have been an assumption on my part given that it’s true for the first couple of examples.
I did integrate hyperspheres and derive a formula for n-dimensional spheres. It’s interesting that you get an extra factor of pi at every even dimension. I’ve wondered if that has anything to do with the number of independent axes of rotation you can have.
I feel like I should study math again. Don’t think I could derive that formula now.
@@zemoxian I think there is a recent video of 3B1B exactly on that extra Pi
@@csDiablo1 yeah, absolutely sure! It's easy to see that that sum equals 1 in the case of a n-dimensional tetrahedron for example.
If you didn't know, the n-th row of Pascal's triangle describes the number of i-dimensional objects that form an n-dimensional tetrahedron ( for example, a 3-dimensional pyramid has 4 V 6E 4 F and 1 Pyramid, 4-6-4-1) and the 1 left over in the equation is the first 1 in Pascal's rows (it is another well-known result that the alternating sums of the numbers in the rows equals zero)
This is really beautiful. It's even more beautiful than the theorem itself, which was hard to beat.
Man, this channel is awesome. Keep up the great work!
This is great. You're a good teacher and I appreciate the time you spent making it
Okay, this is actually one of the most beautiful things I've seen in math.
Thank you for presenting this. Haven’t had a math class in more than 40 years but I did have formal logic which helped a bit when following this video. If I had seen this in high school I might have had a whole different career path.
Omg. I wish more ppl were interested in math to appreciate things like this, and your vid itself. Great edit job too, congrats the team. Perfect job man. +1 sub for sure.
Well, 200k and counting, not bad I'd say :)
I remember the Numberphile video and I'm amazed that such a simpler proof is available now! Thanks for sharing it.
I am very very fascinated by
1) How hardworking you are with all these presentations
2) How kind, positive and interested in math you are.
It's perfect that you make these videos, it literally makes me much happier because i fall in love with math more and more.
P. S. Sorry for my english, it's not my language.
Glad you like the videos. It's a lot of work but it's also very rewarding to then get comments like this that show people really appreciate what I am doing :)
"Step back and squint your eyes."
Brilliant guide to this insight!
Merry Christmas!!! What a beautiful proof. Amazing.
Wow. That's a lot to take in. I get the idea, but I feel like to truly get an intuitive grasp, I would need to take some time to think it all over.
Amazingly well explained. Well done!
It was me, I discovered this proof back in grade school when making arts & crafts. I wrote a note in my journal of discovering the proof, but I had to also go back and watch Power Rangers.
7:02 yes it can. It's sufficient to look at the last two digits of a number to check if it's divisible by 4 since 4 divides 100. The last two digits were 81 which is one above a multiple of four.
Thank you kind sir
Yes, but can you prove it is prime : P That would be the issue in this case.
@@maulaucraw1209 😆😆
Brilliant! I admit I didn't follow every step of all this on first viewing, but I know there's nothing there beyond my ability to understand. I will watch it again (maybe several times), because it's easy to see that it's truly beautiful!
Minecraft villager be like: 5:30
when the paper is worth 2 emeralds
Speaking of Minecraft... 33:13 first PayPal supporter 🤔
Drake Tungsten notch agrees 😂
I wonder if there is a villager sound expansion mod that includes this take of the sound in the variety or if it will have it included now.
mathologer is a gamer confirmed
7:00 yes it can. The number ends in 81. That's a multiple of 4 + 1.
To elaborate a bit, 6513...46381 = 6513...46300 + 81. The number on the left obviously has no remainder when divided by 4 (being a multiple of 100), leaving only 81 to be considered.
@@keyboard_toucher Thx captain abvious, but "multiple of 4 depends of last 2 digit " is a tool given at school before the age of ten, just like " sum up digits of a number to know if you can divide it by 3 "
I'm surprised there aren't more comments about how your shirt literally says "To infinity and beyond" in math geek. At least, I think it does?
Yeah I noticed his shirt too
Great video. I actually had a project in my number theory class to verify the one sentence proof. Very fun, but this is way more enlightening.
When you do Euler's polyhedron formula, here is an interesting bit you could include. For any polyhedron*, the angular deficits at the vertices sum to 720 degrees (4 pi steradians.) This can be very quickly proved via Euler's polyhedron formula, using for a polygon sum-of-angles = 180 x number-of-vertices - 360. The appeal is that this is about a 30 second proof.
For example, consider a square pyramid with regular triangles. The 'top' vertex has 4 triangles, so the deficit is (360 - 4x60)=120 degrees. The other four vertices have a square and two triangles so the deficit is (360-90-2x60)=150. The sum of the deficits is 4x150+120=720.
I expect (I haven't looked into it) that this is a special case of a theorem which says integrate-curvature-over-a-topologically-spherical-surface = 4 pi, and in turn gives surface area of a unit sphere = 4 pi. And probably integrate-curvature-over-any-surface = 4 pi (1 - number of holes in surface)
* Not self-intersecting, topologically equivalent to a sphere.
2^2+ i^2=3
Veeery funny :)
Wait a minute, does that mean that if we extend the domain of x and y into the complex numbers, it works for any (real) prime? 4^2+(3i)^2=7, for example
@@JMairboeck Yes. As he mentions at the end of the video, any odd number can be written as x^2-y^2. So any odd prime p has p=x^2-y^2=x^2+(iy)^2
Yes Joachim. looks like.
And so 6^2 + 5i^2 = 11
Or we can simply say that
ANY PRIME NUMBER CAN BE WRITEN AS a^2 + b^2
or a^2 - b^2 (and we dont need imaginary numbers) 10^2 - 9^2 = 19 12^2 - 11^2=23. 16^2 - 15^2=31
Only now I notice:
10+9=19
12+11=23
16+15=31
He says 4k+3, and that's equivalent to 4k-1.
The simple part: any odd number n that can be written as the sum of two squares must be the sum of an even square a^2 and an odd square b^2. Now a^2=0 (mod 4) and b^2=1 (mod 4), so that n must be 1 (mod 4).
For an easier understanding I'd like to add that every odd b^2 can be expressed as (x+1)^2, with x being an even number.
Now obviously that makes b^2 equal to x^2 + 2x + 1.
As x is even, both x^2 and 2x are always divisible by 4, so any b^2 must be of the form 4k+1.
(therefore obviously any a^2 + b^2 with a being even and b being odd has to be of the form 4k+1 as well...)
@Šimon Rada good point! I changed to the good old "x" to avoid confusion with the original "a".
@Šimon Rada yes, that was part of the first statement (not mine): "any odd number n *that can be written as the sum of two squares* [...must be of the form 4k+1]" :-)
I really appreciate the work you are doing. I wouldn't find(look for) this nice proof on my own and if you didn't post the video I would spent this limited time I had today on something useless... Your videos boost my inspiration and thus make me feel better. Keep going!
That's great :)
A year ago I left a comment on one of these video's saying I was so inspired I was going to make my own math education you tube video's. I have something very special for everyone coming very soon, it's a free software project that I created while working on a tool to make animations for my video's and is almost ready to be released. I just published the first video on my channel, check it out!
Everyone like this comment lmao its TheOneThreeSeven :O
You've got to learn to use apostrophes correctly!
The numbers what do they mean?
I wanna learn python. Make the UI of the software user friendly. I wanna try the software. I saw your video and that was great.
I had subscribe your channel
Mathologer’s Theorem: π is the sum of two squares. 21:19
Impossible. For a degree greater than 2 .
It's actually a simple corollary of the theorem that a circle cannot be transformed into *one* square.
guys relax, he was referring to the fact that he pronounced "P" as π (pie)
@@federico6416 😜
The movie "Fermat's Room" is indeed excellent, I'm glad it's getting a bit of publicity!
Just finished watching it... thanks for the recommendation!
I see eye to eye with you! Totally!
Huh, I vaguely remember watching it a while ago and sort of liking it, but not thinking it especially awesome? I should rewatch it I guess?
I've seen many beautiful 4K videos on RUclips, but out of *4k+1* videos, this is definitely the best :)
32:20
Any odd number can be written as x² - y².
We first factor x² + y² as usual, leaving us with:
k = x² - y²
k = (x + y)(x - y)
We want to get rid of the y term and cancel it into 1 so that k can simply be represented as 2x + 1 (or in this case, 2x - 1). To do this we set y = x - 1.
The rest of the computation is as follows:
k = (x + (x - 1))(x - (x - 1))
k = (2x - 1)(1)
k = 2x - 1
Therefore every odd number can be written as the difference of two squares by using consecutive x and y.
32:30
All odd primes have a unique way of being represented as a difference of two squares.
We have already proved above that all odd numbers can be represented as the difference of two squares regardless of whether or not the numbers themselves are prime. To prove that there are no other possible choices for prime numbers we may look at the difference of squares a bit closer.
The expression x² - y² can be factored into (x + y)(x - y).
In this case any composite number ab (in this case, 15) can be expressed multiple ways because we can write it as 1*ab (1*15) or a*b (3*5), both of which can be converted into difference of squares, one for each pair of factors.
1*15 = (8-7)(8+7) = 8² - 7²
3*5 = (4-1)(4+1) = 4² - 1²
1*ab = (((ab+1)/2) - ((ab-1)/2))(((ab+1)/2) + ((ab-1)/2)
= ((ab+1)/2)² - ((ab-1)/2)²
a*b = ((a+b)/2 - (b-a)/2)((a+b)/2 + (b-a)/2)
= ((a+b)/2)² - ((b-a)/2)²
However, there is only one factorization for any prime p, namely:
1*p
Therefore, since we can only factor primes in one way, there must also be exactly one way to represent p as a difference of two squares.
Thank you very much..I wrote this to my assignment in university.Thank you.Thank you.❤️❤️❤️❤️
You can write these proofs much more succintly.
1) Any odd number can be written as 2k+1. Obviously 2k+1 = (k+1)^2 - k^2, so 2k+1 can be written as the difference of two squares.
2) Given the above, we know that for any integer k there always exist integers p, q such that 2k+1 = p^2 - q^2 = (p+q)(p-q). Both (p+q) and (p-q) must be odd, since 2k+1 is odd. So if 2k+1 is also prime, one of (p+q) and (p-q) must be 1 -- it's obvious that it's the latter.
Thank you for this great video. A long time ago I heard that Zagier did a one-sentence-proof without knowing what it was until two weeks ago. I did a bit of thinking on my own and want to share what I found (probably not as the first one) because it might be interesting.
In his original paper Zagier states that his proof is not constructive. In itself both involutions (the trivial t:(x,y,z) --> (x,z,y) and the zagier-involution z as discribed in the video) don't give many new solutions starting from a given one. But combined they lead from the trivial solution to the critical, from the fixpoint of the zagier-involution F := (1,1,k) to the fixpoint of the trivial involution t.
Proof (sry no latex here): Let n be the smallest integer with (z*t)^n(F) = F. So t*(z*t)^(n-1)(F) = F (multiply by z on both sides). And therefore (t*z)^m * t * (z*t)^m (F) = F with m = (n-1)/2. Bringing (t*z)^m to the other side proofs that (z*t)^m (F) is a (the) fixpoint of the trivial involution, ie a critical solution.
Note that n is always odd, assuming n is even results in a contradiction: If n is even we have t*(z*t)^k * z * (t*z)^k * t(F) = F with k=(n-2)/2. So again we see that (t*z)^k*t(F) is a fixpoint, this time of z, and therefore equals F. Multiplying by z gives us (z*t)^(k+1)(F) = F contradicting the choice of n.
>One person assigned each theorem a score of 0, with the comment, “Maths is a tool. Art has beauty”; that response was excluded from the averages listed below, as was another that awarded very many zeros, four who left many blanks, and two who awarded numerous 10s.
lol
hey!! your videos are really helpful ..please keep uploading such stuff. please do not stop.
You are a godsent angel, I've had my mouth open the whole video, I wish I could subscribe twice
In his 1940 book “A Mathematician’s apology” the mathematical superstar G.H. Hardy writes: “Another famous and beautiful theorem is Fermat’s ‘two square’ theorem... All the primes of the first class” [i.e. 1 mod 4] ... “can be expressed as the sum of two integral squares... This is Fermat’s theorem, which is ranked, very justly, as one of the finest of arithmetic. Unfortunately, there is no proof within the comprehension of anybody but a fairly expert mathematician.”
My mission in today’s video is to present to you a beautiful visual proof of Fermat’s theorem that hardly anybody seems to know about, a proof that I think just about anybody should be able to appreciate. Fingers crossed :) Please let me know how well this proof worked for you.
And here is a very nice song that goes well with today’s video:
ruclips.net/video/qKV9bK-CBXo/видео.html
Added a couple of hours after the video went live:
One of the things that I find really rewarding about making these videos is all the great feedback here in the comments. Here are a few of the most noteworthy observations so far:
-Based on feedback by one of you it looks like it was the Russian math teacher and math olympiad coach Alexander Spivak discovered the windmill interpretation of Zagier's proof; see also the link in the description of this video.
-Challenge 1 at the very end should be (of course :) be: an integer can be written as a difference of two squares if and only if it is odd or a multiple of 4.
-one of you actually some primality testing to make sure that that 100 digit number is really a prime. Based on those tests it's looking good that this is indeed the case :)
-one of you actually found this !!! 6513516734600035718300327211250928237178281758494417357560086828416863929270451437126021949850746381 = 16120430216983125661219096041413890639183535175875^2 + 79080013051462081144097259373611263341866969255266^2
- a nice insight about the windmill proof for Pythagoras's theorem is that you can shift the two tilings with respect to each other and you get different dissection proofs this way. Particularly nice ones result when you place the vertices of the large square at the centres of the smaller squares :)
-proving that there is only one straight square cross: observe that the five pieces of the cross can be lined up into a long rectangles one of whose short side is x. Since the area of the rectangle is the prime p, x has to be 1. Very pretty :)
-Mathologer videos covering the various ticked beautiful theorems:
e^i pi=-1 : ruclips.net/video/-dhHrg-KbJ0/видео.html (there are actually a couple of videos in which I talk about this but this is the main one)
infinitely many primes was mentioned a couple of times already. This video has a really fun proof off the beaten track:ruclips.net/video/LFwSIdLSosI/видео.html
pi^2/6: again mentioned a couple of times but this one here is the main video: ruclips.net/video/yPl64xi_ZZA/видео.html
root 2 is irrational: one of the videos in which I present a proof: ruclips.net/video/f1yDExNAEMg/видео.html
pi is transcendental: ruclips.net/video/9gk_8mQuerg/видео.html
And actually there is one more on the list, Brower's fixed-point theorem that is a corollary of of what I do in this video: ruclips.net/video/7s-YM-kcKME/видео.html
-When you start with the 11k windmill and then alternate swapping yz and the footprint construction, you'll start cycling through different windmill solutions and will eventually reach one of the solutions we are really interested in. Zagier et al talk about this in an article "New Looks at Old Number Theory" www.jstor.org/stable/10.4169/amer.math.monthly.120.03.243?seq=1
4k+1, 4k-1
"Very nice song" is a link back to this video
Prof. Hardy's life appears to be increasingly anticlimactic. Always overshadowed or outdone, it seems.
Windmill summary is 404ing
The link to the "very nice song" is incorrect. It simply links right back to this video.
@@kenhaley4 Fixed the link :)
6:59 the primes of the form 4k + 1 can be written as the sum of two integer squares. We only need to check the last two digits to determine a numbers modulo 4. This yields 81 which is 20*4 + 1 ⚀
This is wrong. Here's why:
Although what you claim might be correct in most scenarios, it isn't in this one specifically; the fact that 4k is divisible by 2 and the 1 is prime* means that the aforementioned theorem cannot be extrapolated unto said value. In other words, the theorem doesn't "fit" for the equivallence we are trying to prove.
Beautiful beautiful explanations. Every student deserves a professor like you
This is yet again a gem of a video and I hope I one day will be able to teach this to someone. It must be such a thrill to see people get it!
Thanks an enormous lot for the time taken and it is so helpful for making maths fun for so many! (Well at least me!)
To make the video more perfect I would like to point a possible mix up of words:
28:53 : you said a2 = b2 but I think it's a2 = c2.
I point it out for all those like me who must constantly rewind and listen to every single word many times to grasp it.
Many thanks! 💛
Awsome video! In the x^2-y^2 problem at the end, all solutions divisible by 4 are also possible (if you assume that x and y are coprime then you can get all odd numbers as well as numbers divisible by 8).
5:38
All primes that can be written as a sum of two squares are primes
:)
I got that... and then remembered what I was watching and felt silly.
@@Mathologer I mean, it's not wrong, is it? ;)
This reminds us of the old saying that mathematics is a giant truism (or tautology) that reduces to something like 1+1=2. In Physics, Dirac said: The world of elementary particles would be much more scarce if not for so many imaginative physicists.
(p^q)->p , yes
Beautiful proof, beautifully explained!
I loved this video. I was able to follow it, and learned as well. Very interesting.
This is the proof found in "Proofs from the Book"! Don Zagier condensed this into one (not easily understood) sentence.
Must be a more recent edition than the one on my bookshelf :) Maybe also have a look at the links in the description of this video :)
20:16
This is brilliant!
That's the very reason this theorem is about primes.
eliya sne It's crazy to think about it that way, but you're totally right. The proof wasn't very "primey" until that key moment.
But, because of the famous identity, known to the ancient Greeks, any number that is a product only of primes of the form 4k + 1 (and possibly including 2) will also be a sum of two squares.
Things get more complicated if you allow primes of the form 4k + 3. The simplest way to describe it (YMMV) is that in the Gaussian integers (that's numbers that can be written in the form a + bi, where a and b are integers), primes are exactly the numbers that are either of the form a + bi where (a + bi)(a - bi) = a^2 + b^2 = p (prime in the [regular] integers) or p prime in the integers, with p = 4k + 3.
That's one reason this theorem is important. It tells us how to factor complex integers.
THIS VIDEO IS FANTASTIC!!! THANK YOU
I'm studying mathematics right now nad I really love integer numbers, they have so many interesting properties and you really need to stretch your mind to find them. I find calculus, topology, geometry and all that stuff seemingly complicated, but actually easy (the proofs are very often similar), but number theory is always fascinating. At first glance it may seem the easiest part of mathematics, but it's probably the hardest one to understand deeply.
21:20 And therefore pi is a Sum of Two Square. That Excitement Nearly Killed me.
:)
Me2
p=x^2-y^2=(x+y)(x-y) => if p-prime, then x=y-1 => p=2x+1 (proof of the unique)
Amazing video as always!
I see some commenters sharing their favorite theorems. In the theme of counting how many objects can be created in a certain way I recently learned about Kurotowski's closure-complement problem. It asks: given any subset of any topological space, by taking successive closures and complements how many different sets can be created? The answer turns out to be 14 ! What a strange number. It seems too high, but if you smush together enough weird subsets of R you can achieve it.
Lovely explanation and illustrations.Really a nice proof.
I gasped out loud when he pointed out that the windmills pair up with each other. That was amazing
Cisco Ortega what does gasped out mean
@@shatter6012 audibly
@@thomassabino5440 oh thanks now it makes sense
This man is straight up a beast.
Great! Solved the two embedded problems which made me feel good! You are a clever youtuber as well as a good mathematicvian! :))
I love Professor Polster's geometric approach for this proof. It is genius! Great job, Mathologer!
21:15 "and therefore pi is a sum of two squares" 🤔 now that is some mathologer magic I missed in between the lines
Time to watch it one more time. Double the fun :)
The crux of it is that he had x^2 + 4y^2, and 4y^2 is the same as (2y)^2, so that's a square, and x^2 is obviously a square number, so that's the sum of two squares.
In fact, this is how he started out this section of the video, go back to 10:38 and watch that bit. He starts out by defining p this way (since this is what he was trying to show), then he split the y^2 into y(y) and replaced one of the y with z to make a more general formula, and then from there he proved that there is always a case where y and z are equal.
I'm a mathematician myself so I know how the proof works. "pi" in my comment is not a typo since it *sounds* like he says that pi (3.1415....) is the sum of two squares 🤔
Technically true if we don't consider integer squares
Yeah it does sound like he's saying "pi is the sum of two squares", but I assume he just mispronounced "p"?
I'm wondering if the original proof uses the function pi(n), referring to the nth prime number? He swapped out for p, but misspoke once after all his research.
I’m pretty sure I got the 4k+3 proof. Might need corrections:
1st claim: to get an odd number as the sum of two numbers, they must have opposite parity (one even one odd).
Proof: by exhaustion. Even+even=even, odd+odd=even.
2nd claim: The square of a number has the same parity as the number itself.
Proof: (2k)^2 = 4k^2 = 2(2k^2). (2k+1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1.
Therefore, for an odd number to be the sum of two squares, it must be an even number squared plus an odd number squared.
Consider (2k)^2 + (2m+1)^2. Using previous working, this is equal to 4k^2 + 4m^2 + 4m + 1 = 4(k^2 + m^2 + m) + 1. Therefore, the sum of the squares of two numbers with opposite parity is always one more than a multiple of 4. There is no other way to get an odd number as the sum of two squares, so getting any number of the form 4k+3 is impossible.
Yes, but you have not proved the hard part: that a number of the form 4K+1 can indeed be written as the of two squares.
Marvellous!!!
U r an excellent teacher. U know the nuances of voice modulation while teaching. Excellent write up.
Brilliant. And explained with amazing clarity!
7:05
yes
if a number has last two digits which can be divided by 4, the whole number can be
because 100 can be divided by 4
so any multiple of 100 can be, like 83500
and you can check by delete all the other digits
like 83516
it will be 83500+16
83500 can be divided and you have to check 16
now with the prime number
it end with 81 which is 80+1
4(20)+1
:)
the rest don't not matter because they can be divided by 4 any way
That's it and that's the answer I was expecting :) I was actually quite surprised by this answer by ben1996123: 6513516734600035718300327211250928237178281758494417357560086828416863929270451437126021949850746381 = 16120430216983125661219096041413890639183535175875^2 + 79080013051462081144097259373611263341866969255266^2
I see Mathologer's new upload. I just literally drop anything else I do, and watch. Cat video after this, maybe? :)
As always, thank you for your videos !
Great visualization of the proof!!
Fermat was where "The proof is left as an exercise" started.
:)
I have the most excellent documentation of who came up with the windmill interpretation of this proof, but there isn't enough space to place it into this youtube comment.
This is fabulous!! What a great video.
Thank you for this fantastic video!
Note that the footprint-preserving involution defined in 18:01 does not need the special form of the prime p, and in fact the conclusion in 20:30 is: the footprint-preserving involution has exactly one fixed point if p=4k+1, and none if p=4k+3. Thus the number of windmills is odd if p=4k+1 and even if p=4k+3.
The argument in Chapter 6 also still works if we do not assume the form of the prime p, but the conclusion reads: "there is at most one way of writing p as a sum of two squares".
So if we like this video actually also includes the trivial case 4k+3:
p=4k+1: odd number of windmills, exactly one fixed point of yz, p writes uniquely as a sum of two squares.
p=4k+3: even number of windmills, no fixed points of yz, p is not a sum of two squares.
At 24:37, instead of cutting the tiles, why not consider the whole plane, which is covered by "equally many" blue+green and red squares. Of course, one has to consider a proper limit, but it's still easier to see what's going on than with the cut-and-rearrange procedure.
I have a question regarding 32:19, the challenge at the end.
You claim that the existence of integers x,y with x^2 - y^2 = n (> 0, for simplicity) leads to n being odd.
As i found the counter example x = 4, y=2 and therefore n=16 - 4 = 12 being not odd , I probably misunderstood you.
Any help is kindly taken.
Greetings from Germany.
Yes, well spotted, of course that statement is wrong. The correct statement is: an integer can be written as a difference of two squares if and only if it is odd or a multiple of 4 :)
Really interesting and entertaining at a time. Thanks. You're very good
I love the friendly rivalry between you and numberphile. I also love your visualizations.
Videos are back :D
"If there are any parts of this video that you struggled with, just ask"
Yes. Where do you get your T-shirts from?
It took me a few seconds to figure this one out, but hey - you've got a friend :-)
Edit: Greetings from AUstria to Australia
Haven't figured out. Can you tell? 🤔
Asking for a friend..
I get my t-shirts from all over the world. The one in this video I actually made myself :)
Mathologer does it mean “to infinity and beyond”?
We are struggling with figuring out what the symbols mean 😅
First thing I noticed in the video hehe. Nice Toy Story reference :p
I watched this video on Christmas morning 2020. At the risk of goading, this is a stunning video and I'm tremendously grateful for it.
Thank you for your explanation. Really enjoyed it.
Makes me wonder just how much of mathematics can be reduced to stuff that's easier to understand.
TheOneThreeSeven. I love the fine structure of his name
I think that he's one 37 year old man who likes math. I know it goes deeper, but that's my impression.
That number is a constant surprise to me.
HA!! You nailed it =) If I get enough subscribers on my new channel I just launched yesterday I will do a username backstory reveal
And he drives an Alfa?
@@LukeSumIpsePatremTe lmao this is actually how old I am =)
32:30 That is true for any 2 consecutive integer squares; since, as established, in your ”sequence calculus” -video, the differences of 2 consecutive integer squares are the odd numbers; and since one of the squares is even, and the other one is odd. However, skipping an odd number of integer squares results in 1 of 2 cases: Either both of the squares are even, or both are odd; and, in both cases, their differences are even (and mostly non-zero). For example: 5²-3² = 25-9 = 16 | 2. Or 10²-6² = 100-36 = 64 | 2.
Loved this video...! And the little challange was... Kinda easy to be honest! Stay awsome!
I must say, I love your shirt!
Typical Fermat. Claiming he has proofs but not delivering. *Unlike* Mathologer of course 😜
I used to do my math homework in Myriad Pro so I'm happy to see you using that font for math
I rarely use the word but this is an honestly elegant proof. I do so love geometric proofs!