Pi hiding in prime regularities

A story of pi, primes, complex numbers, and how number theory braids them together.
Mathologer on why 4k + 1primes break down as sums of squares: • Why was this visual pr...
Help fund future projects: / 3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/leibniz-thanks
Home page: www.3blue1brown.com/
For those of you curious about the finer details, here's a writeup from the viewer Daniel Flores justifying the final approximation: www.overleaf.com/read/wdzkfjb...
The fact that only primes that are one above a multiple of four can be expressed as the sum of two squares is known as "Fermat's theorem on sums of two squares": goo.gl/EdhaN2
Music by Vince Rubinetti:
vincerubinetti.bandcamp.com/a...
Timestamps
0:00 - Introduction
1:39 - Counting lattice points
5:47 - Gaussian integers
10:30 - The lattice point recipe
17:50 - Counting on one ring
20:14 - Exploiting prime regularity
25:19 - Combining the rings
28:36 - Branches of number theory
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with KZbin, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: 3b1b.co/recommended
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Website: www.3blue1brown.com
Twitter: / 3blue1brown
Patreon: / 3blue1brown
Facebook: / 3blue1brown
Reddit: / 3blue1brown

[@rgbatom5145]

Pi is hiding everywhere. This is a prime example

[@BrendonGreenNZL]

That is a complex pun.

[@rubensf7780]

This pun is irrationally funny

[@thenoobalmighty8790]

prime - rem = pi

[@Mathfan002]

It transcendents the real plane

[@sootangel]

fighting for my life trying not to yell “SHUT UP” because it’s 3:00 AM 🫡

[@yerivalpolanco1448]

Being able to watch this kind of content so easily and for free is probably the best thing about living on this time.

[@kennedystapleton2279]

Underrated comment on an underrated channel

[@lunakid12]

@@kennedystapleton2279 ~4M subscribers is quite the opposite of underrated.

[@ogmakefirefiregood]

I wonder if we could get KZbin college credit?🤔

[@P-nk-m-na]

@@lunakid12 still deserves more, given how much effort goes into this stuff and the value it gives.

[@maxwellsequation4887]

Yes

[@smogy001]

8:16 "this might seem needlessly complex" I see what you did there

[@sanscipher9166]

One would imagine math puns would be more often.

[@jadenruanes1858]

L0l

[@achtsekundenfurz7876]

28:03 "And the bigger _R_ is, the more accurate both of these estimates _R_ "

[@achtsekundenfurz7876]

25:05 "And I think, holy ______" Looks like I guessed the next word horribly wrong... ;)

[@cedricdb]

I paused the video to look for this comment and I’m so happy I found it

[@Leyrann]

26:34 The most beautiful thing in the world is when half an hour of complicated math comes together in something so simple and direct.

[@LeoLokoII]

i bet he's a jojo fan

[@achtsekundenfurz7876]

Something else that's beautiful: it's easy to see that numbers of the type 4n-1 can't be the sum of two squares of integers. Because if n is an integer, n-1 is too, and you can write those numbers as 4(n-1)+3. Now all squares of integers are either some 4n or 4n+1 types, so if you add two, you can never get 4n+3, only 4n, 4n+1 or 4n+2 for some n. BUT: what about 21, which is 4n+1? The reasoning about the ki function says that it cannot, because there's a factor of some number 4n+3 in it that's not squared. Now, 21 is easy; if it's the sum of two squares, one of the squares must be >= 10.5, i.e. 16, and 5 is not a square. But the result using the ki function applies to all numbers you can factor. If you find a factor of the 4n+3 kind that's not squared, the answer is no.

[@zxinc123]

My jaw just fell when watching this part

[@sajalkmittal]

It definitely is the most beautiful thing in the world

[@polygongeometry3743]

@@sajalkmittal Most beautiful things is the thing which is beautiful according to your nature of mind

[@MrDaanjanssen]

23:50: 'Make sure everything feels good up to this point' *Starts the video all over again for the 4th time*

[@learningsuper6785]

Seriously? I watched it on 1.25 speed...

[@theflaggeddragon9472]

we gotta badass over here

[@sage5296]

Daan Janssen watch it on 3.14 times sped

[@mitchellloren3568]

/r/iamverysmart ? Get off your high horse you pretentious fuck.

[@XxSteamStreamxX]

I did too xD but that's just because i like to fit in more videos in a day.

[@adjoint_functor]

26:37 genuinely yelled in shock and awe at this part. This right here is why I love math: those moments where all the complexity and intricate patterns collide and combine into one clean and beautiful whole. This video made me more excited and intrigued than most movies. Bravo, Grant, bravo.

[@rubinkatz9850]

yes, an OMG moment

[@TechnicallyLogical2009]

Math: is math This guy: 💯💯💯💯💯

[@alegitnolife]

100@@TechnicallyLogical2009

[@akuma.00]

i literally screamed: “OH MY FUCKING GOD” then started laughing until i saw your comment

[@isaacnewtonstolemyjoy]

Truly the best feeling you can get in math 😅

[@danielc1112]

This method of teaching, it's a revolution in pedagogy really. A calm warm voice explaining the hidden structure of number theory, with a detailed and colourful animation that is perfectly in sync. I've never seen anything quite like this channel.

[@btf_flotsam478]

Skimming over the really hard part? Quick note: he never shows that all primes of the form 4k+1 can be expressed as the sum of two squares, other than by shitting his pants and hoping no-one smells it.

[@Jesin00]

​@@btf_flotsam478He explicitly pointed out he wasn't explaining that part here, so it's not like he was dishonest or hiding. If you know where I can find a good explanation, I'd love to see it.

[@Benny_Blue]

I never in my *life* thought I’d understand where that pi infinite series came from. This video is spectacular - I followed along the whole time. Then afterwards, although it was tough, I successfully redid all the steps mentally, and they made perfect sense. Thank you so much for this - you did a great job!

[@pauselab5569]

pretty sure it is just a version of the taylor series for tangent

[@andrewkarsten5268]

@@pauselab5569that is the typical calculus proof but Grant provided a geometric intuition behind that. Yes, logically this is more or less the same as the calc proof shown at the beginning.

[@sandyjr5225]

Yeah particularly inverse tangent.

[@ahoustonpsych]

Hey I haven't commented before but I just wanted to say that I am absolutely in love with what you're doing. It's clear how passionate you are about this stuff; your passion and the care you put into these videos an extreme pleasure to watch. The animations are downright astonishing in how well you've managed to make all the right visual connections to complement the explanations that you're giving. Not to mention the extremely high quality of the animations is far above what I've seen anywhere else. Your voice is very calming and you speak very clearly in an approachable and inviting way which really helps hit the nail on the head. There's no question that you understand what you're saying at a deep level. Really everything is freaking awesome and dead-on. I can't really express the impact you've had and are continuing to have on me, especially as of late. I had been passionately studying machine learning over the last few months, putting almost every bit of my free time into reading something about the software or the math involved within. It occurred to me recently that I need to go back to college to learn some of the high-level stuff so I had been specifically studying into basic calculus for the placement test. I was floored when essence of calculus came out, and doubly so when I saw that you're looking to do a series on probability which is heavily involved in machine learning. This excites me in ways that I haven't felt before, and has been adding fuel to a deep passion for this stuff. Not to mention other videos like these ones which goes deep into subjects that you definitely wouldn't expect to see. It sounds a bit dramatic and a bit rambley now that I actually type it all out, but I am serious. It's hard to truly put it into words which is why I'm glad you have a patreon. Everything down to the quality of your ads is top notch (which is actually what prompted me to post this since I'm checking out remix). This was all super gross but I just had to let you know how much I appreciate you and what you do, and how much I enjoy the content that you're making here.

[@3blue1brown]

Wow, I don't even know how to respond here. Thank you so much. Best of luck with your machine learning pursuits. You might enjoy taking a look at Welch Labs videos on the topic, if you haven't already.

[@learningsuper6785]

If you are going to be a machine learning engineer, you don't need to go back to university to achieve that.

[@EmadGohari]

3Blue1Brown really appreciate your content and effort. keep doing it. you are inspiring many people to pursue math and cs related subjects. Thanks for awesome quality in your work.

[@anandchangani3122]

@@3blue1brown thanks for suggesting welch lab. i am also on same state as TheGreekBrit is. without wasting your time i wanted to let you know that mate you are changing the life of a young boy who is belonging from small village of india and always wanted to learn math as the way you teach!

[@kaziaburousan166]

@@3blue1brown I was also able to proof it some time before this...but I am just curious that can we approach the same counting the lattice problem in any difference way??.. ( I already have done method using polygon, so any difference method??)

[@MrDaanjanssen]

3Blue1Brown, you should know that someone just stood up from his chair and gave you a standing ovation

[@General12th]

Not necessarily you... just someone.

[@timpunny]

I would get up but the toilet is comfy

[@aashaypandharpatte8102]

Same here

[@TigersTalons]

Timothy Gunter or are your legs just numb? Jk lol

[@CECItheMATOS]

Two

[@DEWILL]

Seriously, we don't deserve this quality of explanation. And we cannot express how grateful these videos are. You deserve my loudest clap ever. Thank you.

[@Masterfusion]

Wow hello there friend. Tbh I am not surprised to see you there.

[@rohanshah6178]

I am just startled by the way you put together concepts of Gaussian Integers, Prime regularities, and Multiplicative functions to obtain such an amazing result. Was just amazed to see all these boil down to such a beautiful equation. Thank you so much for making this video. I really appreciate your efforts to bring such complex yet beautiful results to math enthusiasts like me. Thank you very much.

[@AnyVideo999]

Shortest 30 minute video I've seen in a while. I just cannot believe how high quality your videos are.

[@SpySappingMyKeyboard]

Seriously, every time I see the time on one of these videos and go "I don't have time for that"! Then I want to rewatch.

[@SSGranor]

I don't think I've seen a 30 minute video that felt this short since Vi Hart's Twelve Tones.

[@iustinianconstantinescu5498]

XxWolF-_-PicKxX For me that's 2nd place.

[@fossilfighters101]

+

[@arbitrage2141]

XxWolF-_-PicKxX Me either, I hope this guys ad revenue is off the chain. I can definitely imagine that this content is viewed at universities. Id pay heavily to advertise to advertise to this demographic.

[@nUrnxvmhTEuU]

7:47 "They're called Gaussian integers, named after Martin Sheen“ :D

[@DavidRichfield]

Luapix apparently they resemble each other? pod.kneedrag.org/posts/8193

[@earthbjornnahkaimurrao9542]

There was a mathematician named Martin Sheen who was a student of Gauss. He first popularized the concept of the Gaussian integers having named them after his mentor.

[@spastikatenpradikat4163]

Michal Grňo Well, Martin Sheen was once a voice actor in a flatland short animation. :D

[@levmatta]

Michal Grňo. My reaction was: Ok... wait WHAT!!! rewind (think: good joke)

[@EebstertheGreat]

Earthbjorn is pulling your leg. The joke is just that they look vaguely similar.

[@kjyhh]

0:11 Wow, the PI guy jumped into the formula.

[@rangerocket9453]

Yes haha

[@matthewkellman1166]

This proof and the explanation was truly incredibly elegant. Thank you for putting in the time to create these fantastic videos. Your channel is truly unparalleled in this universe!

[@tornadoreaper]

nice Dono

[@AssasinMilo]

3B1B: "so by factoring prime numbers" Me : He's a witch!

[@GamingKing-jo9py]

where?!? timestamp pls. still a witch

[@Skiddla]

@@GamingKing-jo9py 9:27

[@insomnia20422]

@@GamingKing-jo9py like everywhere and he means factoring into complex numbers not real ones, and of course only the non-gaussian primes, but thats trivial anyway... ;)

[@angelmendez-rivera351]

InSomnia DrEvil Technically, no number is a prime number unless the integral domain with respect to which factorizations are defined is specified, since primehood is a relationship between an element and the ring it lies in, not an inherent property of elements themselves absolute. For example, we can consider the ring of real-valued polynomials of degree m < n or m = n. In this ring, the prime elements are all the linear polynomials. We can also talk about sets of algebraic integers of a particular degree, and we can form unique factorization domains, in which we have well-defined prime elements which are irreducible.

[@gunhasirac]

In a ring (a algebraic structure with commutative addition and multiplication), prime (ideal) is defined w.r.t to the ring itself. For primes in a ring R1 (here is integer), when you look at it in ring R2 (here Gaussian integer) containing R1, it can either remain prime in R2 or admit a non-trivial factorization into primes in R2. An early theorem in algebraic number theory states that: prime number p can be written as sum of squares of two integers (i.e. admits non-trivial factorization in Gaussian integer), iff p = 1 (mod 4) (i.e. has remainder 1 when divided by 4). This theorem is not hard to understand, but as a simple problem like this, you already need an extend tool in order to work out the proof. Concepts in number theory are just abstract as hell, and I totally gave up after seeing ramification in valuation theory.ヾ(:3ﾉｼヾ)ﾉｼ

[@anselmschueler]

"This might seem *needlessly complex*" Nice joke there! (8:15)

[@ebigunso]

wow totally missed that one

[@Gayuha]

I don't get it. Can you explain?

[@tobiasmattsson2794]

Amazing video, please keep up the good work, you really made my day with this video. :)

[@ajnelson1431]

Imaginary numbers are more generally called "complex numbers"

[@ebigunso]

To be more exact, "complex numbers" are all the real numbers, all the imaginary numbers, and anything with those two combined. Imaginary numbers are just "0 + Ai" so yes it is a complex number but that could get some people confused.

[@AroundTheBlockAgain]

Was NOT expecting the "hey why don't we sort these into columns" at the end but D U D E That was BEAUTIFUL I thought you were going to pull a fancy "and here's how to add up every integer intersection on every single circle this radius or lower" but that was even wilder than I was expecting. Super fun ride. Math Good.

[@elmahjourhamza3042]

He puts such effort to explain very advanced math and makes it "cute" ... You contribute in fighting MATHOPHOBIA ! Congrats on that.

[@SerranoAcademy]

You, sir, are an absolute genius. Thank you for what you're doing.

[@sivaburra7428]

Hi Luis, You are also a great teacher, I love the way you teach machine learning . By the way I bought your book and I am loving it.

[@rubenvela44]

@ford fairlane 1

[@twentytwentyoneishvkmemory7430]

@ford fairlane It's clearly window.

[@ben_jammin242]

I wished to say something similar. 3b1b keeps saying "this may be weird" or "you might not think of it this way" but for someone fascinated by the maths of each of the "core" concepts, with little background in formal maths beyond high school, its really refreshing having these different ends of two seemingly different pieces of string, come together. I couldn't imagine where or when something like this presentation might be taught.. I like when he says "using this recipe" etc, it owes to the "creative" nature underlying how maths becomes proven in the first place. From my own perspective though, prime numbers and pi seem to be very dissimilar concepts, only similar in the air of mystery they carry. Still watching, but to see them come together seems to give both "observations" credibility/usefulness in the universe as a whole. Thanks for the video! Holding reservations until the end. Edit: clarifying a statement.

[@korwi7373]

You sir, too.

[@TheAgamemnon911]

I have never seen math so beautifully explained.

[@lukapopovic5802]

Agamemnon I have watched 3blue1brown's serie about calculus. The best math explanations I have ever found on KZbin and it's not even close.

[@milley9126]

ME to but I'm not ther yat because we haven't learn it

[@Number_055]

2:27 seeing that and immediately recognising the cosine wave was really mind blowing.

[@stv3qbhxjnmmqbw835]

Mc Lauren series of cosine 😏 Cool....! I noticed after your comment.

[@gallium-gonzollium]

hmm, cosine use pi so we use pi to find pi-

[@spawn142001]

This visual explanation for the formula for pi is mind blowingly genuis. This is one of your best videos to date and using geometry to explain seemingly complex formulas creates the intuition in understanding that these formulas on their own lack. For someone very interested in math who doesnt have alot of prerequisite knowledge on something like formulas for pi. This video takes an individual from going oh thats equals pi because other people say so to it equals pi because were essentially doing something like taking the area of a circle to infinity sequentially by counting the latice points of intersections, noticing a pattern correlating the radius of the circle and its complex factors. Creating a formula for this for any given radius so that we dont literally have to count every point by hand using some number/algebraic theory thrown in to create the formula. And summing this formula to infinity for every integer value of radius. Then divide out the radius to give an infinite sum for our unknown constant PI. And of course because of how complex number factors and how factoring numbers in general work all of this correlates to prime numbers, because we have to factor our radi to get our intersection points. Quite possibly ive said some of that wrong or have used the wrong words, but this is mind blowingly genuis, and of course pi would be very elegantly derived this way using complex numbers as complex numbers mathematically are a perfect system for describing two dimensional rotations, and rotations around an origin have everything to do with a circle

[@stellar.s.stellar]

This is a beautiful method. I was honestly left speechless when you organized the chi function into a table that brought about the 1/n factor of R^2 for all n. It just popped out instantly. And it's incredible how all of these concepts are related.

[@BigDBrian]

doctor: you have 31 minutes to live me: *loads up this video*

[@turolretar]

more like you have 5*Tau minutes to live

[@thepowerful7593]

@@turolretar lol

[@Bryan-qd4fk]

But then it'd go by too fast

[@thelanavishnuorchestra]

I'm not very good at maths, but always wished I were. I've spent a lot of time in my life struggling to better my capabilities and have found programming and graphics for mathematical exploration has been my best road to better understanding. Your videos are so perfect in this regard and you explain and illustrate them so well. Thank you for another great video.

[@jacksonstarky8288]

Thought at 26:30: I'm still patiently waiting for a video focusing on prime numbers and how they connect this video to your Riemann hypothesis video. I would also love to see a video on L-functions and one on the Gamma function and the Euler-Mascheroni constant. It's criminal that this video has only half as many views as your Riemann video, because this one is at least as beautiful.

[@quarksify]

3b1b is my favorite anime

[@FacultyofKhan]

>Bored out of my mind >Phone notification bell rings >O boy 3blue1brown is back! >At this moment, I am euphoric, not because of some generic math lecture, but because, I am enlightened by 3b1b Great video, as usual!

[@FacultyofKhan]

Forgive me senpai, for I have transgressed

[@dorpeled4768]

Same here XD

[@staudinga]

>meme arrows Are you serious?

[@vhavahgmh]

ｓｅｎｐｉｅ～ , WE NEED MORE *MEME ARROWS* AND MAKE MEXICO PAY FOR THEM

[@CvnDqnrU]

You guys are cancer

[@PhoenixClank]

I'm not really watching this video to find out how to calculate Pi, but I did ask myself "how on earth do I figure out which divisors a number has", and when you organized them into columns it blew my mind! It just makes so much sense all of a sudden! It seems so trivial!

[@fernandoduarte950]

Wow, this was completely outstanding great! I have to give you my congrats to bring us maths in such an interesting, intuitive, illustrative manner. And for making us to think about them more and more, and not only in the same way they are commonly presented. Salutations of a Mexican follower.

[@tomconti-leslie7089]

Wow. Like seriously, wow. I've been watching your videos for a while and I've seen them getting longer and longer, but as I saw someone else say in the comments, this is the "shortest 30 minute video I've seen in a while"! Please keep going with these sorts of videos, they are totally worth it. The visual aspect is absolutely perfect, and that beautiful coming together of everything you'd spent the last 20 minutes setting up at the end of the video was just amazing - I started grinning around 26:50 when I saw it all happening ;) I should probably be studying for my exams instead of watching this but this is so much more rewarding... Thank you so much!

[@ExaltedDuck]

Half an hour of this and I'm left feeling like the final summation is just kind of... Irrational...

[@astr0nox]

Transcendental in fact

[@smeggerknee2448]

I thought it was a piece of cake,or......possibly sum pi😸

[@SirNobleIZH]

I love how this guys is talking all professionally about complicated math and then on the outro screen it says "clicky stuffs"

[@nanenikodem]

For the first time in my long life, math makes sense. I am a musician, writing a dissertation about the circle of fifths, working with the Coltrane circle of 60 chromatic notes (5 octaves). The perfect fifths and fourths of two opposing tonics are prime numbers here, unless this position is part of the circle of fifths, and with the exception of nr 49, which should be prime. I feel that this lecture is actually proving that the circle of fifths is the same as the unit circle in the plain of complex numbers and that perfect harmony is the reason for primes, and also explains the twin prime conjecture.

[@kamilazdybal]

Wow, this is a truly remarkable video! There were two things that I couldn't stop admiring while I was watching it: 1. You've convinced me one more time that behind every formula and every concept that looks like a jibber-jabber at first is a logical, elegant, and sometimes even simple way of understanding it, it's just a matter of how you will present it to someone else (or to yourself). The effort of finding the right way to present is always worth it. 2. It's amazing to think how many dependencies between numbers there are, that we don't realize in "everyday-life-maths", or even ones that we haven't discovered yet. It's amazing how they are flawlessly connected. Thank you for sharing your way of understanding maths and thank you for all the effort to make this, and all the other videos. Keep up the great work! :)

[@dappermink]

You are the only guy out of there who makes me think "wow this is the best video I've ever seen of my whole life" every time I'm done with watching any of your videos. Really, you never cease to amaze me at a point that it became unbelievable. I hope so hard I'll never loose that feeling I have to discover more maths, thank you so much for all you are doing, keep it up you are amazing!!

[@quinn7894]

I just thought of terminologies for numbers modulo 4! 0 mod 4 = Spring 1 mod 4 = Summer 2 mod 4 = Autumn/Fall 3 mod 4 = Winter

[@josephgroves3176]

27:00 holy sheet that was close to smth I worked on in sixth form: to find the fraction below N divisible by a certain number of primes. I had no idea pi or complex nums were just around the corner --I was just colouring in pie-charts and measuring...

[@xyz.ijk.]

Absolutely love your work. So appreciative of the time you put into these. I'm a lawyer and, were I as talented as Leibniz, your videos would persuade me to become a mathematician. In the meantime, I'm just happy learning this all at the feet of an obvious master.

[@goofyrice]

My first 3Blue1Brown video... "hmm interesting." '... named after Martin Sheen.' *pause, exit full screen, like, subscribe, full screen, play*

[@Irondragon1945]

Wait i didnt get that one

[@blagos8150]

You must be approximately 30 or above years to enter the joke

[@bradypostma5167]

@@Irondragon1945 - What I got out of it is: of couse it's named after Gauss. "Martin Sheen" is just absurd. But I guess Gauss and Martin Sheen kinda look alike, as 3B1B once noted on Twitter. I guess it's also a bit of an inside joke about that. twitter.com/3blue1brown/status/804168776761503744?lang=en

[@RicardoCalderon821]

HAhaha suddenly went: “did he just...?” This guy makes you love math

[@swarnadeepsaha7686]

You could have done it without exiting fullscreen. All you had to do is click the logo on the bottom right corner. Although considering this comment is 2 years old, it makes me wonder if that was a feature then.

[@moshemeirkatz944]

OMG. This is the second maths video ever that has made me laugh, when you understand something, and see the pure symphony that is the delicate and beautiful interaction of the description of the world, it is a soul experience so profound, the physical expression can only be laughter. Well at least for me. Thank you so much.

[@alexandregermain8011]

That kind of randomly deterministic stuf and the very good visual & litteral explanations you give just makes me want to learn maths again.

[@aSeaofTroubles]

Thank you so much! Like a lot of viewers, I encountered various chi-like functions while self-learning on Wikipedia and felt overwhelmed. Finally, thanks to your great video, I have a clear, useful mental model for why these types of functions exist and how they are actually utilised. I really love the idea of a multiplicative function. Thank you for a gentle introduction into this special class of functions. I feel like I have been searching for them for a long time! I love how logarithms convert multiplication to addition, and now there seems to be more pieces to the problem converting puzzle :) It sort of feels like a coincidence that the powers we sum up to for each factor just happen to enumerate all the divisors we have (24:51)... but at the same time this reveals something deep that is happening when we "count" how many options we have. I'll have to give this more thought, although it definitely makes sense from an after-the-fact book-keeping standpoint. Lastly, I was blown away by the very simple re-arrangement of divisors at 26:36. I feel like I should have known this general fact about the sums of all divisors of all numbers up to N much earlier! It makes sense now how we can account for ALL divisors in a meaningful way, opening up new ways to solve problems. What a beautiful way to expand everything out into a series :)

[@AnmolSahu]

There should be a 'love' button. 'Like' is not enough!

[@Perririri]

There actually _is:_ [insert heart sign]

[@alphax1049]

Janeen Phayne ❤️ here’s one. You’re welcome

[@lyrimetacurl0]

these days only 3B1B can "love" the comment...

[@engineeringconcepts138]

This is the best math on KZbin for sure.

[@henrikljungstrand2036]

Wow, this is a really cool visualisation of the connection between 4n+k numbers (and thus 4n+k primes because of multiplicativity), Gaussian integers and pi. I'm sure there must be a similar connection between the 6n+k numbers, Eisenstein integers and pi. Remember that Gaussian integers is just what you get when you adjoin a primitive 4th root of 1 to the ordinary integers, closing it up under addition and multiplication. So it is the integral domain resulting from factoring the monic polynomial x²+1 into grade 1 polynomials. Similarly the Eisenstein integers is what you get when you adjoin a primitive 6th root (or 3rd root, since -1 is already present) of 1 to the ordinary integers, closing it up properly. So it is the integral domain resulting from factoring the monic polynomial x²-x+1 (or x²+x+1) into grade 1 polynomials. Both these integral domains are grade 2 extensions of the integers (thus 2 dimensional geometrically), they are unique factorization domains (so the counting method works without problems of choice), and they are "locally Elliptic" in the sense of having a finite number of units i.e. factors of 1 (unlike the golden integral domain a+b*phi, resulting from the monic polynomial x²-x-1, which is unique factorization, but "locally Hyperbolic" in the sense of having infinitely many units and thus infinitely many numbers of almost any particular norm, in fact for every norm except 0, we have either no numbers or infinitely many numbers of that norm). Thus we should be able to approximate pi by building circles of ever growing norms in these two integral domains. It is an interesting question whether we can find monic polynomials of say degree 3, that when split into grade 1 polynomials (or perhaps only factored into one polynomial of grade 1 and one of grade 2) gives us a unique factorization domain with a finite number of units, corresponding to a 3 dimensional discrete geometry with a cubic norm behaving similarly to the ordinary quadratic norms? Another interesting question is whether we can make sense of any particularly well behaved finite section of an integrals domain with infinitely many units, for the purpose of counting area. Like e.g. in the golden integral domain (which is 2 dimensional), phi = (1+5^½)/2 is of norm -1 since (5^½)² = 5, the conjugate of a+b*5^½ is a-b*5^½ and thus |1/2+5^½/2| = (1/2+5^½/2)(1/2-5^½/2) = (1+5^½)(1-5^½)/4 = (1²-5)/4 = - 4/4 = -1. So in particular phi² is of norm 1, thus counting the area of the infinite hyperbola of norm n, for n positive, down to the "hyperbola" of norm 0, we can take all numbers of the form x*(phi²)^k as "equivalent to" x, since they all have the same norm (say n, if x has norm n), thus the hyperbola is periodic when "rotated" by a multiplication of phi², which gives us a similar factor to the factor 4 for Gaussian integers or the factor 6 for Eisenstein integers (or the factor 2 for ordinary integers). Which means we can actually get a sensible finite sum by counting classes of numbers equivalent under multiplication by phi². And doing this, we get a new way of approximating some constant for the hyperbola (is it pi or something else?), provided the summation of phi² equivalence classes of numbers (golden integers) of norm n, over all positive integer n, divided by the squared radius of the latest hyperbola, actually converges in the usual Archimedean/Euclidean norm i.e. the real number norm. Provided this works, we may then tackle the problem of 3d "volumes"/"areas" using not the ordinary quadratic norm, but instead a cubic norm given by a grade 3 monic polynomial, e.g. x³+x+1, regardless of whether we get infinitely many units (locally Hyperbolic) or finitely many units (locally Elliptic), as long as we have unique factorization. Locally Elliptic here is roughly the same as Euclidean metric, while locally Hyperbolic is roughly the same as Minkowsky metric, although this is only valid for quadratic norms, plus we don't necessarily get any more numbers of norm 0 even when using a "Minkowsky metric" like norm. One of the beauties of working over rational numbers and integers rather than real numbers, is that there are extensions of dimension larger than 2 that gives rise to fields and integral domains, no pesky zero divisors! Whether this gives us non-convergence in the real norm using infinite sums, and possible convergence in certain p-adic norms, i don't know. It is sure worth a try, to find out what would be a good cubic norm and what would be a good degree 3 conjugation based on the monic polynomial x³+x+1, provided we split it in 3 grade 1 polynomials. It is obvious that unless it immediately splits into three grades of degree 1 when factoring one out, we will also have three conjugates of degree 2, swapping just two of the roots. Not sure if this always happens over the ordinary integers (the formula for solving the general cubic should give us a hint!), so we would have to use Gaussian or Eisenstein integers as a base integral domain instead, because i think that having a Galois group of order 6 instead of order 3 might be problematic, especially since Sym(3) is non-commutative unlike Sym(2) and Alt(3). I'm sure that we should be able to construct a monic polynomial that gives us a specific version of the general cubic solution, where the term inside the square root sign in the sum within the cube root sign is made into a rational number (or better integer), thus making that grade 2 conjugation invalid, WITHOUT making the whole polynomial reduce into a product of grade 1 polynomials over the rational numbers (or integers), thus giving us the desired property of our monic polynomial's splitting field being of dimension 3 rather than dimension 2. But i am currently too busy to calculate the details so i'll have to return to this later on.

[@kyanilcauli9002]

This is precisely the properties of the ring of Gaussian Integers - being an unique factorization domain, where all factorizations are unique upto associates (units in the ring, here it's only these four -> 1, -1, i and -i) - and Grant succesfully explains everything perfectly - so well that not even the slightest requirement of Ring Theory is needed for the viewer to understand. You're a genius, Grant.

[@cricket6363]

Someone might have already said this, but there's a numberphile video about the theorem on primes of the form 4k+1 being able to be expressed as the sum of 2 squares, it's called something like "1 sentence prime proof". Also amazing video :)

[@ashes2ashes3333]

This is actually such an amazing video - I've been learning about some of this stuff at uni and you've explained it incredibly well. That's a really difficult thing to do!

[@someone-en1gg]

Bhaiya me 12th me hu mujhe aadha b smjh ni aaya

[@zhang_han]

This really should be the way math is taught. I'm just imagining as someone who took math in university what this would have taken to get into the mind of the median student in a math major if the only teaching tool was a chalkboard, and it would definitely take much longer than 30 min.

[@dananskidolf]

There's something really wonderful about having something explained so well, so engagingly and at just the right pace that you can go "ah I see where you're going with this" in the exact moment before it is revealed.

[@Jodabomb24]

26:37 was the moment when it clicked. The biggest smile came across my face and I remembered why it is that I watch your videos.

[@ZacWut]

"The Gaussian integers, named after Martin Sheen" The memes are too strong

[@jamesfrake]

Can you explain for the uninitiated

[@humancentibeat6955]

@@jamesfrake Gauß never died and has renamed himself . Jk Martin Sheen is a Gauß look-alike, so that's where that comes from

[@ryant3541]

Every time I watch your videos I'm genuinely surprised at how well I follow along. It makes me feel pretty smart for a 15 year old. Keep up the amazing videos man

[@ifukill7538]

Getting the gist of this as I've watched other math, I am really enjoying learning this. I never passed any exams at school, this being taught so simply while being complicated. Thank you. 😷😃

[@MrBebopbob]

Beautiful! Even serious students of mathematics can gain insight from your videos. Your creativity and hard work are much appreciated!

[@lakshaymd]

DAMN. This was just amazing. I am still kind of freaking out over how it all came together in the end. Thank you for reminding me why I love math.

[@spetsnatzlegion3366]

Random maths thing: exists Pi, phi and e: yo wassup I’m just gonna slide on in here don’t mind me

[@lfelipefrcoelho]

LOL So accurate kkkkkkkk I have this exactly pov about this 3 numbers

[@nathanielconrad4188]

Great job on this video! Your clear explanations allowed me to follow along the whole time without a pause 🙂

[@aidenstrasser3268]

@8:16 the pun is real "This might seem needlessly *complex*..."

[@EntropicalNature]

Extremely well done sir! I'm so in love with these video's! The way you present advanced maths is refreshing and the animations are supreme! Keep up the excellent work!

[@WhattheHectogon]

Gosh it startles me how long certain amazing videos have been out, as I've watched it so many times and it feels new every time...but welp it's been over a year and a half D: i remember seeing this exact video pop up on my feed too...crazy how fast time goes. Still an amazing video as always

[@nicholasheilig3694]

Probably your best video ever for coming up with such an interesting proof on this Leibniz summation.

[@jamesleng8210]

Great video! Really helping me understand the geometric aspects of algebraic number theory. Thanks for the fascinating content.

[@jonashammerich3552]

Your explaining style is absolutely amazing! I´m in 9th grade and understood nearly everything. Great job!

[@dezenterrier]

I'm only 18, so I barely even started my journey through mathematic. I do my best to understand everything, but things like complex numbers, higher dimentions or analitic stuff still seems odd and not clear for me. I've no idea how you are doing it, but (at least I think so) i understand almost everything you are talking about (even though english isn't my primary language, what you've propably noticed :p), everything is very logical and clear, I really appreciate your job!

[@cringy7-year-old5]

i just turned 16 and i feel the same

[@joshuatinyforest1204]

Math, as the general set of other subsets of math, is my favorite subject. Thank you for what you do. Even though I do not understand the videos at the PreCalc level of math & above, they are still enjoyable for me & to try to understand.

[@haavmonkey]

For anyone wondering why we consider factorization to be unique, this is because actually unique factorization is defined up to multiplication by units. And units are the elements of the ring that have multiplicative inverses, in the case of the integers we have 1 and -1 being the only units, where as in the Gaussian integers 1 , -1, i, and -i are the units. Units generally have very different properties in rings, this is also why we also don't consider 1 or -1 to be prime in the integers.

[@melody_florum]

I love how the broken shape tween in adobe flash has become an almost signature look of your videos. Keep it up!

[@NomadicVoxel]

I know this is old but it's not actually flash, he wrote his own video engine in Python specifically for animating formulas and graphs. It's opensource and called Manim.

[@NatanielRiver]

My mind was blown as I saw the rearrangement of the Chi functions leading to the infinite sum! Really informative and interesting video :)

[@rutujtatu6666]

Never seen an ingenious and prudent person like you 😀The way you accentuate and articulate is just impeccable and emaculate, love the way you teach ❤️

[@martingleich2748]

These videos are just great, they tell and explain things I would never look up by myself. And it's explained really well, I'm sitting there grinning like an idiot for 30 minutes.

[@DheerajBhaskar]

I feel satisfied watching this video. It's hardwork but it pays of at each and every step (you learn to simplify something complex and you get a dopamine hit). This like the best kind of videos. Way to go narrator :-D👍

[@lebgdu17pktudorpa23]

This is the single most crazy math video I have ever seen in my life. Your videos are of a rare quality. To say the least. Wow. I'm just... just literraly speechless. You are, like maths, incredible. Huge respects.

[@alchemyphoenix2374]

You are the best teacher I have ever seen. I don't even care about math that much and I just get sucked into every one of these.

[@ultimatedeatrix9149]

Stages of watching a 3B1B video: First quarter: hehe interesting In the middle: wtf is happening why am i here im too dumb T_T Before the final reveal: yassss, i know exactly how its gonna connect together :D Love how he teaches in such a way that we know the answer before he brings in the final equation, even for a topic as complex as this. Makes us to appreciate ourselves even though he did all the work of explaining it. Blessed to be in an era where top quality content like this is free.

[@lukejohnston2390]

25:00 probably the most ridiculously excessive and beautiful way to find the factors of a integer.

[@FTFP1300]

Having seen many 3Blue1Brown videos multiple times, I was surprised that I hadn't seen this one yet (that, or I did see it and simply forgot the whole thing). Also, it's very cool to see that even the older videos still hold up incredibly well, which gives me hope on the impact these videos will have in the years to come. This video also reminded me of a problem I had in undergraduate ring theory about the probability distribution of the gaussian integers (I don't completely remember, but it was some subset of gaussian integers having infinitely many members, but ultimately being a very small number with some limit being equal to zero), and I used a similar method of the circles approach, but instead with squares, so it was very fun to remember my thought process and struggles with solving it. These videos are so cool, and the sense of wonder and ingenuity is helpful for me as I go through new challenges in graduate school.

[@Mrhennayo]

One of the best channels for discovering new perspectives related to maths : big thumb up and thanks a lot for theses contents

[@MrAwesomesize]

My god, that was amazing! Your videos really make me appreciate mathematics in a completely different way

[@CharlesPanigeo]

21:25 some author's reserve multiplicative functions to mean functions where GCD(a,b)=1 implies that f(ab)=f(a)*f(b). That is, f(ab) = f(a)*f(b) if a and b are relatively prime or coprime.

[@alexismiller2349]

I guess it should've been called a strictly multiplicative function but that would be splitting hairs

[@CharlesPanigeo]

@@alexismiller2349 yeah true true. Its not really necessary here to make the distinction so I understand why he left it out.

[@hybmnzz2658]

In number theory that is standard.

[@andrewatlarge2368]

Nice video! Probably the most elementary way to see that an odd prime can be written as a sum of two integer squares follows from Euclid's classification of Pythagorean triples.

[@nintorws]

A beautiful proof for a beautiful result. I enjoyed every minute of this video, and now I know a very simple infinite sum to express pi. Wonderful!

[@pedrohenriquecontente332]

"I know this looks like it is getting needlessly complex" HAHAHA

[@official-obama]

When?

[@official-obama]

@@XeNoX_off thank you

[@ejetzer]

2:10 that's also a link to some quantum mechanics and thermodynamics (how many modes are possible in momentum space)

[@Ai-Tube]

6

[@kgallchobhair]

You have a very soothing voice, so occasionally I make the mistake of thinking your videos will be good to fall asleep to. Every time, by a few minutes in I'm just like [I don't need sleep, I NEED ANSWERS!] 10/10 love it, keep up the great work

[@FrazerKirkman]

That was done of the most beautiful math I've seen, and so well presented. Thank you so much.

[@ahmidii]

"Pause and Ponder..." Thanks 3Blue1Brown, you are incredible!

[@xelaxander]

Even the ads are good...how does the world deserve this!?

[@lukapopovic5802]

Alexander Nenninger Well, this type of ads can help the people who watch this video. If I understood correctly, you will get paid to work for that company/organisation, or whatever it is.

[@totheknee]

It's a counter to Trump. We need a reason to live.

[@briancannard7335]

This thing has changed my life. Thank you, @3blue1brown!

[@armantookmanian1938]

I graduated college as a math major. I spent four years taking many advanced course, but I never had an instructor or professor teach me the beauty of math, or the way the various disciplines can intersect and overlap, reinforcing each other, to solve and confirm solutions to complex (pun intended) problems. Sadly, I fear this is true for many who attend higher education. Thank you for rekindling the embers of my love and enjoyment for math.

[@CacchiusMan]

This has been the best and most awesome way to introduce the Gaussian integers. Also it's nice to see the role of the invertible elements {+1,-1,+i,-i} on the complex unit circle for the ring of Gaussian Integers; Now I understand better the phrase "unique factorization up to invertible elements" during the Algebra course, which i did not understand at the time. Thank you very much! :D

[@zuloo37]

If you're up for a challenge, try finding another formula for π using lattice counting of the Eisenstein integers (so a equilateral triangular lattice, by adding a cube root of unity for example). Here's a hint: whether an integer prime is an Eisenstein prime depends on whether it is congruent to 1 or 2 mod 3, and 3 is the only ramified prime (i.e. it behaves kind of like 2 does in the Gaussian integers)

[@CacchiusMan]

Hi! I did not forget about your suggestion, but i was busy. I did not know about Eisenstein numbers, so i made a tiny research about that. Unfortunately i was not able to find a closed formula for pi with these numbers, but i feel like It has to do with the Mengoli series, or with a series with pi/6 as a limit.... what is the answer?

[@zuloo37]

A prime in the integers is an Eisenstein prime if and only of it is congruent to 1 mod 3. In the same way that you have the norm of a Gaussian integer a+bi as N(a+bi) = a^2 + b^2, the norm of an Eisenstsin integer a+bω (where ω = -1/2 + i sqrt(3)/2) is N(a+bω) = a^2 + b^2 - ab. So to count the Eisenstein numbers on a circle of radius sqrt(n), you'd be counting the number of pairs (a, b) such that a^2 + b^2 - ab = n. In the same way that this video did things, you can find a nice formula for this count based on the prime factorization of n. Just as powers of 2 didn't affect it for Gaussian integers, powers of 3 don't affect it for Eisenstein integers. Then for primes congruent to 1 mod 3, you can just have a factor of the power on that prime plus 1 just as in the video, and for primes congruent to 2 mod 3 the whole thing will be 0 if they have an odd power and unchanged if they have an even power. From this you can define the corresponding character (yeah, they're actually called that, despite the video making a very nice pun on it) χ where χ(n) = 0 if n is 0 mod 3, χ(n) = 1 if n is 1 mod 3, and χ(n) = -1 if n is 2 mod 3. This defines a multiplicative function. Then for each prime factor you can add up the values χ(1) + χ(p) + ... + χ(p^k) as in the video and get the same counting function. Note that there are 6 units in the Eisenstein integers (points on the unit hexagon) so the whole thing wil be multiplied by 6 for the total count for each circle. Again, just as in the video, the products of all of these sums for each prime factor ends up just being a sum over all divisors d of n of χ(d), due to this character being multiplicative. Then there is one last slight difference: for the Gaussian integers, the lattie points actually gave unit squares which contributed 1 to the area, but with the Eisenstein integers, if you want to tile the plane, you have to use hexagons centered at each lattice point, which no longer have unit area. Find the area of this hexagon and multiply it by the same kind of sum as in the video (but with a factor of 6) and you'll find that as R approaches infinity, you get the following formula for π, alternating and skipping multiples of 3: π = 3 sqrt(3) (1 - 1/2 + 1/4 - 1/5 + 1/7 - 1/8 + 1/10 - 1/11 + ...) (This works out because L(χ, 1) = π/(3 sqrt(3)) is the infinite sum in parentheses here)

[@CacchiusMan]

Awesome! :) i was able to see by hand that you need the six-roots of unity, to derive the norm by multipling (a+bw)(a+bw^2) (since w^2 is the coniugate of w) and noticed that perfect squares all cross 6 points, that powers of three "change nothing", and some examples of prime numbers that are =1mod3 that cross 12 points but i was unable to calculate the characters of the divisors. Also i don't know about characters yet, seems like advanced algebra/Group theory (still an undergrad here!) Thank you very much for the answer! :D

[@zuloo37]

Oh, sorry, I wrote the first sentence there backwards. I meant to say that a positive integer prime splits (into a product of two distinct Eisenstein primes) if and only if it's congruent to 1 mod 3. For example, 7 = (3+ω)(2-ω). (This is a product of complex conjugates.)

[@ALNYTheGreatScientist]

Soooooo coooooooooooooool. This is my new answer for people who ask me why I like math so much.

[@ANJA-mj1to]

After a vaste range of latice points applications, You give great illustrated queastion for applied mathematicas - "... Where we are (between two mathematical well known branches) ? " what is also basic theme of the video. Brief but invaluable way of introduction and meaning that describe letter "chi" such a main subject of importance, throughout in final chapter discussion of Gaussian prime and forms with all examples that can be relay on!

[@canyoupoop]

You have no idea how much this is helpful to me. This might just change my life and I am not even joking. Grant, you truly are spectacular. ❤️

[@Khari99]

Okay maybe I might be going crazy but is it a coincidence that the Chi function lines up with sin and cos functions? It just goes back and forth between -1, 0 and 1 which is the range of cos and sin and they are also the values we pay attention to the most? 1 and -1 are the inflection points y value inflection points and if im not mistaken, the 0's can be written as some expression of pi.

[@PeterJavi]

Khari99 chi function

[@Khari99]

PJ Vis edited thank you

[@PeterJavi]

Yw

[@AuroraNora3]

I'm also wondering about this

[@themodernshoe2466]

That's the first thing I thought of too. It'd be cool to see if it really is related or just a coincidence.

[@Enliden]

I am studying Euclidean Rings, Unique Factorization Domains and principial ideal domains at University using gaussian integers as a typical example, naturally. Yet I've never been introduced to this use of abstract algebra... I'm fascinated, and find it stunning. It's hard to listen without trying to interpret this in the language of abstract algebra.

[@Raffael-Tausend]

i'm impressed how primes, gaussian integers, leibnitz pi formula (and arctan(1)) are connected!

[@hexeddecimals]

I've always been confused by pi. What does 3.14159... have to do with circles? I understood the reasons behind the intresting properties of other cool numbers like phi and e, but pi always eluded me. Watching this video and changing the narrative slightly, everything clicked. I viewed it instead as you know the formula for the area of a circle is the radius squared times some constant, and by using complex numbers and number theory, you can find the value of this constant! I view the definition of pi as the infinite alternating sum of the odd reciprocals, not the ratio between a circle's circumference and diameter, as that is just an emergent property. This change in perspective made me finally appreciate pi. Thank you 3Blue1Brown!