Pi hiding in prime regularities

  • Published on May 19, 2017
  • A story of pi, prime numbers, and complex numbers, and how number theory braids them together.
    Brought to you by you: 3b1b.co/leibniz-thanks
    Home page: www.3blue1brown.com/
    And by Remix: www.remix.com/
    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:
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Comments • 1 454

  • Sarthak Girdhar
    Sarthak Girdhar Hour ago

    Wait. Is this content for free? Are you kidding me?

  • Galileosays
    Galileosays 10 hours ago

    Incredible. How many cubic lattice points in a sphere?

  • 풋볼리딩
    풋볼리딩 Day ago

    한국어 번역해주신 분 감사합니다.

  • *Zls one*
    *Zls one* 3 days ago

    26:35 can you see binery?

  • markovic69
    markovic69 3 days ago

    thank you mate, just thank you .

  • Vrajesh Vyas
    Vrajesh Vyas 3 days ago

    This was discovered by Madhava
    As if that was not enough he coded it in a verse with following strictly the meter. Great Combo of Right and Left brain

  • Shahar S.
    Shahar S. 3 days ago

    Your proof is fabulous, BUT…
    You ignore a very important part:
    You say that the number of lattice points on √N circle is 4(x(1)+x(p1)+…)(x(1)+x(p2)+…)∙∙∙
    How can you prove that all these points are distinct ? If you don’t prove it, the number of points may be smaller.

  • 왕고구마
    왕고구마 4 days ago

    At last 3B1B is beginning to touch the deepest secret of the universe.

  • Sheldon Wu
    Sheldon Wu 4 days ago

    Oh my god. I watched from the beginning to the end.

  • The Masked Man
    The Masked Man 4 days ago

    What the fuck is χ?

  • Ni Lin
    Ni Lin 5 days ago

    I don't know if you have the same experience: Whenever a math teacher explains to me a math theory, at some point I'll encounter something that the teacher deems trivia. But it's impossible for me to keep going without fully understanding this 'trivial' matter. Is math teaching really just a teacher with arms teaching a roomful of students who may or may not have arms how to arm wrestle?

  • Quwertyn
    Quwertyn 5 days ago

    4:35 can someone tell me what he's saying?

  • Naté Slough
    Naté Slough 6 days ago

    Χ(n) = sin(n * π / 2)

    DEWILL 6 days ago +1

    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.

  • physics crush
    physics crush 6 days ago

    Is lattice point can be of any dimension?

  • débora tayná
    débora tayná 8 days ago +1

    Please, someone who speak english and portuguese fluent translate the subtitles of this video for portuguese. Pleasee

    • Diego FCM
      Diego FCM 4 days ago

      débora tayná
      Coloca em espanhol se vc tem dificuldade com ingles

  • ilprediletto
    ilprediletto 9 days ago


  • Thumbomb
    Thumbomb 9 days ago

    1:46 "Who am I really?"

  • Mats Leandersson
    Mats Leandersson 10 days ago

    The best explanation I’ve ever received (to anything I’ve ever wanted to know). 👏🏻👏🏻👏🏻

  • Harambe McHarambeface
    Harambe McHarambeface 11 days ago

    Ok, that is super cool

  • Sylwester Bogusiak
    Sylwester Bogusiak 12 days ago

    Primes are Fibonacci dependants.... in a halfway ;)

    Look this...

    New formula for prime numbers? ... something about density of primes in 2D space...

    SP= (((P1+P2+...+Pn)) / QP)/GF
    SF < Pn

  • Sylwester Bogusiak
    Sylwester Bogusiak 12 days ago

    Primes are Fibonacci dependants.... in a halfway ;)

    Look this...

    New formula for prime numbers? ... something about density of primes in 2D space...

    SP= (((P1+P2+...+Pn)) / QP)/GF
    SF < Pn

  • Aryan Mishra
    Aryan Mishra 13 days ago +1

    My friend asked me that what's this 3b1b channel and i told him that its the π lovers channel

  • Jinil Kaiser
    Jinil Kaiser 14 days ago


  • LOL Science
    LOL Science 14 days ago +1

    Wow I actually understood it from the first time...

    Give me a Nobel prize!

  • Thái Sơn Lương
    Thái Sơn Lương 15 days ago

    Who came here for studying English? 😉😉😉

  • 로직 LogiQ
    로직 LogiQ 16 days ago

    I’m korean middle school student. I’m really surprised by your unique and amazing idea. I wonder what your job is. I really enjoy your videos. Thank you for making these videos.

  • Emmanuel R.
    Emmanuel R. 18 days ago

    HEy 1:10 he made the illusion of black dots of intersections, black dots only appear when you see them.

  • R. Mo
    R. Mo 19 days ago

    I'm a "spreadsheetdologist". I love writing functions and using large lists of numbers as inputs to see what happens. I guess I'm some sortof number tinkerer. I'm sure that I have discovered a few things in number theory using my unconventional ways of messing about with spreadsheets, but at 53 I'm a bit old and hessitant at revealing how I probably re-discovered things others already know and can describe in a mathematical way (which I am trying to get to grips with using my limited knowledge of the subject). I'm using spreadsheets as an anvil to hammer out and see what comes out. hahahaha. Love this video and others made by a talented guy and his team. Thanks and keep it up.

  • kuba feder
    kuba feder 21 day ago

    truly amazing, thank you for explaining beaty of math to ordinary people like me, u are doing it great

  • Hakan Çakıcı
    Hakan Çakıcı 22 days ago +1

    Yep, it's big brain time.

  • Attornado Andrea
    Attornado Andrea 24 days ago

    1 is not prime. 😭

  • Aayush Patel
    Aayush Patel 24 days ago

    my brain hurts😓😓

  • Skittus
    Skittus 25 days ago

    2:56 3,6 roentgen. Not great, not terrible.

  • Ashley Stone
    Ashley Stone 29 days ago

    :O damn we should be taught this form of math from a very young age. Imagine having an intuitive grasp of 2-dimensional numbers the same way one has of a number line.

  • nikki.creeps around.on.holidays

    Soooo.... Who also thinks quantum physics is easier?

  • wwwjeafcom
    wwwjeafcom Month ago

    So, Pi, after all, is cousins with Prime. Both are a Circle, a Big Ban, based on 12. The Squaring of the Circle is how Pi came about. It has numbers based on 60, which is an ancient Sumerian base number. Easy to Square the Circle. Which means, the Ancients, before the Event 12 thousand years ago knew about the Squaring of the Circle as well as our best mathematicians. They should investigate the ancient structures and documents in order to acquire knowledge from the ancients. I believe those ancient stones were left for us a Time capsule dealing with an event that will occur in their future. They knew they would be wiped out, so they must have had a way to shelter from the “storm,” that would engulf the Earth 12,000 years ago.
    We need matemática to understand the cycle of Betelgeuse, the next event. That star with a cross piercing its fury. An orange supergiant about to collapse and we are in one of its dipoles, too close for comfort.
    We need to understand the cicles that rule the universe, perfectly tuned to the 12 notes of music.
    Are the 12 notes cousins of Pi and Prime.
    It all makes perfect sense.
    Thanks for your math

  • Eduardo Alvear
    Eduardo Alvear Month ago

    Pi is Prime’s cousin. They have the same father, a circle. Which comes out of 12 and 16, and goes on. Ancient knowledge it seems to me you have tapped into. I see the force is strong with you. Thank you! I can finally say the Pi is a prime number intimately related since the Big Bang, in a circle, a two dimensional expression of a three and fourth dimensional expression.
    Thanks, you are very inspiring. Thanks for sharing your brightness. :)

  • gideon yuval
    gideon yuval Month ago

    3B1B - your math videos are so good, I'm wondering if you have a twin brother (or sister) who covers other topics, such as physics?

  • Feras Hamdan
    Feras Hamdan Month ago

    Super nice video

  • Obscurité
    Obscurité Month ago

    The way he places the shameless self promotion is funny

  • Frazer Kirkman
    Frazer Kirkman Month ago

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

  • Eric
    Eric Month ago

    the greatest video i have ever watched,thanks!

  • slayer 1
    slayer 1 Month ago

    Although this is a very good video excellent editing and thought-provoking I would like to see an application show the actual math say like building a bridge or launching a rocket or my most popular thought would be auto mechanics it has electro-mechanics Hydro mechanics thermal mechanics kinetics a world of different math that I have never really used but would like to know about I've been a lot of mechanics would like to know about things like that also not only mechanics engineers in general

  • 문수혁
    문수혁 Month ago

    How incredebile

  • Smogy 001
    Smogy 001 Month ago

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

  • TheFamousArthur
    TheFamousArthur Month ago

    83 and 89 are sexy primes.

  • TheFamousArthur
    TheFamousArthur Month ago

    89 and 97 are octopus primes.

    TARUN AKASH Month ago

    at 9:09, i figured where it is going.

    INDEED of LIFE'S Month ago

    i am u r big fan sir
    ...thanks for everything 😊😇

  • GreatFlamin Zeus
    GreatFlamin Zeus Month ago

    Caption Error: 9:58-10:01
    You say “[multiply by i & -i]” but the caption says i & -1
    Also the captions can’t handle superscripts or the like so exponents look like coefficients.

  • Santiago Pachiarotti

    Excelent video! Congratulations

  • Imagine Existance
    Imagine Existance Month ago

    My favorite pi formula is
    Or a loop of
    it adds the error of the distance from the nearest whole multiple of pi
    if it is below a multiple it adds the sin getting closer to pi
    If the multiple ahead is even, it goes to the previous odd one and if ahead of a odd one it goes forward

  • Nicholaus Uhl
    Nicholaus Uhl Month ago

    There was an optical illusion at ~7:55 where all the dots, besides the ones you were looking at directly, were hollow.

  • Luke Johnston
    Luke Johnston Month ago

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

  • David Vicente
    David Vicente Month ago

    Thank you for your video!
    Is your video inspired by "Geometry and the Imagination" written by Hilbert and Cohn-Vossen?

  • Leif
    Leif Month ago

    You said 3.is a multiple of 4 at 11:07 in the video? Mistake..not on purpose unlike the Martin Sheen comment I'm assuming...neither is 7 for that matter..

  • LeBGDU17 PkTuDorPa
    LeBGDU17 PkTuDorPa Month ago

    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.

  • David Wilkie
    David Wilkie Month ago

    Whenever information is arranged in a crystal type resonance arrangement, ie has a state of "two-ness" and the probability effect that's noticed dropping pins or nails on the parallel lines grid, it's a situation that shows up as Pi, a specific "calculation" that demonstrates a similar consequence in the above example.
    As an interested Observer rather than a practical Scientist/Mathematician, ..watching presentations like these reveals much about the mechanism of the Universal interference connection, e-Pi-i, that I cannot put in intelligible words. (That's why these videos are so very impressive)

  • Kutluhan Bilgeşah Bayzan


    THE POWERFUL 2 months ago +1

    If you add quaternions...

  • so ia
    so ia 2 months ago

    Hablen español pesados de poronga

  • Jaime A Farah
    Jaime A Farah 2 months ago

    Beautiful video !
    it seems all roads lead to π

  • 드라불리
    드라불리 2 months ago

    This vidios support korean... but still I can understand


  • 서명교
    서명교 2 months ago

    Wait a minute... doesn't your last part needs more explanation? There are R^2 terms on left side so we can't be approximate the sum like that isn't it?

  • Michael Lubin
    Michael Lubin 2 months ago

    That was awesome. Thanks! More algebra than most of your videos, which is probably why I understand it...

  • Alessandro Tagliafierro
    Alessandro Tagliafierro 2 months ago +1

    My words can't possibly express the amount of respect and admiration I have for you after watching this kind of videos

  • zehao zhao
    zehao zhao 2 months ago

    what number theory book are you using?

  • Kelsey Phillip Payne
    Kelsey Phillip Payne 2 months ago

    One of my favorite math vids of all time :)

  • Jasper Kole
    Jasper Kole 2 months ago

    "Factors of 2 just don't make a difference"
    *Mercene primes cry in the corner*