Pi hiding in prime regularities

  • Published on May 19, 2017
  • A story of pi, primes, 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 • 1 583

  • jiwon kang
    jiwon kang 7 hours ago

    Mathematics video is beautiful when it explain algebra in geometric way. I'm so glad to watch your video!

  • MFM Gaming
    MFM Gaming Day ago

    Could this be used to find new primes?

  • Leo179
    Leo179 Day ago

    7:50 lol

  • Dmitry Protasov
    Dmitry Protasov Day ago

    Is it possible to make a video with an ideological explanation of proofs of great theorems, such as Fermat’s Last Theorem or Goldbach’s Ternary Problem?

  • Ben Schulz
    Ben Schulz Day ago

    So this is where the Riemann hypothesis comes into play? It shows the divisibles of whole real numbers, and therefore the Primes distribution along the critical line?

  • King Hamlet
    King Hamlet 4 days ago

    Is relative to agent salts Good Plates

  • Michael McDonald
    Michael McDonald 4 days ago

    haha you factor the gausian integers. I do that all the time but go ahead and show everyone else.

  • Gary Barbour II
    Gary Barbour II 6 days ago

    Didn't they just find a legitimate formula for prime numbers?
    And wouldn't that mean that some function of that formula could be used to go from an *approximate* formula for pi, to an *actual* formula for pi?

    • Gary Barbour II
      Gary Barbour II 4 days ago

      @Thomas Pallister isn't it only actual in the hypothetical sense? Don't have to extend the formula as you require more digits? Meaning every application of the formula is a truncated version of the true answer, albeit as accurate as it needs to be. For all practical purposes it works fine, and it really does not need to be used at all, since you can get the value for pi to as many digits as you want and use it as a constant. But I am curious about a more direct approach, such that other interesting characteristics of the formula could be useful or beautiful for unforseen reasons.

    • Thomas Pallister
      Thomas Pallister 4 days ago

      There have been several formulas, but none are efficiently computable. So I guess you could do that, but the formula for pi is already actual. I don't know what you mean by it being approximate

    SMART BLACKPINK 7 days ago

    Or you can just look at the list of Pythagorean triples instead of making and editing then uploading a half an hour long video

    • Antanis
      Antanis 3 days ago

      He's already made a video for visualizing all of the pythagorean triples too. That's a fun watch as well.

  • Jayraj Karelia
    Jayraj Karelia 7 days ago

    In love with ' Pi '
    Looks so innocent.

  • Atharva #breakthrough

    needlessly complex?

  • Nicholas Natale
    Nicholas Natale 8 days ago

    30 minute video, no ads.
    Honestly, I wouldn’t mind 2 or 3 ads in this video, because I know it helps you. To know that you can spend weeks, maybe even months making a video, and choosing not to put any ads in it, just shows that you are doing this for our entertainment and our entertainment only. Thank you, you just earned another like.

  • JackFlashTech
    JackFlashTech 8 days ago

    Was this the first part of Euler’s proof that e^(i*pi) - 1= 0? Or did this come later? Because that infinite series has a relation to cosine and sine, which is where the 1 comes in, right?

  • Rex Rex
    Rex Rex 8 days ago

    Beautiful....in the most gaussy and sheeniest of ways 😍😉

  • Nomekop 777
    Nomekop 777 8 days ago

    7:50 "They're called the Gaussian integers, named after [picture change] Martin Sean."
    I feel like there's a joke there but I don't get it

  • BrakeTheGame
    BrakeTheGame 10 days ago

    5:25 aaaaaah i see a black spot moving inside the yellow dots

  • Irene Adriani
    Irene Adriani 10 days ago

    Boy you are gold

  • Argo Lake
    Argo Lake 11 days ago +1

    Martin Sheen was the voice for A. Square in the movie of Flatland.

  • G2W
    G2W 11 days ago

    We totally understand what you're saying.

  • Caleborg
    Caleborg 12 days ago

    Why do u portray the student pi guys as always angry? we wouldnt be watching these videos if we didn't like math. make us pi guys look happy! ;)

  • Lorenzo Sarria
    Lorenzo Sarria 12 days ago +1

    The reason why there are no lattice points for the numbers that are 3 more than a multiple of 4 is that the numbers that can be factored are the sum of 2 squares (The Pythagorean triples visualized video helps explain this), and numbers that are 3 more than a multiple of 4 can’t be written as a sume of 2 squares because all squares are either 1 o 0 more than a multiple of 4

  • Tato Chan
    Tato Chan 12 days ago +1

    your videos talk about stuff i studied in math class in 9th grade and the next second it sounds like witchcraft

  • Alchemy Phoenix
    Alchemy Phoenix 14 days ago

    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.

  • Miguel Iglesias
    Miguel Iglesias 14 days ago +7

    I wonder what's behind naming the Gaussian integers after Martin Sheen.
    Not sure why, but I really do.

    • Miguel Iglesias
      Miguel Iglesias 8 days ago +1

      @Rex Rex, you're right. They do look alike. I was thinking more of an abstract kind of opposite resemblance than a direct physical one; as if Martin Sheen had a reputation of being not particularly good with numbers. Something like Cheech and Chong as the Surgeon Generals on the use of drugs.

    • Rex Rex
      Rex Rex 8 days ago +1

      Doppelganger....of sorts

  • Joy Godwin William Henry

    You are awesome

  • Joy Godwin William Henry

    Did you find this ?

  • Kenneth Garringer
    Kenneth Garringer 14 days ago

    Advanced witchcrafts

  • Kenneth Garringer
    Kenneth Garringer 14 days ago

    Kings and Queen's pi chart.

  • Kenneth Garringer
    Kenneth Garringer 14 days ago

    Artist ic

  • Kenneth Garringer
    Kenneth Garringer 14 days ago

    Great stuff CSI. Scene

  • Kenneth Garringer
    Kenneth Garringer 14 days ago

    Almost witchcraft

  • Kenneth Garringer
    Kenneth Garringer 14 days ago

    How we advanced NASA. Stellar stealthier

  • Kenneth Garringer
    Kenneth Garringer 14 days ago

    Great for solving crimes

  • Angel Mendez-Rivera
    Angel Mendez-Rivera 14 days ago

    *Theorem:* No natural number n > 0 such n = 3 mod 4 can be written as a^2 + b^2, where a and b are integers.
    If a = 0 mod 2, then a = 2k for some integer k, implying a^2 = (2k)^2 = 4k^2 = 2(2k^2) = 0 mod 2 = 0 mod 4.
    If a = 1 mod 2, then a = 2k + 1, implying a^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 1 mod 4.
    Since a and b are arbitrary, this implies that if b = 0 mod 2, then b^2 = 0 mod 4, and if b = 1 mod 2, then b^2 = 1 mod 4.
    Therefore, if a = 0 mod 2 and b = 0 mod 2, then a^2 + b^2 = 0 mod 4; and if a = 0 mod 2 and b = 1 mod 2, or vice versa, then a^2 + b^2 = 1 mod 4; and if a = 1 mod 2 and b = 1 mod 2, then a^2 + b^2 = 2 mod 4.
    Since every case is covered above, no case can result in a^2 + b^2 = 3 mod 4. Q. E. D.
    *Note:* I know the equality sign is abuse of notation when talking about modulo classes. I lack a mathematical keyboard, so you will have to simply accept the abuse of notation for now.
    What this all implies is that prime integers 3 mod 4 are Gaussian primes.

  • Norma Black9.1.5
    Norma Black9.1.5 18 days ago

    Zero's not even the number dummies

  • Norma Black9.1.5
    Norma Black9.1.5 18 days ago

    That's not even all the numbers 1 2 3 4 5 fire engine helmet six seven eight nine

  • Norma Black9.1.5
    Norma Black9.1.5 18 days ago +1


  • Abbe Nylund
    Abbe Nylund 21 day ago +1

    16:26 he sounds hella mad when someone touched his system
    "What if you introduce a factor like 3?"
    *well that really mucks up the whole system*
    "Oh... okay I'm sor-"
    *there is no way that you can split up this 3, no matter where you put it, it leaves the columns imbalanced.*

  • CabanaCaseda
    CabanaCaseda 21 day ago

    The moment that bastard Pi shows up you spend days searching for the circle, you know that damn thing is somewhere

  • TTKND And Co.
    TTKND And Co. 25 days ago


  • Luke Roberts
    Luke Roberts 25 days ago

    I did six edibles and watched this and I know this sounds clishe but you just blew my fucking mind thank you man

  • Jean-Yves BOULAY
    Jean-Yves BOULAY 26 days ago

    Pi and Golden Number: not random occurrences of the ten digits
    Number Pi and the Golden Section as well as the inverse of these numbers are made up of a series of apparently random decimal places. This paper is on the occurrence order of the 10 digits of decimal system in these fundamental mathematic numbers. It is in fact that the ten digits of decimal system does not appear randomly in the sequence of Pi and in Golden Section. Also, same phenomena operate in many other constants of which the square roots of numbers 2, 3 and 5, the first three prime numbers.
    Read more at //jyboulaypublications.e-monsite.com/

  • Kiên P.S.
    Kiên P.S. 29 days ago

    22:15 I understand that the number choices we have for 5^3 is (3+1), but how do you know that you can also express the number of choices for 5^3 as x(1)+x(5)+x(5^2)+x(5^3) (I've understood the x(n) function)?

  • Tiqerboy
    Tiqerboy 29 days ago

    let's see if I understood what you just presented.
    Those numbers in red are the poison pills. If R of the circle contains just one of these as a factor, it nukes everything and therefore no lattice points are hit. You need them in pairs.
    The numbers in green are good guys. If R contains one of them, you get 8 lattice points in return for each of these. These numbers also look like pythagorean numbers to me, because they represent the hypotenuse of a right angled triangle as that integer, where each leg of the triangle is an integer length.
    The numbers in yellow are neutral. If R has them as factors, the number of lattice points stay the same.
    So, for a circle to hit lots of lattice points, you want R to contain tons of green numbers, NO red numbers and okay to have yellow numbers.

    Now, you'd think 33 would be a GREEN number since it's one higher than 32 (a factor of four). Unfortunately it's not prime as 33 = 11 x 3. Two RED numbers. And since these need to be paired up, 33 would nuke everything so a circle of R = sqrt(33) hits NO lattice points.

  • Sateakersd Ldsalder

    you learn more here!

  • JohnSikes73
    JohnSikes73 Month ago +1

    For people interested in this kind of results (which belongs to a branch called analytic number theory), you can read this book called “Analytic number theory” by Tom.M.Apostol. I was always inclined towards number theory since childhood. When I went to college, this book became my bible. :-)

    MARBLE_ L3MUR Month ago

    You may have been asked already, but where did you get that "as easy as pi" mug?

  • lati
    lati Month ago

    W O A H

    SUOMYNONA Month ago +1

    I'm so baffled by how high the like:dislike ratio on this video is. Usually the randomness of USclip users garuntees at least 5% dislikes. This video 98.8% likes.
    That's awesome.

  • Aerospace Engineering with Brian McNulty

    Dude, you are a maniac! These videos are so good, we use them in linear algebra at my college. How long does it take to do something like this, and which language do you use to program these visualizations? AMAZING!

  • iUFOm
    iUFOm Month ago


  • Konstantin Kouptsov


  • Robert Maklovitz
    Robert Maklovitz Month ago

    This chi function is epic

  • Gwebanget Forever
    Gwebanget Forever Month ago

    After series calculation completed, there is a conclusion that the outer limit area is bigger than the circular area, it doesn't necessarily the outer LENGTH is bigger than circumference LENGTH. the constant "3.14159" is correct for 2D, but for 1D, it's "3.17157" (times 2).

  • 3141592 65358979323846

    I wish I would have had such a math teacher at highschool!!

  • H.M. CHO
    H.M. CHO Month ago +1

    I'm really proud of myself who clicked the video and found out the channel!

  • jhony angarita
    jhony angarita Month ago

    Una línea curva no es una línea recta ni está compuesta por líneas rectas

  • jhony angarita
    jhony angarita Month ago

    3.14159... es la mitad de la sumatoria de los lados de un polígono regular la circunferencia no es un polígono regular es una curva cerrada
    Los cálculos que se hace para calcular los lados de polígonos regulares qué están dentro de la circunferencia lo cual quiere decir que el perímetro de la circunferencia es mayor

  • Jenny -s77
    Jenny -s77 Month ago

    Who s the teatcher? 😂

  • Wild Animal Channel

    I doth my cap to you, Sir.