Prime Spirals - Numberphile

  • Published on Jul 9, 2013
  • Prime numbers, Ulam Spirals and other cool numbery stuff with Dr James Grime.
    More links & stuff in full description below ↓↓↓
    James Clewett on spirals at:
    And more to come soon...
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  • Science & TechnologyScience & Technology

Comments • 1 737

  • Smish 0
    Smish 0 Day ago

    What if we do a Sacks spiral but instead of square numbers we use cubes?

  • james caley
    james caley 5 days ago

    So what is the predictive value of this? If a diagonal has a large number of primes is that true if you extend the diagonal? Or does the probability of finding a prime regress to the mean?

    I have seen similar patterns like this in data before. When I insisted it is non-random other people think I am just seeing things. Maybe I am just autistic or something...

  • Nokia n900
    Nokia n900 10 days ago

    Is it just me, or does this guy's eyes always look like he's on LSD?

  • Osanne
    Osanne 16 days ago

    I wonder if the patterns are still as obvious if you would exclude even numbers. I mean, of course there's all these diagonals; horizontals and verticals can't form if you always come across even numbers on those lines. except for the 2, primes are never even...

  • Metaldetection tube world wide

    In the middle of the random patern was a chinese sign 😱😉

  • Anthony Williams
    Anthony Williams 17 days ago +1

    57 IS a prime number tho 4:49

  • Jonty's Corner
    Jonty's Corner 20 days ago +1

    What if you highlighted numbers which only have a single other factor which is also a prime?

  • HippySlayer3.14
    HippySlayer3.14 26 days ago

    This guy looks like a mixture of doctor Watson and bilbo

  • David A. Yorkson
    David A. Yorkson 26 days ago

    Has anyone tried to do a 3D spiral? Or cubic spiral? Perhaps in higher dimentions different patterns can appear...

  • Klauz Wayne
    Klauz Wayne 27 days ago

    When you created the noise chart visible at 3:20 , did you exclude the "even" diagonals from potential noise?
    And did you put in the same total amount of points in both pictures?
    Also on the primes sheet you see a higher density in the center than on the white noise.

  • Santiago Martinez
    Santiago Martinez 28 days ago

    Spiral from 1 to 12, where 13 takes the place of 1 and continues on to 14 ... and so on and son... and you will see that all primes line up... Nicolas Tesla patent it.

  • Cokeman5
    Cokeman5 29 days ago +1

    As far as I can tell, with the equation 4x^2-2x+1, besides the beginning(3,13,31), you will never get 3 primes in a row.

  • Kalle K
    Kalle K Month ago

    Maybe soon we will see the prime of prime spirals on prime time.

  • Caden vids
    Caden vids Month ago

    What if you go in a Hilbert’s curve?

  • Wardhouse
    Wardhouse Month ago

    Best of Hans Zimmer/One Simple Idea.mp3

  • james boyd
    james boyd Month ago

    How about projecting the numbers from within a sphere?

  • a completely awesome name yay

    what if you use a hexagonal spiral, or not a spiral at all, what if you add in negative numbers?

  • Ed Mark
    Ed Mark Month ago

    What if we spiral only with odd numbers?

  • Osmund Francis
    Osmund Francis Month ago

    7:24 And that big gap is the squares ... and the squares minus one. This is because a square number, minus one, has two factors (x^2 - 1 = (x + 1)(x - 1)).

  • Colby Marsh
    Colby Marsh Month ago

    Woah on the sax spiral, the primes seem to form a sorta of Cardioid!

  • What's on my mind
    What's on my mind Month ago

    If you only circle even numbers you will get diagonal lines as well, also if you do the same thing for odds, you get diagonal lines.

    This is not a pattern, sorry to break it to yall.

  • theodor dimou
    theodor dimou Month ago

    Another weird thing with primes is the space between. From 0 to 100 you have odd space always a pattern of 1,3,5,7 after 100 you have power of 2 space always. 2,4,6,8,10,12,14...

  • John McGuire
    John McGuire Month ago

    Another way to say PRIME is INDIVISIBLE, and another way to say INDIVISIBLE is FASCIST (dun-dun)!

  • roglo
    roglo Month ago

    4x²-2x+1: for x=4, it is 57, Grothendieck's prime number! :-)

  • Michał Marciszewski

    Have you read Douglas Adams books? Try making Ulam spiral starting with 42 (which is "the meaning of life")...

  • Seraphim227
    Seraphim227 Month ago

    Ah, the dirty windowpane spiral...

  • Kenji Gunawan
    Kenji Gunawan Month ago

    For those of you who thinks I was wrong - the exception for the multiples is the first multiple of 2, 3 or 5.

  • David Wilkie
    David Wilkie 2 months ago

    Primes and Cofactors of primes, like the "turtles all the way down" assertion, are and infinite regression arranged fractal-frequency geometry arranged/projected by resonance around the Universal vanishing point, .dt, in probabilities that are naturally occurring conception of e-Pi-i resonance in Time Duration Timing, as potential possibilities derived from the Origin of the Temporal Superposition-point Singularity. We are embedded in the unity of active probability surrounding zero-infinity difference macroscopically and inside the infinite expansion of zero-infinity displacement in eternity, spacing.

    So the superimposed vanishing point distributed connection of modulated QM-Time Principle, is a continuously created multi-phase universal timing statement, drawn in eternal co-existence probability positioning, ..of macro-micro +/-projection, Quantum Operator spirals pivoted on the Supuerspin unit quantization Principle In-form-ation of the Phys-Chem vortices-vertices, of Atomic and Astronomically integrated form-ulae.
    The "wibbly-wobbly timey-wimey vortex" of reversible perspective could be the basis of an amateur researcher's understanding of WYSIWYG. Because log base e spiral spacing of constant potential positioning occurs naturally coordinated at the interference connection of e-Pi-i resonance imaging of multi-phase superimposed frequency interference..(?).
    (Needs some actual Mathematicians to do the work, "numberness" works as the discrete, provable elemental steps of Natural number systems, and in indefinite continuously created connections of modulated Calculus Integration Information fields in QM)
    Fun to Imagine.

  • Simon Coppack
    Simon Coppack 2 months ago


  • Rachel McClain
    Rachel McClain 2 months ago

    Do you think the pattern might be clearer if plotted in more dimensions?

  • Brendan Franklin
    Brendan Franklin 2 months ago

    Random with rules.

  • Ruffi Fuffler
    Ruffi Fuffler 2 months ago

    A demand for exceptions in the rules for ordering symbols, implies that base arithmetic defines the need for the next prime without contradiction to prior order, so writing the transcendental general rule will soon become arbitrarily complex without a co-hog relationship to the physical, so says Hairy Wau?

  • stupid_sleazoid 2
    stupid_sleazoid 2 2 months ago

    Well I definitely see patterns in white noise picture he showed

  • João Rodrigo Souza Leão

    Hi ! I have recently found a way to represent any prime using powers of the golden ratio. Would that be important ? I have all the calculations (algebra) ready! I also wrote programs to further test the results. I still have to write the paper. But to wich journal ? Wich one would be appropriate ? I am a trained astrophysicist and amateur mathematician. Please help !

  • saultube44
    saultube44 2 months ago

    Prime Numbers Equations by James Grim please, make it happen, 8.75 years later watching this but I'm interested in Him doing the video, and yes I'm subscribed

  • MrAffeman
    MrAffeman 2 months ago

    Primes reveal themselves "somewhat" on a 2D plane, what if you made a 3 dimensional cube, could it be more obvious where the primes are? What if you take it one step further to a hypercube, what will that show?

  • Abdul Kareem Barghouthi

    Well, i think it's intuitive that they're appearing on diagonals since the spiral you made has all the odd numbers on the corners of each square as the squares spiral larger.

  • Christopher Kingsland
    Christopher Kingsland 3 months ago

    I was struck by the idea when seeing this prime spiral mapping - Is it possible that the distribution of planetary systems across a galaxy (i.e.ours, since we can now gather stats given the number of discoveries so far), and even more interestingly - super earths - is somehow related to prime distribution? After all, like pi, e, and i that seem to come up everywhere, why not primes? Is there some kind of geometrical relationship between prime (integers), the big two irrationals (pi and e), the golden ratio, and i (perpendicular to our 3D reality)?

  • Joseph Asghar
    Joseph Asghar 3 months ago

    Dizzying and beautiful

  • Truce
    Truce 3 months ago +1

    Primes are always odd so if you took random odd numbers wouldn't it create diagonals as well

  • aud_io
    aud_io 3 months ago +6

    I discovered something very interesting about primes. It seems that all of the prime numbers greater than 3 are either one more or one less than a multiple of 6. I did some googling and apparently this is a known thing, and there's even a way to prove it.
    Any number can be described by one of these 6 categories:
    (1) A multiple of 6
    (2) A multiple of 6 plus 1
    (3) A multiple of 6 plus 2
    (4) A multiple of 6 plus 3
    (5) A multiple of 6 plus 4
    (6) A multiple of 6 plus 5
    For categories (1), (3), and (5), you would always end up with an even number, so none of those numbers can be prime. For category (4), you would always end up with a number divisible by 3, so none of those numbers could be prime either, so all prime numbers must fall within category 2 or 5, which would mean any prime greater than 3 could be represented with 6n+1 or 6n-1. 6n+5 and 6n-5 would also work too.

  • Steve T.
    Steve T. 3 months ago +3

    Writing a number line in a hexagonal style produces some pretty interesting spirals as well. All primes fall on one of two axes, either the 1st, or 5th axis, and you can see where the multiples of inner numbers will "block" because of the patterns of every multiple of every number crossing on to either axis. -Where any multiple of any number crosses the 1st or 5 axis, there will be no prime. Also it's pretty to stare at lol. ((1-6 for the first ring, then 7-12 for the 2nd ring 13-18 for the 3rd ring. -with 7 above 1, 8 above the 2nd side, 9 above the 3rd, 10 above the 4rth side, 11 above 5, 12 over 6, 13 above 7 in the 1st column, 14 above 8 in the 2nd column .... ect.....)) You can see clearly where n mod 6 = 1, and also when n mod 6 = 5. :)

    • aud_io
      aud_io 3 months ago +2

      I discovered something similar by drawing a graph of fractions with the x-axis as the denominator and the y-axis as the numerator (I only did it for fractions less than 1 to make it less tedious). I put dots on the fractions that are in simplest terms and a small circle on all repeated ones, and then I drew lines to connect adjacent and diagonal dots, and afterward I colored in all of the repeating shapes. The denominators that were multiples of 6 had a really amazing looking pattern to them. I kinda want to buy some huge graph paper to make a bigger version (I only had room to go up to 20/20). Definitely gonna try that hexagon thing too!

    • TheDerpy Kitty
      TheDerpy Kitty 3 months ago +1

      Steve T. Now I have to try this

  • Luca Crisi
    Luca Crisi 3 months ago

    Why isn't 1 considered PRIME?!

  • TeeTerTime
    TeeTerTime 4 months ago +1

    G2 wins the invitational and Pengu is over here talking about numbers!

  • mamaboo cee
    mamaboo cee 4 months ago

    I love primes...

  • Mehico2fel
    Mehico2fel 4 months ago

    Are those random numbers was odds?

  • medexamtoolsdotcom
    medexamtoolsdotcom 4 months ago

    I don't see why it would be a surprise that there would be certain lines that are heavier or lighter than others. For instance mark off the lines where the multiples of 3 are, and there will of course be NO primes on those lines. Same with 5's. So rather than being uniform, there will be those lines where there is nothing at all. Well, you superimpose a bunch of things like that together, with things being fainter in some lines and darker in others, and I would expect to get something just like this.

  • Ken Taylor
    Ken Taylor 4 months ago

    I did this with a fibonacci snowflower spiral. There are some spirals here of odd numbers without primes present.

    • Ken Taylor
      Ken Taylor 4 months ago

      I think these are the product of two odd numbers.

  • Daniel A Millar
    Daniel A Millar 4 months ago

    Did the random one exclude evens? I’m not doubting that prime numbers aren’t totally random, but I do wonder if that visual example is disingenuous.

  • Faic Legion
    Faic Legion 4 months ago

    Oh I remember doing this

  • Brian Tepper
    Brian Tepper 4 months ago +1

    Curious if there are any other types of spirals that show other interesting patterns when filled in with primes

  • Wagner Lip
    Wagner Lip 4 months ago

    Trying to make a visual image that justify more patterns for primes, but we don't know primes. In real, we do know what is NOT primes, so we could make a pattern for those, and perhaps, primes start to appear easier. Also, spirals induce to a sequence of logic quantification, primes do not follow that pattern, we already know that, so why follow that path?

  • Neko Master
    Neko Master 4 months ago

    why not fractal

  • Yiyi Wu
    Yiyi Wu 4 months ago +1

    It is obvious why the “stripes” pattern exists. It’s because in this layout odd vs even numbers form sort of a chessboard. Besides 2 every prime is odd. Even though not every odd is prime our brains will notice the pattern created by only numbers of one “color” on the chessboard being illuminated

    • Yiyi Wu
      Yiyi Wu 4 months ago +1

      In other words the only way he has refined the search for primes is to not include even numbers! Also seems silly to compare to randomness

  • Cracked Emerald
    Cracked Emerald 4 months ago

    I think that we need to discover another type of number to fully understand primes

  • Arcadio Arcadio
    Arcadio Arcadio 4 months ago

    Probably if someone would use a 3, 4 or 7-dimensional base the pattern would be just a straight line, the line of truth connecting past with the future, a thread of the unknown realm. Maybe AI automatic algorithms will solve this.

  • Joe Harris
    Joe Harris 5 months ago

    "And ye shall know them by their stripes."

  • Emilio Arroyo Mohamed
    Emilio Arroyo Mohamed 5 months ago

    Try again the spiral without even numbers and see if there are still stripes

  • Omegacat13
    Omegacat13 5 months ago

    Hello from the future! You might want to sit down, I have a lot of things to warn you about. Like a lot, a lot.

  • OceanSky Web Design
    OceanSky Web Design 5 months ago

    You would really love this book. I did. Peter Plichta illustrates how the prime numbers are ordered on concentric circles numbered 1 to 24 and then 25 to 48 and so on; expanding outward like cross shaped rays of sunlight radiating outward. The guy was a genius!

  • Fracmik
    Fracmik 5 months ago

    Maybe plugging primes into the equation is the way to obtain more? Just a random idea from a not-advanced-educated viewer

  • Little Cripple
    Little Cripple 5 months ago

    My favourite pattern is whenever you put all the primes in a spiral, and whenever you highlight all primes, you get a completed spiral. Pretty cool huh

  • Prabhat Soni
    Prabhat Soni 6 months ago

    Great visuals!

  • Kim Welch
    Kim Welch 6 months ago

    So, you're doing a bunch of 2-dimensional spirals. Have you looked at 3-dimensional or 4-dimensional spirals. Yes, it's really hard to do on paper, but some of the 2d stuff you're showing look like projections from a larger dimensional shape.

  • Casey
    Casey 6 months ago

    Has anyone searched for the opposite of the golden diagonals, with the lowest density of primes?

  • Angela Garet
    Angela Garet 6 months ago

    Primes frequency is moving away from perfect squares, cubes, etc.

  • Shruggz Da Str8-Faced Clown

    It also appears that, within the grid of this square spiral, there is a preponderance of contrasting horizontal and vertical lines whereupon non-primes lie.

  • bnkjkdsbklafj hjbvjhbfdasjka

    if you skuint at it you can see stripes in the random patern
    but they aren't as vibrant, dense and long as in ulam's spiral

  • Engineer Asik
    Engineer Asik 6 months ago

    any sequence having general term tn=an²+bn+c where a,b,c are constants is called quadratic sequence

  • Engineer Asik
    Engineer Asik 6 months ago

    that quadratic polynomial is the general term for the quadratic sequence

  • huckbeduck
    huckbeduck 6 months ago

    I found an equation to find the next number, going diagonally or horizontally or vertically. (This is for all the numbers to create an ulam spiral without writing every digit). Just add 8 to the difference between two of the previous consectutive numbers of the direction you want to go. 2,10,26,50 is a diagonal for example; (50-26=24, 24+8=32, 32+50=82. "82" is the next number in the sequence.

  • Soreofhing
    Soreofhing 6 months ago

    1:16 "He was sat...". "He was seated...". There. Fixed it for you.

  • kirigata
    kirigata 6 months ago

    I wonder what those lines would look like if plotted in 3d. instead of using a square for the spiral, try a cube?

  • Simon Shugar
    Simon Shugar 6 months ago

    3:10 Wouldn't it be better to compare the prime spiral to random ODD numbers chosen instead of ALL numbers? Odds are all diagonal from each other in this spiral so it may just be that that we're seeing.

  • Max Musterman
    Max Musterman 6 months ago

    Need help. Is there a way to get the (x,y) position of any number? 1 has (0,0).

  • Supernova
    Supernova 7 months ago

    That's amazing! I love this!

  • Shivam Mishra
    Shivam Mishra 7 months ago

    The line equation is a parabolic eqn , which in the second diagram seems to be a spiral

  • Kids 4Life
    Kids 4Life 7 months ago

    I was doing some math and found that (2n)+(n^2)-1 created primes very well if n is even. Example: (2 x 99922222222220)+(99922222222220^2)-1 is prime. I also saw that up to 200 being n (leaving out odd numbers) it spit out a prime 42% of the time.

  • ryavix
    ryavix 7 months ago

    Now if only we could get these over educated folks to STOP thinking 2 dimensionally.

  • Mondo LeStraka
    Mondo LeStraka 7 months ago

    Love this!!

  • Andriy Makukha
    Andriy Makukha 7 months ago

    Ulam comes from Lviv, one of the most beautiful cities in Ukraine. There are places that he used to visit with his math buddies.

  • Krishaang Kohli
    Krishaang Kohli 8 months ago +5

    James showing his true 'attraction' for primes
    "Look at these curves."

  • BigMan Ollie
    BigMan Ollie 8 months ago

    what would happen if you were to do this with other tessellating shapes? i.e. filling in a spiral on a map of hexagons etc..?

  • saqqaq _
    saqqaq _ 8 months ago


  • Svsnmurty Gattimi
    Svsnmurty Gattimi 8 months ago

    I am working on Composite Numbers factors based on normal Algebra and Geometry( not divide 1,2,3,..). I need One composite number with unknown factors to find factors based on my work. please help anyone.

  • Caden Bintliff
    Caden Bintliff 8 months ago

    i eat children

  • Velma Velvet
    Velma Velvet 8 months ago

    The round one reminds me of the Earth's magnetic field.

  • Toph Morris
    Toph Morris 8 months ago

    4:49. So, when x=4, the result isn't prime? That somehow seems logical to the degree of being obvious, but I suppose it isn't since there's a +1 in the formula itself. I want to experiment with this now and see the values of x that give you prime numbers and those that don't, and compare/contrast. I can't imagine this already hasn't been done, though. Moments like this, I hate being a math pleb.

  • Robert Morgan
    Robert Morgan 8 months ago

    What if you only circle the mersenne primes?

  • Brandon Gammon
    Brandon Gammon 8 months ago

    What would a square spiral of just prime numbers look like???

  • jat green
    jat green 9 months ago

    ok, i'm writing a computer program to go in spirals checking for diagonal lines and predicting primes and checking if they are. i really want to see how many primes it comes up with and how fast it is compared to a simple primes checker that checks every number

  • Steven Wenker
    Steven Wenker 9 months ago

    Please zoom out a little bit

  • Brandon Hamer
    Brandon Hamer 9 months ago

    I wonder what it would look like if you did ulams spiral but coloured numbers according to how many prime factors each number has. All primes would be one colour, then numbers like 6, 10, 14 and 15 another colour and 8, 12, 18 and 20 another and so on. I tried looking to see if someone had done this but couldn't find anything.

  • The Walnut Destroyer
    The Walnut Destroyer 9 months ago


  • Robi_CK
    Robi_CK 10 months ago +14

    0:58 - Kudos for pronouncing Stanisław right, with "ł" not "l".

  • Timothy Hinkle
    Timothy Hinkle 10 months ago

    if you repeated this same experiment in more than 2 dimensions what are the results? 2,3,4...26

  • Jake Mooshian
    Jake Mooshian 10 months ago

    I would like to see Ulam's spiral using only odd numbers.

  • Nikhil Nirmal
    Nikhil Nirmal 10 months ago

    Must watch Channel Nikhil Nirmal
    Prime numbers identification easily .

  • Corpus Crewman
    Corpus Crewman 10 months ago

    I love how the primes graphed along the Archaemedian spiral result in figures that resemble logarithmic graph functions.

  • Luis Padua
    Luis Padua 10 months ago

    I'd like to see a video on the standard model lagrangian density formula.