# Prime Spirals - Numberphile

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**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: thexvid.com/video/3K-12i0jclM/video.html

And more to come soon...

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And "golden line" in this context was made up by Brady!

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Smish 0Day agoWhat if we do a Sacks spiral but instead of square numbers we use cubes?

james caley5 days agoSo 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 n90010 days agoIs it just me, or does this guy's eyes always look like he's on LSD?

Osanne16 days agoI 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 wide17 days agoIn the middle of the random patern was a chinese sign 😱😉

Anthony Williams17 days ago^{+1}57 IS a prime number tho 4:49

Not Secure17 days ago29*3=57

Jonty's Corner20 days ago^{+1}What if you highlighted numbers which only have a single other factor which is also a prime?

HippySlayer3.1426 days agoThis guy looks like a mixture of doctor Watson and bilbo

David A. Yorkson26 days agoHas anyone tried to do a 3D spiral? Or cubic spiral? Perhaps in higher dimentions different patterns can appear...

Klauz Wayne27 days agoWhen 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 Martinez28 days agoSpiral 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.

Cokeman529 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 KMonth agoMaybe soon we will see the prime of prime spirals on prime time.

Caden vidsMonth agoWhat if you go in a Hilbert’s curve?

WardhouseMonth agoBest of Hans Zimmer/One Simple Idea.mp3

james boydMonth agoHow about projecting the numbers from within a sphere?

a completely awesome name yayMonth ago^{+1}what if you use a hexagonal spiral, or not a spiral at all, what if you add in negative numbers?

Ed MarkMonth agoWhat if we spiral only with odd numbers?

Osmund FrancisMonth ago7:24 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)).

MATH GeniusMonth ago😂 yup

Colby MarshMonth agoWoah on the sax spiral, the primes seem to form a sorta of Cardioid!

What's on my mindMonth agoIf 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 dimouMonth agoAnother 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 McGuireMonth agoAnother way to say PRIME is INDIVISIBLE, and another way to say INDIVISIBLE is FASCIST (dun-dun)!

rogloMonth ago4x²-2x+1: for x=4, it is 57, Grothendieck's prime number! :-)

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

Seraphim227Month agoAh, the dirty windowpane spiral...

Kenji GunawanMonth agoMe be like: OF COURSE THERE ARE STRIPES! EVERY EVEN NUMBER IS NOT PRIME, EVERY MULTIPLE OF 3 IS NOT PRIME, EVERY MULTIPLE OF 5 IS NOT PRIME, ETC. WHICH MAKES IT LOOK STRIPE-Y.

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

David Wilkie2 months agoPrimes 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 Coppack2 months agoVihart!

Rachel McClain2 months agoDo you think the pattern might be clearer if plotted in more dimensions?

Brendan Franklin2 months agoRandom with rules.

Ruffi Fuffler2 months agoA 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 22 months agoWell I definitely see patterns in white noise picture he showed

João Rodrigo Souza Leão2 months agoHi ! 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 !

saultube442 months agoPrime 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

MrAffeman2 months agoPrimes 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 Barghouthi3 months agoWell, 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 Kingsland3 months agoI 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 Asghar3 months agoDizzying and beautiful

Truce3 months ago^{+1}Primes are always odd so if you took random odd numbers wouldn't it create diagonals as well

aud_io3 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.

Noah Duller3 months ago^{+1}Nice

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_io3 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 Kitty3 months ago^{+1}Steve T. Now I have to try this

Luca Crisi3 months agoWhy isn't 1 considered PRIME?!

TeeTerTime4 months ago^{+1}G2 wins the invitational and Pengu is over here talking about numbers!

mamaboo cee4 months agoI love primes...

Mehico2fel4 months agoAre those random numbers was odds?

medexamtoolsdotcom4 months agoI 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 Taylor4 months agoI did this with a fibonacci snowflower spiral. There are some spirals here of odd numbers without primes present.

Ken Taylor4 months agoI think these are the product of two odd numbers.

Daniel A Millar4 months agoDid 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 Legion4 months agoOh I remember doing this

Brian Tepper4 months ago^{+1}Curious if there are any other types of spirals that show other interesting patterns when filled in with primes

Wagner Lip4 months agoTrying 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 Master4 months agowhy not fractal

Yiyi Wu4 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 Wu4 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 Emerald4 months agoI think that we need to discover another type of number to fully understand primes

Arcadio Arcadio4 months agoProbably 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 Harris5 months ago"And ye shall know them by their stripes."

Emilio Arroyo Mohamed5 months agoTry again the spiral without even numbers and see if there are still stripes

Omegacat135 months agoHello 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 Design5 months agoYou 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!

Fracmik5 months agoMaybe plugging primes into the equation is the way to obtain more? Just a random idea from a not-advanced-educated viewer

Little Cripple5 months agoMy 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 Soni6 months agoGreat visuals!

Kim Welch6 months agoSo, 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.

Casey6 months agoHas anyone searched for the opposite of the golden diagonals, with the lowest density of primes?

Angela Garet6 months agoPrimes frequency is moving away from perfect squares, cubes, etc.

Shruggz Da Str8-Faced Clown6 months agoIt 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 hjbvjhbfdasjka6 months ago3:20

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 Asik6 months agoany sequence having general term tn=an²+bn+c where a,b,c are constants is called quadratic sequence

Engineer Asik6 months agothat quadratic polynomial is the general term for the quadratic sequence

huckbeduck6 months agoI 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.

Soreofhing6 months ago1:16 "He was sat...". "He was seated...". There. Fixed it for you.

kirigata6 months agoI wonder what those lines would look like if plotted in 3d. instead of using a square for the spiral, try a cube?

Simon Shugar6 months ago3:10 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 Musterman6 months agoNeed help. Is there a way to get the (x,y) position of any number? 1 has (0,0).

Supernova7 months agoThat's amazing! I love this!

Shivam Mishra7 months agoThe line equation is a parabolic eqn , which in the second diagram seems to be a spiral

Kids 4Life7 months agoI 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.

ryavix7 months agoNow if only we could get these over educated folks to STOP thinking 2 dimensionally.

Mondo LeStraka7 months agoLove this!!

Andriy Makukha7 months agoUlam comes from Lviv, one of the most beautiful cities in Ukraine. There are places that he used to visit with his math buddies.

Krishaang Kohli8 months ago^{+5}James showing his true 'attraction' for primes

"Look at these curves."

BigMan Ollie8 months agowhat 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 _8 months agoJames Grime? MORE LIKE JAMES PRIME

Svsnmurty Gattimi8 months agoI 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 Bintliff8 months agoi eat children

Velma Velvet8 months agoThe round one reminds me of the Earth's magnetic field.

Toph Morris8 months ago4:49. 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 Morgan8 months agoWhat if you only circle the mersenne primes?

Brandon Gammon8 months agoWhat would a square spiral of just prime numbers look like???

jat green9 months agook, 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 Wenker9 months agoPlease zoom out a little bit

Brandon Hamer9 months agoI 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 Destroyer9 months agoGreat

Robi_CK10 months ago^{+14}0:58 - Kudos for pronouncing Stanisław right, with "ł" not "l".

Timothy Hinkle10 months agoif you repeated this same experiment in more than 2 dimensions what are the results? 2,3,4...26

Jake Mooshian10 months agoI would like to see Ulam's spiral using only odd numbers.

Nikhil Nirmal10 months agoMust watch Channel Nikhil Nirmal

Prime numbers identification easily .

Corpus Crewman10 months agoI love how the primes graphed along the Archaemedian spiral result in figures that resemble logarithmic graph functions.

Luis Padua10 months agoI'd like to see a video on the standard model lagrangian density formula.