# 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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TeeTerTime3 days ago^{+1}G2 wins the invitational and Pengu is over here talking about numbers!

mamaboo cee3 days agoI love primes...

Mehico2fel6 days agoAre those random numbers was odds?

medexamtoolsdotcom6 days 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 Taylor8 days agoI did this with a fibonacci snowflower spiral. There are some spirals here of odd numbers without primes present.

Ken Taylor8 days agoI think these are the product of two odd numbers.

Daniel A Millar9 days 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 Legion9 days agoOh I remember doing this

Brian Tepper12 days agoCurious if there are any other types of spirals that show other interesting patterns when filled in with primes

Wagner Lip20 days 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 Master22 days agowhy not fractal

Yiyi Wu25 days 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 Wu25 days 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 Emerald26 days agoI think that we need to discover another type of number to fully understand primes

Arcadio Arcadio28 days 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 HarrisMonth ago"And ye shall know them by their stripes."

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

Omegacat13Month 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 DesignMonth 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!

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

Little CrippleMonth 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 Soni2 months agoGreat visuals!

Kim Welch2 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.

Casey2 months agoHas anyone searched for the opposite of the golden diagonals, with the lowest density of primes?

Angela Garet2 months agoPrimes frequency is moving away from perfect squares, cubes, etc.

Shruggz Da Str8-Faced Clown2 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 hjbvjhbfdasjka2 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 Asik2 months agoany sequence having general term tn=an²+bn+c where a,b,c are constants is called quadratic sequence

Engineer Asik2 months agothat quadratic polynomial is the general term for the quadratic sequence

huckbeduck2 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.

Soreofhing2 months ago1:16 "He was sat...". "He was seated...". There. Fixed it for you.

kirigata2 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 Shugar2 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 Musterman3 months agoNeed help. Is there a way to get the (x,y) position of any number? 1 has (0,0).

Supernova3 months agoThat's amazing! I love this!

Shivam Mishra3 months agoThe line equation is a parabolic eqn , which in the second diagram seems to be a spiral

The Fourth Musketeer3 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.

ryavix3 months agoNow if only we could get these over educated folks to STOP thinking 2 dimensionally.

Mondo LeStraka3 months agoLove this!!

Andriy Makukha3 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 Kohli4 months agoJames showing his true 'attraction' for primes

"Look at these curves."

BigMan Ollie4 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 _4 months agoJames Grime? MORE LIKE JAMES PRIME

K GN4 months agoGreat video! Thanks.

Svsnmurty Gattimi4 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 Bintliff4 months agoi eat children

Velma Velvet4 months agoThe round one reminds me of the Earth's magnetic field.

Toph Morris4 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 Morgan5 months agoWhat if you only circle the mersenne primes?

Brandon Gammon5 months agoWhat would a square spiral of just prime numbers look like???

jat green5 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 Wenker5 months agoPlease zoom out a little bit

Brandon Hamer5 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.

Nin compoop6 months agoGreat

Robi_CK6 months ago0:58 - Kudos for pronouncing Stanisław right, with "ł" not "l".

Timothy Hinkle6 months agoif you repeated this same experiment in more than 2 dimensions what are the results? 2,3,4...26

Jake Mooshian6 months agoI would like to see Ulam's spiral using only odd numbers.

Nikhil Nirmal6 months agoMust watch Channel Nikhil Nirmal

Prime numbers identification easily .

Corpus Crewman6 months agoI love how the primes graphed along the Archaemedian spiral result in figures that resemble logarithmic graph functions.

Luis Padua6 months agoI'd like to see a video on the standard model lagrangian density formula.

Jack Kidd6 months agowhat if you write the numbers in triangles, or pentagons, or hexagons....instead of a square

Hamza1477 months agoAmazing ! You remind me some of my discussions about prime numbers with a dear friend of mine when we were at high-school.

Axe7 months agoLong ago, I've found two interesting formule for primes: sqrt(120n+1) and sqrt(120n+49).

For quite a lot of values of n, whenever the formula's output was an integer, it was prime. What's more, the first formula returned primes ending in 1 and 9, and the second - in 3 and 7. I haven't calculated the breakdown point (value of n where the formula returns a non-prime integer) due to lack of experience in number theory, but it seems to be quite high. Could you please look at that? Maybe not in a video, but is there any research done on this already?

Kai Na7 months agoWell primes are odd, so it appears normal to see stripes if you arrange numbers in a spiral, since odd and even numbers are intertwined

Venkatesh babu8 months ago1=√1=√1/1=√1/√1=1/√1=√1×√1=- i^2 = ... , So Fibonacci series is powers of i.

David Tribble8 months agoWhat's so special about the primes?

I've wondered for many years now if primes can be considered just a special case of the set of naturals having 2 divisors, Sd(2) = {2, 3, 5, 7, 11, ...}, where the next set is the naturals having 3 divisors Sd(3) = {4, 6, 9, 10, 14, 15, ...}, and so on for Sd(4), Sd(5), etc. Sd(1) is just {1}, of course. The trick, then, is to find numeric relationships between Sd(i) and Sd(j), and then generalize these to all Sd(n, for all n in N).

S. Smith9 months agoA similar concept I came up with while doodling in school too...

Get grid paper, and do rows, draw lines through primes; they line up at various different angles

He Fr9 months ago"look at those curves" -James Grime

Algorithm9 months agoI used this exact concept in 5th grade when I was just playing around with primes and trying to figure out a pattern. It's interesting to see that there is actually a pattern as I didn't notice anything when I did it. I only went up to 100 though.

AlekVen's stupidface10 months agoYou know that if you write down numbers like that, every diagonal will either contain strictly odd numbers, or even numbers, and they'll follow each other?

Of course you'll see the pattern of primes considering that, excluding 2, every single one of those is odd, thus only lie on certain diagonals.

Nathan 113210 months agoninja

Nathan 113210 months agoil é ou gotaga ?

Nathan 113210 months agogo 1vs1 fortnite

Nathan 113210 months agoez

Frank Harr10 months agoYou know, the primes may or may not have a patter, but the non-prime DEFFINATELY have a pattern.

A. Joe10 months agoThe primes in the Archimedian spiral look very much like parabolic functions rotated 90 degrees to the left. Has anyone investigated whether there may be a complex rotation of a second order polynomial involved in creating this pattern?

Дмитрий Кузнецов10 months agoI wonder if anyone tries to do real math in some other system of calculation different from decimal. may be those patterns could be seen even easier.

Aaron Rotenberg10 months agoWhat do you mean 57 isn't prime? Everyone knows it's the Grothendieck prime!

Denis SEO10 months agoSo on some visual representation you can get a straight line of primes going into infinity?

z10 months agojames prime

thebudkellyfiles10 months agoThank you for so many great and interesting videos.

paperEATER10111 months agoI see Elvis in the "random" picture

Meph11 months agoIt kinda looks like a swastika. I wonder if math is trying to tell us something.

nightmisterio11 months agoDo prime visualization in base 12

Ted Rowell11 months agoCan someone make one of those squares where all the even numbers are in the correct location, and all the odd ones random? I wonder what that would look like.

Leo YohansenYear agoCompare it with the graph for 6x + or - 1.

XaYear agoIt seems to me that the reason the prime numbers form diagonal lines is just because they're odd. After all, if you circle all the odd numbers instead of the prime numbers, you'll get a checkerboard-like grid. Naturally, since prime numbers, besides 2, are odd, they will tend to form random diagonal lines.

Coco PuffsYear agox^2 + x + [button smash your calculator here]

RBWNYear agoa

OsanneYear agoCould the pattern of diagonals (partially) be caused by the fact that prime numbers except two are always odd numbers?

In such a spiral notation the odd and even numbers immediatly form a grid of odd and even number lines, and prime numbers can already only exist on half of those.

So the difference from a completely random pattern is already visible the moment you say "the random numbers cannot be even". I doubt there would be a clear difference between odd random numbers and prime number patterns.

MisterNewOutlookYear agoHave these spirals been tried on a sphere or within a sphere?

Chris LarsonYear agoThe cause for the diagonals appearing in the prime spiral as opposed to the chaos in the randomly generated picture is because the prime spiral creates a checkerboard between evens and odds. Essentially, if you took the random pictures and then removed all of the even, my hypothesis is that it would look similar to the prime spiral with respect to the prevalence of diagonals

Gunihelm SchafYear agoI found a equation who gives u every prime number

Patryk WieczorekYear agoMaths is beautiful!

ScarletFoxYear agoWhat if we're thinking of this wrong.

Every pattern like this creates a line in which no primes are possible.

Maybe the primes are the gaps where every one of these lines from every possible pattern doesn't cover.

Paul WhiteYear agoWanna see how to find primes ?...map them onto a set of concentric circles of 24 sections each . Each section represents one sequential natural number and keep going for each larger circle. You will find them only on 8 " rays " from the centre. All very geometric .See prof. Dr. Peter Plichta's book ...god's secret formula. I believe the formula 6n + 1 or - 1 ,might be worth looking into. Cheers all

SanderYear ago@3:18

I'd like to see that picture of random numbers, but with the added rule that all even numbers other than 2 are excluded. This because the picture is now also filled with horizontal and vertical stripes while in the Ulam's spiral it is by definition only possible to form stripes that are diagonal.

CornerTalkerYear agoTry putting the digits of pi in an Ulam's spiral and then marking the primes.

magnuseeYear agoDont do it in a grid. Try a honeycomb pattern

magnuseeYear agoHexagonales

John PerkinsYear agoI would like to see the spiral with the obvious diagonals removed, then see if other diagonals become apparent.

ROVAKANYear agowhere can i get those pictures of spirals please ?

Ismir EghalYear ago0:23 rap career secure if maths should one day not

work for him anymore