[HN Gopher] Why do prime numbers make these spirals? (2019)
___________________________________________________________________
Why do prime numbers make these spirals? (2019)
Author : nivethan
Score : 351 points
Date : 2023-04-26 00:14 UTC (22 hours ago)
(HTM) web link (www.3blue1brown.com)
(TXT) w3m dump (www.3blue1brown.com)
| smitty1e wrote:
| > These patterns are certainly beautiful, but they don't have a
| hidden, divine message about primes.
|
| As far as we know, for now.
| jacknews wrote:
| Yeah, it's clearly someone's fingerprint:
|
| https://i.imgur.com/yFMPmtW.png
| Poppys wrote:
| My browser did not enjoy that 40MB web page.
| y42 wrote:
| Shameless self-promotion, just because it seems to fit: I build
| an "animation engine" around primes and spirals, just in case you
| need some distraction or you are bored:
|
| https://primes.nickyreinert.de/
| aaron695 wrote:
| [dead]
| anoncow wrote:
| Could there be a way to derive prime numbers based on an
| extrapolated graph which would be less resource intensive as
| compared to current prime identifying methods?
| kfrzcode wrote:
| It's because we live inside a spiral galaxy that's transitioning
| into a new galaxy type so we see through a spiral
| tabtab wrote:
| Doesn't work in Australia.
| zmmmmm wrote:
| Bit of a let down when they reveal that plotting integers in
| general results in the same spiral pattern as primes. So
| naturally primes as a subset of integers also produce spirals.
|
| Seems the title is a bit misleading!
| onethought wrote:
| The whole point of the video is that asking these kinds of
| questions even if at first they seem kind of silly or arbitrary
| often lead to really interesting observations. In this case:
| Dirichlet's theorem on arithmetic progressions [0]
|
| [0]
| https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arith...
| laserbeam wrote:
| I found it insightful, not a let down. It says a lot about
| human nature and our biases towards spotting patterns that may
| be incomplete or inaccurate.
| euroderf wrote:
| In the absence of reliable telepathy, there is indeed
| survival value in spotting patterns in incomplete
| information.
| desertraven wrote:
| A bit misleading I agree. I suppose since prime numbers DO make
| these spirals, it's not inaccurate.
| smitty1e wrote:
| We expect patterns from quotidian integers. The seeming chaos
| of primes, in contrast, brings the intrigue.
| crdrost wrote:
| I don't know, for me the fact that this gives you a
| visualization of a totient is really interesting.
|
| Another really interesting idea is to deliberately change the
| thing which you are rationally approximating; you don't have to
| rationally approximate p if you don't want to, that's just if
| you make steps of 1 radian. Make steps of q radians and you get
| the denominators for rational approximations of q/p.
|
| This is used in the golden spiral algorithm[1] to evenly-ish
| distribute points on a sphere, we choose the most irrational
| number q/p = ph, the golden ratio. Since all of its rational
| approximations suck, the spirals are as inoffensive as they can
| be.
|
| 1. https://stackoverflow.com/questions/9600801/evenly-
| distribut...
| subroutine wrote:
| Something less interesting is testing what the polar plot
| looks like if you plot the angle in degrees instead of
| radians. Or, like in this plot, where I defined 10 degrees as
| exactly one complete turn around the circle:
|
| https://i.ibb.co/59k5dRT/polarplot.png
|
| Slightly more interesting is what the primes look like when
| one complete turn is defined as 1, 2, 3... 30 units:
|
| https://i.ibb.co/F0GvrSY/polarplot2.png
| Bytewave81 wrote:
| It's referencing the Math Stack Exchange question which asked
| the same question:
| https://math.stackexchange.com/questions/885879/meaning-of-r...
| desertraven wrote:
| What's the commentary on prime numbers currently? Is it expected
| that there is some unfound pattern to them? Or will they always
| remain elusive?
| frogpelt wrote:
| No.
| Taywee wrote:
| Maybe.
| timetraveller26 wrote:
| Yes
| mathgenius wrote:
| The answer to these questions is the Riemann hypothesis. John
| Baez just wrote a related paper about "motives" for beginners
| (undergrads?) [1]. It's worth a read even if you squint at the
| equations like i do, there's plenty of interesting commentary
| as well.
|
| [1] https://math.ucr.edu/home/baez/motives.pdf
| smackeyacky wrote:
| Prime numbers are important for cryptography - if there is a
| pattern to them it might have an effect on what we think is
| secure or not.
|
| https://www.abc.net.au/news/science/2018-01-20/how-prime-num...
| C-x_C-f wrote:
| Elusiveness is subjective, but there's certainly plenty of
| patterns to be found (though, to be fair, what constitutes a
| pattern might also be subjective). You should check out Terence
| Tao's blog [0], he's not only one of the most prominent
| mathematicians in the field but he's also excellent at
| explaining things. These slides [1] for example don't require
| any background in math.
|
| [0] https://terrytao.wordpress.com
|
| [1] https://www.math.ucla.edu/~tao/preprints/Slides/primes.pdf
| westcort wrote:
| One of the thrills of studying the primes is the discovery that
| all primes greater than 3 are of the form 6k+1 or 6k-1. And for
| primes greater than 2, all primes are of the form 4k+1 or 4k-1.
| It is something that is commonly rediscovered by students, and
| that new independent finding was quite exciting for me.
|
| The reasoning, which is in the article here, is that you can make
| any whole number you wish if the number is of the form 6k+1,
| 6k-1, 6k+2, 6k-2, 6k+3, or 6k-3. But you cannot make primes with
| the numbers of the form 6k+2, 6k-2 (they would always have to be
| divisible by 2), and you cannot makes primes with numbers of the
| form 6k+3, 6k-3 because they are always divisible by 3. So what
| are you left with? All primes >3 must of the form 6k+1 or 6k-1.
| And that factor 6 is just a bit less than 2 pi (a full turn in
| radians) so you get spirals from the offset. They are also a
| pixel or two off, but that is imperceptible.
|
| The same logic is a nice exercise to apply to the problem of why
| primes >2 can only be of the form 4k+1 or 4k-1. Apply the same
| logic as above.
| bheadmaster wrote:
| I have discovered the same thing while practicing prime sieve
| algorithms back in the day. Such properties of primes are quite
| useful for optimizing both speed and memory of sieve
| algorithms.
|
| More generally, if you take the first _n_ primes _p_1, ...,
| p_n_ and define _P=p_1*...*p_n_ , then all primes bigger than P
| be in the form of _P*k +- a_ , such that _a < P_ and _GCD(P, a)
| = 1_. In case of _n=2_ ( _P=2*3=6_ ), there is this nice
| property that the only such _a_ are _1_ and _5_ (which are
| equivalent), but in principle, the same can be done for any
| _n_. It 's just that the set of all _a_ has the size equal to
| Euler 's totient function of _P_ , which grows pretty fast as
| _n_ increases.
|
| For example, if _n = 3_ , then _P = 2*3*5 = 30_ , so all prime
| numbers bigger than _30_ have to be in the form of _30k +- 1_ ,
| _30k +- 7_ , _30k +- 11_ , _30k +- 13_ , _30k +- 17_ , _30k +-
| 19_ , _30k +- 23_ or _30k +- 29_ (notice that half of these are
| equivalent to the other half and can be ommitted). It is
| interesting that in this particular case, all _a_ are either
| _1_ or a prime less than P: I don 't think that property holds
| for all _n_ , though.
| naniwaduni wrote:
| Indeed it's not, it's just anything coprime to P. This
| explains why there are always +-1s, and in particular why
| e.g. 210+-1/210+-209 are viable, even though 209=11*19.
| qikInNdOutReply wrote:
| Its the dimensionality? 2nd dimension, 3rd dimension and there
| permutation shadows? So by the mathematical principal, there is
| only one interesting permutation, and thats the "natural" one
| within the first dimension, aka 1 and 2s.
| leoff wrote:
| >all primes greater than 3 are of the form 6k+1 or 6k-1
|
| what's the value in using this "formula"? We could also keep
| extending this rule, and say that all primes greater than 5 are
| of the form 30k+-1, 30k+-7, 30k+-11, or 30k+-13. Or go further
| by multiplying coefficient of X with the next primes
| Vvector wrote:
| 30k+-1 = 6x+-1 where x = 5k and so on. Why use four formulas
| when one will suffice?
| kfrzcode wrote:
| > imperceptible
|
| isn't that sort of cheating, when it comes to math?
|
| It's "almost" the thing just not the thing
| WastingMyTime89 wrote:
| Nah, it opens a whole new area of research. Everything is a
| great opportunity for new fun in maths.
|
| You just have to properly define what you mean by
| "imperceptible" and you are good to go. Now you can look for
| a bounded approximation. See how it changes as things get
| bigger. See if it can be improved.
|
| Bounds and approximation are generally a very useful tool.
| Properties simplifications and working on sub parts can also
| yield interesting results.
|
| Generally I would say that if you can't prove something,
| trying something close but simpler is nearly always a good
| idea.
| Traubenfuchs wrote:
| What's k?
| Hendrikto wrote:
| An integer.
| WastingMyTime89 wrote:
| A weird way of writing _n_?
|
| More seriously, it's an integer variable. By convention,
| letters from the middle of the alphabet are used for them
| (generally _n_ then _k_ ).
|
| Here, the commenter uses _k_ because that's what's used in
| the article and that's what's used because _n_ is already
| used to designate the class in the definition of a residue
| class.
| kazinator wrote:
| > _the discovery that ..._
|
| Wouldn't the vast majority of those studying primes learn this
| from their textbook?
|
| Note that 6k-1 is the same as 6k + 5. By writing that way, we
| can focus in positive representations of the modulo 6
| congruence.
|
| 6k + 0 can't be prime, it's divisible by 6, yielding k
|
| 6k + 1 might be prime: we cannot rule it out by division.
|
| 6k + 2 cannot be prime, it's divisible by 2, yielding 3k + 1.
|
| 6k + 3 cannot be prime, it's divisible by 3, yielding 2k + 1
|
| 6k + 4 cannot be prime, it's divisible by 2, yielding 3k + 2
|
| 6k + 5 might be prime again.
|
| That covers all cases of the modulo 6 congruence.
|
| Thus only 6k + 1 and 6k + 5 can possibly be prime.
|
| This is trivial fluff, only a smidgeon more clever than "all
| primes greater than 2 are of the form 2k + 1".
|
| Another line of reasoning:
|
| If a number N is divisible by 6, then it is even. This means
| that N + 2 and N + 4 are also even. Thus none of those numbers
| are prime.
|
| If a number N is divisible by 6, it is also divisible by 3.
| This means that N + 3 is also divisible by 3. Thus, it cannot
| be prime.
|
| That leaves N + 1 and N + 5, whose divisibility doesn't relate
| to 6.
| Arnt wrote:
| My textbook, at least, spent its space on the important
| axioms, theorems and corollaries. There were some easily-
| rediscovered results there, but mostly its pages described
| stuff that wasn't trivial to me.
|
| And that's why it's one of the five books I have kept in the
| decades since.
| rvbissell wrote:
| Care to share the name of that textbook?
| Arnt wrote:
| Authors called Edwards and Penney IIRC, but it's lent out
| this year.
| eternalban wrote:
| That is because 6 is a priomorial. For any primorial p', k*p'
| +/- 1 will result in a number _relatively prime_ to p ', some
| of which are absolute primes.
|
| The key to understanding primes is in relative primes and
| reduced residue sets. All patterns in (higher) primes
| (absolute) are generated by the members of RRS of smaller
| primes. This includes the clusters, such as twins, triple,
| quadruple, ..., primes. RRSs also hint [imo] at intimate
| connection between complex numbers and primes.
|
| https://en.wikipedia.org/wiki/Primorial
|
| https://en.wikipedia.org/wiki/Coprime_integers
|
| https://en.wikipedia.org/wiki/Reduced_residue_system
|
| https://en.wikipedia.org/wiki/Root_of_unity
| hinkley wrote:
| You can also approximate pi as 22/7, which explains some of the
| other patterns that appear farther out.
| ihatepython wrote:
| > You can also approximate pi as 22/7
|
| Except in Indiana, where the legal value of pi is 3.2 by
| mandate.
| viciousvoxel wrote:
| I'm curious why, even if you were going to approximate it,
| it would be rounded _up_ to the nearest decimal rather than
| rounded in the standard way to 3.1?
| jjgreen wrote:
| To be fair, this was proposed but not accepted by the state
| legislature (in 1879)
| quietbritishjim wrote:
| Trivial correction:
|
| > 6k+3, or 6k-3.
|
| These are the same collection of numbers, you meant to put 6k
| instead of one of them :-)
| CGamesPlay wrote:
| This is neat and I never noticed this. For 5, I guess the
| linear component would be (2*3*5)k, but it isn't as interesting
| or useful because the constant component would be +/- 1, 7, 11,
| or 13. This method feels like it's basically a "higher order
| prime sieve".
|
| [append] Oh, and because this pattern (of 1 always being one of
| the constants) carries out for arbitrarily large linear
| coefficients, that also explains the "twin prime" phenomenon:
| https://www.youtube.com/watch?v=QKHKD8bRAro
| oluckyman wrote:
| A "higher order prime sieve" indeed:
| https://en.wikipedia.org/wiki/Sieve_of_Pritchard
| tie_ wrote:
| Isn't 4k+1/4k-1 rule trivial though?
|
| If we take any number K=N*4 divisible by 4 and >2, that'd be an
| even number by definition. The two closest odd numbers on
| either side would be (K-3), (K-1), (K+1), (K+3). As it happens
| (K+3) is the same as K(-1) for the next N, and (K-3) is the
| same as (K+1) for the previous N. So _all_ odd numbers follow
| this rule.
|
| What "4k+1 or 4k-1" says in a roundabout way is that all prime
| numbers (>2) are odd, which isn't much of a surprise.
| naniwaduni wrote:
| So is the 6k+-1 rule: 6k and 6k+-2 are all even, 6k+-3 is
| divisible by 3. You can extend this further: all primes
| greater than 5 must take one of the forms 30k+-1, 30k+-7,
| 30k+-11, 30k+-13. This is much less exciting, but ...
| suggestive. (No, not _that_ suggestion, that one isn 't
| actually true.)
|
| For a certain point of view, most of math is trivial
| corollaries.
|
| (Proof: check.)
| BadOakOx wrote:
| I think this still seems trivial with a 5th grade algebra
| knowledge.
|
| Additionally, for the 4k+1 / 4k-1 topic, it is just a
| complicated way of saying 2k+1 (as parent suggested).
| Spinnaker_ wrote:
| Has anyone approached you about writing a math textbook
| yet? You've already got the tone perfected.
| wholinator2 wrote:
| Well I'm sure it looks trivial to you. But the joys of
| math often aren't in the difficulty but the discovery.
| Would your prefer it not have been stated at all, or did
| you just want to let us know you understood it.
|
| As well, i think the 2k+1 thing is drastically more
| trivial and not at all equivalent being that all you need
| to know for 2k+1 is that 2k+1=odd. 4k and especially 6k
| take a larger generalization and different analytical
| method and often aren't included in the definition of the
| primes we learn like 2k+1 (odd) is.
| jameshart wrote:
| This simple fact blows up into a bunch of even more surprising
| things that are true of all primes when you follow the algebra
| to its logical conclusions. Like:
|
| Since this means every pair of twin primes > 3 must be
| separated by a multiple of 6, so they can be written as 6k+1,
| 6k-1. That means the product of any pair of twin primes will be
| of the form 36k^2-1.
|
| In other words take _any_ pair of twin primes, multiply them
| together, add 1, divide by 36, you are guaranteed to get a
| perfect square. E.g. 11*13=143, +1=144, /36=4 =2^2.
|
| Or (and this one's actually a little more complicated because
| it gets kind of casewise) you can show that the square of any
| prime (>3) is either one more or one less than a multiple of 24
| (which is the product of the 6 and the 4 from the 6k and 4k
| rules)
| waitforit wrote:
| Somehow I find 30k +/- 1, 7, 11, 13 more pleasing. It's 8
| numbers and the primality of each "group" can be encoded in a
| byte.
| pcmaffey wrote:
| Yes, it was such a thrill to discover this in college (not a
| math major). I formulated it as all primes >2 are "factors" of
| 1 and 5 in a base 6 notation.
| MayeulC wrote:
| Well, factor is clearly the wrong technical term here, you
| probably meant the last digit? Like the last digit (remainder
| of modulo 10 division: p%10 in C-notation) of a prime number
| (p>5) in base 10 can be {1,3,7,9}.
|
| And yes, in base 6 that becomes 1 and 5, as p%6 = 1 or 5,
| equivalent to p=6k+-1. Interesting observation, thanks :)
| [deleted]
| harles wrote:
| It could use a (2019) in the title. Interesting, but not recent.
| irrational wrote:
| Does it matter for math articles? It's not like the math has
| changed in 4 years.
| mypornaccount wrote:
| I double checked, 1+1 is still 2 in ios 16.3.1
| jagged-chisel wrote:
| In 16.4.1, you have the opportunity to make it 4.15%
| bigger.
| varjag wrote:
| On the other hand 1+2+3 is 24 in iOS 11.1
| interestica wrote:
| Sometimes it's hardware
|
| https://en.m.wikipedia.org/wiki/Pentium_FDIV_bug
| laserbeam wrote:
| It has less to do with math, and more to do with the
| conventions of posting to HN where people expect dates for
| non-recent links. There's nothing wrong with them otherwise.
| bnegreve wrote:
| The convention exists because for most articles,
| publication year is an important piece of context. I'm not
| sure we have to blindly follow the convention when it
| isn't.
| edgineer wrote:
| There is an aspect of novelty when it comes to YouTube videos
| dylan604 wrote:
| Something has to define when new math came to be /s
| thrdbndndn wrote:
| > Does it matter for math articles
|
| It matters for all the articles, so yes.
| [deleted]
| irrational wrote:
| But, why does it matter for all articles? Is it just a
| blind rule, or can we apply it just when it actually makes
| sense?
| d0mine wrote:
| The year in the title means: it is so good, it is worth
| reposting here.
| rarx wrote:
| a factor, plus one. maybe? so it's the pair of primes in that
| short sequence
| Scubabear68 wrote:
| I'm hardly a math genius, but isn't this basically because he's
| using polar coordinates, where nearly any trend is going to look
| like curves and spirals?
| hn_throwaway_99 wrote:
| Hence, the video...
| CyberDildonics wrote:
| Hence the video what? It's polar coordinates making the
| spiral, not prime numbers.
| tomxor wrote:
| This is pointed out in the video
| hn_throwaway_99 wrote:
| Hence the video making it extremely clear pretty much right
| off the bat that yes, any integer plotted that way will
| form a spiral. The video then goes into a very interesting
| discussion IMO into why you see the _specific_ patterns of
| spirals and rays with the primes.
| westcort wrote:
| The primes >3 are all of the form 6k+1 or 6k-1, so they are
| close to 6k. As k increases, the resulting marks on the
| polar coordinate system form nearly complete turns because
| 6 is close to 2 pi. Hence, spirals. See my other comment
| for a clearer explanation of why primes >3 all must
| necessarily be one more or one less than a factor of 6k
| where k is an integer. That is the crucial piece that the
| author did not make totally clear.
| euroderf wrote:
| So what does it look like if you fudge it by scaling 2 pi
| down to 6 ?
| chrisacky wrote:
| A straight line of pillars.
| [deleted]
| m3kw9 wrote:
| I can also make all integers look like sine waves sin(x)
| mlyle wrote:
| Better divide x by something or you get a pretty crummy sine.
|
| Or multiply by 19.
| quickthrower2 wrote:
| Or use the high-school sin. (Take's degrees as input)
| MayeulC wrote:
| Tangentially, this is directly related to the article, as
| x*sin(x) is the x-component of plotting (x,x) in polar
| coordinates: with more precision (with x a real), you'd obtain
| a single spiral, but with integers where 2pi[?]6, they appear
| as six.
|
| Plotting primes p like this is plotting p*e^(ip), but the
| spirals are an interesting observation. You can straighten them
| out by adding a pi factor to the polar coordinate (so iside the
| sin...) but that's less interesting.
| b33j0r wrote:
| y = (-1)**x
|
| -1 is a circle, brah!
| peter_retief wrote:
| This is one of my favourite mysteries.
|
| https://en.wikipedia.org/wiki/Goldbach%27s_conjecture
|
| How is it that all even number are the sum of 2 primes and odd
| number the sum of 3 primes?
| misja111 wrote:
| The odd number sums follow from the fact about the even
| numbers.
| peter_retief wrote:
| Why though?
| misja111 wrote:
| If you can make any even number with 2 primes, you can make
| an odd number by subtracting one odd prime from the odd
| number first (e.g. 3) and the resulting even number with 2
| more primes.
| mik1998 wrote:
| We don't know whether it's true.
| peter_retief wrote:
| It is true but unproven.
| qumpis wrote:
| Well, then it's a great mystery of up to 10^18: "T. Oliveira
| e Silva ran a distributed computer search that has verified
| the conjecture for n <= 4 x 10^18"
| hn_throwaway_99 wrote:
| I thought this video was fantastic. Just a great intro and
| example of how "playing" with numbers and visualizations can help
| you understand deeper, more profound concepts.
|
| For commenters saying "well, doesn't anything in polar
| coordinates end up like spirals?", do yourself a favor and watch
| the video. He says as much pretty early on that yes, any plotting
| of the integers like that will look like spirals, but goes into
| excellent detail in my opinion explaining why the _specific_
| patterns you see with prime numbers are as they are.
| ShaurAsar wrote:
| [dead]
| gnarbarian wrote:
| I suspect most functions would make a spiral on a polar graph if
| you zoomed out far enough.
| onethought wrote:
| Well, except as the video points out - primes don't make spiral
| when you zoom out far enough, they make a "rays" pattern...
| which then if you zoom out even further becomes a much gentler
| spiral.
| jameshart wrote:
| And the fact that pi is irrational means you will continue
| zooming out getting ever closer to rays, but never actually
| find rays - you'll always have a slight spiral.
___________________________________________________________________
(page generated 2023-04-26 23:03 UTC)