# Prime Number Pattern **URL:** \ **Category:** Factual Questions\ **Created:** [May 12, 2016, 3:16pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610 "2016-05-12T15:16:29Z")\ **Posts on this page:** 19\ **Page:** 1
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [May 12, 2016, 3:16pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/1 "2016-05-12T15:16:29Z")
It has been discovered that, if the number 30 is centered as an origin, prime numbers are symmetrically distributed like ripples. [http://drasticartsandsciences.yolasite.com/picture-gallery.php](http://drasticartsandsciences.yolasite.com/picture-gallery.php) The symmetry seems to be repeated every 30 units. [http://drasticartsandsciences.yolasite.com/picture-gallery-5.php](http://drasticartsandsciences.yolasite.com/picture-gallery-5.php) How well does this aid in the search for unknown primes? ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [May 12, 2016, 3:29pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/2 "2016-05-12T15:29:52Z")
It would aid only if there were a predictive formula it generated. Baker does nothing except create pretty patterns, and everybody knows already [you can create pretty patterns with primes](https://www.google.com/search?newwindow=1&site=&tbm=isch&source=hp&biw=1080&bih=1780&q=prime+numbers+patterns&oq=prime+numbers+patterns&gs_l=img.3...1190.4723.0.4898.22.12.0.10.10.0.123.1024.10j2.12.0....0...1ac.1.64.img..0.21.1046...0j0i30j0i5i30j0i24.e2FemH1FyEw). All he says is “Mathematicians and Prime number enthusiasts are left to further explore the higher numbers that this attached art promises will reveal greater values.” That’s real nice of him. His site has been around for years and nobody has paid any attention to him. My prediction is that nobody ever will. ---
**Author:** ![friedo](https://avatars.discourse-cdn.com/v4/letter/f/8edcca/32.png) [@friedo](https://boards.straightdope.com/u/friedo)\ **Post date:** [May 12, 2016, 3:30pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/3 "2016-05-12T15:30:16Z")
All that’s been “discovered” here is that 30 is a multiple of six. It’s been known for a few thousand years that all prime numbers \>3 satisfy 6n±1 where n is an integer. So if you look at multiples of 30 (which are multiples of six) there is a good probability you will find primes nearby. ---
**Author:** ![MonkeyMensch](https://avatars.discourse-cdn.com/v4/letter/m/82dd89/32.png) [@MonkeyMensch](https://boards.straightdope.com/u/MonkeyMensch)\ **Post date:** [May 12, 2016, 3:58pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/4 "2016-05-12T15:58:11Z")
[QUOTE=friedo] All that’s been “discovered” here is that 30 is a multiple of six. It’s been known for a few thousand years that all prime numbers \>3 satisfy 6n±1 where n is an integer. So if you look at multiples of 30 (which are multiples of six) there is a good probability you will find primes nearby. [/QUOTE] And additionally 30 is the product of the first three primes (30 = 2 \* 3 \* 5). I have no doubt that there are similar “resonances” using 210 (210 = 2 \*3 \* 5 \* 7) as an offset. The margins of my graph paper are not large enough to include such a proof here. [QUOTE=Some stoner geek] This document declares that there is now a pattern and a model to work from in which Prime numbers may be predicted and plotted. [/QUOTE] Ahh, math by declaration… ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [May 12, 2016, 4:12pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/5 "2016-05-12T16:12:51Z")
If by “prime” you mean a number not divisible by 2, 3, or 5, it works perfectly. Notice that the number symmetric with 11 is 49 which is certainly not divisible by 2, 3, or 5, but is the least composite number that isn’t. This is irrevelant for finding actual primes. ---
**Author:** ![Isilder](https://avatars.discourse-cdn.com/v4/letter/i/cc9497/32.png) [@Isilder](https://boards.straightdope.com/u/Isilder)\ **Post date:** [May 13, 2016, 7:24am UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/6 "2016-05-13T07:24:35Z")
Its silly to show patterns in the small numbers. Thats like drawing lines across the times table(s)… Here’s a reason Baker’s small number patterns don’t help. The largest gap below X is AT LEAST log X \* log log X \* log log log log X / (3 \* log log log X \* log log log X ) That means that the gaps just keep getting bigger and bigger. From [http://www.wired.com/2014/12/mathematicians-make-major-discovery-prime-numbers/](http://www.wired.com/2014/12/mathematicians-make-major-discovery-prime-numbers/) ---
**Author:** ![RickJay](https://avatars.discourse-cdn.com/v4/letter/r/bb73d2/32.png) [@RickJay](https://boards.straightdope.com/u/RickJay)\ **Post date:** [May 13, 2016, 1:31pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/7 "2016-05-13T13:31:37Z")
It’s not really a perfect pattern anyway, so I admit I don’t see the point. The “ripples” are not evenly spaced. Uhh… and **he doesn’t even get the prime numbers correct**. In order to make his ripples work he’s left some prime out. For instance, he does not list 307 as a prime number, but it is. If he had 307 as a prime it’d screw up his “ripples” entirely. I don’t have the time to go through and see what other primes he left out, but come on. ---
**Author:** ![gnoitall](https://avatars.discourse-cdn.com/v4/letter/g/bb73d2/32.png) [@gnoitall](https://boards.straightdope.com/u/gnoitall)\ **Post date:** [May 13, 2016, 1:52pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/8 "2016-05-13T13:52:08Z")
> [@MonkeyMensch](#): > > Ahh, math by declaration… Indeed. [This paper](https://en.wikipedia.org/wiki/Indiana_Pi_Bill) declares that π is 3.2. If that doesn’t foot-stomp the limitations of unprovable mathematical declarations, nothing does. ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [May 13, 2016, 3:54pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/9 "2016-05-13T15:54:26Z")
> [@Enola\_Straight](#): > > It has been discovered that, if the number 30 is centered as an origin, prime numbers are symmetrically distributed like ripples. > [http://drasticartsandsciences.yolasite.com/picture-gallery.php](http://drasticartsandsciences.yolasite.com/picture-gallery.php) > > The symmetry seems to be repeated every 30 units. > [http://drasticartsandsciences.yolasite.com/picture-gallery-5.php](http://drasticartsandsciences.yolasite.com/picture-gallery-5.php) > > How well does this aid in the search for unknown primes? > [@](#): > > In some instances, (Gallery 3) a Prime number on the left or right side of center does not have a corresponding Prime on the other side, but the arc fits and enhances the overall pattern structure despite this. These arcs land on composites of Prime numbers on their opposite side. Examples: -49 + 109 = 60, however -49 is not prime. It is a composite; 7 x 7 = 49. -77 + 137 = 60. 77 is a composite of 7 x 11, two Primes. No, it does not “fit”, which is just one reason this is utter nonsense and useless. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [May 13, 2016, 4:59pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/10 "2016-05-13T16:59:00Z")
How many times are we going to go through this? [http://boards.straightdope.com/sdmb/showthread.php?p=10557247](http://boards.straightdope.com/sdmb/showthread.php?p=10557247) [http://boards.straightdope.com/sdmb/showthread.php?p=15549783](http://boards.straightdope.com/sdmb/showthread.php?p=15549783) [http://boards.straightdope.com/sdmb/showthread.php?p=18829806](http://boards.straightdope.com/sdmb/showthread.php?p=18829806) Etc. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [May 13, 2016, 5:16pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/11 "2016-05-13T17:16:25Z")
As to this particular instance, quoting the website in the OP: “You will notice that not all Prime numbers are marked on this number line. Some of the Prime numbers on one side of the line do not have corresponding Primes on the opposite side.” So as **RickJay** says, it’s not even a reliable pattern. What’s so interesting about it? Essentially, the observation is that frequently, when 30 + x is prime, so is |30 - x| (= x - 30 for x \> 30), and vice versa. Sure; to be a prime (\> 5), you mustn’t be divisible by 2, 3, or 5 (among other conditions), and 30 + x satisfies these three particular conditions if and only if |30 - x| does, so the one being prime makes it somewhat more likely than “random” that the other is. But nothing here is particularly special to 30. In fact, we can replace 30 here with any value and we should still expect to find lots of prime pairs centered at that value (it’s conjectured, albeit not proven, that there will always be infinitely many such pairs, no matter what value we replace 30 with, and indeed, much stronger quantitative density results than this are conjectured as well). ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [May 13, 2016, 5:20pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/12 "2016-05-13T17:20:10Z")
“How well does this aid in the search for unknown primes?” No more than the trivial observation that primes cannot be divisible by 2, 3, or 5 already does. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [May 13, 2016, 5:34pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/13 "2016-05-13T17:34:56Z")
> [@Indistinguishable](#): > > In fact, we can replace 30 here with any value and we should still expect to find lots of prime pairs centered at that value For example, replacing 30 with 1 would cause the “ripples” to be located at positions corresponding to the familiar “twin” primes; i.e., primes separated by 2, of which it is among other things conjectured that there are infinitely many, though most primes are not of this form. (The “ripples” in the original diagram, similarly, correspond to primes separated by 60 (once hitting the negative values)). [By the Hardy-Littlewood conjecture, we do expect to see particularly frequent pairs when picking a center with many small distinct prime factors; thus, 30, as 2 \* 3 \* 5, is a particularly good choice. But even better would be 210 = 2 \* 3 \* 5 \* 7, etc.] ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [May 13, 2016, 5:41pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/14 "2016-05-13T17:41:20Z")
Oh, amending that last note: [By the Hardy-Littlewood conjecture, we do expect to see particularly frequent pairs when picking a center with many small distinct odd prime factors; thus, 30, as 2 \* 3 \* 5, is a particularly good choice. But just as good would be 15 = 3 \* 5, and even better would be 105 = 3 \* 5 \* 7, etc.] ---
**Author:** ![D18](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@D18](https://boards.straightdope.com/u/D18)\ **Post date:** [May 13, 2016, 6:48pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/15 "2016-05-13T18:48:14Z")
> [@Indistinguishable](#): > > How many times are we going to go through this? > > [http://boards.straightdope.com/sdmb/showthread.php?p=10557247](http://boards.straightdope.com/sdmb/showthread.php?p=10557247) > [http://boards.straightdope.com/sdmb/showthread.php?p=15549783](http://boards.straightdope.com/sdmb/showthread.php?p=15549783) > [http://boards.straightdope.com/sdmb/showthread.php?p=18829806](http://boards.straightdope.com/sdmb/showthread.php?p=18829806) > > Etc. [GEORGE CARLIN] Six! [/GEORGE CARLIN] ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [May 13, 2016, 7:44pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/16 "2016-05-13T19:44:46Z")
> [@](#): > > In some instances, (Gallery 3) a Prime number on the left or right side of center does not have a corresponding Prime on the other side, but the arc fits and enhances the overall pattern structure despite this. These arcs land on composites of Prime numbers on their opposite side. Well there you go! It’s always either a prime, or a composite of primes! Just like, oh, every other number other than 0 or 1. ---
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [May 14, 2016, 3:13am UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/17 "2016-05-14T03:13:11Z")
> [@Indistinguishable](#): > > How many times are we going to go through this? > > [http://boards.straightdope.com/sdmb/showthread.php?p=10557247](http://boards.straightdope.com/sdmb/showthread.php?p=10557247) > [http://boards.straightdope.com/sdmb/showthread.php?p=15549783](http://boards.straightdope.com/sdmb/showthread.php?p=15549783) > [http://boards.straightdope.com/sdmb/showthread.php?p=18829806](http://boards.straightdope.com/sdmb/showthread.php?p=18829806) > > Etc. Jeez…you make me sound loopy like that **Hyruu** guy infatuated with PHI. 😉 ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [May 16, 2016, 4:16pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/18 "2016-05-16T16:16:00Z")
I was going to let this all go, but it’s been gnawing at me to have told quite the full story here: > [@Indistinguishable](#): > > Essentially, the observation is that frequently, when 30 + x is prime, so is |30 - x| (= x - 30 for x \> 30), and vice versa. Sure; to be a prime (\> 5), you mustn’t be divisible by 2, 3, or 5 (among other conditions), and 30 + x satisfies these three particular conditions if and only if |30 - x| does, so the one being prime makes it somewhat more likely than “random” that the other is. In fairness, it’s also the case that if 30 + x is divisible by, say, 7, then |30 - x| isn’t, and so, conversely, if 30 + x is prime and therefore NOT divisible by 7, then |30 - x| is MORE likely than normal to be divisible by 7. So this divisibility correlation cuts the other way for primes other than 2, 3, or 5. However, we can quantify the two effects and thus their combination, like so: Given that 30 + x is not divisible by the prime p, we find that the probability of |30 - x| satisfying the same property is (1 - m/p)/(1 - 1/p)^2 times as large as usual, where m is 1 or 2 according as to whether p does or does not divide 2 \* 30. This acts independently over separate primes, and the product over all p comes out to a hair over 3.52; thus, assuming this is the only mechanism through which primality of 30 + x and |30 - x| are correlated (this is the Hardy-Littlewood conjecture), we find that they are about 3.52 times as likely to be prime as two randomly selected values of the same general magnitude. So the boost from {2, 3, 5} (accounting for a factor of precisely 3.75) more than outweighs the opposite effect, omitted in my previous post, from all other primes (accounting for a factor of just under 0.94). (It works similarly for any center, not just 30; in looking for primes separated by a distance of 2 \* r, even just the correlation boost from the prime factor of 2 alone outweighs the potential opposite effect from all other primes combined, so that such prime pairs are more likely by some factor than just any random pair of values of similar magnitude (again, assuming the Hardy-Littlewood conjecture)) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [May 16, 2016, 4:27pm UTC](https://boards.straightdope.com/t/prime-number-pattern/754610/19 "2016-05-16T16:27:41Z")
> [@Indistinguishable](#): > > I was going to let this all go, but it’s been gnawing at me to have **not** told quite the full story here: Missing word reinstated in bold