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#1 2015-11-27 04:07:03

blackcap
Member
Registered: 2015-11-20
Posts: 7

Interesting prime gap patterns (with tons of pictures!)

So I started off drawing curves between the primes on a number line
eMf30to.png

I then drew curves between all the other numbers using the following rules:
1) Draw a upwards curve to a number that is not already taken
2) If the next number is also available, skip until you find a number that is taken, and then draw a downwards curve to the number before it
Repeat
OWkdI6S.png

So I did this for a while in geogebra..
iWMdMHD.png
(That's all the primes smaller than 200)

..until I finally got bored and decided to write a program to do it for me.
This is the first 600 primes:
X2E2wWX.png

There is a few interesting things to note here. A circle usually have between 1 and 3 smaller circles inside it:
1) They are next to each other on the number line
2) The two child circles will be of the same size
3) You often get one child circle in the middle with two circles of the same size at either side

There are examples of circles with both 4 and 5 children, but they are rare.


First 1200 primes:
2p9L2FD.png

First 2400 primes:
5T2k3Y4.png

Notice how all the big circles has bumps under their legs? Here's a zoomed in picture of the biggest lump from the 2400 picture:
DBXiYXx.png
It appears as if all big circles are born from these black holes of primality (it's actually the opposite, it's a huge prime gap). Over to the right in the picture you'll see a black hole that hasn't even finished rendering (I only asked for the first 2400 primes, so there was no gap large enough for the remaining curves to go to.
Also interesting to note is that the three curves that outlines the rightmost circle, is the two other big circles and a new one.

Now I am sure that you are sitting at the edge of your chair, shivering in anticipation for how that un-rendered black hole from 2400 is going to turn out. There is already 4 circles for the up and coming mega circle to hold, will it have more than 4? Well, shiver no more, for we need not go further than 2600 to find out:
kA55BIU.png

The black hole is actually not fully sated until 6000:
XCBdASm.png
Nyjnjmg.png
(Is it just me, or does all the circles appear to "point" in the same angle?)

I'm leaving you with this svg (infinitely zoomable vector image) of the first 10k primes:

Last edited by blackcap (2015-11-27 06:48:21)

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#2 2015-11-27 06:24:53

blackcap
Member
Registered: 2015-11-20
Posts: 7

Re: Interesting prime gap patterns (with tons of pictures!)

A few more curiosities.

h7fCnZu.png
^ This is a rainbow bat. He is commonly found lurking next to, or more often, in between black holes.

Here's a pair of twin rainbow bats:
eaRsB3V.png

Also, the smaller a circle is, the more densely populated with primes it is going to be:
4IHQjeO.png
(the tacks at the bottom are primes, note the rainbow bat in the center)

Last edited by blackcap (2015-11-27 06:27:12)

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#3 2016-01-06 00:08:34

zetafunc
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Registered: 2014-05-21
Posts: 2,108
Website

Re: Interesting prime gap patterns (with tons of pictures!)

Lovely pictures, thanks for sharing.

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#4 2016-02-12 02:03:08

mathaholic
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From: Earth
Registered: 2012-11-29
Posts: 3,251

Re: Interesting prime gap patterns (with tons of pictures!)

That's cool! big_smile


Mathaholic | 10th most active poster | Maker of the 350,000th post | Person | rrr's classmate
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