Look ma, I've solved the Prime Numbers! ;.)

Hum. I either broke the matrix or I've stumbled across nature's secret morse code.

In any case this looks very interesting I must say. Again what we see here is the outcome of my playing with the Ulam Prime Number Spiral concept. In my variation I have arranged the prime numbers up to 1000000 in a fermat spiral using an angle increase of 2.41379035550925 (henceforth called the "Klingemann Constant" ;-))

As usual I probably did not do my homework properly and others have come across this pattern before. Only neither on Wikipedia nor on Mathworld Wolfram I could find any traces of it. Also google image search did not turn up anything except the usual ones. If you know of any prior sightings please leave a link in the comments.

My next step will be to try to find the formulas for those rays, since if my totally unscientific hunch holds true that any prime number must lie on one of those rays one could devise a simple test for primality.

[UPDATE 19:30]
Holy Crap, something is at work here... Now guess what this is:
41
2441
12041
13241
15641
18041
20441
32441
33641
34841
40841
45641
50441
57641
60041
66041
70841
[...]
792041
800441
801641
805241
811241
814841
823241
830441
832841
841241
844841
853241
855641
856841
860441
867641
868841
872441
873641
876041
883241
884441
890441
892841
895241
903641
908441
915641
916841
918041
920441
924041
925241
931241
932441
937241
942041
948041
949241
951641
957641
961241
962441
966041
969641
978041
982841
992441

These are all prime numbers that lie on a single ray of this diagram.

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Comments

Quasimondo says:

Another ray:

7,3607,6007,7207,12007,14407,15607,19207,204 07,22807,24007,26407,28807,34807,36007,39607, 43207,46807,49207,51607,52807,55207,63607,780 07,80407,88807,90007,93607,98407,99607,105607 ,108007,112807,127207,129607,130807,135607,13 8007,140407,145207,146407,147607,152407,15360 7,154807,156007,157207,158407,160807,162007,1 66807,172807,174007,178807,180007,186007,1884 07,190807,192007,199207,200407,204007,206407, 208807,214807,217207,219607,220807,222007,223 207,231607,234007,236407,237607,240007,241207 ,248407,249607,250807,254407,262807,264007,26 5207,272407,276007,279607,283207,284407,28800 7,292807,297607,300007,304807,312007,313207,3 14407,318007,322807,331207,338407,345607,3492 07,354007,360007,362407,367207,373207,375607, 376807,379207,381607,382807,398407,405607,406 807,412807,416407,421207,422407,426007,432007 ,433207,434407,441607,442807,444007,448807,45 1207,456007,458407,459607,463207,469207,47160 7,475207,476407,481207,482407,488407,492007,4 94407,502807,507607,510007,517207,523207,5256 07,534007,535207,536407,543607,544807,548407, 549607,553207,558007,561607,565207,567607,568 807,578407,580807,583207,589207,590407,595207 ,601207,603607,609607,615607,619207,624007,63 4807,648007,656407,657607,658807,662407,66840 7,669607,673207,675607,676807,679207,681607,6 84007,692407,693607,702007,708007,712807,7188 07,720007,721207,723607,724807,726007,729607, 748807,751207,760807,762007,766807,769207,771 607,787207,793207,794407,800407,801607,804007 ,807607,811207,819607,822007,826807,828007,83 4007,835207,838807,841207,842407,843607,85080 7,854407,855607,864007,870007,874807,879607,8 85607,886807,897607,900007,901207,903607,9060 07,910807,912007,920407,922807,933607,936007, 937207,948007,954007,958807,963607,978007,979 207,984007,987607,997207

Rule: last two digits are 07 and the rest of the number if 07 is chopped off is divisible by 12
Posted 5 months ago. ( permalink )

Quasimondo says:

11,2411,6011,7211,12011,14411,16811,19211,204 11,21611,22811,27611,30011,32411,36011,46811, 49211,50411,54011,61211,63611,64811,67211,744 11,75611,81611,82811,84011,88811,90011,94811, 98411,99611,100811,108011,109211,111611,11521 1,116411,120011,126011,128411,130811,136811,1 40411,142811,150011,153611,156011,157211,1620 11,163211,165611,175211,181211,183611,187211, 190811,199211,201611,205211,206411,210011,212 411,213611,214811,220811,222011,223211,225611 ,231611,232811,235211,240011,246011,259211,26 0411,266411,267611,268811,271211,272411,27481 1,276011,280811,282011,283211,285611,301211,3 02411,306011,308411,313211,319211,321611,3240 11,332411,338411,340811,348011,349211,350411, 355211,357611,358811,361211,363611,368411,373 211,376811,378011,386411,393611,394811,397211 ,405611,406811,408011,410411,411611,418811,42 4811,426011,428411,430811,434411,436811,44761 1,450011,459611,465611,468011,470411,478811,4 81211,483611,487211,493211,495611,499211,5040 11,508811,511211,517211,518411,519611,524411, 531611,532811,537611,543611,550811,552011,553 211,556811,559211,560411,571211,579611,582011 ,584411,586811,588011,591611,600011,602411,60 4811,608411,612011,621611,627611,628811,64201 1,644411,645611,652811,654011,655211,667211,6 69611,670811,675611,680411,682811,697211,7008 11,702011,703211,708011,709211,716411,722411, 727211,730811,734411,738011,746411,747611,753 611,754811,756011,758411,766811,781211,789611 ,801611,802811,817211,822011,825611,826811,82 8011,829211,830411,837611,852011,853211,85681 1,862811,864011,865211,872411,876011,878411,8 85611,888011,889211,894011,895211,901211,9048 11,906011,907211,909611,915611,918011,920411, 921611,926411,930011,934811,939611,949211,952 811,954011,955211,957611,966011,974411,978011 ,979211,986411,993611,994811,996011,998411,99 9611
Posted 5 months ago. ( permalink )

Quasimondo says:

13,1213,3613,4813,7213,9613,18013,19213,21613 ,28813,30013,32413,33613,36013,40813,42013,45 613,51613,52813,54013,55213,60013,67213,74413 ,82813,85213,86413,87613,88813,92413,96013,97 213,102013,105613,108013,114013,126013,128413 ,133213,135613,136813,141613,144013,145213,14 7613,151213,160813,164413,168013,170413,17641 3,178813,181213,186013,189613,192013,194413,1 98013,204013,205213,206413,211213,213613,2196 13,225613,226813,228013,229213,231613,242413, 243613,244813,250813,252013,254413,255613,256 813,259213,260413,264013,267613,268813,273613 ,277213,278413,279613,284413,285613,286813,28 9213,294013,297613,303613,304813,315613,32881 3,331213,337213,338413,339613,352813,357613,3 61213,366013,370813,372013,373213,382813,3864 13,387613,388813,394813,399613,410413,411613, 414013,415213,416413,418813,427213,441613,456 013,457213,460813,462013,463213,464413,470413 ,478813,480013,482413,487213,489613,492013,49 4413,495613,496813,498013,500413,505213,51121 3,520813,523213,524413,528013,529213,531613,5 34013,544813,550813,559213,562813,564013,5664 13,570013,574813,576013,579613,580813,582013, 583213,586813,589213,597613,603613,607213,609 613,613213,614413,620413,622813,625213,633613 ,642013,643213,650413,660013,666013,681613,68 8813,692413,698413,705613,716413,718813,72121 3,724813,726013,729613,746413,752413,760813,7 65613,766813,768013,783613,786013,788413,7944 13,796813,805213,807613,810013,813613,818413, 822013,824413,825613,828013,830413,834013,835 213,836413,838813,841213,843613,847213,852013 ,856813,859213,860413,861613,864013,865213,87 0013,874813,876013,877213,878413,880813,88321 3,898813,901213,902413,903613,906013,907213,9 09613,915613,922813,928813,931213,932413,9336 13,939613,940813,942013,943213,949213,952813, 962413,966013,970813,973213,985213,990013,994 813,999613
Posted 5 months ago. ( permalink )

Quasimondo says:

17,1217,2417,3617,4817,13217,21617,22817,2641 7,27617,28817,33617,36017,37217,42017,44417,4 6817,48017,50417,52817,55217,56417,60017,6241 7,63617,64817,67217,75617,78017,84017,88817,9 0017,96017,103217,104417,117617,120017,127217 ,130817,134417,135617,140417,146417,147617,14 8817,152417,157217,159617,160817,162017,16561 7,169217,171617,174017,176417,178817,182417,1 87217,188417,189617,196817,198017,202817,2112 17,214817,218417,223217,226817,229217,240017, 246017,252017,255617,260417,266417,268817,271 217,272417,273617,274817,277217,280817,297617 ,298817,300017,302417,303617,315617,316817,32 0417,321617,325217,330017,331217,332417,33721 7,339617,344417,348017,352817,354017,361217,3 80417,384017,391217,394817,397217,398417,3996 17,406817,409217,411617,414017,416417,417617, 423617,424817,438017,439217,442817,446417,447 617,453617,470417,471617,472817,474017,480017 ,488417,492017,493217,495617,496817,500417,50 1617,504017,506417,508817,516017,517217,51841 7,522017,538817,541217,543617,546017,548417,5 54417,556817,558017,559217,564017,566417,5724 17,574817,582017,584417,598817,606017,609617, 610817,614417,615617,621617,627617,630017,634 817,636017,643217,649217,651617,657617,658817 ,661217,668417,675617,680417,684017,686417,69 7217,698417,699617,702017,703217,708017,70921 7,711617,722417,723617,728417,735617,736817,7 39217,742817,748817,751217,753617,759617,7620 17,766817,770417,774017,777617,781217,786017, 787217,790817,799217,800417,801617,804017,819 617,831617,835217,842417,848417,853217,854417 ,860417,861617,865217,866417,867617,868817,87 3617,876017,879617,882017,883217,884417,89161 7,892817,896417,907217,908417,910817,913217,9 25217,928817,932417,940817,942017,944417,9552 17,960017,962417,970817,972017,974417,976817, 978017,980417,984017,986417,991217,992417,993 617,994817

There is indeed a rule: the last two digits are 17 and the remainding number when you chop of the 17 is divisible by 3 and 4
Posted 5 months ago. ( permalink )

davebollinger says:

i wonder if the Klingemann Constant ;) isn't a factor of something else...
if you examine the pattern of rays around the circle, you'll find repeating groupings of 1,2,2,1..2.. rays. i haven't bothered to count how many times that grouping repeats, but... what would happen if you multiplied your angular constant by the number of groupings? would you be left with a single instance of that 8-ray grouping? (or in general, what happens to those groupings when you multiply your constant by some integer n, say 2, for example?)
Posted 5 months ago. ( permalink )

Prehensile Eye says:

try floating this to a number theorist in some university. i'm sure it would be interesting to them. :)
Posted 5 months ago. ( permalink )

Quasimondo says:

Dave - now that I have cooled down a bit I've finally understood what you mean - yes that pattern stings in the eye, gosh - it might be only 8 angles after all. 160 rays divided by 8 is 20 groups, maybe I can go down to 18° segments.
Posted 5 months ago. ( permalink )

Quasimondo says:

Yes, there are indeed 20 groups. When I clean up the roundoff errors I get a series of distances that are spaced out approximately 31,21,10,21,11,21,31,11
Posted 5 months ago. ( permalink )

Quasimondo says:

@drauh - yes at the University they will be as happy about is as the scientists in Douglas Adams' Hitchhikers Guide:

"Then, one day, a student who had been left to sweep up after a particularly unsuccessful party found himself reasoning in this way: If, he thought to himself, such a machine is a virtual impossibility, it must have finite improbability. So all I have to do in order to make one is to work out how exactly improbable it is, feed that figure into the finite improbability generator, give it a fresh cup of really hot tea... and turn it on!

He did this and was rather startled when he managed to create the long sought after golden Infinite Improbability generator. He was even more startled when just after he was awarded the Galactic Institute's Prize for Extreme Cleverness he was lynched by a rampaging mob of respectably physicists who had realized that one thing they couldn't stand was a smart-ass."
Posted 5 months ago. ( permalink )

lennyjpg says:

:)
Posted 5 months ago. ( permalink )

Quasimondo says:

Discussion continues here: www.flickr.com/photos/quasimondo/3785322473/
Posted 5 months ago. ( permalink )

Quasimondo says:

In case you came here first - check this: incubator.quasimondo.com/flash/wheel_of_prime s.php
Posted 5 months ago. ( permalink )

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Look ma, I've solved the Prime Numbers! ;.)

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