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

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    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.

    mímesis, philipplaeubli, and 9 other people added this photo to their favorites.

    1. Quasimondo 57 months ago | reply

      Another ray:

      7,3607,6007,7207,12007,14407,15607,19207,20407,22807,24007,26407,28807,34807,36007,39607,43207,46807,49207,51607,52807,55207,63607,78007,80407,88807,90007,93607,98407,99607,105607,108007,112807,127207,129607,130807,135607,138007,140407,145207,146407,147607,152407,153607,154807,156007,157207,158407,160807,162007,166807,172807,174007,178807,180007,186007,188407,190807,192007,199207,200407,204007,206407,208807,214807,217207,219607,220807,222007,223207,231607,234007,236407,237607,240007,241207,248407,249607,250807,254407,262807,264007,265207,272407,276007,279607,283207,284407,288007,292807,297607,300007,304807,312007,313207,314407,318007,322807,331207,338407,345607,349207,354007,360007,362407,367207,373207,375607,376807,379207,381607,382807,398407,405607,406807,412807,416407,421207,422407,426007,432007,433207,434407,441607,442807,444007,448807,451207,456007,458407,459607,463207,469207,471607,475207,476407,481207,482407,488407,492007,494407,502807,507607,510007,517207,523207,525607,534007,535207,536407,543607,544807,548407,549607,553207,558007,561607,565207,567607,568807,578407,580807,583207,589207,590407,595207,601207,603607,609607,615607,619207,624007,634807,648007,656407,657607,658807,662407,668407,669607,673207,675607,676807,679207,681607,684007,692407,693607,702007,708007,712807,718807,720007,721207,723607,724807,726007,729607,748807,751207,760807,762007,766807,769207,771607,787207,793207,794407,800407,801607,804007,807607,811207,819607,822007,826807,828007,834007,835207,838807,841207,842407,843607,850807,854407,855607,864007,870007,874807,879607,885607,886807,897607,900007,901207,903607,906007,910807,912007,920407,922807,933607,936007,937207,948007,954007,958807,963607,978007,979207,984007,987607,997207

      Rule: last two digits are 07 and the rest of the number if 07 is chopped off is divisible by 12

    2. Quasimondo 57 months ago | reply

      11,2411,6011,7211,12011,14411,16811,19211,20411,21611,22811,27611,30011,32411,36011,46811,49211,50411,54011,61211,63611,64811,67211,74411,75611,81611,82811,84011,88811,90011,94811,98411,99611,100811,108011,109211,111611,115211,116411,120011,126011,128411,130811,136811,140411,142811,150011,153611,156011,157211,162011,163211,165611,175211,181211,183611,187211,190811,199211,201611,205211,206411,210011,212411,213611,214811,220811,222011,223211,225611,231611,232811,235211,240011,246011,259211,260411,266411,267611,268811,271211,272411,274811,276011,280811,282011,283211,285611,301211,302411,306011,308411,313211,319211,321611,324011,332411,338411,340811,348011,349211,350411,355211,357611,358811,361211,363611,368411,373211,376811,378011,386411,393611,394811,397211,405611,406811,408011,410411,411611,418811,424811,426011,428411,430811,434411,436811,447611,450011,459611,465611,468011,470411,478811,481211,483611,487211,493211,495611,499211,504011,508811,511211,517211,518411,519611,524411,531611,532811,537611,543611,550811,552011,553211,556811,559211,560411,571211,579611,582011,584411,586811,588011,591611,600011,602411,604811,608411,612011,621611,627611,628811,642011,644411,645611,652811,654011,655211,667211,669611,670811,675611,680411,682811,697211,700811,702011,703211,708011,709211,716411,722411,727211,730811,734411,738011,746411,747611,753611,754811,756011,758411,766811,781211,789611,801611,802811,817211,822011,825611,826811,828011,829211,830411,837611,852011,853211,856811,862811,864011,865211,872411,876011,878411,885611,888011,889211,894011,895211,901211,904811,906011,907211,909611,915611,918011,920411,921611,926411,930011,934811,939611,949211,952811,954011,955211,957611,966011,974411,978011,979211,986411,993611,994811,996011,998411,999611

    3. Quasimondo 57 months ago | reply

      13,1213,3613,4813,7213,9613,18013,19213,21613,28813,30013,32413,33613,36013,40813,42013,45613,51613,52813,54013,55213,60013,67213,74413,82813,85213,86413,87613,88813,92413,96013,97213,102013,105613,108013,114013,126013,128413,133213,135613,136813,141613,144013,145213,147613,151213,160813,164413,168013,170413,176413,178813,181213,186013,189613,192013,194413,198013,204013,205213,206413,211213,213613,219613,225613,226813,228013,229213,231613,242413,243613,244813,250813,252013,254413,255613,256813,259213,260413,264013,267613,268813,273613,277213,278413,279613,284413,285613,286813,289213,294013,297613,303613,304813,315613,328813,331213,337213,338413,339613,352813,357613,361213,366013,370813,372013,373213,382813,386413,387613,388813,394813,399613,410413,411613,414013,415213,416413,418813,427213,441613,456013,457213,460813,462013,463213,464413,470413,478813,480013,482413,487213,489613,492013,494413,495613,496813,498013,500413,505213,511213,520813,523213,524413,528013,529213,531613,534013,544813,550813,559213,562813,564013,566413,570013,574813,576013,579613,580813,582013,583213,586813,589213,597613,603613,607213,609613,613213,614413,620413,622813,625213,633613,642013,643213,650413,660013,666013,681613,688813,692413,698413,705613,716413,718813,721213,724813,726013,729613,746413,752413,760813,765613,766813,768013,783613,786013,788413,794413,796813,805213,807613,810013,813613,818413,822013,824413,825613,828013,830413,834013,835213,836413,838813,841213,843613,847213,852013,856813,859213,860413,861613,864013,865213,870013,874813,876013,877213,878413,880813,883213,898813,901213,902413,903613,906013,907213,909613,915613,922813,928813,931213,932413,933613,939613,940813,942013,943213,949213,952813,962413,966013,970813,973213,985213,990013,994813,999613

    4. Quasimondo 57 months ago | reply

      17,1217,2417,3617,4817,13217,21617,22817,26417,27617,28817,33617,36017,37217,42017,44417,46817,48017,50417,52817,55217,56417,60017,62417,63617,64817,67217,75617,78017,84017,88817,90017,96017,103217,104417,117617,120017,127217,130817,134417,135617,140417,146417,147617,148817,152417,157217,159617,160817,162017,165617,169217,171617,174017,176417,178817,182417,187217,188417,189617,196817,198017,202817,211217,214817,218417,223217,226817,229217,240017,246017,252017,255617,260417,266417,268817,271217,272417,273617,274817,277217,280817,297617,298817,300017,302417,303617,315617,316817,320417,321617,325217,330017,331217,332417,337217,339617,344417,348017,352817,354017,361217,380417,384017,391217,394817,397217,398417,399617,406817,409217,411617,414017,416417,417617,423617,424817,438017,439217,442817,446417,447617,453617,470417,471617,472817,474017,480017,488417,492017,493217,495617,496817,500417,501617,504017,506417,508817,516017,517217,518417,522017,538817,541217,543617,546017,548417,554417,556817,558017,559217,564017,566417,572417,574817,582017,584417,598817,606017,609617,610817,614417,615617,621617,627617,630017,634817,636017,643217,649217,651617,657617,658817,661217,668417,675617,680417,684017,686417,697217,698417,699617,702017,703217,708017,709217,711617,722417,723617,728417,735617,736817,739217,742817,748817,751217,753617,759617,762017,766817,770417,774017,777617,781217,786017,787217,790817,799217,800417,801617,804017,819617,831617,835217,842417,848417,853217,854417,860417,861617,865217,866417,867617,868817,873617,876017,879617,882017,883217,884417,891617,892817,896417,907217,908417,910817,913217,925217,928817,932417,940817,942017,944417,955217,960017,962417,970817,972017,974417,976817,978017,980417,984017,986417,991217,992417,993617,994817

      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

    5. davebollinger 57 months ago | reply

      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?)

    6. Prehensile Eye 57 months ago | reply

      try floating this to a number theorist in some university. i'm sure it would be interesting to them. :)

    7. Quasimondo 57 months ago | reply

      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.

    8. Quasimondo 57 months ago | reply

      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

    9. Quasimondo 57 months ago | reply

      @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."

    10. anabela montes 12 months ago | reply

      how did you get the number 2.41379035550925, the increase angle of your nice spiral?

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