New Spin On Prime Sequence Spiral Sieves?

[Hemiboso]

Hello,

I'm seeking one or more number theorists who would kindly (or unkindly--I can take it :-) critique an algorithmic prime sieving spiral (and related rectangular array) that I believe has novelty (perhaps due to delusions of grandeur--oh well; it's fun being deluded!).

 

Just this week I purchased a URL and posted my work (the result of 15 years attempting to penetrate the mysteries of this noble sequence using my favorite heuristics, trial-and-error and brute force), and some explanatory text here:

 

http://www.primesdemystified.com/

 

Please excuse the site's name: While it's true that I have demystified the prime number sequence to my own satisfaction, I doubt very much if a "bonafide" mathematician would agree.

 

Best regards,

Gary Croft

[email protected]

[Farsight]

I'm into physics Gary, and I can't offer much. But your algorithmic rectangular sieve reminds me a little of the way I see primes in my very simplistic way. You have n frictionless marbles on a flat table, within a sliding rectangular enclosure which can range from a 2-marble-wide shape like this | thru this □ to this ―. If you can slide the enclosure around to exactly fit the n balls, then n is not prime. You then repeat for n+1 etc and essentially derive patterns. Then you use these patterns to improve tractability within say cryptography because you cannot contrive this massively-parallel frictionless slider in a computing environment.

 

Your spiral-sieve looks interesting. I wonder if you could depict it on a 3-dimensional "bullet" shape with a circumference of 30 to show a rifling pattern with the cookie-cutter gaps. But regardless, it does very much sound like primes demystified. Do you have a smaller version of the image below anywhere? It would be nice to be able to post it up in a nutshell so it's right there in front of the casual reader.

 

[Qfwfq]

I'm seeking one or more number theorists who would kindly (or unkindly--I can take it :-) critique an algorithmic prime sieving spiral

The spiral arrangement is pretty much ad hoc, it seems that what you are really doing is making each next round a range of 30 numbers. One thing you might be interested to know:

The first sentence is an example of the Euler heuristics. The possible remainders that you specify are those that have no factors in common with 30 (which is [tex]2\cdot 3\cdot 5[/tex]). This goes for any number besides 30, it gives an infinity of necessary and not sufficient conditions (some more useful, others less) for finding primes. Obviously the remainder mustn't be 0 and if it shares any factor(s) with the divisor then the given number will be a multiple of those factor(s).

 

Try fiddling with [tex]210=2\cdot 3\cdot 5\cdot 7[/tex] and [tex]2310=2\cdot 3\cdot 5\cdot 7\cdot 11[/tex]

[phillip1882]

i've found several interesting relationships with the primes. here's my personal favorite.

if p1 is prime >2, and p1+c is prime, then there is another prime, p2 with p2+c also prime, such that p2+c < 2*(p1+c).

for example with twin primes, 5,7; 11,13; 13 < 14. 17,19; 19 < 26. and so on.

this works with any even c, as far as i can tell.

[Hemiboso]

Farsight, Thank you for your feedback, and for taking the time to "get it." The 3-dimensional shape you describe is very much the way I visualize the sequence, firing and swirling down a virtual shaft. I leave it to others more skilled than I to 3-dimensionalize or animate the spiraling motion.

 

I'll post a smaller version of the sieve, as requested (Need to shrink it down from 1.9MB!)

 

Thanks again!