

Prime Number Patterns - sebg
http://www.jasondavies.com/primos/

======
Houshalter
See also divisor plot:
[http://www.divisorplot.com/](http://www.divisorplot.com/)

This guy plotted the table of numbers and their divisors and found lots of
patterns in it.

~~~
austin_y
Seriously, thank you for posting that! I enjoyed his site immensely and will
definitely revisit it in the future.

~~~
jxramos
I concur, his site is amazing, he's definitely on to something, great
abstractions there and relations to factorials. Love it.

------
floatrock
> The prime numbers are those that have been intersected by only two curves:
> the prime number itself and one.

This is merely a prettier but somewhat obfuscated Sieve of Eratosthenes:
[https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes](https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes)

~~~
panic
It's a richer visualization than that. The Sieve of Eratosthenes only marks
which numbers are composite -- this visualization actually shows each of the
factors.

------
jrapdx3
For all the time and energy already devoted to studying prime numbers, it's
amazing how we keep finding compelling new things to say about the subject.

I agree this presentation shows considerable grace and beauty, quite effective
at showing the numeric relationships among positive integers in an intuitive
clear way. As the scale varies, the "zoom" feature very nicely allows the
viewer to appreciate the fractal nature of the numerical order. Educational
and fun, it's a creative success.

Just yesterday on HN there was a thread concerning favored books, mentioning
"Proofs Without Words", which this site reminds me of. Where it works it is a
wonderful concept, and here it works well.

------
halosghost
This is beautiful! Not just is it really quite cool to explore with, but is
truly gorgeous. It is true as another user put that this is a graphical way of
looking at a sieve of Eratosthenes, but to say that it is is “only” suggests
to me a lack of appreciation for the beauty (plus, it also maps other non-
prime patterns: deficient, perfect and abundant).

------
cypherpunks01
Not sure if it's mathematically related to the OP patterns, but the prime
number patterns apparent in the Ulam Spiral have always been my favorite
example of this:

[https://en.wikipedia.org/wiki/Ulam_spiral](https://en.wikipedia.org/wiki/Ulam_spiral)

------
stillbourne
1/2 (6 n-cos(pi n)+3)

This can be expanded into a prime sieve that shows what he has here.

    
    
      #prime.pl
    
      use Math::Complex;
      use Math::Trig;
      use POSIX;
      # P = A = 1/2 B (6 n-cos(pi n)+3) \ B = 1/2 (6 n-cos(pi n)+3)
    
      my $pi = pi;
    
      my @k;
      my $count = "100";
      my $n = "1";
      my @c;
    
      ### given two sets A, B returns B \ A (ie all elements that are in B but NOT in A)
      sub complement
      {
          my @set1 = @c;
          my @set2 = @k;
          my @ans=();    
    
          ## find intersection of A and B
          my @intersection=(@set1);
    
          for(my $i=0;$i<@set2;$i++)
          {
             for(my $j=0;$j<@intersection;$j++)
             {
                if($set2[$i] eq $intersection[$j])
                {
                   $set2[$i]="?";
                }
             }
          }
    
          for(my $i=0;$i<@set2;$i++)
          {
             push(@ans,$set2[$i]) if $set2[$i]!~m/\?/;
          }
          return @ans;
      }
    
      for ($count = 100000 ;$count >= $n; $n++){
      	my $k = ((3 * $n) - (0.5 * cos($pi*$n))) + 1.5;
      	push (@k, $k);
      }
    
      my $c; 
      my $x = $#k;
      my $y = $k[$x];
    
      foreach (@k){
      	my $b = $_;
      	my $n = "1";
      	$c=1;
      	next if ($b >= floor(sqrt($y + 1)));
      	for ($count = $y + 1 ;$count >= $c; $n++){
      		$c = (((3 * $n) - (0.5 * cos($pi*$n))) + 1.5) * $b;
      		next unless ($c <= $y + 1);
      		push (@c, $c);
      	}
      }
    
      print "stophere";
    
      %hashTemp = map { $_ => 1 } @c;
      @c = keys %hashTemp;
      @c = sort { $a <=> $b } @c;
    
      #@P = &complement;
    
      print "stophere";

~~~
jawilson2
If n=8, then this function equals 25. Am I missing something?

~~~
stillbourne
Yeah, its a sieve, this block:

    
    
      foreach (@k){
      	my $b = $_;
      	my $n = "1";
      	$c=1;
      	next if ($b >= floor(sqrt($y + 1)));
      	for ($count = $y + 1 ;$count >= $c; $n++){
      		$c = (((3 * $n) - (0.5 * cos($pi*$n))) + 1.5) * $b;
      		next unless ($c <= $y + 1);
      		push (@c, $c);
      	}
      }
    

Allows you to iterate of the next set and find the none prime and remove them
from the SET of 1/2 (6 n-cos(pi n)+3) Thats how a sieve works. 1/2 (6 n-cos(pi
n)+3) is only good up to n^2 then you need to multiply 1/2 (6 n-cos(pi n)+3)
by i as you noted the first non-prime result was @ 5^2 therefore you need to
sieve the rest of the numbers up to 7^2 by doing (1/2 (6 n-cos(pi n)+3)) * 5
Then after 7^2 seive off with (1/2 (6 n-cos(pi n)+3)) * 7 you continue to
repeat this for all numbers in the original set. See the sieve of
eratosthenes. Mine though shows primes as a set of sin waves kind of like what
the op has. As a matter of fact the post is basically a visual representation
of what I coded above.

------
delinka
I'm seeing patterns in the factoring of non-prime integers, but nothing to
highlight any patterns that exist among prime numbers.

------
kristopolous
absolutely stunning design. nothing of new mathematical interest. beautiful
though - simply gorgeous.

------
techbio
72 is so pretty

