Thanks Pi man.  The rest of the pattern (of non-pairs) is 4,16,28,...(intervals of 12) and   8,20,32,... (also intervals of 12).
These all seem to follow what you've described.

I'm running this into the millions for the first time with python and glad to know what they show.
I'll post some pictures when I can.

Indeed they are. I think pi man's proof can be applied to any even number that isn't a multiple of 3, so as well as the numbers you've mentioned, you also won't find any pairs of 10, 22, 34... (in intervals of 12) or 14, 26, 38... (in intervals of 12).

You also won't find any pairs of 2 apart from the pair that start off the sequence. pi man's proof still works there, and the pair of 2's exists because they describe the differences between 3, 5, 7. pi man's proof says that one of those is a multiple of 3, which 3 is, but it is also prime.

Incidentally, whenever you see a 2 in that sequence of differences, it means that you've found a pair of twin primes (two consecutive odd numbers that are both prime). No one knows whether the twin primes go on for infinity or not. There's probably a vastly huge sum of money being offered somewhere for anyone who can prove either way whether they do.

Anyway, a very warm welcome to the forum. Have lots of fun here.

Last edited by mathsyperson (2007-01-02 05:28:00)

...Ooops. It's a shame this forum doesn't have an embarassment smilie.

I've always been fascinated with primes.   There's got to be some sort of pattern in there somewhere.   Many, many, years ago I tried using different bases (binary, hex, etc.) to see if any patterns jumped out.   Then the use of the internet began to growth quickly and I discovered that primes were fascinating to many.   Brilliant minds have spent countless hours on this and I realized that finding a pattern is not going to be as simple as me converting the numbers to a different power-based system.

I still think that a completely different numbering system might be the answer.   For example, instead of a power-based system, how about a prime-based system?   So instead of expressing a number in ones, tens and hundreds..., express them in terms of 2's, 3's, 5's, 7's, etc.

Decimal    PrimeBased 
0                        0         
1                        1         
2                      10         
3                    100         
4                      20                     
5                  1000
6                    110
7                 10000
8                      30
9                    200
10                 1010
11              100000
12                   120

Basically this numbering system is expressing the prime factorization in a numeric form.   From right to left, the columns are the prime numbers: 1, 2, 3, 5, 7, 11.   So 6 in this system (3^1) * (2^1) * (1^0).  12 is (3^1) * (2^2) * (1^0).     This system has not  helped much and it has its flaws.   Although the columns are prime-based, the actual digits  are decimal.   So 4 is represented as 20 but "2" in this system is 10.  So 4 should really be 10\0 with a "" separating the digits.   Make any sense?

I would like to write more but our New Year's guests have arrived.   Happy New Year's to all you mathematicians!

Last edited by pi man (2007-01-02 16:11:57)

Regarding prime pairs.  There is a way to plot this info in 2D that makes that prime pairs visible.

Take the list of differences (2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6 ...)
and plot them (n1,n2) (n3, n4)
basically take each pair and turn them into coordinates.

This will show that not only do prime pairs (primes 2 aparts) continue to happen over and over, but so do many other combinations (minus certain holes).  These holes in the scatter graph that forms are very constant as the graph fills.

You can also start "late" in the sequence and find it fills in both directions.

It's as if to say "all distances between primes exists at all sizes" minus of course the holes I originally posted about.
I have images, is there a way I could post them?

In a scary form of ascii art, here is the pattern as it seems to fall, all odd spaces are removed so this grid represents even numbers only with the origin in the lower left.

ooooooooooooooooooooo
o oo oo oo oo oo oo o
 oo oo oo oo oo oo oo
ooooooooooooooooooooo
o oo oo oo oo oo oo o
 oo oo oo oo oo oo oo
ooooooooooooooooooooo
o oo oo oo oo oo oo o
 oo oo oo oo oo oo oo
ooooooooooooooooooooo
o oo oo oo oo oo oo o
 oo oo oo oo oo oo oo
ooooooooooooooooooooo
o oo oo oo oo oo oo o
 oo oo oo oo oo oo oo

The actual scatter graph looks more like this....

oo 
o o
 oo     o
oooo
o oo 
 oo oo     o (very random placement)
ooooooo
o oo oo o
 oo oo oo o
oooooooooooo     o
o oo oo oo oo
 oo oo oo oo oo
oooooooooooooooooo
o oo oo oo oo oo  
 oo oo oo oo oo oo oo

^
Here at origin, pattern gets very solid very quickly.

Aha! Posted image as my avatar!

Last edited by BluemanSteele (2007-01-03 22:23:03)

Ok I've figured out how to post images, I'll edit this post and put up a more interesting one.
While this post is up, the picture below is much like my avatar but color and height coded to show repeated "hits" on that spot.

Higher and brighter = more repetition.

Uploaded Images

Last edited by BluemanSteele (2007-01-04 04:03:01)

Are there any interested C programmers who could take pseudo code and optimize it?

Yep.  I'd be one of those.

Great to hear Ricky!

I'm going to describe it similar to text processing. I'm sure there is a much more efficient way to do it, but this will be easier to describe. My algorithms are less than text-book quality so I'll leave them out.  This is a bit of a repeat from above but I'll post it that way for clarity.

1. Create (or read from) a list of prime numbers starting at 3

      (3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61...)


2. Make a list of the differences between on number and the next.

      3 -> 5 = 2
      5 -> 7 = 2 ...

      (2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6 ...)

3. Take the first two numbers in the list (n1 and n2) and plot them as coordinates.

4. Do the same for either a. (n3 and n4) or b. (n2 and n3), I've found both produce interesting results.

   That is take... 
      (2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2)

   And plot points
      (2,2) (4,2) (4,2) (4,6) (2,6) (4,2)

    OR. plot to points
      (2,2) (2,4) (4,2) (2,4) (4,2) (2,4) (4,6) (6,2) (2,6) (6,4) (4,2)

5. Since all numbers are even,optionally divide all numbers by two to collapse the resulting shape of negative empty areas.

Let me know if you have any questions if I have not been clear.  I'd post my Python but eek do I have to show the world what a horrible coder I am???  :-)

BluemanSteele wrote:

Great to hear Ricky!

I'm going to describe it similar to text processing. I'm sure there is a much more efficient way to do it, but this will be easier to describe. My algorithms are less than text-book quality so I'll leave them out.  This is a bit of a repeat from above but I'll post it that way for clarity.

1. Create (or read from) a list of prime numbers starting at 3

      (3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61...)


2. Make a list of the differences between on number and the next.

      3 -> 5 = 2
      5 -> 7 = 2 ...

      (2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6 ...)

3. Take the first two numbers in the list (n1 and n2) and plot them as coordinates.

4. Do the same for either a. (n3 and n4) or b. (n2 and n3), I've found both produce interesting results.

   That is take... 
      (2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2)

   And plot points
      (2,2) (4,2) (4,2) (4,6) (2,6) (4,2)

    OR. plot to points
      (2,2) (2,4) (4,2) (2,4) (4,2) (2,4) (4,6) (6,2) (2,6) (6,4) (4,2)

5. Since all numbers are even,optionally divide all numbers by two to collapse the resulting shape of negative empty areas.

Let me know if you have any questions if I have not been clear.  I'd post my Python but eek do I have to show the world what a horrible coder I am???  :-)

I made the same.
Only for optimisation: there are many "fast" tests for finding the next prime. I recommend you download some library for computing primes, instead of computing them using Erastritenes' sieve of brute force.
That will save much time for big primes.

I don't know.
But here's some results:
There's a theorem, stating that for every number k, there exists infinity couples of consecutive prime numbers, such that the gap between them is greater or equal to k. Graphically that means, that when we have enough prime numbers, this graphic will cover some square, no matter what the square side is. More basically, this structure grows.
Not so helpful, but it's proven to be true.