Hi,
I found a pattern in the prime numbers, I used an encoded binary sequence, and the pattern became clearly visible:
That's all zeros and ones, depending on if a prime is at the positions given by the formulas:
y = 30x + 1 y = 30x + 7 y = 30x + 11 y = 30x + 13 y = 30x + 17 y = 30x + 19 y = 30x + 23 y = 30x + 29
The pattern can be seen if the zeros and ones are displayed as a bitmap, but only if there is the correct number of columns in the bitmap, and that is the pattern I found, the column count to show the pattern seems to be this for primorial30:
8*prime (where 8 is how many coprimes 30 has)I tested prime 7 so far, but I think the pattern seen might hold for larger primes too.
When 8*7=56 columns are used ie here are the first 20 rows, all having columns of zeros:
0,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,0,1,1,1,0,1,1,1,1,1,1,0,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,1,0,1,1,1,1,0,0, 1,0,0,1,1,1,1,1,1,0,1,0,1,0,1,1,1,1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,1,1,0,0,1,1,1,1,0,1,0,1,0,1,1,1,0,1,1,0,0,1,0,1, 1,0,1,1,0,1,1,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,1,1,0,1,1,1,0,0, 1,0,1,1,1,0,1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,1,1,1,1,0,1, 0,0,0,1,1,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0,1,1,1,1,0,1,1,0,1,0,1, 1,0,1,1,0,1,0,0,0,1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,1,1,1,1,0,0,0,1,0,1, 0,0,0,0,1,1,1,1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,1,0,1,0,0,0,1,1,1,1,1,0,1,1,1,0,1,0,0, 1,0,1,1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0,1,0,1,1,1,0,1,1,0,0,1,1,0,0,1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,1,1,0,0, 0,0,0,1,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0,0,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,1,1,1,1,1,1,0,1, 0,0,1,0,1,0,1,0,0,0,1,1,0,0,0,1,1,0,0,0,0,0,1,1,0,1,0,1,1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0,0,1,1,0,0,1,1,1,1,0,1, 0,0,1,1,0,0,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0,0,0,0,0,1,1,0,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,1,1,0,0,1, 1,0,0,0,0,0,1,1,1,1,1,0,1,0,0,0,1,1,1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0,0,0,0,1,0,0,0,0, 1,0,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,0,1,1,0,0,0,1,0,1,1,0,0,0,1,0,0,1,0,0,1,1,1,0,1,1,0,1,1,1,1,1,0,1,1,1,0,1,0,1, 1,0,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0,0,1, 0,0,0,1,1,0,1,1,1,0,0,0,0,0,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,0,1,1,1,0,0,0,0,0,1,0,1,1,0,0,0,1,0,0,0, 0,0,0,1,1,1,0,0,1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0,0,0,1,1,1,1,0,0,1,0,0,0,0,0,0,1,1,1,0,1,0,1,1,1,1,0,0,1,1,0,0,1, 1,0,1,1,0,0,0,1,1,0,0,0,1,0,1,0,0,0,0,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,1,1,0,0, 1,0,1,1,0,0,1,0,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0,1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,1,0,1,1,0,1, 0,0,0,1,1,0,1,0,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,0,1,1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0,0,0,0,1, 0,0,1,1,1,0,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,0,1,1,0,0,1,1,0,1,0,0,1,0,0,0,1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0,0,0,0,0,The columns of zeros match up to give these patterns that can be verified:
30x+7 is always NOT prime every seventh x 30x+19 is always NOT prime every seventh x (offset one from x for 30x+7) 30x+17 is always NOT prime every seventh x (offset two from x for 30x+7) and continuing for other zero columns.So I can increase the compression of this prime binary sequence further.
Here is a picture showing the pattern of zero columns that occur for
8*7 columnsHere is a picture showing there are also zero columns (narrow white vertical lines) for 8*83, so it is probable that for any 8*prime, there are zero columns that show a further pattern of spacings that are not primes.
I guess if you add up all these not prime rules you can get a formula for prime numbers :D
cheers, Jamie