Tuesday, 11 January 2011

Fractal Primorial Patterns - a few basics

Lots of people have noticed primorial patterns in the primes. Just a quick note on one way of thinking about these:

Each time we go up through the sieve of Eratosthenes we get a primorial length symmetrical pattern of multiples. This pattern is endlessly repeated.

So, at a very basic level, after we have put 2 and 3 in the sieve we have the pattern

1 2 3  4   5   6
7 8 9 10 11 12

This is symmetrical (with 3 as a centre of symmetry of the first row) and will endlessly repeat.

The same is true after we sieve 5. This leaves a symmetrical pattern of 1, 7, 11, 13, 17, 19, 23, 29, which repeats in each group of 30 numbers.

If we make a sieve of 210 columns for the sieve up to 7, or 2310 columns for the sieve up to 11 we see the same kind of pattern - a symmetry of gaps which repeat on each subsequent iteration.

This process creates a fractal pattern, a overlaid pattern of self-similar elements. But it is not strictly a fractal pattern of primes or composites, because some primes fall within the pattern. It is a fractal pattern in which the gaps indicate "numbers that are not multiples of the primes {2....p}" (including 1 x 2, 1 x 3, 1 x 5 .... 1 x p).

Each time a primorial length pattern is reiterated it is also important to note what happens to each repeating gap (for instance 29, 59, 89, 119, 149, 179, 209). Within the prime pattern, the next prime p in the sieve will only remove 1/p of each particular gap (above you can see that 7 sieves out 119 from this particular set of gaps).

The overall pattern of gaps sieved out by the next prime is also symmetrical. Each pattern is symmetrical around a multiple of the previous primorial (so at the stage 7 is added, the symmetry is around multiples of 30, when 11 is added, the symmetry is around multiples of 210 etc). So numbers sieved in the first half of each iteration of the pattern are also sieved in a mirror pattern in the second half.

My failed twin prime proof relied on showing that we get the same kind of symmetrical pattern of gaps in the layout of twin prime pairs, and that the next prime only removes 2/p of the pairs in each line of gaps. This is correct but it only proves that when we sieve with any finite set of primes, there is an infinite, repeating pattern of "numbers that are not multiples of those primes".

1 comment:

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