Saturday, 5 November 2011

The composite 6n+1/6m-1 grid

To help look at prime patterns, I used a spreadsheet to create a grid of multiples of numbers in the form (6n+1) and (6n-1). The reason for this is that all numbers of this form are either prime numbers or multiples of smaller primes of the same form. So for instance, the first few composite numbers of this form are 25 (5*5), 35 (5*7), 49 (7*7), 55 (5*11) and so on. At it's most basic the grid looks like this.

If you imagine this continuing to the right and upwards, it is a small part of the grid of all numbers generated by multiplying together 6n+1 numbers (on the axis to the South and East) and 6n-1 numbers (on the axis to the North and West). The North-East quadrant contains numbers that are of the form (6n+1)(6m-1), the North-West contains (6n-1)(6m-1), the South-East contains those of the form (6n+1)(6m+1). (The whole pattern would mirror across the bottom diagonal row, which contains all the square numbers of this form, thus I leave it blank for clarity).

Next I coloured in stripes of composites with factors of n. So, for instance, I coloured in multiples of 5 yellow.

 Overlaying multiples of 7 (brown), 11 (pink) and 13 (blue), you start to see the rather obvious fractal style pattern, a bit like a Sierpinski carpet (except we are eliminating squares of growing size within an infinite plain), but with the centre of the squares off-centre:

(You'll see that where a later prime is on the same line, I've overlayed the colour, so the 55 line is coloured pink for 11, not yellow for 5).

Here is a view of a larger section, also completed up to the multiples of 13.

And one more long shot, to show how the grid extends.

Note that all numbers that appear in the composite area (eg not the central axes) will also appear later on in those central axes as they are higher numbers in the form (6n+/-1). Also numbers with 3 or more factors will appear several times in the composite area, for instance 245 appears as 7*35 as well as 5*49. An infinitely extended version of the grid would (in theory) contain all composite numbers of the form (6n+/-1) as well as all prime numbers of this form (restricted to the central axes).

The problem with the composite 6n+/-1 grid

This grid gives an interesting visual way to observe the build up of symmetrical primorial patterns - basically you go up through the Sieve of Eratosthenes and gradually colour in the map and the patterns that build up are the same patterns I (and others) have observed in the distribution of composite numbers (and, consequently, the remaining numbers that are still "prime candidates", in other words numbers that are either prime or multiples of higher primes).

However there is a problem, which is that the grid of "composite numbers", as it expands, will also include the prime numbers in the central axes. So, when we go on to colour in multiples of 17, the 17 square will be coloured in as though it is a composite. This is one of the basic problems of sieve theory (as I understand it) as this sieve fails to clearly distinguish primes. Extended to infinity, the entire grid would be coloured in as "composite", even though there are an infinite number of primes.

(This is a serious problem for any attempt to understand twin primes and other related problems using these primorial patterns. It is clear that any set of prime numbers up to Pn will leave an infinitely repeating symmetrical pattern of "prime candidates" in subsequent Pn# length primorials. But we aren't actually looking for "prime candidates". We are looking those candidates that get sieved out as "composite" even though they are actually one of a pair of twin primes).

In the next post I will look at a slightly different grid.


  1. Hi I've done some prime finding research myself, and if I can give you a sort of advice is that looking only at the odd numbers may by misleading.. There are patterns in the primes...
    And as far as I hed been natural numbers seem to Me as some sort of a fractal. Hope I can load You som pics to make It a bit clear.

  2. I tend to agree. What I've been looking at more recently is grids of all numbers expressed as the difference between two squares which is very interesting and shows more fractal kinds of patterns.