Sunday, 22 January 2012

Composite numbers seen as the difference between two squares Part 3

We've been looking at composite numbers adjacent to multiples of 3x2^n as n increases (alternatively I could say adjacent to successively doubling 6, so we get 12, 24, 48, 96.) The reason I am doing this will become more clear shortly as we see the next few rounds of sieving this way.

We've seen how the density of composite numbers in these diagrams halves each time we double the multiple of 3x2^n (eg move from 6 to 12 to 24 to 48...)

Next I want to narrow the search down from composites adjacent to 24 to composites adjacent to 48. This results in half of the lines in the previous diagram being eliminated:

The yellow lines contain composites adjacent to a multiple to 48, the white lines are eliminated.

Note that the pattern of yellow lines is symmetrical, repeating every 8 lines. We will see more complex symmetrical eliminations on subsequent stages, but at this stage the symmetry is very simple.

Pink squares are composites adjacent to the first 12 multiples of 48, blue squares are the next 12 (13 to 24). There are 15 of the former and 17 of the latter, 32 in all. This would be more than enough to eliminate all of the first 24 pairs if they were spread across all numerical values, but there are some values that appear several times, for instance 6 (because 289 and 287 are both composites) and 8 (which appears three times because 385 is a multiple of 5, 7 and 11 as explained in the previous post).

Note that collectively these are in a "kite-shaped" band similar to the region which we saw for multiples of 24. Eventually I will give a more formal description of this area, at this point it is just the similarity that interests me.

As a result of instances where both the 6N+1 and 6N-1 number are composite, and instances of composites with distinct factorisations, there are gaps left in the pattern, which indicate twin primes.

In the next few posts we will see a more obviously fractal pattern develop and I will also give some explanations based on modular arithmetic as to why these patterns develop in a rather self-similar manner.

Part Four

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