Wednesday, 25 January 2012

Composite numbers seen as the difference between two squares Part 5

The next step of this slightly laborious exploration is to move on to composite numbers that are adjacent to multiples of 96. Here is the chart adapted for this purpose. It was getting fiddly changing the settings so from now on the second column indicates which multiple of 48 the composite number is adjacent to - so the even multiples of 48 are multiples of 96 and odd multiples are eliminated (these have been left in yellow).

First let's look at a fairly small section, up to 96 and 95 on the horizontal and vertical axis.

Pink indicates a composite number adjacent to a mutliple of 96.

At this stage we start to see a more complex symmetry. The outlined boxes indicate the areas which reflect. Vertically there is a symmetry from 1 to 47 around the line between 23 and 25, also from 1 to 95 around the line separating 47 and 49. Horizontally, the symmetry is around the 24 line and the 48 line. Within the boxes formed by the axes of symmetry, there is a group of four composite numbers that repeats in each - the first four in the 0 and 12 columns vertically and the 1-23 columns horizontally. This group of four reflects vertically. Horinzontally it repeats (or you can see it as a symmetry with the 24 column as the axis). This builds up into predictable numbers of composites within these boxes - in the area before the 96 column there are 16 groups of 4, 64 composites altogether. I'm pointing this out here because it is a pattern that repeats in later charts.

There are some more complex issues about these symmetries (based on modular arithmetic) which explain why successive eliminations leave a very similar pattern in place, but  I'll come back to that.

Here is a longer view of the pattern.

You can't really see the numbers on this one, however you can see the kite-shaped pattern recurring in the blue numbers, which is all the composites adjacent to multiples of 96 up to 24 x 96. A similar pattern would apply (at twice the size) to multiples up to 48 x 96. 

We find 39 instances of composite numbers up to 24 x 96 (including repeats). Because of the repeats, there are 3 gaps in the pattern, at 2 x 96 (twin primes 191/193), 12 x 96 (twin primes 1151/1153) and 22 x 96 ( twin primes 2111/2113).

1) There is a predictable number of composite numbers in the boxes I have indicated - for instance within the largest box here (not counting the composites within the 192 column which has a line either side of it) there are 4 x 64 = 256 composites.

2) We can see the area within which it is possible for there to be multiples up to a certain threshold. There are limits on this area, which I will look at in more detail later. The choice of 24 as the threshold is a slightly arbitrary one - I've just chosen it in order to show the self-similarity as we move up through these charts and narrow down the area in which we are looking for composite numbers.

More on this to follow in later posts.

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