**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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