Wednesday, 25 January 2012

Composite numbers seen as the difference between two squares Part 6

Here is a section of the pattern of composite numbers adjacent to multiples of 192. I know it's getting hard to read the numbers, and I won't keep picking out the kite-shaped pattern I've previously noted. The main thing to look at here is how everytime we go up to the next multiple, we lose half the composites from the previous pattern and a similar pattern develops.

And here is the pattern for composites adjacent to 384 (numbers in pink/purple) - in this one I've left the composites adjacent to 192 in blue as it helps to see how one pattern turns into the next one. (The squares with heavy black outlines are all part of the pattern for composites adjacent to 384 and coloured pink).

And this is a longer view of the same pattern (getting ridiculously cramped now...) The black squares are composites adjacent to multiples of 384 up to 24 x 384 - note the same kind of kite-shaped pattern (though I didn't quite have space to fit this all in).

OK, there's no point going on reiterating larger versions of this pattern as it is already impossible to read. The next thing is to go back and look at some of the reasons (based on modular arithmetic) why we keep seeing the same kind of pattern recurring.

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