Saturday, 11 February 2012

Self-similarity in composite patterns (expressed as difference of two squares) Part 2

OK, the thing I want to look at next is the sequences that develop within this grid when we look at N^2 - P^2 for a fixed value of N-P (for instance 12^2 - 7^2, 24^2 - 19^2, 36^2 - 31^2 etc where N - P = 5)

In the image below, the coloured squares are composite numbers adjacent to multiples of 48.

The (non-blue) squares are coded thus:

N-P = 5 Yellow
N-P = 7  Orange
N-P = 11 Green
N-P = 17 Red
N-P = 23 Mauve
N-P = 41 Purple

You can see that each of these series progresses down a diagonally repeating position within the pattern. It is also evident that the series for N^2 - P^2 moves up and down in steps of P, for instance, for N - P = 5  the pattern is -18, -13, -8. -3, 2, 7, 12, 17, a difference of 5 each time.

This is the basic reason why, in the grids we have been looking at in previous posts, there is a self-similarity in the pattern as we move up to composites adjacent to multiples of 48, 96, 192 etc. For instance to get the pattern for N^2 - P^2 numbers adacent to multiples of 96, we halve any even numbers in the sequence above and eliminate the odd numbers, and we get a pattern of -9, -4, 1, 6. Then for numbers adjacent to multiples of 192, we do the same thing and get a pattern of (-7), -2, 3, (8).

These series repeat with regular periodicity. In terms of twin prime candidate pairs (in other words pairs of 6N+1 and 6N-1 numbers that are 2 apart), the pattern for N^2 - 5^2 repeats in a cycle of 2 (you can see above that the pattern for 192 hits the same pairs as the pattern for 48 - the signs are reversed meaning that it is a negative version of the same pattern but the same pairs are nonetheless present).

The specific reason for this can be seen when we look at the results in terms of mod 4 arithmetic. For N^2 - 5^2 the pattern in mod4 can be seen as -2 (=2) -1 (=3), 0, 1, 2, 3, 0, 1, increasing by one each step. For N^2 - 7^2 the pattern in mod 4 will decrease by one each step. And since any odd number must also be a 4N+/-1 number, the pattern will always increase or decrease by one at each step.

This means that when we double the target by going up from composites adjacent to 48 to those adjacent to 96, 192 etc, we eliminate every other number in the series. If you look at the yellow squares above, there are eight in the larger square, and four in the medium square. Out of these numbers 4 are divisible by 2, 2 are divisible by 4 and one is divisible by 8.

I'll talk more about these series in later posts.

The only other thing to point out at this stage is how the pattern for N - P = 41 (purple) behaves. There are none in the top left square of the diagram, so does this mean it is a discontinuous series?

Well no - instead you can see it like this:

48^2 - 7^2 = 2255
24^2 - (-17)^2 =  287 (which is 41*48 less than 2255)
0^2 - (-41)^2 = -1681 (which is 41*48 less than 287)

And 24^2 - 17^2 is already in the pattern in the N - P = 7 diagonal. So this is effectively filling the gap in the N - P = 41 series also.

(Just a reminder, the minus numbers are there because of the way the grid works, meaning we get 0^2 - 41^2 rather than 41^2 - 0^2. You can basically ignore the minus sign, or bear in mind that the minus numbers are 6N+1 numbers while the positive ones are 6N-1 numbers).

Of course this is rather obvious since the series above can be rewritten as

41 x 55
41 x 7
41 x 41 (when you ignore the minus signs)

So effectively we are just seeing that the series of multiples of 41 (or any other odd prime) will only hit a number adjacent to a multiple of 48 twice in every 48 steps (once on a 48N+1 number and once on a 48N-1 number). This isn't terribly surprising, but I still find it interesting to view it as the difference between two squares rather than as a series of multiples.

In the next post in this series I want to talk about the way that composite numbers are distributed as a result of the modular patterns mentioned above.

(First I am going to break off and take a look at the Ulam Spiral though).

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