Tuesday, 14 February 2012

Ulam Spiral (Gann Square) and differences between square numbers Part 2

I just want to look at the sequences of composite numbers expressed as the difference between two squares in an Ulam spiral.

In this one I've coloured in odd composites as follows:

N^2 Blue
N^2 - 1 Yellow
N^2 - 4 Beige
N^2 - 9 Pink
N^2 - 16 Pale green
N^2 - 25 Orange
N^2 - 36 Purple
N^2 - 49 Green

I haven't gone any higher than this, so not all composite odd numbers are coloured.Even numbers are grey and make up half of the diagonals on the diagram.

A few could have been coloured in more than one way (for instance 9 is 5^2 - 4^2 but also 3^2).

Bear in mind that the first in each sequence may not be composite, for instance 3, 5, 7, 11 are primes - but all subsequent instances are composites.

The thing I like in this image is that it picks out clearly what I was saying in the previous post. Each of the coloured routes above has essentially the same form - it eventually settles down into a consistent diagonal, prior to that it spirals around, and follows more oblique diagonals for spells (this behaviour will be extended for larger squares).

I'm also interested to see that where the patterns are spiralling out before settling down, each colour tends to hit the same diagonals, for instance in the N^2 - 49 pattern, 35 and 735 are on the same diagonal, so are 627 and 51, 527 and 147, and 207 and 351.

Also there is an alignment on the diagonals in the oblique sections of the N^2 - 4^2 pattern and the N^2 - 6^2 pattern - 105 and 405, 153 and 493, 209 and 589, and 273 and 693 are on the same diagonals. The same is true of the oblique section of the N^2 - 3^2, N^2 - 5^2 and N^2 - 7^2 patterns. It would be interesting to see how consistent this alignment is with higher numbers as it may also help to concentrate composite numbers on certain diagonals.

I'll give a partial explanation of these last two alignments in a later post, having done a bit of reading about polynomials and the Ulam spiral. Meanwhile I do find this diagram at least starts to help me understand why primes concentrate in particular diagonals on the Ulam spiral.

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