Saturday, 28 August 2021

Enumeration of Grid Points

I've been reading all my books that include some Number Theory, and doing some Study of Numbers, which I'm thinking of publishing under the title "Numerology: The Wisdom and Folly of Numbers".

Here is a little item that is probably not new, but I can't find it in any of my sources. Can anyone locate it in some publication? I've a vague idea I've seen something like it somewhere. 

I call numbers of the form (2^r)x(3^s) "Basals". In other words they are any numbers that exclude prime factors greater than 3. .The sequence runs: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 98, 108, 128, ...

The corresponding powers of 2 and 3 in the sequence run: (0,0), (1,0), (0,1), (2,0), (1,1), (3,0), (0,2), (2,1), (4,0) and so on. Every pairing occurs and they are listed in a unique order.

This scheme thus provides an enumeration of the grid cells of an endless board, as partially illustrated in this tour diagram, from (0,0) = 1 to (9,0) = 512. 

Enumeration of Coordinate Points

It is interesting that the path appears never to cross itself. 

Further: This diagram shows the similar result obtained using (2^r)x(5^s) to determine the sequence. This goes up to 5^4 = 625 at (0,4). The moves in the upward direction tend to be knight moves. 

Enumeration using primes 2 and 5.

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