My Integer Factorization Study: The Right Triangle


I was sitting on my zafu (zazen cushion) today when the MIRIAM* process knocked me clean off of it. Hey, I’d been sitting for twenty minutes, and I didn’t mind bringing my zazen to a halt. Plus, this was a good one.

What I got in the way of suddenly-delivered insight was the idea that I’d been going about the visualization process for the odd positive integers all wrong. I had been enamored by the structure of the odd integers between the perfect squares, both even and odd. I had built (and shared here) charts with differently-colored cells for primes and discrete semiprimes.

And I was doing it all wrong. I should have suspected this when one of the properties kept bugging me: Every second diagonal line, I would of course need to leave out the odd perfect square value. This gave a zigzag effect to the remaining numbers, defying simpler formulaic description of their arrangement on the chart.

What I got from the MIRIAM process today was this: Create a new chart of odd integers, but this time, let it be ALL the odd integers. Arrange them in triangular fashion, and the odd perfect squares will take care of themselves, not by disappearing, but by all lining up on the main diagonal.

I started arranging numbers after I put my zafu away, and could see that the chart became that of a simple bivariate quadratic function:

If x is the column number, starting with 1, and y is the row number, starting with 1, then the number in the cell at row x and column y is given by

f(x,y) = (x+y)2-x-3y+1.

Wonderful! A simple 19-line Perl script gave me values for x, y less than 32 in comma-separated-values format, and I imported it to Numbers. I then colored the values less than 1000 as follows: yellow for perfect squares, green for prime numbers, and blue for discrete semiprimes. Then I took my list of discrete semiprimes whose Ascent is zero (see this paper for an explanation) and made the numbers in those cells be in bold red type.

Already this is showing me interesting structure. Note that odd integers of the form x2-1, x2-4, x2-9, x2-16, etc. are in rows which, except perhaps at the beginning, contain no prime values.

I also took a couple of the columns well past the <1000 boundary, in which the entire set of values charted up to the odd perfect square value were prime, and most of the discrete semiprime values that followed also fell on the x2-a2 lines for some a, and had zero Ascent. This is happening, I observed, most likely on columns (x values) divisible by 3.

I plan to examine more where the discrete semiprimes of varied values of the Ascent fall on this table, by way of investigating these now more sensibly arranged numbers for useful structure.

* – MIRIAM = Moving Inspiration Rapidly Into Accepting Minds

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