## Integer Factorization Update 9/21/2023: The Zone Grid, The Trivial Jump, Etc.

It’s been a week of inundation by ideas. I’ve battled headaches and all of the struggles of balancing potentially good mathematical revelations with all of life’s necessary other concerns. I am in a play – It’s a small role, but I am also the Musical Director for some gospel song interludes the director wants handled according to his vision for “The Trip to Bountiful.” This is above and beyond helping my lovely wife prepare for a trip to visit the grandchildren who are out of state, and of course all of the normal housekeeping chores, which often involve our seven pets, and on which I am always woefully behind.

Oh, and did I mention having fallen in love with Ruth Ozeki’s novel “The Book of Form and Emptiness,” whose main character, author, storyline, and very existence mesh so well with my inner life and experience, it’s amazing?

But the math is getting so good … and that’s what this post is really getting into. The space bus I call MIRIAM (Moving Inspiration Rapidly Into Accepting Minds) is dropping me off at some simply wonderful revelatory places. Numbers are supremely beautiful, behind so much of the supreme beauty we see, and this work, this week, has been wonderful.

Let’s start with the Zone Grid, whose concept I posted here, and whose behaviors in connection with a Characteristic Polynomial and Characteristic Line, both of whose equations I base on the Ceiling Square and Remainder of an arbitrary odd positive integer m, are beguiling.

Let’s start with the Remainder Checkerboard: When I lay out the Zone Grid systematically by the construction method, and look at the possible starting points for typical odd discrete semiprimes whose Ceiling Square and Remainder don’t suggest special cases for factorization (Remainder is a perfect square, Remainder and Ceiling Square have a non-zero GCD, and so forth), I see that the remainders, which are all congruent to 0 or 1 modulo 4, form a nice checkerboard pattern on the area to the left of the wedge of the Zone Grid between Zone Boundaries 0 and 1. The boundaries are the red numbers in the screenshot below, and the cyan numbers in what I’m calling the “jump launch zone” form the checkerboard of possible remainders, the f(0) values for the Characteristic Polynomials f(x)=x2+2cx+r uniquely corresponding to m=c2-r.

This has a prettiness and neatness to it I like. Also, notice that each row of non-perfect-square remainders in the jump launch zone is symmetric around its center in terms of the white/cyan design. That also is nice.

But I was looking at some other, albeit much more random, prettiness a day or two ago, which led me indirectly back to the Zone Grid: the behavior I posted before when one fixes a Remainder value and looks at the factors of m for the Ceiling Square values in a particular range. This led to some fun graphs with Apple Numbers, which looked a little like noisy audio spectrographs to me:

The dots correspond to something quite obvious, the distribution of the factors of x^2-13208. Still, it pushed me onward to find more patterns.

What I am seeking now is pattern and pathways relating to how the starting point on the Zone Grid for a particular m and its Trivial Jump Point, which one can calculate easily and which corresponds to the m = s2-(s-1)2 trivial Fermat factorization solution that exists for every odd positive m. I am looking at the rectangles made by these values on the Zone Grid, their x coordinates and differences, and, only inasmuch as might help me see a pattern or formula, the smaller rectanglesh and their coordinates created by known non-trivial factorizations.

It’s still an amazing adventure. It still pushes me to get out of bed and explore it in my “copious spare time” between line memorization, cooking, cleaning, and being as good a husband as I can with all considered.

And it will still produce revelations I will share here as I come to them.

How about I close with some new Peter Gabriel music?