It occurred to me tonight to start looking at the successive remainders on the Zone Grid in a different way. During my studies of this arrangement of number lines vertically shifted by quadratic amounts on the Cartesian Plane, I discovered that a “skew” linear transformation of the zone lines and the Remainder Line preserves the x values of the intersection points, since the skew I am using is in the y direction only. Also, the family of zone lines appears to have skew-symmetric properties: Once it has skewed to where the slope of “zone n” becomes the slope “zone n-1” had been, or vice versa, the complete family of lines (including “negative” zone lines) is essentially the same. So in a mathematical/formulaic sense, it does not much matter in which zone we choose to place the first remainder: The x-distance one travels from there to each zone line, and to the zone line that gives the factorization solution, will be the same.
See my previous jcsbimp.com posts and papers about Integer Factorization from the past several months for explanation of the concepts and thoughts that led me to all of this. I am necessarily being compact in this progress report because I don’t want to take much time explaining before I begin working on the idea that follows.
It occurs to me to begin thinking about an Initial Trapezoid whose diagonal one would form on the Zone Grid by drawing the remainder line calculated by the previously explained construction method, from a zone boundary that passes through the greatest perfect square less than what I have been calling remainder 0 or r0, the difference between the Ceiling Square of m and m itself, to the following zone boundary that passes through the Ceiling Square on the line where one would find r0 between the two chosen zone boundaries. For simplicity’s sake, to make an Initial Trapezoid with lateral symmetry, I would choose what I’ve been calling the Z0 and Z1 boundaries, locating the (x,y) point for where r0 falls between them.
What I want to do with this is determine the (x,y) points of the corners of the Initial Trapezoid, and then calculate the distance between the top left and bottom right corner of the Trapezoid, the distance between the top left corner and the point for r0, and work out the proportion in a general form. I believe the proportion will be a rational number, and hope to find a relationship between that number and the “interzone distances” that come after, that will quickly suggest a calculation method for the Ascent, the amount to increase the Ceiling Square in order to create a perfect-square difference between it and m.
I hope to have an update not too very long after supper, but I wanted to share right away what I’m up to right now.