This update refers to concepts defined in earlier posts, without explaining them extensively here. I have placed links at the bottom of this post to recent explanatory posts.
In designing my most recent approach, I produced a two-dimensional travel path taken by successive remainders c2-m, (c+1)2-m, … (c+n)2-m, … where m is the odd integer we wish to factor and c is its Ceiling Root, making c2 its Ceiling Square. I did this by lining up shifted integer number lines to assign integer values to Cartesian Grid points so that straight-line boundaries existed of perfect square values between “zones” of other values, through which a straight line produced in such a grid by the successive remainders given above would pass.
What I have discovered, and am investigating further for possible usefulness in speeding up factorization, is that the distance between the location on the “jump grid” of the first remainder, i.e., the starting point along the remainder line, and what I call the Zone 0 Boundary, and all the distances between the zone boundaries, have integer ratios to each other. This surprised me a little, though I suspected it might be the case for the line segments between zone boundaries. The first jump, from a location determined by an initial remainder which, if we’re bothering to do this at all, won’t be on a perfect square line, fitting in this way with all the boundary-to-boundary jumps, well, that’s some extra prettiness on which I had not counted.
I’ll be sure, as always, to share any exciting revelations that seem to come from further exploration of this.