First of all, I want to write down on my blog some of the nitty gritty, in a fashion I hope will be clear, of how I construct and use the Zone Grid I have been using in my calculations and explorations, and my discovery of how to plot thereon the Remainder Line corresponding to the Characteristic Polynomial f(x)=x2+2cx+r for a positive odd integer m where c is the Ceiling Root and r the Remainder of m so that m=c2-r constrained by the rules of construction I briefly albeit inadequately described here.
First of all, the numerical basis for the Zone Grid is a function on the points of the Cartesian Plane: z(x,y)=x(x+1)+y, so that each vertical line for integer points y=0, 1, … has as its grid points a vertical shift of some amount of the number line. I look at these grid points by turning the Cartesian Plane 90 degrees clockwise, so that the x axis points down for increasing x values, while the y axis points to the right for increasing y.
I decided this was an arrangement of copies of the integer number line in equally spaced rows that interested me when I discovered that one could arrange the non-negative integers in a triangle as follows, with the perfect squares appearing on the sides of a symmetric area so that each shifted number line was the bottom of an isosceles triangle, as in the illustration below:
This discovery had arisen from my examining what I called “Beehive plots” of successive remainders rn which satisfy m=(c+n)2-rn when c is the Ceiling Root of m, and r0 is the Remainder, the difference between m’s Ceiling Square c2 and m. As explained in posts over the last couple of years, I have been examining this series of remainders to see if I can determine formulaically when the series would have perfect square values.
Notice that the sides of the triangle pictured above have the successive perfect squares falling along straight lines. Each successive perfect square appears one line lower, and in a straight line with, the previous perfect square. If one were to extend the shifted number lines of the Zone Grid on the left and right sides so that their numbers go from 0 to positive infinity, one could notice that successive perfect squares on successive rows (y-values) of our clockwise-turned Cartesian Plane fall along a family of straight lines y=(2n-1)x+n2, n ranging over the integers.
When I looked at the series of remainders rn and realized they satisfied rn=n2+2cn+r, I noticed something else, which linked the “Beehive plots,” the “Bee lines”, and the Zone Grid together: Successive values of rn, located on successive rows (y values) of the Zone Grid, fell along straight lines as well. This seemed promising to me after some more investigation, because I also knew that perfect square values of rn corresponded to factorizations of m. This meant that each remainder series corresponded to a straight line that would intersect some of the straight lines of perfect squares. It would do so at the points of the factorization solutions I was seeking. It also would do so at points corresponding to the trivial factorization m = 1 * m = ((m+1)/2)^2-((m-1)/2)^2.
The Zone Grid chart has grown and developed: I set a starting point methodology for the various possible values of r0, and in jumpfactor.pl, I use that fixed starting point to determine the fixed Remainder Line for subsequent calculations. I even got artsy, or at least stylistic, with some of the properties and structures I observed:
And my mathematical research is still not done here, not by a long shot. Nor, apparently, is my need to update my math study explanations to keep them clear and current.