ADVICE: Go back a post or three in my blog from this one to find a link to the construction method referenced in the terms below that are unique to my factorization work.

When one plots the Characteristic Polynomial for an odd positive integer m=c^{2}-r as a straight line connecting the integer points of f(x)=x^{2}+2cx+r on the Zone Grid, one produces a line that passes through a grid point to the left of the Zone 0 boundary with the relationship between the line’s slope and its y-intercept as follows:

intercept = -(a*slope+b), where

a = ceil(sqrt(r)), and

b = a*(a+1)-r.

The first several non-perfect-square remainders, their multiplicative coefficient a, and their additive coefficient b, appear as follows, with a scatter plot of a vs. b.

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