## Integer Factorization: A Seemingly Vital Derivation

For brevity, this post presupposes familiarity with material from earlier posts, namely this one and this one.

I have just derived the precise quantities for the following: The line on the Zone Grid for a Remainder Series that represents odd positive integer m=c2-r has a linear equation with slope 2c-2(ceil(sqrt(r))-1. This formula makes this slope an odd positive number for the values of m of interest in factorization. This Remainder Line will intersect a finite number of the zone boundary lines in the x-positive, y-positive quadrant of the Zone Grid, and will be parallel to exactly one of them.

The operations by which I derived this give me values, relationships, and expressions for further study, and my derivation calculations went as follows

``````f(x) = x^2+2cx+r.

Note: For non-negative integer x, f(x) is the xth number in the remainder series given by (c+n)^2-m for n=0,1,2,... .

r0 = f(0) = r, the original difference between m's ceiling square (c^2) and m.
r1 = f(1) = 2c + r + 1.
x0 = ceil(sqrt(r0).
y0 = r0-x0(x0+1).
x1 = x0+1.
y1 = r1-x1(x1+1) = r1-(x0+1)(x0+2)
= 2c+r+1-(x0+1)(x0+2).
y1-y0 = 2c+r+1-x0^2-3x0-2
-r+x0^2+x0
= 2c+1-2x0-2
= 2c-2x0-1.``````