# Tag: number theory

• ## Integer Factorization: Patterns in the Weeds

Advertisements When m is an odd positive integer, c2 is the least perfect square greater than or equal to m that is odd or even as (m+1)/2 is odd or even, and r=c2-m, we can construct a Zone Grid and compute the slope of a Remainder Line passing through appropriately chosen values of r0=r and…

• ## Integer Factorization: Deez Numbahs Iz Kwazy!

Advertisements Truly I find myself in the weeds here, folks. And that’s a good thing. It’s good because the process of getting myself out of the tangle of weeds could involve untangling the principal tangle with which I’ve concerned myself for this past almost-year. I wrote the Perl script for the Initial Trapezoid I talked…

• ## Integer Factorization: The Initial Trapezoid

Advertisements Not all of the work is done yet, and the thought-work has just begun, but I wanted to update with a screenshot of the concept of the Initial Trapezoid: the quadrilateral bounded by the green and red lines where the Remainder Line in orange, passing through a typical pair of remainders r0 and r1,…

• ## Integer Factorization: Playing With the Zones

Advertisements 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…

• ## Integer Factorization: Update for 21 March 2023

Advertisements 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…

• ## A Method of Construction and a Theorem for Integer Factorization

Advertisements By J. Calvin Smith, B.A., Mathematics, Georgia College, Milledgeville, Georgia, United States of America (1979) – Retired Member, American Mathematical Society Written 14 March 2023 at Mountain River Chalet (the author’s home), Talking Rock, Georgia, USA. The following definitions, theorem, and construction are the end result of research by the author into factorization of…

• ## Integer Factorization: An Informal Report

Advertisements Here is an informal article, in PDF, about the work I’ve done lately, including some of the new revelations about which I’ve blogged.

• ## Integer Factorization: Going Forward

Advertisements What do I do next? Well, I’m inclined to think I should look at how I’m making the Zone Jumps, which don’t get bigger as rapidly as I’d like, into multi-Zone jumps that do. This will involve understanding the quadratic increase of the numerators of the intercept points with Zone boundaries, and the linear…

• ## Integer Factorization: Revision to jumpfactor.pl, March 8, 2023

Advertisements I added a usage line, and also programmed the loop to look only at numerators which are even, cutting the number of iterations of the main loop in half. Thus repurposed variable “idx” as the Zone number, and added another variable to serve as the iteration counter. The revised code will be, and will…