Tag: number theory

Integer Factorization: An Overdue Update
Advertisements New revelations in my study of factorization, Ceiling Squares, and the Zone Grid have not been coming lately, but I have indeed been looking at the data and trying out new ways to visualize their relationships, every day, and my work is still underway, despite no evidence of these facts in this blog. Right…

Integer Factorization: Update, Saturday 15 April 2023
Advertisements The most tantalizing current avenue in my study is to continue examining the relationships between m’s trivial factorization arising from its expression as ((m+1)/2)2((m1)/2)2, its remainder line on the Zone Grid, and the zone boundaries thereon. I am looking at 427 of the odd discrete semiprimes with the smallest prime factors in hopes of…

Integer Factorization: This Morning’s Delivery
Advertisements It is a rainy Saturday of Holy Week, 2023, smackdab between Good Friday and Easter/Resurrection Sunday on the religious calendar which I personally celebrate. It is also a year, give or take a week, since inspiration started coming to me in the early morning hours, usually after our chugs woke me up to have…

Integer Factorization: The Great Trapezoid
Advertisements In an effort to elicit potentially useful patterns from my mathematical study, and particularly from the Zone Grid idea I discovered, it would seem that I have come up with another way to help me visualize what is going on with the Remainder Series associated with the Ceiling Square and its Ascents. This one…

Integer Factorization: What’s Up Lately (2 April 2023)
Advertisements What’s got me all excited lately mathematically has been further exploration of the odd positive integers and the way their remainder series behave when translated into Remainder Lines on the Zone Grid. I established the construction of such a grid and the lines in question in a post a few weeks ago, and now…

Integer Factorization: My Latest Findings, 30 March 2023
Advertisements “It’s amazing what a relatively lackluster Junior and Senior year in college math, followed by 33 years of working for the government bureaucracy, will do to one’s expectations of oneself in one’s major field. “I am actually startled, especially in retrospect, that I’m sitting here now, at age 65, working out new results, and…

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=c2m, 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 almostyear. 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 thoughtwork 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…