• ## Integer Factorization: A Ceiling Square Spread and a Math QR Code!

Advertisements I am once again trudging into the deep forest of how prime numbers, discrete semiprimes, and other composite numbers distribute themselves among the positive odd integers, looking for patterns. The illustrations below are from the beginning steps, taken yesterday and today, of this stage of my math study. The table I’ve built using Apple…

• ## Integer Factorization: A Relationship on the Zone Grid

Advertisements 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=c2-r as a straight line connecting the integer points of…

• ## Integer Factorization: Assessing Recent Revelations

Advertisements Once again, the findings, though pretty, are seemingly of little moment, at least in terms of discovering something new. But wait. Breathe. Think. Here is what I know now, in the form it has most recently taken: Let m be an odd positive integer, and c2 be the smallest perfect square greater than m…

• ## Integer Factorization: A Larger Remainder Form Table Value

Advertisements I decided to look at a Remainder Form Table and the resultant graph for the Remainder r=115625 taken from the Ceiling Square and Remainder of the discrete semiprime m=1427061659. Making sure I selected values in the correct ranges for my graph (see my previous post’s warning edits), I looked at x2-115625 for x ranging…

• ## Integer Factorization: Behavior with Fixed Remainders

Advertisements EDITED TO ADD – WARNING! – Do not yet take any of the following formulae or ideas as verified, not just yet. I am having a problem with larger values plugged into the approach. I have to investigate and debug, or at least understand better, what is going on. EDITED TO ADD, PART TWO:…

• ## Integer Factorization: Looking at Data in Many Ways

Advertisements The following are scatter plots of Remainder vs. slope of Characteristic Line vs. Ascent vs. Solution Zone Boundary, plotted against each other in the six possible pairings, for the first hundred or so discrete semiprimes whose Ascent is greater than zero. Certainly the distributions are far from random, but the patterns are also far…

• ## Integer Factorization and Discrete Semiprimes: A Family of Seven

Advertisements Somewhere between 212 and 232 lies a set of seven equally spaced numbers which are consecutive odd discrete semiprimes. They spread out in uninterrupted uniformity, all the integers congruent to 1 modulo 4 from 481 to 505. Their prime factors, a pair each, bounce around with predictable unpredictability (so far!), as do their unique…

• ## Integer Factorization: Back in the Mess, 9/2/2023 Edition

Advertisements The difference between the beauty and symmetry of the chart at the bottom below, and that of the charts above it, is as great as the difference between the amount I have discovered about integer factorization, and the amount left for me to figure out. The bottom chart I explained in my post yesterday.…

• ## Integer Factorization: Some Newly Observed Relationships

Advertisements If odd integer m is a discrete semiprime, then the search for the least positive integer whose square when added to m produces another perfect square is the problem addressed by the admittedly slow process of Fermat’s factorization method. Despite its lack of advantages over trial division as a factorization method, Fermat’s approach fascinates…

• ## Integer Factorization: This Avenue Is Still Interesting.

Advertisements Today’s work feels fun and hopeful. I am on a clear liquid diet for medical testing reasons, and that might be helping my mental clarity a bit. I am exploring the relationship between the Characteristic Polynomial and the Characteristic Line on the Zone Grid, and have discovered, possibly, a new relationship between the smaller…