Tag: number theory

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 x2115625 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…

Integer Factorization: Chasing Elusive Characteristics
Advertisements I have computed a chart for all the odd discrete semiprimes less than 250 (I think the count is 45 of them), by way of teasing out characteristics and properties that I hope I can tease out formulaically in days/weeks/months to come. It’s the current tree up which I’m barking, so to speak. By…

Integer Factorization: A New Article on iCloud
Advertisements https://www.icloud.com/pages/0a8XdZPFDHf0wcjBTTkuTlxfg#Equivalent_Problems_to_Integer_Factorization I am trying, above, a new way to share my results.

Integer Factorization: Current Study Avenue, with a New Chart
Advertisements I’ve been building a new table of the positive odd discrete semiprimes. My objective is to consolidate observations about all such among the integers less than 20,000, but right now I’m not even up to 2000, so there’s plenty of work to be done, and it’s all aimed at the objective of finding something…