Tag: number theory

Integer Factorization: Brief Remainder Line Observation
Advertisements The slope and yintercept of the Remainder Line for an odd integer m=c2r with Ceiling Square c2 and Remainder r on the Zone Grid increase linearly over all values of m having the same r, according to their c value. This is something I’ve already programmed in to jumpfactor.pl: It calculates the Remainder Line…

Integer Factorization and the Zone Grid: Constructing and Using Remainder Lines
Advertisements First of all, I want to write down on my blog some of the nitty gritty, in a fashion I hope will be clear, of how I construct and use the Zone Grid I have been using in my calculations and explorations, and my discovery of how to plot thereon the Remainder Line corresponding…

Integer Factorization: Update for 25 September 2023
Advertisements An incidental quirk i noticed in jumpfactor.pl outputs led me to scurry to program a jumpfact2.pl based thereon, but which produced a speedup in only one out of the fifty test numbers (10 to 12 digits) of my discrete semiprime data suite. What that exciting/disappointing work caused me to pause has resumed: a more…

Integer Factorization Update 9/21/2023: The Zone Grid, The Trivial Jump, Etc.
Advertisements It’s been a week of inundation by ideas. I’ve battled headaches and all of the struggles of balancing potentially good mathematical revelations with all of life’s necessary other concerns. I am in a play – It’s a small role, but I am also the Musical Director for some gospel song interludes the director wants…

Integer Factorization: The Least Ceiling Square LCS(r)
Advertisements Let m be an odd positive integer. Let c be the least positive integer for which c2 is greater than m, which is odd if (m+1)/2 is odd and even otherwise. c thus defined is the Ceiling Root of m, c2 is m’s Ceiling Square, and the difference between m’s Ceiling Square and m…

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=c2r 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 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:…