Tag: integer factorization
Integer Factorization: New Charts
Advertisements These charts are from my Numbers file for a recent study of Ceiling Squares, Remainders, and Ascents for the odd integers 1 through 999, with definitions of the quantities indicated as follows: If m is a positive non-prime integer: Let c be the integer ceiling of the square root of m. Adjust the value…
Integer Factorization: Series of Ceiling Squares, Take 3 (or 4)
Advertisements I embarrassed myself twice this week, having to take down insufficiently debugged algorithms and Perl code for not testing thoroughly enough. However, I now have an algorithm I’ve once again tested, and will test some more (yes, opening the possibility that a bug will require me to correct/retract), and this one has a symmetric…
Integer Factorization: Drop back and punt.
Advertisements I had to backtrack and delete it post a couple of days ago, sharing my system for calculating Series of Ceiling Squares. It was not that the program was coded improperly, but rather that the algorithm’s math did not get rid of the “infinite telescoping” of the series for some value or another. I…
Factorization Study: Graph in Motion
Integer Factorizations: Notes for the week of Jan 8-14, 2023
Advertisements Consider odd positive composite integer m, with adjusted ceiling square c2 and remainder r. There will be a least positive integer s such that m=s2+t2 for some positive t<s. This value of s will be greater than or equal to the ceiling root c corresponding to the ceiling square. Define the ascent of m…
Integer Factorization: The Continuing Refinement of Square One
Advertisements At the risk of being repetitive, the “beehive plots” and other recent work bring me back to the basic problem to be solved in order to advance the ability to factor large discrete semiprimes: Let m be an odd positive composite integer, and let c be the integer ceiling of the square root of…
Integer Factorization: Building a Better Beehive?
Advertisements I updated the “hivegraph.pl” script to give the “storeys” of the beehive plots y-values that made their slopes linear, and thus hopefully cause patterns to stand out. I wanted to create hive plots with plenty of bees to test it out, and I wasn’t disappointed in the results. The first discrete semiprime I created,…
Integer Factorization: The Hive Mind, Now Automated
Advertisements I’ve written a perl script to produce the “Beehive Plots” that I am currently using to search for patterns and relationships between odd positive integers, their (Adjusted) Ceiling Squares, and their Fermat factorizations, if they have them. Remember that to each possible factorization of a positive odd integer m=ab, where a and b are…
Integer Factorization: The Hive Mind and the Bee Line!
Advertisements In my study of how to factor an arbitrary odd discrete semiprime m=pq, I have begun looking at the series of remainders I get when I subtract m from its Adjusted and Ascended Ceiling Square (c+2n)2 where n, the ascent, ranges over non-negative integers. (See my earlier Integer Factorization blog posts for explanation of…