Tag: mathematical visualization

Integer Factorization: Math Graphs du Jour
Advertisements These are graphs of the behavior of the terms of the Series of Ceiling Roots for m=11882081 starting with the basic Ceiling Root of 3449 and increasing by 2. (See past posts for explanation and discussion of Ceiling Roots.) This is the crazy sort of pattern of numbers I hope to tame and characterize, […]

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 nonprime integer: Let c be the integer ceiling of the square root of m. Adjust the value […]

And now, a message from the odd positive integers, my grandson Keegan, and myself.
Advertisements Part I: I decided simply to graph the values of the (Adjusted) Ceiling Square and its remainder for all of the odd positive nonprime integers I’ve put in my 3D graph so far (1 through 441). The result jumped out at me with its nearsymmetry of dots, which appear in the graph below. But […]

Factorization Study: Graph in Motion
Advertisements https://share.icloud.com/photos/056cVxRRrMUfrlOf5Q4nWKBKg

Integer Factorizations: Notes for the week of Jan 814, 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: Building a Better Beehive?
Advertisements I updated the “hivegraph.pl” script to give the “storeys” of the beehive plots yvalues 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 nonnegative integers. (See my earlier Integer Factorization blog posts for explanation of […]

“Approximation” by JCSBimp
Advertisements An Arrangement of Squares Based on an Arrangement of Not Quite Regular Pentagons more fun with online graph paper art Rows of squares sometimes overlap, but columns of squares do not. And now, a little bit more … and the key. The squares mark the complete vertices of the graph formed by the packed […]