# Tag: mathematical visualization

• ## 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…

• ## 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 non-prime integers I’ve put in my 3D graph so far (1 through 441). The result jumped out at me with its near-symmetry of dots, which appear in the graph below. But…

• ## 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: 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…