# Tag: number theory

• ## 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…

• ## Integer Factorization: The Adjusted Ceiling Square, and Newly Deriving an Old Characteristic Polynomial

Advertisements Consider the problem of factorization of m, an odd discrete semiprime (the product of two distinct odd primes). m can be expressed as the difference between two perfect squares in two different ways, giving the “Fermat factorization” of m corresponding to its expression 1*m and p*q, with p and q being m’s distinct prime…

• ## Integer Factorization: Look at the Data

Advertisements I am returning my attention, amid dealing with holiday decoration and winter coldness precaution, to the numbers both small and largish, in examining discrete semiprimes for patterns that may give clues to factorization. The table/scribble-sheet below, completed without automated computation beyond a calculator, is my current view of the behaviors of the smallest ones…

• ## “Telescoping” the Positive Integers: A Systematic Approach

Advertisements I’ve systematized, somewhat, the recursive expression of an arbitrary positive integer as a recursive sequence of differences of systematically adjusted ceiling squares with their remainders. I am hoping this will allow me to continue on a path to greater revelations about integer factorization. As outlined below, slightly surprisingly to me this morning, I’m not…