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 these terms.) This study has produced a few interesting results, and some interesting metaphorical images in the process of visualizing the results.
Here you see what I am calling a Beehive, a centered arrangement of rows showing, in this case, for n=0 through n=7, where m falls in the range between the two values of (c+n)2-t2 for the positive integer values of t that produce the greatest such value less than m, and the least such value greater than m. Those values are marked in the Beehive by X, and m by an asterisk.
To make this clearer, here is the narrative description of what is happening with the calculations for this Beehive’s discrete semiprime, which is m=177:
177 = 15^2-48; bounds: 15^2-8^2 = 161 and 15^2-6^2 = 189 = 17^2-112; bounds: 17^2-12^2 = 145 and 17^2-10^2 = 189 = 19^2-184; bounds: 19^2-14^2 = 165 and 19^2-12^2 = 217 = 21^2-264; bounds: 21^2-18^2 = 117 and 21^2-16^2 = 185 = 23^2-352; bounds: 23^2-20^2 = 129 and 23^2-18^2 = 205 = 25^2-448; bounds: 25^2-22^2 = 141 and 25^2-20^2 = 225 = 27^2-552; bounds: 27^2-24^2 = 153 and 27^2-22^2 = 245 = 29^2-664; bounds: 29^2-26^2 = 165 and 29^2-24^2 = 265.
The Beehive here is only drawn as far as the eighth storey down, so to speak, because at the next stage, 312 differs from m by 282, giving m as the difference of two squares and permitting Fermat factorization. Also, I am adhering to the convention of the Adjusted Ceiling Square and its Ascents always being even or odd, as (m+1)/2 is even or odd. In the case of 177=s2-t2 for some s and t, s must be odd and t must be even. (It has to do with the value of m and of any perfect square modulo 4. Again, see previous Integer Factorization posts.)
I am a little excited about this because the Beehive above is showing a typical behavior, one that repeats for other Beehives of other integers. And I am amused because it can be described in a metaphorical fashion. If I were to take larger and larger values of m, with larger and larger numbers of times I need to “ascend” the Adjusted Ceiling Square for a particular m in order to find a remainder that is a perfect square, I would watch my asterisk symbol – which I consider the Bee in the Beehive, jump around for a while, erratically but possibly predictably. However, as the Bee (and the Hive Mind) knows what number that is (I assume a number contains knowledge, so to speak, of how it is going to behave), it knows when it will have a positional value equal to one of the walls of the Beehive: the way out of it!
Think of bee hives in all the old classic cartoons you’ve seen: the old-fashioned kind, shaped kind of like the rounded form in the illustration. The door is at the bottom, just a round opening for the lucky bee to get through.
And when after jumping around, bouncing off walls, the Bee here gets low enough, it sees its clear path to the door of the Beehive, and it makes a Bee Line for it!
The simple and consistent predictability, and the ease of visualizing the path to the answer, do indeed have me thrilled that I might have something I can use to simplify factorization, however metaphorical. I was already speaking metaphorically earlier this year about my discoveries in terms of barking up trees and trying to find squirrels. Perhaps my dogged intellect should have been sniffing around for Beehives instead!