Tag: integer factorization

Another Excerpt from “Ceiling Squares, Central Squares, and Factorization”
Advertisements Now consider an arbitrary odd positive integer m, its trivial factorization as 1*m, and its corresponding expression as the difference of two squares [(m+1)/2]2[(m1)/2]2. Since m is odd, either the minuend (the first term) is even and the subtrahend (the second term) is odd, or vice versa, as follows: When m is congruent to […]

Math: An Excerpt from InProgress Report
Advertisements Ceiling Squares and Discrete Semiprimes A discrete semiprime is the product of two distinct prime numbers. A discrete semiprime has exactly four divisors. If m=ab is a semiprime, with a and b the prime factors of m, and ordered so that a<b, then m has exactly four divisors which are, in numerical order: 1, […]

Wed Jun 8 3:43 AM – MIRIAM wakes me with a Vision
Advertisements Yes, dear reader, it was bound to happen sometime, I guess! I slept soundly, having gone up to the bedroom before my wife got sleepy, and the dogs came up to settle in with me, too. And I slept, and I dreamed, and it had to do with ceiling squares and integer factorization, but […]

My Integer Factorization Study: The Hard Problem Remains
Advertisements Of course the hard problem of integer factorization remains! Should I succeed where so many others, with so many more well developed analytical tools have failed? And yet, I press on. In short, this is . . . The Hard Problem, in a(n Impenetrable) Nutshell Ceiling squares have gotten me a nice little efficient […]

The Mathematical Tree Up Which I Am Barking Now (June, 2022)
Advertisements It’s the same old tree as before, pretty much, but hey… When m is any odd positive integer, choosing a=1 and b=m gives a factorization of m into ab. In this case, the perfect squares of the appropriate range whose difference gives the equation m=ab come from observing that solving the system of equations […]

Cute Li’l Number
Advertisements In building my current math study tables – from which the graphs in the previous post came – I encountered the number 385, which, as a direct consequence of its factoring into two factors in exactly four different ways, also has four different expressions as the difference of two squares. This in itself is […]

Curious Hieroglyphics
Advertisements … which our intrepid blogger has found inscribed on the positive odd integers.

Mathematical Insights, an Update: Ceiling Squares and Central Squares
Advertisements But First: A Very Personal Psychological/Spiritual/Mystical/IReallyDon’tKnowWhatExactlyItIs Side Trip My mathematical insights are still coming fairly predictably almost every night, trains of thought that feel inspired and that push me to think them to completion, or at least to some kind of form. My Primary Care Physician sees my state as being similar to that […]

AfterMath: Even Numbers Are Just Odd.
Advertisements The thoughts I still get in the early morning hours about numbers and factorization are not so imperative now that I have programmed and published that new, simple algorithm. Sometimes I can just smile at them and go back to sleep. And other times … they’re just WEIRD. This morning’s stayawakeandthinkaboutit notions concerned the […]

Factorization Perl Script Update
Advertisements I cleaned up the code for cf_exam.pl just now. Making the script better also made it a little shorter, but I doubt that it is any faster. Anyway, have at it!