# Tag: number theory

• ## The Ascent of m: A Concept Whose Properties I Shall Explore Soon

Advertisements The minuend of any such square-difference expression of m will then be greater than m’s ceiling square. Define the ascent of such a factorization or corresponding square-difference expression as the difference between the square root of the minuend in that expression and the ceiling root of m. EXAMPLES: The ascent of the trivial 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-[(m-1)/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 In-Progress 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/I-Really-Don’t-Know-What-Exactly-It-Is 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…

• ## My Math Mind: Mistaking the Pedestrian for the Profound Since, er, Coupla Months Ago, Maybe?

Advertisements I keep getting these insights that make me go “Wow!” and then two seconds later, or sometimes two days later, depending on the complexity of the idea in question, I go “Waitaminnit… that’s not really profound at all. It’s closer to pedestrian, and almost PROFANE.” Okay, I don’t go that in so many words.…