Tag: integer factorization

Integer Factorization: The (Currently) Incalculable Property of the RemainderRemainder Grid
Advertisements The Usual Definitions, Restated for Current Explorations: Let m be an arbitrary odd positive integer whose factorization into powers of primes is not something we necessarily know. Determine whether (m+1)/2 is odd or even, or, equivalently, whether m is congruent modulo 4 to 1 or 3, respectively. Consider the possible expressions of m as…

Integer Factorization FYI: Current “scratch paper” worksheet
Advertisements Copied from my text editor app, just so show lots of recent cogitation evidence.

Integer Factorization: The RemainderRemainder Charts (Brief)
Advertisements Okay, so the charts are filled out as fully as I’m filling them out tonight, and I promised an explanation, so this is a quick one. (It’s late.) I took the interesting number above, 101010101, which has more than two prime factors, and looked at the results of my characterize.pl script, which looked at…

Integer Factorization: Pretty Exciting, Right Now
Advertisements To me, pretty mathematical properties are always exciting. They do not always lead to new discoveries of great mathematical moment, however. Right now, I’ve discovered something pretty, that may lead to other discoveries, which is pretty exciting in itself to me, whether momentous or not. I am developing something new I call RemainderRemainder Tables,…

Integer Factorization: A Possibly Not So Big Idea, but Who Knows?
Advertisements The reason I took that recent sidetrip into looking at the “Fermat factorization” of multiples of 4, further using the fact that 4n = (n+1)2(n1)2 to decide upon the even/odd “polarity” to determine an appropriate Ceiling Square and its Root, was that I had gotten a bit excited about the ability to use quadratic…

Integer Factorization: A Brief Insight Prefatory (Possibly) to a Big One
Advertisements When expressing a number in terms of its Ceiling Square and Remainder, some time ago I decided it would be good to come up with a systematic technique that would work with even integers as well as odd. I have what might be a refinement of the part of that technique that deals with…

Integer Factorization: Pushing Forward
Advertisements [To the regular reader: I do apologize for a bit of repetition of definitions, but I do this mainly for someone just jumping in to my posts for the first time, so that they’ll get up to speed quickly.] Let m be a discrete (squarefree) semiprime, c its Ceiling Root and r the difference…

Integer Factorization: Update for February 9, 2023
Advertisements Let m be an odd discrete (squarefree) semiprime whose prime factors are p and q, with p<q. Determine whether m is congruent to 1 or 3 modulo 4, or, equivalently, whether (m+1)/2 is odd or even, respectively. This must match whether (p+q)/2 is odd or even. Let the Ceiling Root of m be the…

Integer Factorization: Something Interesting (Tuesday, Feb. 7, 2023)
Advertisements As phenomena of factorization go, this one is fascinating to me, though actually only a minor insight: Consider the discrete semiprimes (i.e., squarefree composite numbers with only two prime factors) within the odd positive integers, and the prime factors of each one of them, p and q, ordered so that p<q. Last year, I…

Integer Factorization: Math Graphs du Jour
Advertisements These are graphs of the behavior of the terms of the Series of Ceiling Roots for m=11882081 starting with the basic Ceiling Root of 3449 and increasing by 2. (See past posts for explanation and discussion of Ceiling Roots.) This is the crazy sort of pattern of numbers I hope to tame and characterize,…