Tag: integer factorization

Integer Factorization: The Jump Chart for the Trip to 111000011
Advertisements … As a tool, to help me figure out how to do it in ONE JUMP… i.e., a formula for the general case so I can look at the Ceiling Root, along with the first few remainders (or just one of them if it’s a really good formula!), and come up with the magic…

Integer Factorization: My Mycelial Mathematical Meanderings?
Advertisements It’s fun when I find a metaphor from the world of fiction, especially from the world of popular fiction, and incorporate it into my thinking about something real (or, more accurately, practical) with which I am dealing, and then I discover that the metaphor was, or I could easily make it, a more appropriate…

Integer Factorization: Tonight’s Prettiness
Advertisements Some notes from tonight’s method of turning the numbers around and around. 365 is easy to factor, but just ignore that for a while. Using the Ceiling Squares method, the Ascent is 18, so there’s a pattern we can look for, see, and interpret. Looking for beauty we can generalize . . .

Integer Factorization: 26 February 2023 Update and Screen Capture
Advertisements What you see here is an attempt to look for patterns in the distribution of remainders that result from subtracting the first several smallest composite numbers from their Ceiling Squares, an evergrowing triangle of numbers made by taking everincreasing values of s, with values of t less than s, relatively prime to s, even…

Integer Factorization: Rabbit Holes, Rabbit Holes
Advertisements What have I been jumping down into this week? After making and sharing parts of some studies into how odd positive integers with fixed remainders from their Ceiling Squares distribute themselves, and constructing companion lists of factorizations and Ascents for differences of even/odd and odd/even pairs of perfect squares relatively prime to each other…

Integer Factorization: The Current Exploration, 20 February 2023
Advertisements Information updated on 22 February 2023. When the Ceiling Square c2 for a positive odd integer m leaves a remainder r that is not a perfect square, that r still must be congruent to 0 or 1 modulo 4, corresponding to c being odd or even. I have started exploring how odd positive integers…

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,…