Tag: integer factorization

Integer Factorization: Making the Crooked Straight
Advertisements My current approach to integer factorization, elements of which I have described in the last several days on this blog, involves a process I discovered last night that surprised me with its beauty and simplicity: It is a twodimensional method, straightforwardly described and probably easily programmed, for visualizing quadraticallydistributed numbers, namely consecutive perfect squares…

Integer Factorization: Plotting the Bee Line
Advertisements I came up with an idea late last night (after midnight turned March 3 to March 4) that is a bit exciting, although I know enough from the past months of study to temper my excitement with some realism as to the chance of it translating into an actual algorithmic breakthrough. So far, the…

Congruence Consequences I’d Forgotten
Advertisements My integer factorization study has me looking a great deal at differences of perfect squares, one even and one odd. This is an easily justified path to take, since powers of two will be apparent and can divide out as soon as we start, regardless of the numerical base we are using. But I…

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…