Tag: mathematical visualization
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Advertisements Factor, factor, burning bright, In some wavelength beyond light, For you I seek, frustrated. Is all my work ill-fated?
Advertisements Sometimes, to look at fearful symmetries, I just need typeset graphics, if you please.
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 two-dimensional method, straightforwardly described and probably easily programmed, for visualizing quadratically-distributed 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…
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: 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,…
Integer Factorization: New Charts
Advertisements These charts are from my Numbers file for a recent study of Ceiling Squares, Remainders, and Ascents for the odd integers 1 through 999, with definitions of the quantities indicated as follows: If m is a positive non-prime integer: Let c be the integer ceiling of the square root of m. Adjust the value…
And now, a message from the odd positive integers, my grandson Keegan, and myself.
Advertisements Part I: I decided simply to graph the values of the (Adjusted) Ceiling Square and its remainder for all of the odd positive non-prime integers I’ve put in my 3D graph so far (1 through 441). The result jumped out at me with its near-symmetry of dots, which appear in the graph below. But…
Factorization Study: Graph in Motion
Integer Factorizations: Notes for the week of Jan 8-14, 2023
Advertisements Consider odd positive composite integer m, with adjusted ceiling square c2 and remainder r. There will be a least positive integer s such that m=s2+t2 for some positive t<s. This value of s will be greater than or equal to the ceiling root c corresponding to the ceiling square. Define the ascent of m…