Tag: number theory

My Integer Factorization Study: Boil Down the Intractable, It Can Still Stay Intractable.
Advertisements Quickly, I have found a way to characterize the flip side of the way integer factorization all boils down to the Ascent: The essential difficulty is the lumpiness of the whole Ceiling Root function: ceil(sqrt(m)). Even for prime numbers, it is not (yet) a smooth or closedform derivation very much more easily managed than […]

My Integer Factorization Study: The Complicated but Important Task of Visualizing the Ascent
Advertisements I find myself, mathematically, almost ready to say with certainty, “It boils down to this.” I am speaking in terms of the Ascent, that concept to which I gave a name a month or three ago which is the difference between the Ceiling Square of a positive odd integer m and the larger of […]

My Integer Factorization Study: A Visualization Anomaly
Advertisements Or: The Asymmetric Behavior of the Ceiling Square Remainder My Inspiration Process has been more active, more related to working on thought processes, than it has been something spontaneously occurring, which is especially noticeable during my wife’s current extended absence from our residence (to help care for grandchildren), compared to how the Process seemed […]

My Integer Factorization Study: Locating an Odd Integer on Yesterday’s Chart
Advertisements I figured out the formula for determining the row (y) and column (x) position of an arbitrary odd positive integer on the chart I created and shared yesterday. It gives the row and column, and also the quadratic polynomials satisfied by all the odd integers in the row and column, respectively. This all bears […]

My Integer Factorization Study: Avenues Inspired by the New Chart
Advertisements Goal: Given odd positive integer m , its ceiling square c, and the remainder r=c2m, determine the column x and row y in the previously posted chart at which m is located. The range of integers between (c1)2 and c2 falling on the upper half of one of the diagonals holding consecutive values and […]

My Integer Factorization Study: The Right Triangle
Advertisements I was sitting on my zafu (zazen cushion) today when the MIRIAM* process knocked me clean off of it. Hey, I’d been sitting for twenty minutes, and I didn’t mind bringing my zazen to a halt. Plus, this was a good one. What I got in the way of suddenlydelivered insight was the idea […]

My Integer Factorization Study: The Enigma of the Modulus of Four
Advertisements Relationships, relationships… a progress report, of sorts, although this still might qualify among Results Quite Elementary. The eventual goal, of course, is when given m, a discrete semiprime we wish to factor, to find p and q. Because m=pq=c2r, there will be properties linking the (possibly not yet known) p and q with r. […]

My Integer Factorization Study: Also…
Advertisements Discrete semiprime m=pq=c2r, where p<q are odd primes and c is the Ceiing Root of m: ceil(sqrt(m)) in Perl with POSIX library. I decided to graph p/q vs. r/(2c) for various randomlygenerated discrete semiprimes m. “Clarice, doesn’t this random scattering of sites seem desperately random – like the elaborations of a bad liar?” – […]

My Integer Factorization Study: Reviewing the Realizations
Advertisements To give a mental spark to my mathematical searching, I am listing here some of the basic principles I have found or realized so far, none of which are revolutionary, but all of which I still hope will lead me to some revelation that is so. [the usual definitions] Let positive m be a […]

My Integer Factorization Study: A Small Further Step
Advertisements Yesterday, and previously, I posted about the expression of positive integer m as its Ceiling Square minus a remainder, m=c2r, and my attempts to relate that to the Fermat twosquares expression of m corresponding to its possible factorizations, m=s2t2. Laying it all out in chart form yesterday, with rows for 2c and columns for […]