
Integer Factorization: Quick Summary of Last Night’s Simplification “Revelation”
Advertisements Let m be an odd positive integer we wish to factor. Let c be the smallest integer (also positive) so that c2>m. (If m is a perfect square, we are done.) Set r=c2m. The least nonnegative value of x for which x2+2cx+r is a perfect square is a value of x which produces s=c+x…

Integer Factorization: 9/18/2022 Update (short)
Advertisements It would seem, after reviewing factor_it_2.pl, that that code and the mathematics behind it are not due for an update – because they are beyond hope! 😀 No, really, I’ve moved far beyond those ideas, and they no longer seem fruitful enough to upgrade them. It would be like grafting a car onto a…

Integer Factorization: Recent Work
Advertisements Ideas kept me up when the dogs got me up around 3 a.m. this morning for a walk and some water. That’s good, even though the ideas did not pan out to much. I had church to attend in a few hours, after all. But it occurs to me to share where it’s all…

Integer Factorization: WebPublish in Haste, Repent at Leisure! 😀
Advertisements All right, I am very sorry… I did happen to see that huge error in arithmetic in computing the characteristic function, but not until late tonight. I have fixed it and placed the updated code in my original characterize.pl post. Again, I do apologize for that. But now I am excited again because until…

Integer Factorization: “characterize” Script Updated
Advertisements Here is the link to the post from yesterday with the script in the form I updated today.

Integer Factorization: characterize.pl Update Forthcoming
Advertisements I found (occurring to me in the early morning hours) a way to correct some of the “lumpy” properties of the characterize.pl output. When the algorithm adds 1 to the Ceiling Root, as needed, to produce the Adjusted Ceiling Square, I had forgotten to make sure also that the points I want the characteristic…

Integer Factorization: Postalgorithmicadjustment Thoughts
Advertisements It is plain to me that, after making the straightforward modifications to make sure that the characterization algorithm deals with adjusted ceiling squares of the correct “polarity” (according to m being congruent to 1 or 1 modulo 4), I need to look again at how I am choosing nice, smooth, linearly increasing ranges. These…

Integer Factorization – New Code for characterize.pl
Advertisements I believe this is running correctly. I’ll certainly correct it if it is not. EDIT: I have indeed updated the code. I suspect it’s almost if not entirely in its final form. A couple of small values in the entire set of discrete semiprimes whose primes are less than 100 showed negative discriminants, but…

Integer Factorization: The Adjusted Ceiling Square and Characteristic Polynomial
Advertisements I’ve got a good bit of housework to get going today, and some necessary but necessarily small amount of shopping to do as well, but I want to get this idea down first. The Adjusted Ceiling Square of odd positive integer m is the least perfect square greater than m that is odd if…

My Integer Factorization Study: Update – 9/14/2022
Advertisements I’ve changed the platform for composing my mathematical reports – above and beyond blogging them here – to use Apple’s Pages program on my MacBook Pro. It’s taking some orientation – equations must be in an appropriate mathematics markup language (I am choosing LaTeX) – and I need to export the end product in…