Tag: number theory

Integer Factorization: Reestablishing Some Basic Steps
Advertisements I wrote this in an effort to restate some recent basic observations plainly. I am confident that this is no newlydiscovered set of mathematical truths, but I am enjoying making these derivations piece by piece and seeing what may follow from each successive one. Perhaps someday I will end up somewhere new. Right now,…

Mathematical Explorations: Maximally Composite
Advertisements It’s time for some fun and games with some squarefree “maximally composite” odd numbers and all their representations as the difference of perfect squares. It’s kind of like following a growing pristine wave with patterns I didn’t expect to encounter, until it smashes into foam and chaos against a rocky shore.

Integer Factorization: Quick Observation
Advertisements Examining the series of real values of x for which f(x)=x2+2cx+r is a perfect square, when m=c2r is the number we wish to factor, c is the Ceiling Root, and r is the difference between m and the Ceiling Square, is not evidently any better – and is in fact a slower and unenlightening…

Integer Factorization: When x^2+2cx+r Yields a Perfect Square
Advertisements For odd positive integer m=c2r, define the characteristic polynomial f(x)=x2+2cx+r. We are looking for the least nonnegative integer x for which f(x) is a perfect square. Completing the square of the polynomial may give us a way to find values of x given f(x). I decided to try to determine when f(x) = 0,…

Integer Factorization: A More Uniform Test, and a Similar Result
Advertisements For 100 pairs of randomlychosen p and q less than 100,000,000: the Perl routine factor_it_4.pl performed as shown in the following scatter plot of m=pq vs. the number of iterations it required to factor m: Notice, as I mentioned in my blog post about the previous, less uniform test, that the number of iterations…

My Integer Factorization Study: Leading Me Ever Back
Advertisements As I’ve previously shared, my math study seems to lead me in kind of a circular path, between periods of great excitement at potentially profound discoveries, and “letdown” periods where I realize what I “discovered” is something not only already evident, but blindingly obvious. I still want to remain happily agnostic about which of…

Integer Factorization: Performance Stats/Graphs for factor_it_4.pl
Advertisements I ran factor_it_4.pl tonight on the collection of 10digit and 12digit discrete semiprimes I had generated earlier in my study. An image of the table I generated is below, along with scatter plots of particular behaviors. If someone wanted, I could certainly supply them with the raw data used to generate the graphs. I…

My Fastest Factorization Script So Far
Advertisements This won’t set any algorithmic speed records, but I am happy with it, for now. It comes in at 48 lines of Perl code. factor_it_4.pl is definitely faster than factor_it_3.pl, and routinely gets its answer in fewer than half the iterations of factor_it_2.pl. Once again, I am not experienced with writing arbitraryprecision arithmetic scripts…

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…