Tag: integer factorization

Integer Factorization: Looking Again at the Data
Advertisements Research report follows, the first after a bit of silence, by way of a current example. Consider the integers m1=3317, m2=4061, m3=5461, and m4=6077. They all have adjusted ceiling squares – the least perfect square with oddness/evenness the same as one half of one more than their value that is greater than each of…

Today’s Mathematical Visualization
Advertisements This is an artistically arranged gridbased drawing of the first 24 odd integers greater than 1, expressed as differences of squares – to include the first odd positive integer to have two different “nontrivial” factorizations, and thus two different squaredifference representations.

Adjusted Scatter Plot for Integer Factorization Study
Advertisements The plot below is similar to the one I put in my previous post. However, the data points are, I think, a better set for exposing properties, since I am using the concept of Adjusted Ceiling Square as follows: Let m be an odd positive integer. Define the Adjusted Ceiling Square of m as…

My Integer Factorization Study: Hahaha!!
Advertisements I am laughing at the sheer trickiness, almost by design, of the mathematical relationships I am uncovering in my number study, and the most recent one in particular. I shared yesterday the VERY SMOOT GRAPH, suggesting to me a parabola, charting out the plot of odd integer m vs. the difference between (m+1)/2 and…

Integer Factorization: Update, 10/12/2022
Advertisements Lately, I’ve just been switching back and forth between a kind of macrovision and microvision in examining properties related to my factorization study. I’ve gone back to my table of large (but not very large) discrete semiprimes, and I’ve also looked at, and added information to, my table of the smallest discrete semiprimes, all…

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,…