Tag: integer factorization

“Telescoping” the Positive Integers: A Systematic Approach
Advertisements I’ve systematized, somewhat, the recursive expression of an arbitrary positive integer as a recursive sequence of differences of systematically adjusted ceiling squares with their remainders. I am hoping this will allow me to continue on a path to greater revelations about integer factorization. As outlined below, slightly surprisingly to me this morning, I’m not […]

Integer Factorization and Related Work: An Encapsulation of “Trivial” Representations
Advertisements I belive I now have all the information cobbled together sufficiently to give a concise characterization of all positive integers’ “Trivial” characterizations as differences of squares. For any positive integer m, let the Ceiling Square of m be the least perfect square greater than m. For odd m, let the Adjusted Ceiling Square of […]

Some Math Notes
Advertisements These are from studies ancillary to my integer factorization explorations. Let m = 2n be an even positive integer corresponding to positive integer n, which may or may not be even. Let s be the Ceiling Root of m, the integer ceiling of m’s square root. s squared is the Ceiling Square of m, […]

Integer Factorization: Update, 14 November 2022
Advertisements Bimp’s Decreasingly Frequent Progress Report I am still working daily on my mathematics study, but want to get better at dropping a word now and then about my progress. Below is a text file created (without Perl script help, except in running my “gimme the next prime” wrapper routine) in just the past couple […]

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 […]