Tag: number theory

“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 […]

Just a Teeny Observation…
Advertisements Let m be an odd positive integer, c its Adjusted Ceiling Root, and r the difference between c squared and m. By the Adjusted Ceiling Root of m, I mean the least positive integer whose odd/even parity is the same as (m+1)/2 whose square is greater than m. I call that perfect square the […]

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 […]

The Odd Integers 3 Through 129
Advertisements Arranged as Red Squares on a Square Grid Artistically Arranged According to Their DifferencesofSquares Expression Based on Their Ceiling Squares

More Visualization
Advertisements Here are icons for the odd integers from 3 to 65, based on the differences of each with their ceiling squares (the smallest perfect square greater than their value). The integer corresponding to an icon is the number of red 1×1 squares in that icon.

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 […]