Tag: mathematics

“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 […]

“Approximation” by JCSBimp
Advertisements An Arrangement of Squares Based on an Arrangement of Not Quite Regular Pentagons more fun with online graph paper art Rows of squares sometimes overlap, but columns of squares do not. And now, a little bit more … and the key. The squares mark the complete vertices of the graph formed by the packed […]

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 […]

The Eye
Advertisements Another QuasiPenrose Tiling With Holes, Based on an Irregular but Symmetric Pentagonal Tile . . . . . . and my usual “ringbased” concentric construction method.

Another Graph Paper Tiling
Advertisements Below is the beginning of a second QuasiPenrose tiling I’ve made using virtual graph paper and an irregular pentagon created and flipped thereon.

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