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 with the objective of seeing patterns in the not yet characterized relationship between the ceiling root of an odd positive integer m, the number (m+1)/2, whose square differs from that of (m-1)/2 by exactly m, and the difference between them. No pattern is jumping out at me yet. However, there is a curve that seems to describe almost exactly, if not exactly, a functional relationship between (m+1)/2 minus the ceiling root, and the value of m itself. (See the image below. Nice smooth curve, right?) I’ll sit and scribble after a while, and it may boil down to something really obvious. But that’s the latest on my study.
Integer Factorization: Update, 10/12/2022