I find myself, mathematically, almost ready to say with certainty, “It boils down to this.”
I am speaking in terms of the Ascent, that concept to which I gave a name a month or three ago which is the difference between the Ceiling Square of a positive odd integer m and the larger of the two perfect squares in a Fermat factorization (i.e., of the form s2-t2) of m.
Visualizing this Ascent has proven a challenge to me so far. On the one hand, my “triangular” arrangement of the positive odd integers will be quite helpful in drawing together and making apparent properties of the Ascent, especially for discrete semiprimes. On the other hand, the actual Ascent values’ behaviors do have to do with the actual integer factors of m. Interspersed with those nice discrete semiprimes on that chart are the prime numbers and the integers with more than two prime factors. From what I have seen, both of these kinds of numbers make the possible Ascents jump around and multiply. Even the “Trivial Ascent,” the one corresponding to the factorization m equals 1 times m, yes, it is well defined, formulaic, even, … but it may have little or nothing to do mathematically with the other values for possible other factorizations of m.
Still, I hope to be able to visualize it properly. Even characterizing it formulaically for the odd prime numbers could help me understand how Ascent behaves among the discrete semiprimes.
Stay tuned, folks. ChrIIstopher, my MacBook Pro, is getting an important OS upgrade, but after that, and some outside yard work and inside cleaning I need to do, I’ll keep pursuing this problem, and will report back here.
After all, like I said, I’m pretty sure understanding integer factorization better boils down to this.