Along with my direct mathematical study of data and ways to gain insight into what is going on with Ceiling Squares, their remainders and ascents, and so on, I have been reading the free e-book I got with my AMS Membership: The Joy of Factoring by Samuel S. Wagstaff, Jr. It basically amounts to a good discussion of the number theory behind the classical and newest advances in factoring large integers.
It occurs to me, looking at the material so far, which is somewhat familiar to me but also quite challenging in terms of wrapping my head around it all and working with its intricacies, that when and if I gain an insight that is new about integer factorization, it seems probable to me that such an insight will be on my terms: It might be a departure from the complexity of the material I am reading in the Wagstaff book. That is my current approach to the remark my Federal instructor gave me years ago, that nobody knows how simple or complicated the next breakthrough will be. It might require a new technique and approach similar to the known tools with names Legendre, Moebius, Thue, and so forth. I can stand on the shoulders of giants and attempt such findings, but I might instead find a flight of inspiration that takes my feet off of them, or, equivalently, removes their support from under me, at least somewhat. I’ll still be grateful for the altitude boost all this good work gives my own attempt at new discoveries.
And now, regarding my own studies today: I am compiling a Table of Ascents, the numbers that indicate how far above the Ceiling Squares for odd integers are the actual values of s corresponding to expressions of those integers as s2-t2. The number of different such expressions for an odd integer, I’ve explained before, is the number of different ways to express that integer as a product of two integers; i.e., the number of divisors divided by 2. I’ve chosen the column and row values simply as well: These are the values of (n+a)2-n2 where n, the row number, takes consecutive positive integer values, and a, the column number, takes consecutive odd positive integer values. The resultant integer followed a simpler formula after reduction, and it is that linear value I’ve used for the column headings.
Here’s a picture. Prime numbers are only in column 1. Discrete semiprimes are in the blue-shaded boxes. What you don’t see is that the next column will have no blue-shaded values, the common factor being 9=3*3, of course increasing the number of divisors.