Consider an odd positive integer m for which one is seeking factorization. Consider the more or less typical case where the Ascent of the Ceiling Square of m is large, is turning out to be in the thousands or more; that is, a value of m for which the iterations of my Fermat-based factorization are large.
If I look at the values for r in m=c2-r as c begins at the Ceiling Root of m and increases by two, and eventually but, in this case, r slowly reaches a perfect square value permitting Fermat factorization, I see a jagged, jumpy graph, but one that looks at least a little bit like a sampling of a function with a periodic component but for which the period is slowly changing. Will its similarities with a down-chirp or up-chirp signal in signals processing provide clues as to where m=s2-t2?
My idea here is that I might gain information from looking at some of the values of the Series of Ceiling Roots for successive remainders, and taking the FFT of some of those values for a relatively small set of remainders, say 128. But I need to make sure I know at what I am looking, precisely, and whether I can make an interpretation and/or prediction based thereon. It’s a way of thinking of that series of remainders as a signal uniquely associated with the integer value of m and how it relates to its Ceiling Square.