The latest revelations, which came almost unbidden despite my expressed desire to take a break from this study, turned out to be exciting ones, and I am trying to remain calm.
I have found relationships between the value of r (which is the difference between the square of s (the integer ceiling of the square root of m) amd m, and the value of c1 (which is the difference between s and p, with p being the smaller of the two prime factors of m). I am looking at the small odd positive integers starting at 3, and charting out how they behave. There are factorization exceptions – short cuts – when either r=0 or r is a positive perfect square.
I have also seen that some of the trees hindering me from seeing the forest become more manageable if, instead of the r and s above, I look at a similar family of polynomials suggested by s2, the integer floor of the square root of m, and the difference m-s22.
It is all very exciting, but there are still strange kinks in the relationship between these quantities and n (or n2), the index, so to speak, of the polynomial in one family or the other that will factor.
Maybe this won’t have any more promise than the other trees up which I have barked, to belabor that tired metaphor of mine. But maybe it will. Stay tuned.