Yesterday, and previously, I posted about the expression of positive integer m as its Ceiling Square minus a remainder, m=c2-r, and my attempts to relate that to the Fermat two-squares expression of m corresponding to its possible factorizations, m=s2-t2. Laying it all out in chart form yesterday, with rows for 2c and columns for r, does not at the current moment make anything clearer than known mathematical findings already have done. However, I do believe I can express the Ascent, the positive difference between the s corresponding to a factorization and the Ceiling Square c, in a closed form. Again, this is not groundbreaking, but I am holding on to it as a piece of the puzzle, a provider of an angle on the perspective we already have.
One can always factor m = m times 1, and this gives s=(m+1)/2 and t=(m-1)/2 for the two=squares form, as well as a maximal Ascent. This maximal (trivial) ascent exists for prime and composite numbers, and if we know the Ceiling Square c and the corresponding remainder r, we have:
Trivial Ascent = 0.5 * [(c-1)2-r].
I will be looking at how the Trivial Ascent relates to an odd discrete semiprime’s factorization’s non-trivial Ascent, and to the Ceiling Squares, remainders, and Ascents of the semiprime’s prime factors. Wish me good insight!