Yesterday, and previously, I posted about the expression of positive integer m as its Ceiling Square minus a remainder, m=c^{2}-r, and my attempts to relate that to the Fermat two-squares expression of m corresponding to its possible factorizations, m=s^{2}-t^{2}. 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!

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