I’ve got a good bit of housework to get going today, and some necessary but necessarily small amount of shopping to do as well, but I want to get this idea down first.

The Adjusted Ceiling Square of odd positive integer m is the least perfect square greater than m that is odd if m is congruent to 1 modulo 4, and is even if m is congruent to 3 (or -1) modulo 4.

When constructing ranges in the algorithm most recently discussed, then, one would set the greater square for the first range lower bound, c_{0}^{2}, to be the Adjusted Ceiling Square of m. Then, for the next two ranges, the greater squares will be (c_{0}+2)^{2} and (c_{0}+4)^{2}, and the values of where m falls within the three ranges thus generated will be the values of f(x), the characteristic polynomial, at x=0, 2, and 4.

I simply thought it was much more sensible to base the characteristic polynomial on ranges of differences of perfect squares that matched the even/odd polarity of the actual difference-of-squares form of any solutions of m, based on its being congruent to 1 or -1 modulo 4. Why base a characteristic polynomial algorithm on range values that are immediately apparently not candidates for “Fermat factorization”?

I shall give a report of my working through this when I actually do so – much later today!

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