The values of f(x)=x^{2}+2cx+r, for c being the Ceiling Root and r being the Remainder for an odd positive integer m we wish to factor, show patterns in their distribution that are useful. In particular, I have been looking at the function values for x=0,1,2,… in hexadecimal notation rather than decimal, and eliminating whole columns of candidates when the function values appear in 8-long rows.

This is a known fact but worth mentioning here: Out of the 64 possiible values for the low-order byte of an integer, only 11 can serve as the low-order portion of a perfect square. This helps in, as I said, eliminating from consideration whole columns of candidate values of f(x).

But of course, I am also looking at the bit patterms of these columns of values for the elusive pattern that will permit me to calculate Ascent.

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