Integer Factorization: Still At It …


. . . but otherwise, little to report.

I’m filling in spaces where values are needed for the various columns in my nearly-complete chart of all the discrete semiprimes between 0 and 20000 whose Ascents are greater than 0; i.e., whose Ceiling Squares do not differ from them by exactly a perfect square, which enables immediate Fermat factorization.

There’s this, but it’s mainly review.

Characteristic polynomial corresponding to m
when m = c^2 - r,
and c = ceil( sqrt( m ) )
    plus 0 or 1, as c is or is not congruent
    to (m+1)/2 modulo 2, respectively:

f(x) = x^2 + 2cx + r
     = x^2 + 2cx + c^2 - c^2 + r
     = (x + c)^2 - m.

The least non-negative x for which f(x) is a perfect square
gives m = (x + c)^2 - n^2 for some a,
which also gives x+c+n and x+c-n as factors of m.

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