## JSH: Confusing research

Thought I should mention that examples I give from my research program testing surrogate factoring may be confusing, as you will see results that seem to defy what I say is the theory!!!

Like a prime p too large by the rules I've given will work, or f_1 mod p and f_2 mod p won't match what the theory says they should be for rational solutions.

And that's where the confusion can come in, as I solved out the rational theory of surrogate factoring, but the equations can factor with non-rationals, which behave, non-rationally!!!

For those who need an objective feel of that versus me just babbling, the full set of equations are:

x^2 = y^2 mod p

z^2 = y^2 + nT

2ax = k

and

z = x + ak

and the rational solutions I look for will satisfy that full set of equations with all integers. But the non-rational solutions can give you integers z and y, and factor T, while x, a, and k are non-rational, though their RESIDUES modulo p will be integers.

(The curious can just get some composite T, and try to find integer solutions that satisfy all those equations.)

My guess is that 50% of the solutions from surrogate factoring will come from non-rationals, which don't follow the rational rules, but I haven't tested that as I see it as irrelevant at this point to getting a practical algorithm going.

But it is a wild quirk of the mathematics. The non-rationals will factor for you, but they follow different rules in doing so from the rationals, so, um, you kind of have a crazy side to surrogate factoring as it's, well, not rational.