Surrogate family

jstevh_at_msn.com
Date: 02/19/05 

Date: 19 Feb 2005 06:16:10 -0800

I've been musing more about surrogate factoring, and about how T is
actually related to a family of numbers.

That is, given that M^2 - j^2 = T, there is a finite set of integers a
and b, such that a^2 - b^2 = T, so you have this family.

Just thinking generally on that, it turns out that you can relate y to
that family in an easy way.

yx^2 + Ax - M^2 = 0

and

yz^2 + Az - j^2 = 0

and for some other member of the family, say with y', M' and j', it can
be shown that

M^2 j^2 y = M'^2 j'^2 y'

so you can pull out the rational y's that way, and I say rational as y
will be a fraction.

If that is correct, then it's the first easy closed form solution (yes,
I've said I found the closed form before and been wrong), which would
force the factorization of two more numbers, found from using T.

But, you know? I think it might actually work. If so, then surrogate
factoring has an easier approach than I realized.

There may be many such easy approaches, only not seen now, mostly
because people don't believe it's possible, or, someone else has
figured it out, and isn't talking so publicly.

James Harris

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