### Wednesday, March 11, 2009

## Factoring problem solution simplified

After weeks of debate on the mathematical proof showing I've solved the factoring problem using Pell's Equation, I've finally gone ahead and worked out the simple equations that result from it, which solve the factoring problem.

A critical equation that posters have bugged me about has only two variants, where one is:

r(v) = (D+1)*f_2*v^2 - 2(D-1)v + (D-1)*f_1

If D is the target to be factored, and f_1 and f_2 are integers where f_1*f_2 = D-1, I have proven that if rational v can be found such that r(v) is an integer and abs(r(v)) < D, then D MUST be non-trivially factored. The non-trivial factorization is then forced, as in a mathematical absolute!

Finding v is as simple as solving for r(v) < D, with the quadratic formula. That gives you v:

v < ((D-1) +/- sqrt((D-1)^2 - [(D-1)*f_1 - D][(D+1)*f_2]))/(D+1)

Using calculus it's easy to see the lowest bound for v is:

v >= (D-1)/[(D+1)*f_2]

To get the non-trivial factorization you just need one more equation called t(v), where:

t(v) = (D-1)*(f_1 - f_2*v^2 - 2v)

as non-trivial factors of D MUST be available from r(v) + t(v), or r(v) - t(v), by the gcd.

ALL of the arguing over this research is needless, pointless and dumb, as if I'm correct, I just gave you enough algebra to factor an RSA public key. Just make it D.

I really didn't want to just give the damn thing, but I hoped by now that the NSA would have contacted me, as the algebra is easy.

They have not. No one in the US Government or any other friendly or hostile or any government has contacted me.

All has been scarily and depressingly silent. Our intelligence services all failed.

I don't quite know why posters argued about this result as if it were so complicated, why others on the newsgroups let them, nor why no one just used the easy equations to factor.

The mathematical proof says they do work, though I have one more hedge as there is one more equation for r(v) as there are two variants, just in case the one above doesn't work.

A critical equation that posters have bugged me about has only two variants, where one is:

r(v) = (D+1)*f_2*v^2 - 2(D-1)v + (D-1)*f_1

If D is the target to be factored, and f_1 and f_2 are integers where f_1*f_2 = D-1, I have proven that if rational v can be found such that r(v) is an integer and abs(r(v)) < D, then D MUST be non-trivially factored. The non-trivial factorization is then forced, as in a mathematical absolute!

Finding v is as simple as solving for r(v) < D, with the quadratic formula. That gives you v:

v < ((D-1) +/- sqrt((D-1)^2 - [(D-1)*f_1 - D][(D+1)*f_2]))/(D+1)

Using calculus it's easy to see the lowest bound for v is:

v >= (D-1)/[(D+1)*f_2]

To get the non-trivial factorization you just need one more equation called t(v), where:

t(v) = (D-1)*(f_1 - f_2*v^2 - 2v)

as non-trivial factors of D MUST be available from r(v) + t(v), or r(v) - t(v), by the gcd.

ALL of the arguing over this research is needless, pointless and dumb, as if I'm correct, I just gave you enough algebra to factor an RSA public key. Just make it D.

I really didn't want to just give the damn thing, but I hoped by now that the NSA would have contacted me, as the algebra is easy.

They have not. No one in the US Government or any other friendly or hostile or any government has contacted me.

All has been scarily and depressingly silent. Our intelligence services all failed.

I don't quite know why posters argued about this result as if it were so complicated, why others on the newsgroups let them, nor why no one just used the easy equations to factor.

The mathematical proof says they do work, though I have one more hedge as there is one more equation for r(v) as there are two variants, just in case the one above doesn't work.