### Tuesday, March 10, 2009

## JSH: r(v), s(v) and t(v)

I've proven that Pell's Equation can be generally solved in rationals. So given

x^2 - Dy^2 = 1

I have x and y as functions of an independent variable v, where further x = r(v)/t(v), and y = s(v)/t(v)

where with integer f_1 and f_2 where f_1*f_2 = D-1, I have two sets of equations:

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

s(v) = [2Dv + (D-1)(f_1 - f_2*v^2 - 2v) -(f_1 + f_2*v^2)]

and

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

And the second set:

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

s(v) = [-2Dv - (D-1)(f_1 - f_2*v^2 - 2v) +(f_1 + f_2*v^2)]

and

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

I have proven that if abs(r(v)) < D, and a rational v gives an integer r(v), then a non-trivial factorization of D must be found from

r(v) + t(v), or r(v) - t(v).

For several weeks a group of posters have mounted a rather effective campaign to hide the truth about these equations.

That truth is, they solve the factoring problem.

The posters may be part of a gang of international hackers, looking to lengthen the time until the truth is known, so they can infiltrate computer systems around the world, or, I could be completely wrong.

I don't know. The situation is beyond bizarre. But if it's the worst case, then the Internet has been thoroughly compromised by highly intelligent individuals who can break RSA encryption at will.

x^2 - Dy^2 = 1

I have x and y as functions of an independent variable v, where further x = r(v)/t(v), and y = s(v)/t(v)

where with integer f_1 and f_2 where f_1*f_2 = D-1, I have two sets of equations:

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

s(v) = [2Dv + (D-1)(f_1 - f_2*v^2 - 2v) -(f_1 + f_2*v^2)]

and

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

And the second set:

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

s(v) = [-2Dv - (D-1)(f_1 - f_2*v^2 - 2v) +(f_1 + f_2*v^2)]

and

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

I have proven that if abs(r(v)) < D, and a rational v gives an integer r(v), then a non-trivial factorization of D must be found from

r(v) + t(v), or r(v) - t(v).

For several weeks a group of posters have mounted a rather effective campaign to hide the truth about these equations.

That truth is, they solve the factoring problem.

The posters may be part of a gang of international hackers, looking to lengthen the time until the truth is known, so they can infiltrate computer systems around the world, or, I could be completely wrong.

I don't know. The situation is beyond bizarre. But if it's the worst case, then the Internet has been thoroughly compromised by highly intelligent individuals who can break RSA encryption at will.