email@example.com wrote: > Solving the Factoring Problem > > > Consider a relation between two integer factorizations: > > f_1 f_2 = k + g_1 g_2 > > and a solution with four unknowns w, x, y and z, as they are determined > by four linear equations: > > L_1(w,x,y,z) = f_1 > L_2(w,x,y,z) = f_2 > L_3(w,x,y,z) = g_1 > L_4(w,x,y,z) = g_2 > > What I have found is that remarkably you can use only two linear > equations and k itself to find > g_1 and g_2, through a process I call surrogate factorization.
Indeed you can (subject to the corrections below), but as a factoring algorithm it sucks big time. Remember, to be of any worth whatsoever, a factoring algorithm must not only work, but it must be efficient. Yours is worse than trial division. > > More specifically I use the system of equations > > > (w + x - 2z)(w + 3x + 2y + 2z) = k + (w + x + y + z)(3w + x - y - 3z) > > where > > > k = 2x^2 + 2xy + y^2 - 2w^2 - z^2 - 2xz > > as then I can use > > > w + x - 2z = f_1 > > w + 3x + 2y + 2z = f_2 > > > to find > > x = (f_2 - f_1 - 2y - 4z)/2, w = (3f_1 - f_2 + 2y + 8z)/2 > > and with > > f_1 f_2 = T+k > > where T = (w + x + y + z)(3w + x - y - 3z) > > I have that > > 9(2y + 10z + 5f_1 - f_2)^2 = (18z + 6f_1 - 2f_2)^2 - 18T - 54k + > 45f_2^2 - 99f_1^2 > > (But it's a tedious calculation where it's easy to make a mistake. > Note that k, x and w above have been carefully verified and I tried my > best with the calc, but may have gotten it wrong.