Harristotelian Logic

I've discovered that integer factorization can be generalized using the following equations, which link every factorization to another secondary factorization:

With T the target composite to be factored, letting

x^2 = y^2 + aT and 2xk = k^2 - bT

you can solve to find

(x+k)^2 = y^2 + 2k^2 + (a-b)T

so that it is clear that 'a' and 'b' need not be determined as you need a-b, which is chosen to be some non-zero integer, as is k.

Trivially, if you introduce integer factors f_1 and f_2, you can have

4f_1*f_2 = 2k^2 + (a-b)T

then y = f_1 - f_2 and

x = f_1 + f_2 - k

and an example, which took a single iteration, so it took one try, where I arbitrarily chose to let k = floor(T/30) and I picked a-b=2, consider

T = 732367903, k=floor(T/30) = 24412263, a-b = -2

2k^2 + (a-b)T = 1191915704826532 = ( 2^2 )( 7 )( 73 )( 583129014103 )

f_1 = 7/2 and f_2 = 85136836059038

y=-170273672118069/2 and x=170273623293557/2

so, x+y=-24412256, which has 223 as a factor.

T = 732367903 = (223)(3284161)

and you can see that it can work where you factor this one number to get to your target, but what is the probability that it WILL work?

Good question. I've talked about these simple methods for a few months now and gotten a lot of dismissive replies which never answer that question.

My own research indicates that if 2k^2 + (a-b)T is roughly as hard to factor as T, then you have about a 67% chance of factoring your target, so if you try to make it smaller so that it's easier to factor that probability drops and you need more iterations, so you can go to smaller numbers but would need to check more possibilities as the probability drops.

I state that without proof because it seems to me that at a minimum with the factoring problem, people willing to dismiss a simple method that can be shown to work, should be forced to present SOME mathematics in their rebuttals—like walk through an argument giving the probability of factoring.

Now I think that the modern math world has long gotten away with bluffing its way, and that modern mathematicians learned style over substance—the appearance of proof versus actual proof—so that they can lie.

Yup, I think that modern mathematicians as a group routinely lie having learned that they can get away with it.

Their motivation in lying about factoring is that the factoring problem being hard means more jobs for mathematicians and a dodge from embarrassment because the Internet is currently secured based on it supposedly being a hard problem, while the simple equations I've presented here may show a way that smart people can break that
security system.

RSA has taken away the money prizes for its challenges since I started presenting this research. Coincidence? I hope so, but mathematics is not just about hope.

Modern mathematicians I feel are capable of ignoring such a result based on a hope and a prayer because the PR disaster of the truth coming out is great enough that they will just wait, and hope that no one notices, unless, of course, I'm wrong!

As, after all, I've made it very clear that I will tell the world that mathematicians routinely lie if my research breaks through. If they do lie routinely about their research, why wouldn't they hide a stupid mistake where they gave the world a security system breakable with some simple algebra?

You think they are "beautiful minds" and I think they are con artists, but I could be wrong.

But if I'm wrong, then cannot a supposedly very advanced mathematical community present a mathematical argument that shows what the probability of factoring with the system of equations I've presented actually is?