### Tuesday, June 03, 2008

## But how easy? Fundamental factoring congruence

Given

z^2 = y^2 + nT

where z is a positive integer divisible by 3, and T is a target composite to be factored, it can be shown that

z^2 = 8^{-1} (9nT) mod p

where p is an odd prime less than 2z/3, for which the quadratic residue exists, and n is a control variable chosen such that z has 3 as a factor so it's n = 1 if T mod 3 = 2 and, can be n = 5, if T mod 3 = 1, to force that situation.

If true that relation is one of the most fundamental relations in number theory showing a relationship between prime numbers and every integer factorization.

Yet it turns out it is trivially derived using a technique of adding variables that I first used in order to find a proof of Fermat's Last Theorem, as you can derive that relation by using;

x^2 = y^2 mod p

2x = k and z = x+ k

so you just multiply both sides by k with 2x = k to get 2xk = k^2 and add it to x^2 = y^2 mod p to get

x^2 + 2xk = y^2 + k^2 mod p

and then add k^2 to both sides to get

x^2 + 2xk + k^2 = y^2 + 2k^2 mod p

and now you have

(x+k)^2 = y^2 + 2k^2 mod p

so with T = 2k^2 mod p, and the other equations you are back to

z^2 = y^2 + nT

and the circle is complete.

But now you know that k^2 = 2^{-1}(nT) mod p, and doing the other substitutions you can get to

z^2 = 8^{-1} (9nT) mod p

and with slightly more complicated algebra you can prove that p is an odd prime less than k, so finally you have the p is an odd prime less than 2z/3, where the minimum z if z is a positive integer is sqrt(nT).

Easy math. Trivial. But oh what an impact.

The door is now open to solving for z modulo a succession of odd primes p, and in doing so, factoring nT, and in so doing, factoring T, non-trivially with easy algebra and basic math.

What a revolution.

But how easy is it to understand?

That question is what will decide the fate of mathematicians around the world where if it is very easy and the current delay in acknowledgment continues then mathematicians could face sharp questions about why they sat quiet.

My position as I have multiple major discoveries going back over 6 years is that mathematicians keep quiet because they're fakes fearing that knowledge of this will kill their cash cows and force them to work for a living like most people versus stealing from a public that pays many of them to do worthless research that they must know is false as I have proven it's false.

You lied thinking you'd never get caught. You betrayed humanity thinking the truth didn't matter.

You tried to stop the progress of mathematics which is why you lost.

The only question now is, how big of a price will you pay?

Judgment Day.

z^2 = y^2 + nT

where z is a positive integer divisible by 3, and T is a target composite to be factored, it can be shown that

z^2 = 8^{-1} (9nT) mod p

where p is an odd prime less than 2z/3, for which the quadratic residue exists, and n is a control variable chosen such that z has 3 as a factor so it's n = 1 if T mod 3 = 2 and, can be n = 5, if T mod 3 = 1, to force that situation.

If true that relation is one of the most fundamental relations in number theory showing a relationship between prime numbers and every integer factorization.

Yet it turns out it is trivially derived using a technique of adding variables that I first used in order to find a proof of Fermat's Last Theorem, as you can derive that relation by using;

x^2 = y^2 mod p

2x = k and z = x+ k

so you just multiply both sides by k with 2x = k to get 2xk = k^2 and add it to x^2 = y^2 mod p to get

x^2 + 2xk = y^2 + k^2 mod p

and then add k^2 to both sides to get

x^2 + 2xk + k^2 = y^2 + 2k^2 mod p

and now you have

(x+k)^2 = y^2 + 2k^2 mod p

so with T = 2k^2 mod p, and the other equations you are back to

z^2 = y^2 + nT

and the circle is complete.

But now you know that k^2 = 2^{-1}(nT) mod p, and doing the other substitutions you can get to

z^2 = 8^{-1} (9nT) mod p

and with slightly more complicated algebra you can prove that p is an odd prime less than k, so finally you have the p is an odd prime less than 2z/3, where the minimum z if z is a positive integer is sqrt(nT).

Easy math. Trivial. But oh what an impact.

The door is now open to solving for z modulo a succession of odd primes p, and in doing so, factoring nT, and in so doing, factoring T, non-trivially with easy algebra and basic math.

What a revolution.

But how easy is it to understand?

That question is what will decide the fate of mathematicians around the world where if it is very easy and the current delay in acknowledgment continues then mathematicians could face sharp questions about why they sat quiet.

My position as I have multiple major discoveries going back over 6 years is that mathematicians keep quiet because they're fakes fearing that knowledge of this will kill their cash cows and force them to work for a living like most people versus stealing from a public that pays many of them to do worthless research that they must know is false as I have proven it's false.

You lied thinking you'd never get caught. You betrayed humanity thinking the truth didn't matter.

You tried to stop the progress of mathematics which is why you lost.

The only question now is, how big of a price will you pay?

Judgment Day.