### Saturday, June 07, 2008

## JSH: Some basics about factoring

It occurs to me that some of you may not understand how my research changes the landscape with factoring so I want to explain simply and give some factoring basics.

First off, if you have z^2 = y^2 + nT

where T is your target to factor, and you also have z mod p_1, z mod p_2, and z mod p_3, where z is for a non-trivial factorization and those are odd primes with a minimum value greater than 2sqrt(nT)/3, then necessarily you can calculate z exactly and with z you can factor nT from

(z-y)(z+y) = nT.

That is just an absolute in terms of basic algebra.

So it IS a big deal for me to present equations that allow you to just calculate z mod p.

Most modern factoring methods in some way or another use an equation like z^2 = y^2 + nT, or more familiarly you often have x^2 = y^2 mod N, where N is the target composite, so it's just about variable names and if you know any math at all you know that shifting letters is not a big deal.

Even the Number Field Sieve is a lot about using x^2 = y^2 mod N, as in trying to find x and y (I think it uses two congruences of that type), so the result I have has implications for the most advanced factoring techniques known.

But, you may then naturally wonder, if it's such a big deal to find z mod p, then how can it be something argued out on newsgroups without experts in the field caring?

One simple possible answer to that question is that I must be wrong. REMEMBER, if p_1, p_2 and p_3 can be found in the size range necessary then it is an ABSOLUTE that you can factor non-trivially.

I claim to have a method that gives z mod p, so if that claim is correct and you can get z mod p in the necessary range for just three prime numbers then ABSOLUTELY you will factor non-trivially.

So theory says one thing, absolutely. Where notice I still haven't answered the question of whether or not I must be wrong.

Well, there's the derivation which you can look over, and there is doing examples and you might wonder if maybe with a big target composite T, maybe it IS really hard to find odd primes p that will work, and you can muddle along with those questions believing there must be something wrong somewhere or top people in the field would acknowledge this result!

I think this situation for some of you is a test of your trust in people versus your trust in mathematics and it's probably not fair, but I think some of you wrongly believe that you have mathematical ability, when you do not.

Short of it is that how the newsgroups react doesn't matter. No matter what if the research is viable that will be known and probably in a rather short amount of time as we have a world today that consumes information. But what you cannot forget later, or I don't want you to forget it, is if you couldn't resolve the issue on your own despite the algebra being easy and the problem being hugely significant.

As if you cannot evaluate easy algebra and get the right answer when it's handed to you because you're waiting on some other people or trusting that someone else out there has the judgment you need, then you are NOT a mathematician, no matter what you tell yourself when you look in the mirror.

You are then a social person who relies on other people who really do know mathematics to tell you what is true or not.

First off, if you have z^2 = y^2 + nT

where T is your target to factor, and you also have z mod p_1, z mod p_2, and z mod p_3, where z is for a non-trivial factorization and those are odd primes with a minimum value greater than 2sqrt(nT)/3, then necessarily you can calculate z exactly and with z you can factor nT from

(z-y)(z+y) = nT.

That is just an absolute in terms of basic algebra.

So it IS a big deal for me to present equations that allow you to just calculate z mod p.

Most modern factoring methods in some way or another use an equation like z^2 = y^2 + nT, or more familiarly you often have x^2 = y^2 mod N, where N is the target composite, so it's just about variable names and if you know any math at all you know that shifting letters is not a big deal.

Even the Number Field Sieve is a lot about using x^2 = y^2 mod N, as in trying to find x and y (I think it uses two congruences of that type), so the result I have has implications for the most advanced factoring techniques known.

But, you may then naturally wonder, if it's such a big deal to find z mod p, then how can it be something argued out on newsgroups without experts in the field caring?

One simple possible answer to that question is that I must be wrong. REMEMBER, if p_1, p_2 and p_3 can be found in the size range necessary then it is an ABSOLUTE that you can factor non-trivially.

I claim to have a method that gives z mod p, so if that claim is correct and you can get z mod p in the necessary range for just three prime numbers then ABSOLUTELY you will factor non-trivially.

So theory says one thing, absolutely. Where notice I still haven't answered the question of whether or not I must be wrong.

Well, there's the derivation which you can look over, and there is doing examples and you might wonder if maybe with a big target composite T, maybe it IS really hard to find odd primes p that will work, and you can muddle along with those questions believing there must be something wrong somewhere or top people in the field would acknowledge this result!

I think this situation for some of you is a test of your trust in people versus your trust in mathematics and it's probably not fair, but I think some of you wrongly believe that you have mathematical ability, when you do not.

Short of it is that how the newsgroups react doesn't matter. No matter what if the research is viable that will be known and probably in a rather short amount of time as we have a world today that consumes information. But what you cannot forget later, or I don't want you to forget it, is if you couldn't resolve the issue on your own despite the algebra being easy and the problem being hugely significant.

As if you cannot evaluate easy algebra and get the right answer when it's handed to you because you're waiting on some other people or trusting that someone else out there has the judgment you need, then you are NOT a mathematician, no matter what you tell yourself when you look in the mirror.

You are then a social person who relies on other people who really do know mathematics to tell you what is true or not.