Saturday, January 19, 2008
JSH: Now we're golden!
Wow! The poster Enrico noticed cases where my factoring congruences weren't working, and I was able to work out yet another fascinating feature, as well as explain something that might have been missed up until now which is that you get one 'a' and k pair for each unique factorization.
So, yes, now it's mathematically proven that I found a solution to the factoring problem, and I want to definitely thank Enrico and would like to give him mention in the paper.
I will say now that an early draft of the paper is at the Annals of Mathematics, but I ended up sending two revisions after I rushed them my early research and have thankfully sat for a while as I worked out the issues.
There is no doubt I'd think that a paper solving the factoring problem deserves publication in the Annals.
Some of you may have tested out the congruences and found they did not always work, which gave you doubts, but now the answer is that there is a quirk of prime numbers, as there are two basic types: those for which the negative of the quadratic residue is a quadratic residue and those for which it is not.
So, for instance, with p=17, Enrico noticed that the congruence relations would not work to non-trivially factor 15. I dove into the underlying equations and found that they would not work, and figured out the answer as to why.
It helped that I had the existence equation for 'a' handy.
Rather wild though. All integer factorizations are controlled by the factoring congruences, but some primes exclude themselves but only if they are the primes for which the negative of their quadratic residue is a quadratic residue.
My celebration should be somewhat muted though as it is now definite that the factoring problem is SOLVED.
Governments around the world and institutions affected should behave accordingly.
I am directing EMC2 to give a press release as soon as possible on the situation through its RSA division.
So, yes, now it's mathematically proven that I found a solution to the factoring problem, and I want to definitely thank Enrico and would like to give him mention in the paper.
I will say now that an early draft of the paper is at the Annals of Mathematics, but I ended up sending two revisions after I rushed them my early research and have thankfully sat for a while as I worked out the issues.
There is no doubt I'd think that a paper solving the factoring problem deserves publication in the Annals.
Some of you may have tested out the congruences and found they did not always work, which gave you doubts, but now the answer is that there is a quirk of prime numbers, as there are two basic types: those for which the negative of the quadratic residue is a quadratic residue and those for which it is not.
So, for instance, with p=17, Enrico noticed that the congruence relations would not work to non-trivially factor 15. I dove into the underlying equations and found that they would not work, and figured out the answer as to why.
It helped that I had the existence equation for 'a' handy.
Rather wild though. All integer factorizations are controlled by the factoring congruences, but some primes exclude themselves but only if they are the primes for which the negative of their quadratic residue is a quadratic residue.
My celebration should be somewhat muted though as it is now definite that the factoring problem is SOLVED.
Governments around the world and institutions affected should behave accordingly.
I am directing EMC2 to give a press release as soon as possible on the situation through its RSA division.
# posted by James Harris @ 20:04 
