Saturday, July 22, 2006
Factoring, basic research versus politics
For years I have been doing basic research on factoring, looking for a solution to the factoring problem, with a lot of failures and SPECTACULAR failures along the way. To me that's just the reality of basic research that you fail more than you succeed, but for others that is a tool to control the discussion and push other people in various directions.
However that political behavior is running into the reality of basic research as an evolving process with unpredictable outcomes, where a long run of failure can end with success, as I now clearly have a new factoring method. I'll recap the current research with the initial parts being a repeat of what has gone before as the research builds forward:
The factoring problem can be easily approached using simple algebra.
Start with
x^2 - y^2 = S - 2*x*k
where all are integers, as notice then you trivially have
x^2 + 2*x*k + k^2 = y^2 + S + k^2
so
x+k = sqrt(y^2 + S + k^2)
and finding y is just a matter of factoring (S+k^2)/4.
Now with just the explicit equation you end up with nothing but trivialities, but turning to congruences, you can now simply let
x^2 - y^2 = 0 mod T
which—this is important—now forces
S - 2*x_res*k = 0 mod T
where I put in x_res to emphasize that now it's congruences, so there is loose connectivity and an explicit value of x is not needed—just a residue.
But now I can just solve for k, assuming 2, S and x are coprime to T:
k = S*(2*x_res)^{-1} mod T
where (2*x_res)^{-1} is the modular inverse of (2*x_res) mod T.
That the modular inverse makes an appearance is critical, but more importantly I now have a way to find all the variables!!!
That can be done by simply picking a residue for x_res and then picking S, like x_res = 1, and S =1, to get k.
However, it can be shown that you will not get a solution for your target composite T, unless there exists a number Q such that
Q^2 + k^2 = (x_res^2 - S) mod T
which is the latest crucial result, without which, someone might stumble around trying different values and failing, think the approach was useless.
But that's why an open mind in basic research is important, and why politics should be left to other areas instead of mathematics and the sciences.
The problem I think today is that the current mathematical field is dominated by politicians masquerading as mathematicians, to the detriment of real research.
However that political behavior is running into the reality of basic research as an evolving process with unpredictable outcomes, where a long run of failure can end with success, as I now clearly have a new factoring method. I'll recap the current research with the initial parts being a repeat of what has gone before as the research builds forward:
The factoring problem can be easily approached using simple algebra.
Start with
x^2 - y^2 = S - 2*x*k
where all are integers, as notice then you trivially have
x^2 + 2*x*k + k^2 = y^2 + S + k^2
so
x+k = sqrt(y^2 + S + k^2)
and finding y is just a matter of factoring (S+k^2)/4.
Now with just the explicit equation you end up with nothing but trivialities, but turning to congruences, you can now simply let
x^2 - y^2 = 0 mod T
which—this is important—now forces
S - 2*x_res*k = 0 mod T
where I put in x_res to emphasize that now it's congruences, so there is loose connectivity and an explicit value of x is not needed—just a residue.
But now I can just solve for k, assuming 2, S and x are coprime to T:
k = S*(2*x_res)^{-1} mod T
where (2*x_res)^{-1} is the modular inverse of (2*x_res) mod T.
That the modular inverse makes an appearance is critical, but more importantly I now have a way to find all the variables!!!
That can be done by simply picking a residue for x_res and then picking S, like x_res = 1, and S =1, to get k.
However, it can be shown that you will not get a solution for your target composite T, unless there exists a number Q such that
Q^2 + k^2 = (x_res^2 - S) mod T
which is the latest crucial result, without which, someone might stumble around trying different values and failing, think the approach was useless.
But that's why an open mind in basic research is important, and why politics should be left to other areas instead of mathematics and the sciences.
The problem I think today is that the current mathematical field is dominated by politicians masquerading as mathematicians, to the detriment of real research.
# posted by James Harris @ 17:55 
