Hi Jeff,
You wrote:
> What do you mean you have no idea? You just said in the paragraph
> above that you have built a program and tested it for p up to 1000
> billion. Surely then you must have an idea of how long it takes for
> your software to run with a specific p value compared to a
> Lucas-Lehmer test
This program that I have built is a program of FACTORIZATION with a
size of the divisor factor less than 2^63. To made a single division
of a Mersenne number with a p of 1000 billion (to be clear: (2^p-1) / div)
it take about 100 hundred hours on one CPU of a SGI Origin 3400 (700Mhz).
This is a << classical division >> without the use of FFT and made with
an arithmetic program with arbitrary number of digits that I have made.
But the test issued of my theory that can say: THIS NUMBER CANNOT DIVIDE
2^p-1, OR THIS NUMBER IS A POTENTIAL DIVISOR, is about a million times
faster for divisors less than 2^63 (and it's not optimized).
At that point of developpement of my theory, I'm really not interested
to find optimization of this test. My goal is to understand completely
the theory behind Mersenne numbers. When we will reach this point, it
will become interesting (and probably much more easy) to do that.
It's maybe interesting to develop a little bit my global vision.
9 years ago I have bet that there is what I have called a << super theory >>
of Mersenne numbers.
Today, I cannot answer if what I have found is the ultimate theory.
This new theory is FULL of symetries, EVERYTHING (factorization, test of
primality of Mersenne number, link with 2^p+1, test of primality of a new
kind of numbers [not of the Mersenne form], ...) are DIRECTLY related.
So it's a really good encouraging sign of GLOBAL UNIFICATION, but
unfortunately for factorization, I don't know today if I have reached
what I call << The Graal >>.
SIEVING DIVISORS CAN BE DIVIDED INTO 3 STEPS:
1) Old very classic restrictions that for some, we know for centuries:
div=2*p*j+1
mod(div,8)=1,7, for 2^p-1
if 2p+1 is prime then it divide
...
2) NEW TEST ISSUED OF MY NEW THEORY. (STEP 2 and STEP 3 can be clearly
and easely divided by some kind of symetry).
STEP 2 IS FULLY DETERMINISTIC!
3) TODAY this step 3 remain fully probabilistic, I have still not found
any relations. 3 possibilities for this third step:
-a] It's maybe a <<wall>> that we will never can go beyond. Like
Heizenberg uncertainties in quantum mechanics.
-b] partially DETERMINISTIC.
-c] fully DETERMINISTIC.
SO, IF I CAN SOLVED THE STEP 3, THE PROBLEM OF FACTORIZATION COULD BECOME
FULLY DETERMINISTIC! IF IT'S THE CASE, THEN FOR THE BUILDING BLOCS, IF WE
HAVE A PRIME NUMBER THAT VERIFY ALL THE CONDITIONS OF THESE 3 STEPS, WE WILL
BE ABLE TO SAY WITH CERTAINTY IF THIS NUMBER IS A DIVISOR OF THE MERSENNE
NUMBER, OR NOT, WITHOUT HAVING TO COMPUTE THE DIVISION.
Regards,
Dr Olivier LATINNE
Departement of Applied Meteorology,
Royal Institute of Meteorology of Belgium,
Belgium
Tel: + 32 2 373 67 45 (work)
Mobile: + 32 478 344 340
e-mail [EMAIL PROTECTED]
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