Hi,
In solidarity with all the thousands of people looking for the same Mersenne numbers,
I had a look at my Masters, where I was developing Mersenne like tests. I see that for
instance for numbers J(m,n) = 2^m * 3^n - 1 ( I call them Jacobi nunmbers, since proofs
use Jacobi reciprocity), there are criteria simple as Mersenne. Here is one of them:
Let (m,n) denote the pair of remainders ( m mod 3, n mod 6). For (m,n) \in
\{ (0,1), (1,5), (2,3), (0,5), (1,3), (2,1) \} the Lucas sequence L(5,7) is such that
the zeroes of the generating polynomial x^2 - 5 * x + 7 generate
F^2_{J(m,n)}* / F*_{J(m,n)}, i.e. they have order J(m,n)+1, is J(m,n) is prime.
One has thus the following test:
1. Let r_0 = u_{2^m}, and q_0 = u_{2^{m-1}} mod J(m.n)
(the L(5,7) u sequence), easy with the Lucas doubling theorem.
2. Let f_3(u_n) = u_n*( (P-4Q)u_n^2 + 3Q^n) ( = u_{3n} ) be the trippling rule for
the Lucas sequence L(P,Q) and
r_{s+1} = f_3(r_s), q_{s+1} = f_3(q_s} mod J(m,n).
D(m,n) is prime iff r_n = 0 and q_t \neq 0 \forall t <= n.
Another way to put it is:
v_{2^{m-2}3^n} \neq 0 and v_{2^{m-1}3^{n-1} \neq 0 but
v_{2^{m-1}3^n} = 0.
There are many similar exact tests that can be derived and they may offer
highly larger chances for finding the first Megaprime. The computational
effort migth maybe double or tripple, arithmetic mod 2^m.3^n-1 is easier
than general but harder than mod Mersenne primes, etc.
This is just to ask whether you think that the GIMPS community might be
interested in more details along these lines ?
Sincerely
Preda Mihailescu
