## Re: Lucas mersenne ?

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• mikeoakes2@a... wrote: 05/01/2002 ... Mersenne with (prime or composite) p
Message 1 of 12 , Sep 30, 2002
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mikeoakes2@a... wrote:
05/01/2002
>
> > Thanks for the clarification, Mike.
> > But Shane's question was not answered by either of us:
> > is there a Lucas mersenne pseudoprime,
> > M(p)=2^p-1 with prime p and composite M(p)
> > and one of the two tests satisfied?
> > It's a nice question!
> >
> I have obtained a partial answer to this question: there is no such
Mersenne with (prime or composite) p <= 4500. [I'm sure you'll
appreciate the number of CPU cycles that went into this result,
David -- L(2^4499) is quite big! -- did I compute it? -- "Were
there but world enough and time", as the poet put it...]
> However, I expect there actually to be infinitely many such
Mersennes, for the following reason:- about one in 100,000 integers
are "Lucas pseudoprimes" (as defined in my previous posting), the
first being 15251=101*151, the 23rd
> being 1970299=199*9901 - and no, they don't ALL have just 2 primes
factors:-). So, if there is nothing special about the form (2^n-1)
(a big IF), and if the density of these pseudoprimes doesn't
decrease too much with increasing size (another big IF), then one
would expect about 1 in 100,000 Mersennes to be Lucas pseudoprimes.
> Anyone up to shedding light on these IFs, and/or extending the
search range
> above p=4500?
> Mike Oakes

Hello Mike, You had sent me a program for this, could you send it
again?

I am wondering if PRP/Newpen can be verified first by:
L(2^n-1) mod (k*2^n +/-1)=0
Then if positive execute PRP, and finally proth.

Does the 1/100,000 probability still hold?