mikeoakes2@a... wrote:

05/01/2002

> d.broadhurst@o... writes:

>

> > 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?

What do you think about this variation?