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Re: Lucas mersenne ?

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  • Shane
    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:
      > 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

      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?
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