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Re: Lucas-Lehmer like test

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  • Paul Underwood
    ... No. Note N+1 == 3^n-2+1 == 3^n-1 and N-1 == 3^n-2-1 == 3*(3^(n-1)-1 So given enough factors of N+-1 the combined classical tests can be used:
    Message 1 of 2 , Mar 1, 2006
      >
      > Let n= 1, 2, 3, etc. and Nn= 3^n -2.
      >
      > Does the sequence 1, 7, 25, 79, 241, 727, 1185, etc. have a test for
      > verifying the primality of Nn similar to the LL test for Mersenne Mp=
      > 2^p -1?
      >

      No.

      Note N+1 == 3^n-2+1 == 3^n-1
      and N-1 == 3^n-2-1 == 3*(3^(n-1)-1

      So given enough factors of N+-1 the combined "classical tests" can be
      used:
      http://primes.utm.edu/prove/index.html

      These have been implemented in PFGW:
      http://groups.yahoo.com/group/primeform/

      At the cutting edge, a less than 33.33 % factored percentage could
      lead to a proof: "Konyagin-Pomerance" (KP) or "Coppersmith
      Howgrave-Graham" (CHG).

      If the number is sub-gigantic (<10,000 digits) it can be proven with
      Marcel Martin's ECCP implementation, Primo:
      http://www.ellipsa.net/

      Wojciech Florek has some good pages covering 3^n-2:
      http://perta.fizyka.amu.edu.pl/pnq/

      > I tried for several weeks to find one and couldn't.
      >

      You aren't the first to try ;-)

      Paul
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