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cubic pseudoprimes

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  • paulunderwooduk
    Hi I have run: http://groups.yahoo.com/group/primeform/files/Probable% 20Primes/cubic_mr.c on a 1GHz Athlon for about ten days; for x = 2 upto 10,000,000,001
    Message 1 of 2 , Sep 20 10:21 AM
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      Hi

      I have run:
      http://groups.yahoo.com/group/primeform/files/Probable%
      20Primes/cubic_mr.c

      on a 1GHz Athlon for about ten days; for x = 2 upto 10,000,000,001
      and found 527,345,506 probable primes and no "counterexample".

      The program tests "f|x^f-x=>5 times Miller-Rabin probable prime where
      f=x^3-x-1 and x>1". The probability of a composite f:f|x^f-x getting
      through 5 Miller Rabin rounds is less than 4^-5.

      Some statistics:-

      341 is the smallest base 2 Fermat pseudoprime.

      561 is the smallest Carmichael number.

      705 is the smallest pseudoprime n for the test "the sum of the nth
      powers of the roots of x^2-x-1 is congruent to 1 modulo n".

      124 is the smallest x for which the test "f|x^f-x where f=x^2-x-1"
      produces a pseudoprime f.

      19802 is the smallest x for which the test "f|x^f-x where f=x^2-x-1"
      produces a Carmichael number f.

      271441 is the smallest Perrin pseudoprime. (The nth term in the
      Perrin Sequence is equal to the sum of the nth powers of the roots of
      x^3-x-1.)

      27664033 is the smallest symmetric pseudoprime relative to the
      elementary symmetric functions of the roots of x^3-x-1.

      7045248121 is the smallest Perrin pseudoprime that is also a
      Carmichael number.

      What is the smallest composite f:f|x^f-x where f=x^3-x-1? David
      Broadhurst reckons Erdos' Carmichael conjecture implies x ~ 10^13 for
      a Carmichael number ;-)

      Paul
    • paulunderwooduk <paulunderwood@mindless.
      please see OP: http://groups.yahoo.com/group/primenumbers/message/8856 I have tested 1
      Message 2 of 2 , Jan 1, 2003
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        please see OP:

        http://groups.yahoo.com/group/primenumbers/message/8856

        I have tested 1<x<=10^11 (4,772,369,646 PrPs). Many thanks to Michael
        Angel who tested 10^11<x<=223,490,000,000. We found no composite x-
        PrP x^3-x-1. The computations took some 250 GHz days in total.

        Paul
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