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Re: sufficient test for primes with certificate

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  • bhelmes_1
    Dear David, thank you for your friendly help. ... 1. You are right with your counterexample that the certificate is not right. I hope you will enjoy this
    Message 1 of 33 , Jul 16, 2012
      Dear David,

      thank you for your friendly help.

      > Here is a counterexample with
      > prime g, prime a, and composite p,
      > for which every part of your 3-selfridge test
      > is applied, including your recent desperate "wriggle".

      1. You are right with your counterexample that
      the certificate is not right.
      I hope you will enjoy this statement.

      2. You did not verify the point 4.
      http://109.90.3.58/devalco/suf_helmes.htm#1
      if i am right in understanding your program
      [1+sqrt (2969)]^(p-1)=14278+18966*sqrt (2969)=/=1 mod 29539
      which indicate that p is not a prime.

      Last but not least, there remains a claim that
      the test 1.-4. is a sufficient test for p=3 mod 4, slowly with in average 8 selfridges. This would be an improvement versus the AKS-test.

      If you have fun and time, try to find one counterexample
      which destroy the remaining test :-)

      I think i have 40 years in the future in order to find some nice
      algorithms and i hope that you will participate in some.

      Greetings from the primes
      Bernhard
      >
      > g = 21283;
      > a = 2969;
      > p = 29539;
      >
      > {if(p%4 == 3 &&
      > !issquare(a) &&
      > kronecker(a,p) == 1 &&
      > Mod(a,p)^((p-1)/2) == 1 &&
      > Mod((1+x)*Mod(1,p),x^2-a)^((p-1)/2) == g*x &&
      > Mod(g,p)^2*a == 1 &&
      > !isprime(p), print(" Counterexample!"));}
      >
      > Counterexample!
      >
      > David
      >
    • paulunderwooduk
      ... I compiled a better version of my code with gmp 5.0.5, on a different box running at 3.6GHz and got some better timings { 17.505s pfgw (3-prp) 1m1.986s
      Message 33 of 33 , Sep 21, 2012
        --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:
        :
        > http://tech.groups.yahoo.com/group/primenumbers/files/Articles/

        > I ran various "minimal \lambda+2" tests on Gilbert Mozzo's 20,000 digit PRP, 5890*10^19996+2^66422-3 (x=1), using a 2.4GHz core:
        > {
        > 0m32.374s pfgw64 (3-prp)
        > 1m9.876s pfgw64 -t
        > 1m53.535s pfgw64 -tp
        > 3m0.483s pfgw64 -tc
        > 5m12.972s pfgw64 scriptify
        > 4m4.811s gmp (-O3/no pgo)
        > 4m9.148 pari-gp
        > 1m15s theoretical Woltman implementation
        > }
        >

        I compiled a better version of my code with gmp 5.0.5, on a different box running at 3.6GHz and got some better timings
        {
        17.505s pfgw (3-prp)
        1m1.986s pfgw -tp
        1m13.789s gmp (-O3/no pgo)
        }

        Paul
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