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Re: double frobenius with gcds

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  • djbroadhurst
    ... After I worked out to how to forge one counterexample, the gremlins were able to find more than 21,000: {tst(n,x,a)=2
    Message 1 of 8 , Dec 4, 2012
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      --- In primenumbers@yahoogroups.com,
      "paulunderwooduk" <paulunderwood@...> wrote:

      > > {if(tst(429749641620836200211,69208918734832269040,
      > > 200208504884456069347),print("fooled"));}
      > > fooled
      > Well done. Thanks

      After I worked out to how to forge one counterexample,
      the gremlins were able to find more than 21,000:

      {tst(n,x,a)=2<x&&x<(n+1)/2&&1<a&&a<(n+1)/2&&
      kronecker(x^2-4,n)==-1&&gcd(a^3-a,n)==1&&gcd(a,x)==1&&
      kronecker(x,n)==-1&& \\ added wriggle from Paul
      Mod(2,n)^(n-1)==1&&n%12==11&& \\ by construction
      Mod(Mod(1,n)*(L+a),L^2-x*L+1)^(n+1)==(a^2+1+a*x)&&
      Mod(Mod(1,n)*(L-a),L^2-x*L+1)^(n+1)==(a^2+1-a*x)&&!isprime(n);}

      {F=readvec("underwdf.txt");
      \\ http://physics.open.ac.uk/~dbroadhu/cert/underwdf.txt
      print(sum(k=1,#F,tst(F[k][1],F[k][2],F[k][3]))" counterexamples");}

      21728 counterexamples

      It will be observed that, in each case, n = 11 mod 12 is a
      base-2 Fermat pseudoprime. That was for convenience and
      is not an intrinsic limitation.

      David
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