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25575Re: Quad Frobenius test

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  • djbroadhurst
    Jul 7, 2014
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      At the PrimePage
      http://primes.utm.edu/bios/page.php?id=181
      Paul Underwood advertises a 12-selfridge test:

      N coprime to 6, for any x, a and b such that
      gcd(a*b*x,N)==1 and gcd(a^2-b^2,N)==1 and
      JacobiSymbol(x^2-4,N)==-1, test
      (L+a)^(N+1)==1+a^2+x*a and (L-a)^(N+1)==1+a^2-x*a and
      (L+b)^(N+1)==1+b^2+x*b and (L-b)^(N+1)==1+b^2-x*b all
      (mod N, L^2-x*L+1).

      The Gremlins respond as follows:

      {quad(a,b,x,N)=
      gcd(6*a*b*x*(a^2-b^2),N)==1&&kronecker(x^2-4,N)==-1&&
      (Mod(1,N)*Mod(L+a,L^2-x*L+1))^(N+1)==1+a^2+x*a&&
      (Mod(1,N)*Mod(L-a,L^2-x*L+1))^(N+1)==1+a^2-x*a&&
      (Mod(1,N)*Mod(L+b,L^2-x*L+1))^(N+1)==1+b^2+x*b&&
      (Mod(1,N)*Mod(L-b,L^2-x*L+1))^(N+1)==1+b^2-x*b;}

      {N=1198232097432195323;x=520808885440135009;
      a=89895558195639956;b=834898099310681853;
      if(quad(a,b,x,N)&&!isprime(N),
      print(" Gremlins Mühlen mahlen langsam,");
      print(" mahlen aber trefflich klein."));}

       Gremlins Mühlen mahlen langsam,
       mahlen aber trefflich klein.

      David

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