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24809Re: mod quartic composite tests

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  • paulunderwooduk
    Jan 9, 2013
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      --- In primenumbers@yahoogroups.com, "paulunderwooduk" wrote:
      >
      > Hi,
      >
      > I have devised a new composite test for odd n with x:
      > gcd(x^3-x,n)==1 (mod n)
      > kronecker(x^2-4,n)==-1
      >
      > and the sub-test:
      > (L+1)^n==-L^3+(x^2-2)*L+1 (mod n, (L^2-x*L+1)*(L^2+x*L+1))
      >
      > Can you find a counterexample?

      n=287051 and x=3988 is a counterexample.

      > If this is too easy for you, please try:
      >
      > (L+2)^n==-L^3+(x^2-2)*L+2 (mod n, (L^2-x*L+1)*(L^2+x*L+1))
      >

      This test is (1)+(1)+(2)+(2)+9 selfridge for small x, where (1) is a fermat test and (2) is a frobenius test over L^2-x*L+1 or L^2+x*L+1.

      I already have:
      "Non-square N>1 is prime if and only if for any integer x such that
      KoneckerSymbol(x^2-4,N)==-1
      then both (L+-2)^(N+1)==5+-2*x (mod N, L^2-x*L+1) (Both L+-2 needed.)
      Verified for odd N< 10^7."

      Paul
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