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Re: Paul Underwood's "Selfridge conjecture"

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  • paulunderwooduk
    ... I should use Kronecker! The revision is: 4.X Selfridge Conjecture Non-square N 1 is prime if and only if for any integer x such that
    Message 1 of 46 , Feb 5, 2012
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      --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:
      >
      >
      >
      > Hi,
      >
      > the conjecture now has gcd(30,N)==1 dropped.
      >
      > 4.X Selfridge Conjecture
      > Non-square N>1 is prime if and only if for any integer x such that
      > JacobiSymbol(x^2-4,N)==-1 then both (L+-2)^(N+1)==5+-2*x (mod N, L^2-x*L+1) (respectively.) (Both L+-2 needed.)
      >

      I should use Kronecker! The revision is:

      4.X Selfridge Conjecture
      Non-square N>1 is prime if and only if for any integer x such that
      KroneckerSymbol(x^2-4,N)==-1 then both (L+-2)^(N+1)==5+-2*x (mod N, L^2-x*L+1) (respectively.) (Both L+-2 needed.)

      Paul
    • paulunderwooduk
      ... Combining fails with the composite counterexample n=256999 and x=32768, However, I have tested the 1+1+1+2 conjecture up to n
      Message 46 of 46 , Apr 14, 2012
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        --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:
        >
        >
        >
        > --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@> wrote:
        > >
        > > Hi,
        > >
        > > I have added a Fermat test to make a 1+1+1+2 selfridge test:
        > >
        > > For N>5, with gcd(6,N)==1, find an integer x:
        > > gcd(x^3-x,N)==1
        > > kronecker(x^2-4,N)==-1
        > >
        > > and check:
        > > (x+2)^((N-1)/2)==kronecker(x+2,N) (mod N) (Euler)
        > > (x-2)^((N-1)/2)==kronecker(x-2,N) (mod N) (Euler)
        > > x^(N-1)==1 (mod N) (Fermat)
        > > L^(N+1) == 1 (mod N, L^2-x*L+1) (Lucas)
        > >
        >
        > Note: I should say gcd(30,N)==1 because gcd(x^3-x,N)==1 and kronecker(x^2-4,n)==-1.
        >
        > Re: http://tech.groups.yahoo.com/group/primenumbers/message/24090?l=1
        >
        > Now consider combining the 2 Euler tests with the Lucas test:
        >
        > (L*D)^((n+1)/2)==D (mod N, L^2-x*L+1) (D=x^2-4.)
        >
        > with the restriction kronecker(x+2,N)==-1.
        >
        > These together with the Fermat test makes for a 1+2-selfridge test.
        >
        > Can you find a counterexample?
        >
        > So far the near-refutation from Pinch's carmichael list is:
        > N,x,gcd(x^2-1)
        > ------------------
        > 1909001 884658 1909001
        >
        > Paul
        >

        Combining fails with the composite counterexample n=256999 and x=32768, However, I have tested the 1+1+1+2 conjecture up to n<10^7,

        Paul -- restoring symmetry
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