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Re: number of selfridges?

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  • paulunderwooduk
    ... 5. fails for some numbers in Pinch s 2-PSP list (n
    Message 1 of 46 , Dec 2, 2011
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      --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:
      >
      >
      >
      > --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@> wrote:
      > >
      > > Sorry about the SN ratio....
      > >
      > > I now have the following composite tests for which I would like to know their selfridge values:
      > >
      > > 1. for kronecker(5,n)==-1:
      > > Mod(Mod(1,n)*(l^2-2),l^2-3*l+1)^(n+1)+9==0
      > >
      > > 3. for n==9 (mod 10):
      > > Mod(Mod(1,n)*(l^2-2),l^2-3*l+1)^((n+1)/2)+3*l-3==0
      > >
      > > 4. for n==9 (mod 10) or n==11 (mod 20):
      > > Mod(Mod(1,n)*(l^2-2),l^2-3*l+1)^((n+1)/2)+3*l-3==0
      > >
      > > (gcd(n,30)==1)
      > >
      >
      > 5. kronecker(5,n)==1 and n!=1 (mod 60)
      > Mod(Mod(1,n)*(l^2-2),l^2-3*l+1)^((n+1)/2)+3*l-3==0
      >
      > 1. and 5. have been checked against Richard Pinch's Carmichael numbers list for n less than 10^16,
      >

      5. fails for some numbers in Pinch's 2-PSP list (n<10^12) which are 41 mod(60) ... so that leaves 11,21,31,51 mod(60)?

      Paul (intending to post no more speculation tonight)
    • 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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