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Re: generalised 6-selfridge double fermat+lucas

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  • paulunderwooduk
    ... Back to the drawing board: n=41159; x=3547; y=3225; ... Paul
    Message 1 of 5 , Sep 14, 2010
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      --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:
      >
      > Hi,
      >
      > I will use capital letters to represent 2 by 2 matrices and lower case for integers.
      >
      > Consider:
      >
      > R^2-r*R+1==0
      >
      > I call this "trivial" if r=0 (mod d) or r=+-1 (mod d) for some proper divisor "d" of a given "n", because the equation is cyclic.
      >
      > Now on to the double equations:
      >
      > M^2-x*M+1==0
      > N^2-y*N+1==0
      >
      > I do not want x=+-y (mod d) because they will be identical for that divisor of "n".
      >
      > The composite test for "n" is:
      >
      > First find x and y:
      > gcd(x^3-x,n)==1
      > gcd(y^3-y,n)==1
      > gcd(x^2-y^2,n)==1
      > jacobi(x^2-4,n)==-1
      > jacobi(Y^2-4,n)==-1
      >
      > Secondly, check
      > x^(n-1) == 1 (mod n)
      > y^(n-1) == 1 (mod n)
      > M^(n+1) == I (mod n)
      > N^(n+1) == I (mod n)
      >
      > I have checked n<2*10^4 with gcd(30,n)==1,
      >

      Back to the drawing board:
      n=41159;
      x=3547;
      y=3225;

      :-(

      Paul
    • djbroadhurst
      ... n = q*r with (r-1)/(q^2-1)=1/12 and each kronecker getting its minus sign from the smaller prime factor q. I do recommend such semiprimes for your study
      Message 2 of 5 , Sep 14, 2010
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        > Back to the drawing board:
        > n=41159;
        > x=3547;
        > y=3225;

        n = q*r with (r-1)/(q^2-1)=1/12
        and each kronecker getting its minus sign from
        the smaller prime factor q.

        I do recommend such semiprimes for your study :-)

        David
      • paulunderwooduk
        ... I found one that was not a semiprime, but the ratio rule still applies: n=41159=79*521;ratio=12 n=45629=103*443;ratio=24 n=64079=139*461;ratio=42
        Message 3 of 5 , Sep 15, 2010
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          --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
          >
          > > Back to the drawing board:
          > > n=41159;
          > > x=3547;
          > > y=3225;
          >
          > n = q*r with (r-1)/(q^2-1)=1/12
          > and each kronecker getting its minus sign from
          > the smaller prime factor q.
          >
          > I do recommend such semiprimes for your study :-)
          >

          I found one that was not a semiprime, but the ratio rule still applies:
          n=41159=79*521;ratio=12
          n=45629=103*443;ratio=24
          n=64079=139*461;ratio=42
          n=96049=139*691;ratio=28
          n=197209=199*991;ratio=40
          n=228241=181*(13*97);ratio=26
          n=287051=151*1901;ratio=12
          n=330929=149*2221;ratio=10
          n=638189=619*1031;ratio=372
          n=853469=239*3571;ratio=16
          n=875071=241*3631;ratio=16

          Note: the above "n" are all +-1 (mod 5)

          Paul
        • djbroadhurst
          ... Here the kronecker is (-1)^3 = -1 and we are into Arnault territory: {pu3(n,x)=gcd(x^3-x,n)==1&&kronecker(x^2-4,n)==-1&&
          Message 4 of 5 , Sep 15, 2010
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            --- In primenumbers@yahoogroups.com,
            "paulunderwooduk" <paulunderwood@...> wrote:

            > n=228241=181*(13*97)

            Here the kronecker is (-1)^3 = -1 and we are into Arnault territory:

            {pu3(n,x)=gcd(x^3-x,n)==1&&kronecker(x^2-4,n)==-1&&
            Mod(x,n)^(n-1)==1&&Mod((x+s)/Mod(2,n),s^2+4-x^2)^(n+1)==1;}

            {pu6(n,x,y)=gcd(x^2-y^2,n)==1&&pu3(n,x)&&pu3(n,y);}

            {n=13*97*181; x=218; y=824; if(pu6(n,x,y),print("refuted"));}

            refuted

            Note that there are no "matrices" above: a double mod is enough.
            Congratulations again, Paul, on testing your ideas to destruction.

            Best regards

            David
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