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Re: 4 Fermat and 1 Lucas [freely admitted by its author to be hopeless]

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  • djbroadhurst
    ... Here are 10 counterexamples to your latest vain idea: {tst(n,x)=local(P=x^8-1,Q=1-x^8); kronecker(P^2-4*Q,n)==-1&&gcd(x,n)==1&& Mod(x-1,n)^(n-1)==1&&
    Message 1 of 11 , Aug 1, 2013
      --- In primenumbers@yahoogroups.com,
      "paulunderwooduk" <paulunderwood@...> wrote:

      > Breaking my Trappist vow...

      Why, Paul? You have freely admitted that there is no point:

      > one parameter Lucas plus N Fermat/Euler/M-R PRP test
      > can be counterexampled

      Here are 10 counterexamples to your latest vain idea:

      {tst(n,x)=local(P=x^8-1,Q=1-x^8);
      kronecker(P^2-4*Q,n)==-1&&gcd(x,n)==1&&
      Mod(x-1,n)^(n-1)==1&&
      Mod(x+1,n)^(n-1)==1&&
      Mod(x^2+1,n)^(n-1)==1&&
      Mod(x^4+1,n)^(n-1)==1&&
      Mod(Mod(1,n)*L,L^2-P*L+Q)^(n+1)==Q;}

      {F=[
      [7750135694869, 822096191222],
      [23723039862349, 1323013054084],
      [90273119893069, 5862741794270],
      [264256506403909, 38817437399213],
      [8955652979403079, 1851456656424086],
      [4574665869143389, 885331489130492],
      [5266652551034509, 988874992567097],
      [8618233825140949, 584166437019905],
      [9541864502273629, 720345160544763],
      [10245855908959669, 226701623305716]];

      c=0;for(k=1,#F,n=F[k][1];x=F[k][2];if(!isprime(n)&&tst(n,x),c++));
      print(" fooled "c" times");}

      fooled 10 times

      NB: Please, Paul, no more wriggles, sign tests, gcds, extra Fermats,
      new choices of [P,Q], this August. The Gremlins are sunning
      themselves and find it irkesome to tool up for such vain tests.

      David (their overheated minder)
    • paulunderwooduk
      ... Those are good counterexamples, with some passing Euler PRP tests. I notice all have kronecker(x^4-1,n)==-1, but I give up on this trail, knowing the
      Message 2 of 11 , Aug 1, 2013
        --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
        >
        >
        >
        > --- In primenumbers@yahoogroups.com,
        > "paulunderwooduk" <paulunderwood@> wrote:
        >
        > > Breaking my Trappist vow...
        >
        > Why, Paul? You have freely admitted that there is no point:
        >
        > > one parameter Lucas plus N Fermat/Euler/M-R PRP test
        > > can be counterexampled
        >
        > Here are 10 counterexamples to your latest vain idea:
        >
        > {tst(n,x)=local(P=x^8-1,Q=1-x^8);
        > kronecker(P^2-4*Q,n)==-1&&gcd(x,n)==1&&
        > Mod(x-1,n)^(n-1)==1&&
        > Mod(x+1,n)^(n-1)==1&&
        > Mod(x^2+1,n)^(n-1)==1&&
        > Mod(x^4+1,n)^(n-1)==1&&
        > Mod(Mod(1,n)*L,L^2-P*L+Q)^(n+1)==Q;}
        >
        > {F=[
        > [7750135694869, 822096191222],
        > [23723039862349, 1323013054084],
        > [90273119893069, 5862741794270],
        > [264256506403909, 38817437399213],
        > [8955652979403079, 1851456656424086],
        > [4574665869143389, 885331489130492],
        > [5266652551034509, 988874992567097],
        > [8618233825140949, 584166437019905],
        > [9541864502273629, 720345160544763],
        > [10245855908959669, 226701623305716]];
        >
        > c=0;for(k=1,#F,n=F[k][1];x=F[k][2];if(!isprime(n)&&tst(n,x),c++));
        > print(" fooled "c" times");}
        >
        > fooled 10 times
        >
        > NB: Please, Paul, no more wriggles, sign tests, gcds, extra Fermats,
        > new choices of [P,Q], this August. The Gremlins are sunning
        > themselves and find it irkesome to tool up for such vain tests.
        >

        Those are good counterexamples, with some passing Euler PRP tests. I notice all have kronecker(x^4-1,n)==-1, but I give up on this trail, knowing the Gremlins will outwit me in any event,

        Paul
      • djbroadhurst
        ... So, Paul, my old friend, you have a month to read my secret-spilling tutorial http://tech.groups.yahoo.com/group/primenumbers/message/25241 to understand
        Message 3 of 11 , Aug 2, 2013
          --- In primenumbers@yahoogroups.com,
          "paulunderwooduk" <paulunderwood@...> wrote:

          > > fooled 10 times
          > >
          > > NB: Please, Paul, no more wriggles, sign tests, gcds, extra Fermats,
          > > new choices of [P,Q], this August. The Gremlins are sunning
          > > themselves and find it irkesome to tool up for such vain tests.
          >
          > Those are good counterexamples

          So, Paul, my old friend, you have a month to read
          my secret-spilling tutorial
          http://tech.groups.yahoo.com/group/primenumbers/message/25241
          to understand this one line forger's recipe:

          print(subst(algdep(2*cos(2*Pi/5),2),x,x^8))
          x^16 + x^8 - 1

          Of course were you to add gcd(x^16+x^8-1,n)==1, in September,
          the Gremlins would work with different cosines.

          David
        • WarrenS
          Tao, Harcos, Englesma, et al seem to have stalled trying to improve Zhang s upper bound of 70,000,000. They claim to have confirmed they got it down to 5414
          Message 4 of 11 , Aug 4, 2013
            Tao, Harcos, Englesma, et al seem to have stalled trying to improve Zhang's upper
            bound of 70,000,000. They claim to have confirmed they got it down to 5414 but look
            like they aren't going to be able to go much further (perhaps can push it a bit below 5000
            if combine all their juice?).


            http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes
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