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

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  • 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 1 of 11 , Aug 1, 2013
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      --- 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 2 of 11 , Aug 2, 2013
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        --- 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 3 of 11 , Aug 4, 2013
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          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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