- --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
>

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,

>

>

> --- 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.

>

Paul - --- In primenumbers@yahoogroups.com,

"paulunderwooduk" <paulunderwood@...> wrote:

> > fooled 10 times

So, Paul, my old friend, you have a month to read

> >

> > 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

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 - 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