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

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• ... 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
"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)
• ... 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
>
>
>
> "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
• ... 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
"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
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
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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