--- In

primenumbers@yahoogroups.com,

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

> > {if(tst(429749641620836200211,69208918734832269040,

> > 200208504884456069347),print("fooled"));}

> > fooled

> Well done. Thanks

After I worked out to how to forge one counterexample,

the gremlins were able to find more than 21,000:

{tst(n,x,a)=2<x&&x<(n+1)/2&&1<a&&a<(n+1)/2&&

kronecker(x^2-4,n)==-1&&gcd(a^3-a,n)==1&&gcd(a,x)==1&&

kronecker(x,n)==-1&& \\ added wriggle from Paul

Mod(2,n)^(n-1)==1&&n%12==11&& \\ by construction

Mod(Mod(1,n)*(L+a),L^2-x*L+1)^(n+1)==(a^2+1+a*x)&&

Mod(Mod(1,n)*(L-a),L^2-x*L+1)^(n+1)==(a^2+1-a*x)&&!isprime(n);}

{F=readvec("underwdf.txt");

\\

http://physics.open.ac.uk/~dbroadhu/cert/underwdf.txt
print(sum(k=1,#F,tst(F[k][1],F[k][2],F[k][3]))" counterexamples");}

21728 counterexamples

It will be observed that, in each case, n = 11 mod 12 is a

base-2 Fermat pseudoprime. That was for convenience and

is not an intrinsic limitation.

David