--- In

primenumbers@yahoogroups.com, "paulunderwooduk" wrote:

>

> ? {tst(n,x)=kronecker(x^2-4,n)==-1&&

> gcd(x^3-x,n)==1&&

> gcd(x^2-2,n)==1&&

> Mod(Mod(1,n)*(L+x^2-2),(L^2-x*L+1)*(L^2+x*L+1))^(n)==-L^3+(x^2-2)*L+x^2-2;}

>

> ? {tstfile("underw297.txt");}

> 2/297 counterexamples left in underw297.txt

> ? {tstfile("underw65.txt");}

> 5146/12846 counterexamples left in underw65.txt

>

Adding gcd(x^2-3,n)==1 makes sense because x^2-2==1 (mod x^2-3). So the resurrected test is:

{tst(n,x)=kronecker(x^2-4,n)==-1&&

gcd(x^3-x,n)==1&&

gcd(x^2-2,n)==1&&

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

Mod(Mod(1,n)*(L+x^2-2),(L^2-x*L+1)*(L^2+x*L+1))^(n)==-L^3+(x^2-2)*L+x^2-2;}

Then checking David's files at:

http://physics.open.ac.uk/~dbroadhu/cert/
{tstfile(file)=local(c,n,x,v=readvec(file));

for(k=1,#v,n=v[k][1];x=v[k][2];

if(tst(n,x)&&!isprime(n),c++));

print(c"/"#v" counterexamples left in "file);c;}

? {tstfile("underbh4.txt");}

0/33445 counterexamples left in underbh4.txt

? {tstfile("underbh6.txt");}

0/308619 counterexamples left in underbh6.txt

? {tstfile("underw97.txt");}

0/97 counterexamples left in underw97.txt

? {tstfile("underw297.txt");}

0/297 counterexamples left in underw297.txt

? {tstfile("underw65.txt");}

0/12846 counterexamples left in underw65.txt

? {tstfile("underw65x.txt");}

0/10220 counterexamples left in underw65x.txt

? {tstfile("underwg.txt");}

0/100000 counterexamples left in underwg.txt

Paul