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## 24824Re: mod quartic composite tests

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• Jan 17, 2013
--- In primenumbers@yahoogroups.com,
"paulunderwooduk" wrote:

> Is it a comprehensive list for all n <= 97847746461047271599?

By no means. However the updated file
http://physics.open.ac.uk/~dbroadhu/cert/underwqd.txt.gz
now has more:

{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(file)=local(v=readvec(file),c);
c=sum(k=1,#v,tst(v[k][1],v[k][2])&&!isprime(v[k][1]));
print(c"/"#v" counterexamples left in "file);c;}

tstfile("underwqd.txt");

422355/422355 counterexamples left in underwqd.txt

Challenge: Find a composite 10^10-smooth positive
integer, n, such that:
1) there exist an integer x that passes tst(n,x),
2) n is not in underwqd.txt.

Comment: I do not know of any such integer. Nor do I know
how to search for one. So I guess that means my gremlins
are comprehensively exhausted, though the question is not.

David
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