## 24738Re: single frobenius and double fermat

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• Dec 10, 2012
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>
>
> "paulunderwooduk" <paulunderwood@> wrote:
>
> > gcd(x^3-x,n)==1
> > kronecker(x^2-4,n)==-1
> > (x-2)^((n-1)/2)==kronecker(x-2,n) (mod n)
> > (x+2)^((n-1)/2)==kronecker(x+2,n) (mod n)
> > (x*L^3+1)^(n+1)==(x^2-1)^2 (mod n, L^2-x*L+1)
>
> {tst(n,x)=gcd(x^3-x,n)==1&&kronecker(x^2-4,n)==-1&&
> Mod(x-2,n)^((n-1)/2)==kronecker(x-2,n)&&
> Mod(x+2,n)^((n-1)/2)==kronecker(x+2,n)&&
> Mod(Mod(1,n)*(x*L^3+1),L^2-x*L+1)^(n+1)==(x^2-1)^2;}
>
> print(sum(k=1,#v,tst(v[k][1],v[k][2]))" counterexamples");}
>
> 19959 counterexamples
>

Thanks again, David. I had an mistake in one of my equations that caused erroneous checking of your files.

I have another test. For small x, and choosing where possible x==3 or x==6, the test is on average 3.25 selfridge. For n co-prime to 30 find any x:

gcd(x^3-x,n)==1
kronecker(x^2-4,n)==-1
(x-2)^((n-1)/2)==kronecker(x-2,n) (mod n)
(x+2)^((n-1)/2)==kronecker(x+2,n) (mod n)
(x^3-x)*(L^2-4)^(n+1)==(x^3-x)^2*(25-4*x^2) (mod n, L^2-x*L+1)

I this tested against your files:

{tst(n,x)=kronecker(x^2-4,n)==-1&&gcd(x^3-x,n)==1&&
Mod(x-2,n)^((n-1)/2)==kronecker(x-2,n)&&
Mod(x+2,n)^((n-1)/2)==kronecker(x+2,n)&&
Mod(Mod(1,n)*(x^3-x)*(L^2-4),L^2-x*L+1)^(n+1)==(x^3-x)^2*(25-4*x^2);}

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
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