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Old 2021-10-13, 19:13   #1
paulunderwood
 
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Talking Single Parameter Frobenius test -- 1+1+1+1+2 Selfridges

Code:
{
tst(n,a)=kronecker(a^2-4,n)==-1&&
gcd((a^3-a)*(a+4),n)==1&&
Mod(a-1,n)^(n-1)==1&&
Mod(a,n)^(n-1)==1&&
Mod(a+1,n)^(n-1)==1&&
Mod(a+4,n)^(n-1)==1&&
Mod(Mod(x+2,n),x^2-a*x+1)^(n+1)==2*a+5;}
I am in the early stages of testing this test. Can you beat me to a counterexample?

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Old 2021-10-14, 01:56   #2
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I can save a few Selfridges by using the weaker form of Fermat's Little Theorem:

Code:
{
tst(n,a)=kronecker(a^2-4,n)==-1&&
gcd(a+4,n)==1&&
Mod(a-1,n)^n==a-1&&
Mod(a,n)^n==a&&
Mod(a+1,n)^n==a+1&&
Mod(a+4,n)^(n-1)==1&&
Mod(Mod(x+2,n),x^2-a*x+1)^(n+1)==2*a+5;
}
Now "a" can be zero. So that the test is a base 4 Fermat test plus the Frobenius. The latter has been tested to 2^50.

"a" can also be +/-1 which means bases 2 and 5, bases 2 and 3 respectfully plus the Frobenius. The Frobenius has also been tested up to 2^50 (excluding the cases for the previous paragraph).

If "a" is +/-3 then a-1 and a+1 are the same Fermat PRP test, resuting in bases 2, 3 and 1 or 7 Fermat PRPs plus Frobenius.

In summary:
a=0 => 1+2 Selfridges
a=-1 => 1+1+2 Selfridges
a=1 => 1+1+2 Selfridges
a=-3 => 1+1+2 Selfridges
a=3 => 1+1+1+2 Selfridges
a=4 => 1+1+1+2 Selfridges
a=-5 => 1+1+1+2 Selfridges
otherwise => 1+1+1+1+2 Selfrdges


Last fiddled with by paulunderwood on 2021-10-14 at 07:34
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Old 2021-10-14, 22:03   #3
paulunderwood
 
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Semi-primes are being stubborn, but when I feed in Carmichael numbers counterexamples abound such as [n,a]=[19384289, 8494896]

This is yet another test that shows that X Frobenius tests with X parameters can be fooled.
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