Re: A derived 5 selfridge test
- This is a follow-up, but taking y=x+2. Redefine:
where "a" is the mean of "x" and "y".
Redefine the matrices:
The composite test is to choose any "a" such that jacobi(D,n)==-1 for both:
2^n==2 (mod n)
X^n==1/X (mod n)
Y^n==1/Y (mod n)
This will satisfy:
X^n-L(X)^n+2^n==Y^n-L(Y)^n (mod n)
I also need to do two gcd tests. The first is gcd(210,n), to cover N. The second is with "a" because if gcd(a,n)==d with d>1, then:
X==[-1,-1;1,0] (mod d)
Y==[+1,-1;1,0] (mod d)
and so the exponent tests on X and Y would be identical due to symmetry, breaking the "5-selfridge rule".
Note that "x" is a zero of D(Y) and "y" is a zero of D(X).
I plan to run some tests on Richard Pinch's readily available list of fermat 2-PSPs, checking the composite test by the gcd tests and two lucas V sequences -- the traces of the matrices.
ps. apologies for not saying quadratic non-residue instead of non quadratic residue.