## Re: A derived 5 selfridge test

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• This is a follow-up, but taking y=x+2. Redefine: x=a-1 y=a+1 where a is the mean of x and y . Redefine the matrices: X=[a-1,-1;1,0] Y=[a+1,-1;1,0] and
Message 1 of 4 , Jun 9, 2009
This is a follow-up, but taking y=x+2. Redefine:

x=a-1
y=a+1

where "a" is the mean of "x" and "y".

Redefine the matrices:

X=[a-1,-1;1,0]
Y=[a+1,-1;1,0]

and

L(X)=X-x
L(Y)=Y-y

The composite test is to choose any "a" such that jacobi(D,n)==-1 for both:

D(X)=(a-1)^2-4
D(Y)=(a+1)^2-4

and check:

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.

Paul