21812Re: generalised 6-selfridge double fermat+lucas

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• Sep 14, 2010
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--- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:
>
> Hi,
>
> I will use capital letters to represent 2 by 2 matrices and lower case for integers.
>
> Consider:
>
> R^2-r*R+1==0
>
> I call this "trivial" if r=0 (mod d) or r=+-1 (mod d) for some proper divisor "d" of a given "n", because the equation is cyclic.
>
> Now on to the double equations:
>
> M^2-x*M+1==0
> N^2-y*N+1==0
>
> I do not want x=+-y (mod d) because they will be identical for that divisor of "n".
>
> The composite test for "n" is:
>
> First find x and y:
> gcd(x^3-x,n)==1
> gcd(y^3-y,n)==1
> gcd(x^2-y^2,n)==1
> jacobi(x^2-4,n)==-1
> jacobi(Y^2-4,n)==-1
>
> Secondly, check
> x^(n-1) == 1 (mod n)
> y^(n-1) == 1 (mod n)
> M^(n+1) == I (mod n)
> N^(n+1) == I (mod n)
>
> I have checked n<2*10^4 with gcd(30,n)==1,
>

Back to the drawing board:
n=41159;
x=3547;
y=3225;

:-(

Paul
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