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Re: stronger than BPSW?

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• ... I can go one step further: The trace of the square root of a [-1,0;0,-1], where it exists, is 0 (mod n) for prime n. Jon Grantham, in his paper Frobenius
Message 1 of 7 , May 2, 2011
--- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:
>
> Hi,
>
> Pari/GP implements Baillie-PSW with a strong 2-SPRP test and a strong -- where the jacobi symbol is -1 -- test on x^2-P*x-1 with minimal P.
>
> In my paper "quadratic composite tests", available in the files section of this group, I give another 3-selfridge test based on a strong a-SPRP test and a strong lucas test on x^2-a*x+1 with minimal a.
>
> In a way my test is stronger in that writing n+1 as 2^s*d, I can say
>
> V(a,1,<(n+1)/(2^r)-1,<(n+1)/(2^r)>) is either <a,2> for r=s; or is <-a,-2> for some r: 0<r<=s
>
> In matrix terms I am taking the successive square roots of
>
> [a,-1;1,0]^(n+1) == [1,0;0,1] (for a prime)
>
> Neat, eh?
>

I can go one step further: The trace of the square root of a [-1,0;0,-1], where it exists, is 0 (mod n) for prime n.

Jon Grantham, in his paper Frobenius Pseudoprimes, calls this sort of test "Extra strong", which is a bit confusing. I am thinking of strength in terms of the jacobi symbol being -1 and the selfridge-lucas sense. Should I call it "Extra strong strong"?

Paul
• On Mon, 02 May 2011 13:24:56 -0000, paulunderwooduk ... I imagine that it s the moment when you require a number-based measurement of strength, the kind with
Message 2 of 7 , May 2, 2011
On Mon, 02 May 2011 13:24:56 -0000, "paulunderwooduk"
<paulunderwood@...> wrote:
> Jon Grantham, in his paper Frobenius Pseudoprimes, calls this sort of
> test "Extra strong", which is a bit confusing. I am thinking of
> strength in terms of the jacobi symbol being -1 and the
> selfridge-lucas sense. Should I call it "Extra strong strong"?

I imagine that it's the moment when you require a number-based
measurement
of strength, the kind with a stable base unit, like the Richter scale
:-)

> Paul
YG
• ... This extra step is trivial: The trace the square root, where it exists, of a -I is always zero regardless of primality. One more thing I have noticed for
Message 3 of 7 , May 3, 2011
--- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:
>
>
>
> --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@> wrote:
> >
> > Hi,
> >
> > Pari/GP implements Baillie-PSW with a strong 2-SPRP test and a strong -- where the jacobi symbol is -1 -- test on x^2-P*x-1 with minimal P.
> >
> > In my paper "quadratic composite tests", available in the files section of this group, I give another 3-selfridge test based on a strong a-SPRP test and a strong lucas test on x^2-a*x+1 with minimal a.
> >
> > In a way my test is stronger in that writing n+1 as 2^s*d, I can say
> >
> > V(a,1,<(n+1)/(2^r)-1,<(n+1)/(2^r)>) is either <a,2> for r=s; or is <-a,-2> for some r: 0<r<=s
> >
> > In matrix terms I am taking the successive square roots of
> >
> > [a,-1;1,0]^(n+1) == [1,0;0,1] (for a prime)
> >
> > Neat, eh?
> >
>
> I can go one step further: The trace of the square root of a [-1,0;0,-1], where it exists, is 0 (mod n) for prime n.
>

This extra step is trivial: The trace the square root, where it exists, of a -I is always zero regardless of primality.

One more thing I have noticed for primes +-2 (mod 5) and of the form 4*N+3 is:

[3,-1;1,0]^(n+1)/2) == [-1,0;0,-1] (mod n)
[7,-1;1,0]^(n+1)/4) == [-1,0;0,-1] (mod n)

Paul
• ... It is an interesting fact that: trace( [3,-1;1,0]^(2*N) ) = trace( [7,-1;1,0]^N ) If, for my 6 selfridge test for {a-2;a+2}, I first find:
Message 4 of 7 , May 26, 2011
--- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:
>
>
>
> --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@> wrote:
> >
> >
> >
> > --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@> wrote:
> > >
> > > Hi,
> > >
> > > Pari/GP implements Baillie-PSW with a strong 2-SPRP test and a strong -- where the jacobi symbol is -1 -- test on x^2-P*x-1 with minimal P.
> > >
> > > In my paper "quadratic composite tests", available in the files section of this group, I give another 3-selfridge test based on a strong a-SPRP test and a strong lucas test on x^2-a*x+1 with minimal a.
> > >
> > > In a way my test is stronger in that writing n+1 as 2^s*d, I can say
> > >
> > > V(a,1,<(n+1)/(2^r)-1,<(n+1)/(2^r)>) is either <a,2> for r=s; or is <-a,-2> for some r: 0<r<=s
> > >
> > > In matrix terms I am taking the successive square roots of
> > >
> > > [a,-1;1,0]^(n+1) == [1,0;0,1] (for a prime)
> > >
> > > Neat, eh?
> > >
> >
> > I can go one step further: The trace of the square root of a [-1,0;0,-1], where it exists, is 0 (mod n) for prime n.
> >
>
> This extra step is trivial: The trace the square root, where it exists, of a -I is always zero regardless of primality.
>
> One more thing I have noticed for primes +-2 (mod 5) and of the form 4*N+3 is:
>
> [3,-1;1,0]^(n+1)/2) == [-1,0;0,-1] (mod n)
> [7,-1;1,0]^(n+1)/4) == [-1,0;0,-1] (mod n)
>

It is an interesting fact that:

trace( [3,-1;1,0]^(2*N) ) = trace( [7,-1;1,0]^N )

If, for my 6 selfridge test for {a-2;a+2}, I first find:

Mod([3,-1;1,0],n)^(n+1)==I

I can immediately deduce Mod([7,-1;1,0],n)^((n+1)/2) is also I. This deduction can be made since the traces are the same and the determinants are all 1, and remembering that gcd(30,n)==1. If:

Mod([7,-1;1,0],n)^((n+1)/2) = [r,-s;s,-r+2] (for the trace condition) then the determinant is -r^2+2*r+s^2==1. Rearranging terms:

r-1 == +-s

Mod([7,-1;1,0],n)^((n+1)/2+1) = [7*r-s,-7*s+r-2;r,-s]

The "non-trace" diagonal always sums to zero:

-7*s+2*(r-1)==0 (mod n)

Substituting:

-7*s+-2*s == 0 (mod n)

but gcd(30,n)==1. So s is 0 and consequently r is 1.

So a 4-selfridge (PRP) test follows for numbers +-2 (mod 5)

(i) gcd(30,n)==1
(ii) Mod(3,n)^n==3
(iii) Mod(7,n)^n==7
(iv) Mod([3,-1;1,0],n)^(n+1)==1 (*)

As David has pointed out, (*) can be more quickly calculated as:

Mod(Mod(1,n)*l,l^2-3*l+1)^(n+1)==1

Paul
• ... Further, if kronecker(-3,n)==-1, this can be reduced to 3 selfridges by dropping the Mod(7,n)^n==7 test because the composite test is based a-2==-1 and
Message 5 of 7 , May 31, 2011
--- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:
>
>
>
> --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@> wrote:
> >
> >
> >
> > --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@> wrote:
> > >
> > >
> > >
> > > --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@> wrote:
> > > >
> > > > Hi,
> > > >
> > > > Pari/GP implements Baillie-PSW with a strong 2-SPRP test and a strong -- where the jacobi symbol is -1 -- test on x^2-P*x-1 with minimal P.
> > > >
> > > > In my paper "quadratic composite tests", available in the files section of this group, I give another 3-selfridge test based on a strong a-SPRP test and a strong lucas test on x^2-a*x+1 with minimal a.
> > > >
> > > > In a way my test is stronger in that writing n+1 as 2^s*d, I can say
> > > >
> > > > V(a,1,<(n+1)/(2^r)-1,<(n+1)/(2^r)>) is either <a,2> for r=s; or is <-a,-2> for some r: 0<r<=s
> > > >
> > > > In matrix terms I am taking the successive square roots of
> > > >
> > > > [a,-1;1,0]^(n+1) == [1,0;0,1] (for a prime)
> > > >
> > > > Neat, eh?
> > > >
> > >
> > > I can go one step further: The trace of the square root of a [-1,0;0,-1], where it exists, is 0 (mod n) for prime n.
> > >
> >
> > This extra step is trivial: The trace the square root, where it exists, of a -I is always zero regardless of primality.
> >
> > One more thing I have noticed for primes +-2 (mod 5) and of the form 4*N+3 is:
> >
> > [3,-1;1,0]^(n+1)/2) == [-1,0;0,-1] (mod n)
> > [7,-1;1,0]^(n+1)/4) == [-1,0;0,-1] (mod n)
> >
>
> It is an interesting fact that:
>
> trace( [3,-1;1,0]^(2*N) ) = trace( [7,-1;1,0]^N )
>
> If, for my 6 selfridge test for {a-2;a+2}, I first find:
>
> Mod([3,-1;1,0],n)^(n+1)==I
>
> I can immediately deduce Mod([7,-1;1,0],n)^((n+1)/2) is also I. This deduction can be made since the traces are the same and the determinants are all 1, and remembering that gcd(30,n)==1. If:
>
> Mod([7,-1;1,0],n)^((n+1)/2) = [r,-s;s,-r+2] (for the trace condition) then the determinant is -r^2+2*r+s^2==1. Rearranging terms:
>
> r-1 == +-s
>
> Mod([7,-1;1,0],n)^((n+1)/2+1) = [7*r-s,-7*s+r-2;r,-s]
>
> The "non-trace" diagonal always sums to zero:
>
> -7*s+2*(r-1)==0 (mod n)
>
> Substituting:
>
> -7*s+-2*s == 0 (mod n)
>
> but gcd(30,n)==1. So s is 0 and consequently r is 1.
>
> So a 4-selfridge (PRP) test follows for numbers +-2 (mod 5)
>
> (i) gcd(30,n)==1
> (ii) Mod(3,n)^n==3
> (iii) Mod(7,n)^n==7
> (iv) Mod([3,-1;1,0],n)^(n+1)==1 (*)
>
> As David has pointed out, (*) can be more quickly calculated as:
>
> Mod(Mod(1,n)*l,l^2-3*l+1)^(n+1)==1
>

Further, if kronecker(-3,n)==-1, this can be reduced to 3 selfridges by dropping the Mod(7,n)^n==7 test because the composite test is based
a-2==-1 and a+2==3.

All in this thread will be added to the next release of my paper. I am still testing the ideas for small n, n<10^8,

Paul
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