## Re: number of selfridges?

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• ... I have another test: 1d. for *any* distinct x and y: gcd(n,30)==1 0
Message 1 of 46 , Dec 3, 2011
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--- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:

> >
> >
> > > > > >
> > > > > > 1. for kronecker(5,n)==-1:
> > > > > > Mod(Mod(1,n)*(l^2-2),l^2-3*l+1)^(n+1)+9==0
> > > > > >
> >
> > > 1b. for kronecker(5,n)==-1:
> > > Mod(Mod(1,n)*3*(l-1),l^2-3*l+1)^(n+1)+9==0
> > >
> >
> > I now have a generalization:
> >
> > 1c. for non-zero x and kronecker(x^2-4,n)==-1:
> > Mod(Mod(1,n)*(l^2-2),l^2-x*l+1)^(n+1)+2*x^2-9==0
> >
> > So for a given, find an appropriate x and apply 1c,
> >
>
> 1c fails for n=671 and x=61 :(
>
> I could wriggle and insist on two x's, 4 or 2 apart; or a minimal x. Back to the drawing board!
>

I have another test:

1d. for *any* distinct x and y:
gcd(n,30)==1
0<x<=(n-1)/2
0<y<=(n-1)/2
kronecker(x^2-4,n)==-1
kronecker(y^2-4,n)==-1
Mod(Mod(1,n)*(l^2-2),l^2-x*l+1)^(n+1)+2*x^2-9==0
Mod(Mod(1,n)*(l^2-2),l^2-y*l+1)^(n+1)+2*y^2-9==0
gcd(x^2-y^2,n)==1

but how many selfridge units is it?

Paul
• ... Combining fails with the composite counterexample n=256999 and x=32768, However, I have tested the 1+1+1+2 conjecture up to n
Message 46 of 46 , Apr 14, 2012
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--- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:
>
>
>
> --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@> wrote:
> >
> > Hi,
> >
> > I have added a Fermat test to make a 1+1+1+2 selfridge test:
> >
> > For N>5, with gcd(6,N)==1, find an integer x:
> > gcd(x^3-x,N)==1
> > kronecker(x^2-4,N)==-1
> >
> > and check:
> > (x+2)^((N-1)/2)==kronecker(x+2,N) (mod N) (Euler)
> > (x-2)^((N-1)/2)==kronecker(x-2,N) (mod N) (Euler)
> > x^(N-1)==1 (mod N) (Fermat)
> > L^(N+1) == 1 (mod N, L^2-x*L+1) (Lucas)
> >
>
> Note: I should say gcd(30,N)==1 because gcd(x^3-x,N)==1 and kronecker(x^2-4,n)==-1.
>
>
> Now consider combining the 2 Euler tests with the Lucas test:
>
> (L*D)^((n+1)/2)==D (mod N, L^2-x*L+1) (D=x^2-4.)
>
> with the restriction kronecker(x+2,N)==-1.
>
> These together with the Fermat test makes for a 1+2-selfridge test.
>
> Can you find a counterexample?
>
> So far the near-refutation from Pinch's carmichael list is:
> N,x,gcd(x^2-1)
> ------------------
> 1909001 884658 1909001
>
> Paul
>

Combining fails with the composite counterexample n=256999 and x=32768, However, I have tested the 1+1+1+2 conjecture up to n<10^7,

Paul -- restoring symmetry
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