Browse Groups

• Hi, perhaps the gremlins will like this puzzling composite test. For non-square n coprime to 30 find x and a: kronecker(x^2-4,n)==-1 gcd(x^3-x,n)==1
Message 1 of 4 , Mar 14
View Source
Hi,

perhaps the gremlins will like this puzzling composite test. For non-square n coprime to 30 find x and a:
kronecker(x^2-4,n)==-1
gcd(x^3-x,n)==1
gcd((x+a)^3-(x+a),n)==1

and do the sub-test:
(L+(x+a)^2)^n==-l^3+(x^2-2)*L+(x+a)^2 (mod n, (L^2-x*L+1)*(L^2+x*L+1))

Or in pari-speak:
{tst(n,x,a)=kronecker(x^2-4,n)==-1&&
gcd(x^3-x,n)==1&&
gcd((a+x)^3-(x+a),n)==1&&
Mod(Mod(1,n)*(L+(x+a)^2),(L^2-x*L+1)*(L^2+x*L+1))^n==-L^3+(x^2-2)*L+(x+a)^2;}

Paul -- in triple loop hell
• ... Dispelled with n=17261;x=676;a=65 Paul
Message 2 of 4 , Mar 14
View Source
--- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:
>
> Hi,
>
> perhaps the gremlins will like this puzzling composite test. For non-square n coprime to 30 find x and a:
> kronecker(x^2-4,n)==-1
> gcd(x^3-x,n)==1
> gcd((x+a)^3-(x+a),n)==1
>
> and do the sub-test:
> (L+(x+a)^2)^n==-l^3+(x^2-2)*L+(x+a)^2 (mod n, (L^2-x*L+1)*(L^2+x*L+1))
>
> Or in pari-speak:
> {tst(n,x,a)=kronecker(x^2-4,n)==-1&&
> gcd(x^3-x,n)==1&&
> gcd((a+x)^3-(x+a),n)==1&&
> Mod(Mod(1,n)*(L+(x+a)^2),(L^2-x*L+1)*(L^2+x*L+1))^n==-L^3+(x^2-2)*L+(x+a)^2;}
>

Dispelled with n=17261;x=676;a=65

Paul
• ... I am now testing for a=0, having covered n
Message 3 of 4 , Mar 17
View Source
> --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@> wrote:
> >
> > Hi,
> >
> > perhaps the gremlins will like this puzzling composite test. For non-square n coprime to 30 find x and a:
> > kronecker(x^2-4,n)==-1
> > gcd(x^3-x,n)==1
> > gcd((x+a)^3-(x+a),n)==1
> >
> > and do the sub-test:
> > (L+(x+a)^2)^n==-l^3+(x^2-2)*L+(x+a)^2 (mod n, (L^2-x*L+1)*(L^2+x*L+1))
> >
> > Or in pari-speak:
> > {tst(n,x,a)=kronecker(x^2-4,n)==-1&&
> > gcd(x^3-x,n)==1&&
> > gcd((a+x)^3-(x+a),n)==1&&
> > Mod(Mod(1,n)*(L+(x+a)^2),(L^2-x*L+1)*(L^2+x*L+1))^n==-L^3+(x^2-2)*L+(x+a)^2;}
> >
>
> Dispelled with n=17261;x=676;a=65
>

I am now testing for a=0, having covered n<2*10^6. I have a question. I test separately:
(L+x^2)^(n+1)==1+x^2+x^3 (mod n, L^2-x*L+1)
(L-x^2)^(n+1)==1+x^2-x^3 (mod n, L^2+x*L+1)

Is gcd(x,n)==1 implied by the combined test I posted before?

Paul
• ... I meant: (L-x^2)^(n+1)==1+x^2-x^3 (mod n, L^2-x*L+1)
Message 4 of 4 , Mar 17
View Source
--- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:
>
>
>
>
> > --- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@> wrote:
> > >
> > > Hi,
> > >
> > > perhaps the gremlins will like this puzzling composite test. For non-square n coprime to 30 find x and a:
> > > kronecker(x^2-4,n)==-1
> > > gcd(x^3-x,n)==1
> > > gcd((x+a)^3-(x+a),n)==1
> > >
> > > and do the sub-test:
> > > (L+(x+a)^2)^n==-l^3+(x^2-2)*L+(x+a)^2 (mod n, (L^2-x*L+1)*(L^2+x*L+1))
> > >
> > > Or in pari-speak:
> > > {tst(n,x,a)=kronecker(x^2-4,n)==-1&&
> > > gcd(x^3-x,n)==1&&
> > > gcd((a+x)^3-(x+a),n)==1&&
> > > Mod(Mod(1,n)*(L+(x+a)^2),(L^2-x*L+1)*(L^2+x*L+1))^n==-L^3+(x^2-2)*L+(x+a)^2;}
> > >
> >
> > Dispelled with n=17261;x=676;a=65
> >
>
> I am now testing for a=0, having covered n<2*10^6. I have a question. I test separately:
> (L+x^2)^(n+1)==1+x^2+x^3 (mod n, L^2-x*L+1)
> (L-x^2)^(n+1)==1+x^2-x^3 (mod n, L^2+x*L+1)
>

I meant:
(L-x^2)^(n+1)==1+x^2-x^3 (mod n, L^2-x*L+1)

> Is gcd(x,n)==1 implied by the combined test I posted before?
>
> Paul
>
Your message has been successfully submitted and would be delivered to recipients shortly.
• Changes have not been saved
Press OK to abandon changes or Cancel to continue editing
• Your browser is not supported
Kindly note that Groups does not support 7.0 or earlier versions of Internet Explorer. We recommend upgrading to the latest Internet Explorer, Google Chrome, or Firefox. If you are using IE 9 or later, make sure you turn off Compatibility View.