## Re: Prime chains x-->Ax+B

Expand Messages
• ... That s a really neat script, and (after formatting it properly:-) I believe it. But you were clearly running 64-bit pari, since ... gives only 9-digit
Message 1 of 143 , Dec 4, 2010
>
> > N = 3^t+4
> > M = (N-2)*(N-3)+3
> > be simultaneously prime. The Poisson mean for
> > such cases with t > x >> 1 may be estimated as
> > C/(2*log(3)^2)*sum(t>x,1/t^2) =~ C/(2*log(3)^2)/x
> > where C is a constant of order unity.
>
> Sieving to depth 10^8, with the aid of "znorder" and
> "polrootsmod" in Pari-GP, I estimated
> C ~ 6.5, but the convergence is not good:
>
> {mp=nextprime(10^8);default(primelimit,mp);
> default(realprecision,5);n=1;g=1.;forprime(p=5,mp,
> if(p>10^n,print([g*log(p)^2*exp(2*Euler),p]);n++);
> a=znorder(Mod(3,p));b=znorder(Mod(-4,p));c=if(a%b,0,1);
> f=lift(polrootsmod((x+1)*(x+2)+3,p));for(k=1,#f,r=f[k];
> if(r,b=znorder(Mod(r,p));c+=if(a%b,0,1)));g*=1-c/a);}
>
> [7.6000, 11]
> [7.4962, 101]
> [7.4031, 1009]
> [7.5940, 10007]
> [7.2959, 100003]
> [7.0154, 1000003]
> [6.7590, 10000019]
> [6.5402, 100000007]
>
> David (puzzled by this slow convergence)

That's a really neat script, and (after formatting it properly:-) I believe it.
But you were clearly running 64-bit pari, since
> default(realprecision,5);
gives only 9-digit precision in 32-bit pari, and that's not accurate enough.
To make the code portable, you should have written something like
\p 18
default(format,"g0.5");

I have gone up to nextprime(4*10^9) and have 2 new points for you:-
[6.35205959664247940, 1000000007]
[6.25064108108829438, 4000000007]

Convergence still slow.

Mike
• ... Suppose that we want to start with a square and get a square. Then we must solve the Diophantine equation y^2 = a*x^2 + b For any pair (a,b), Dario will
Message 143 of 143 , Jan 7, 2011
Kevin Acres <research@...> wrote:

> x=a*x+b either is a square or has a square as a major factor.

Suppose that we want to start with a square and get a square.
Then we must solve the Diophantine equation
y^2 = a*x^2 + b

For any pair (a,b), Dario will tell us all the solutions: