--- In

primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:

>

> > asking that

> > 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