--- In

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

>

>

>

> --- In primenumbers@yahoogroups.com,

> "djbroadhurst" <d.broadhurst@> wrote:

>

> > > Let t be the smaller partner of a prime twin (t, t+2)

> > > calculate the infinite product P t/(t-1)

> >

> > Given the best current estimate of Brun's constant,

> > I estimate your product as 3.19617816863...

> > which is significantly greater than Pi.

>

> Off list, I was asked for my method.

>

> I took a log and subtracted half of Brun's constant, B.

> Then one may truncate at modest values of t and use the

> Hardy-Littlewood heuristic to approximate the remainder

> as an incomplete gamma function:

>

> {S(t)=log(t/(t-1))-(t+1)/(t*(t+2));}

>

> {R(t)=3*0.66016181584686957*incgam(-1,log(t));}

>

> {B=1.902160583104;T=134217779;default(primelimit,T);

> default(realprecision,13);e=2^21;s=0;

> forprime(t=3,T,if(isprime(t+2),s+=S(t);

> if(t>e,print([exp(s+B/2+R(t)),t]);e*=2)));}

>

> [3.196178168631, 2097257]

> [3.196178168641, 4194581]

> [3.196178168628, 8388617]

> [3.196178168631, 16777289]

> [3.196178168630, 33554501]

> [3.196178168629, 67109321]

> [3.196178168630, 134217779]

>

> There is no point in using more twin primes, since

> the value of B is uncertain to more that 1 part in 10^13:

> https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;9229e3b7.0208

>

> David

>

Impressing. But my very simple program calculates e.g.for

t = 2097257 product = 2.9190232000697852233 ... and for

t = 4194581 product = 2.931134127297162059 ...

which is very different from your results.

I am currently at

t = 9118078559 product = 3.0173671938991475014 ...

The program is tested several times. I would appreciate if you could also use a simple program without Brun's constant.

Werner