## 24684Re: Twin primes and Pi

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• Nov 26, 2012
>
>
>
>
> > > 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
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