24684Re: Twin primes and Pi
- Nov 26, 2012--- In email@example.com, "djbroadhurst" <d.broadhurst@...> wrote:
>Impressing. But my very simple program calculates e.g.for
> --- In firstname.lastname@example.org,
> "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:
> [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:
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.
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