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I was reading some literature on twin primes and goldbach partitions and noted that Littlewood and Hardy proposed that the total number of twin primes $\pi_{2}(n)$ is approximately equal to

$\pi_{2}(n)\sim2C_{2}\frac{n}{ln^{2}(n)}$

I also read that Hardy and Littlewoods proposed an equation for goldbach partitions $g(n)$, which when simplified for even numbers of the form $n=2^{k}$ where $k$ is a positive integer is also approximately equal to

$g(n)\sim2C_{2}\frac{n}{ln^{2}(n)}$

where $C_{2}$is the twin prime constant. I was surprised with this and actually thought there had to be an error somewhere. To check this, I wrote a program to see if there was some truth to their proposals and was surprised to find there was. I found that the ratio

$g(n)*n/\pi^{2}(n)\sim2C_{2}$

and

$\pi_{2}(n)*n/\pi^{2}(n)\backsim2C_{2}$

for numbers of the form $n=2^{k}$ from k = 10 upto k=30.

In these cases, the total number of twin primes less than the even number $2^{k}$ is substantially the same as the total number of goldbach partitions for that number.

Why is this so? Can someone offer an heuristic explanation why the total number of twin primes and goldbach partitions are substantially the same for numbers of the form $2^{k}$?

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  • $\begingroup$ I am not sure there is a good explanation that fits the format. Do you understand how the constants arise in the twin-prime conjecture and in the extended Goldbach conjecture? If not this seems like a natural place to start. $\endgroup$ – quid Feb 2 '15 at 23:55
  • $\begingroup$ Thank you. I will follow up on how the constants arise. $\endgroup$ – PWM Feb 3 '15 at 0:44
  • $\begingroup$ Let $p,q,r,s$ be prime numbers such as $2n=p+q=r+s$. We find $p-r=s-q$. So if $(p,r)$ and $(s,q)$ are twin prime couples, this identity is true. Maybe that can be the relation. But i'm quite ignorant so nevermind. $\endgroup$ – esege Jan 4 '16 at 12:23

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