Hardy Littlewood Constant C
Since for Goldbach for Gaussian integers, primes of the form n
2+1 matter,
we got interested in them. We also looked at the constant C defined by Hardy-Littlewood,
which gives the ratio between primes on the row Im(z)=1 and the primes on the row Im(z)=0.
We computed the ratio
| { z = a + 1 i | z is Gaussian prime , a ≤ n } |
--------------------------------------------------------
| { z = a + 0 i | z is Gaussian prime , a ≤ n } |
Up to n=157819000000 = 1.5 * 10
11, we see the fraction is 4378609531/3189494198 = 1.37282 ...
In the end, we had to check primes of the order n
2 which is 2.5 * 10
22.
The program can not be simpler. It is a ``one liner".
(But the trick is to get to 10
12 but we hope we will get there eventually ...)
n=m=0;Do[If[PrimeQ[k^2+1],m++];If[PrimeQ[k]&&Mod[k,4]==3,n++;Print[N[m/n,20]]],{k,10^12}]
You see that the convergence is slow. The fluctuations are still of the order 10
-5.
Thats about the same order than what was got 100 years ago or what Shanks and Wunderlich confirmed.
Wunderlich went to n=14'000'000 = 1.4 * 10
7.
This means we go 10'000 higher and have to check primes which are 10
8 times larger.
Wunderlich wrote his paper in 1973 which is 43 years. This illustrates well Moore's law on hardware
advancement!
