Eisenstein Primes



Let w=(1+sqrt(-3))/2 be the third root of -1. The ring Z[w] is a maximal order and so an integer ring in the number field Q[w]. As it is an integral domain one can define primes. They are called Eisenstein primes. An Eisenstein integer a+bw is prime if and only if either (i) p = a^2+b^2+ab is prime and p is 0 or 1 modulo 3, or then that (ii) the square root of p is prime and p is 2 modulo 3. The Eisenstein Goldbach conjecture claims that every integer a+bw with a > 3, b > 3 can be written as a sum of two Eisenstein primes with a ≥ 0,b ≥ 0.

[There is a weak but elegant Eisenstein Goldbach conjecture stating that every Eisenstein integer is a sum of two Eisenstein primes. It pairs with a weak Gaussian Goldbach conjecture stating that every even Gaussian integer is a sum of two Gaussian primes. The weak Gaussian version is due to Holben and Jordan from 1968.]

In the picture to the left, we see the primes in blue or green. The green ones are the ones of the form a+b w with a>0,b>0. The claim is that every red integer is the sum of two green primes. In the picture to the right, we see the analogue Gaussian case. There, we can only reach all even Gaussian integers in the padded quadrant.
Small number can be deceiving. We first thought that the result holds for a > 1, b > 1 but look at this picture. The matrix entries (4, 110),(4, 122),(110, 4),(122, 4) are zero meaning that 3+109w or 3+121w can not be written as a sum of two Eisenstein primes. What is surprising is that the previous row has no such zeros. Every 2+k w appears to be writable as a sum of two Eisenstein primes 1+a w + 1 + bw, meaning that both 1+a2 + a and 1+b2 + b are rational primes.
The Ghosts (Clip from the matrix large)
The matrix (large pictureshowing Eisenstein Goldbach matrix)
Here is a check that the all Eisenstein integers 2+k w are the sum of two positive Eisenstein primes for k < 10000. Our intuition was that this boundary case is the hardest. But no, it was harder to write Eisenstein integers of the form 3+k as a sum of two Eisenstein primes. 
M = 100000; f = Sum[If[PrimeQ[n^2 + n + 1], x^n, 0], {n, 2*M}];
s = CoefficientList[Expand[f*f],x]; Min[Table[s[[k]],{k,3,M}]]
Here is a check that 110 and 122 are the only outliers until 100000. Note that the coefficient list starts with the zero coefficients so that we actually deal with the Eisenstein integers 3+109w and 3+121 w. We call them the Eisenstein twins, really nasty guys. 
M = 100000; f = Sum[If[PrimeQ[n^2 + n + 1], x^n, 0], {n, 2*M}];
         g = Sum[If[PrimeQ[n^2 + 2n + 4], x^n, 0], {n, 2*M}];
s = CoefficientList[Expand[f*g],x]; u=Table[s[[k]],{k,1,M}];
Position[u,0]
It might be that these four entries are the only outliers. We actually believe so. But who knows?


Eisenstein Primes
Eisenstein Primes