As always in this Goldbach setup for Eisenstein primes, we take w as the cube root of -1 in the first quadrant.
It is w=(1+sqrt(3))/2. Positive Eisenstein integers
are Eisenstein Integers of the form a+wb with positive a and b. The are located in the
positive first sextant sector of the complex plane.
The Eisenstein Ghost Twins are examples of positive Eisenstein integers a+wb with a=3, b>2 or b=3,a>2
which can not be written as a sum of two positive Eisenstein integers. It appears that every
Eisenstein integer a+wb with a=2 or b=2 can be written as a sum of two positive Eisenstein Primes.
These two (unique?) counter examples are the reason, why the Eisenstein Goldbach conjecture is formulated as
"Every positive Eisenstein integer a+wb with a and b larger than 3 is a sum of two positive
Eisenstein primes".
It pairs with the Gaussian Prime Conjecture:
"Every even positive Gaussian integer a+i b with a and b larger than 1 is a sum of to positive
Gaussian primes.
But it is the existence of the singular Eisenstein ghosts which prevent stating the
conjecture for a,b larger than 1 also in the Eisenstein case.
