For the case where is transcendental, we see that so no version of goldbach’s conjecture holds. Furthermore, in the case that ,, so the only known version is Goldbach’s weak conjecture.
Now let for some natural number greater than 1. Then the atoms of are of the form where divides a power of and is a prime not dividing . Now take any . Then by goldbach's weak conjecture, can be written as the sum of at most 4 elements of the form for primes . For each term, if , the term is an atom. Otherwise, it can be written as for some . Let be a prime greater than . By Dirichlets theorem, there is some with prime. Because ,. Take some where and . Then for some . So can be wrritten as a sum of atoms. So every element can be written as a sum of at most atoms.