Twin-prime and Goldbach theorems for Z[[x]]

We show that an element f in the ring Z[[x]] of formal power series over the integers is a sum of two irreducible elements in Z[[x]] if and only if the constant term of f is of the form ±p^k±q^l or of the form ±p^k, where p,q are prime numbers and k,l are positive integers. Moreover, if f₀ is of such form, then there exist 2^ℵ₀ pairwise coprime elements g∈Z[[x]] such that both g and g+f are irreducible.