Is a certain sumset derived from primes of a certain form the set of all naturals? OEIS A167055 Numbers n such that $12n + 5$ is prime.
$0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19, 21,...$ are items of  OEIS $A167055$.
I conjecture that the set of the sum of every two items of this sequence is the set of nonnegative integers. i.e.:$0+0=0,\ 0+1=1,\ ...,\ 1+4=5,...$
Further information for $A167055$ not in OEIS
$A167055$: nonnegative integers  that not of the following two forms:
$$
\begin{align*} 
&3x^2+(6y−3)x−y\\ 
&3x^2+(6y−3)x+(y−1) ,\ x,y \in \mathbb{Z^+}
\end{align*}
$$
Is there some clue to solve this problem?
More information see: How come if  i  not of the following form, then 12i+5 must be prime?
 A: Let us denote the set of all the $n$ such that $12n+5$ is prime by $P$. The number of $n\le N$ in $P$ is about $3 N / \log N$. It is conceiveable that the sum of pairwise sums of a set that dense are all positive integers, but to show that should be very hard.
One might compare the problem with the Golbach problem. There one has a set of density $N / \log N$, the primes, and expects to find all even numbers as sums of two (the odd ones being trivially impossible). 
So, the problem at hand is about the same, on the one hand slightly easier as the set is a bit denser $3 N / \log N$ vs $N / \log N$ or rather $2 N / \log N$ (the density among the odd numbers) but then on the other hand  harder as the set might be a bit less direct to approach than the primes. 
Actually problems like this got studied quite a bit, though set up slightly differently at least the presentations I know (see for example K. Halupczok, On the ternary  Goldbach problem with primes in arithmetic progressions having a common modulus for a recent paper). Namely one asks the "Golbach questions" for primes restricted to those in some arithmetic progressions with the same modulus. So in that language your question would be if all numbers congruent $10$ modulo $12$ are the sum of two primes congruent $5$ modulo $12$. This is too hard  to solve at present, as is binary Golbach. 
However, if you change your question to asking for sums of three number of that form then it is known at least all sufficiently large numbers are of that form (paralleling asymptotic ternary Goldbach); this follows from a result of J. Zulauf Beweis einer Erweiterung des Satzes von Goldbach-Vinogradov.
Journal für die reine und angewandte Mathematik, 1952.  
