Today I was taking a look at Coman's book entitled Conjectures on Primes and Fermat Pseudoprimes, many based on Smarandache function (starting from the end, as I often do) and his last conjecture, the only one belonging to Part Two, Section 14 (see the above, p. 81), is as follows.
Coman's Last Conjecture: Let us denote by $\mathbb{P}$ the set of the prime numbers and assume that $p_1, p_2 \in \mathbb{P}-\{2\}$.
If $q \in \mathbb{P}-\{2,3,5,7\}$, then $q = 3 \cdot (p_1 - 1) + p_2$.
Personal consideration: My idea is very simple (I've come up with this in a couple of minutes).
Let $p_1 := 3$. It follows that $q = 6 + p_2$ holds for any pairs $(q, p_2)$ having a gap equal to $6$.
Now, a well-known property of primes is that every prime number is of the form $6 \cdot n \pm 1$ (see proof that all primes above $3$ are congruent to $\{1,5\}\pmod 6$) so that $q = 6 \cdot n \pm 1$ and $p_2 = 6 \cdot m \pm 1$ for some $n,m \in \mathbb{N}-\{0\}$.
Thus, we can say that $p_1 := 3 \Rightarrow q = 6 + p_2 \Rightarrow n = m + 1$ if $p_2 \in \mathbb{P} : q-p_2=6$. Otherwise, let $p_1 := 5 \Rightarrow q = 3 \cdot 4 + p_2$ if $p_2 \in \mathbb{P} : q-p_2=2 \cdot 6$, and so forth.
The point here is that, as far as I know, we cannot say that there are infinitely many pairs of primes with a gap of $6$, $12$, or $18$ (etcetera), since these would be some special cases of Polignac's conjecture, which is still unproved for any prime gap as above).
Am I wrong? Is Coman's conjecture implicit in Goldbach's conjecture, by any chance?
P.S. My first thought is to try to use Chen's (first) Theorem, stating that "every sufficiently large even number (i.e., larger than e^{e^{36}}) can be written as the sum of either two primes, or a prime and a semiprime", since it is trivial to note that $q-3$ is even by construction and $q - 3=3 \cdot p_1 + p_2$ is the sum of $p_2$ and the semiprime $3 \cdot p_1$, the only issue is that $3 \cdot p_1$ is a constrained semprime and not a generic one. Thus, we cannot prove the conjecture for any $q-3 > e^{e^{36}}$.