In a previous question I asked about a counterexample for an observation I did about the Goldbach's comet: it seems that there is always common prime shared between the Goldbach's prime pairs of the even number $n$, $G(n)$, and the even number $n+6$, $G(n+6)$, when $n\ge8$. And for that reason is seems that it is also shared a pair of sexy primes between $G(n)$ and $G(n+6)$, one of the sexy primes is in $G(n)$ and the counterpart sexy prime is in $G(n+6)$. It seems it only happens with distance $d=6$, for other distances I did find a counterexample quickly.
I still did not find any counterexample for $d=6$ because it might be a very big number (my tests with Python initially did not find it, but it could be an error of my code), so I was thinking that if there is no counterexample, there could be a generalization of Goldbach's conjecture as follows:
$\forall n\ / \ n=2t, t\in \Bbb N,\ \exists\ (a_k,b_k) \in \Bbb P\ /\ n = a_k+b_k+6k, k \in [0..(\lfloor \frac{n}{6} \rfloor-1)]$
Where the case "k=0" would be the basic Goldbach's conjecture and $k>0$ would happen if the above mentioned generalization was true.
Does it make sense? If somebody could help me to find a counterexample would be great. Thank you!
The 'generalization' appears to be: for all even $n>2$ and all $0\le k\le \lfloor n/6\rfloor-1$ there is a pair of primes $p,q$ such that $p+q+6k=n.$ But this is just the standard Goldbach conjecture on $n,n-6,n-12,\ldots k$ where $k\in\{6,8,10\}.$ In particular this is equivalent to the Goldbach conjecture by strong induction.