Form of $k$ when both $6k-1$ and $6k+1$ are primes

Question

After a quick glance at sequence A007693 it seems that the following is true:

if $p$ and $p+2$ are prime, then $\frac{p+1}{6}$ is prime.

Questions:

a) Is it the case? If not, what is the smallest counterexample?

b) If true, is the proof: (b.1) elementary? (b.2) involving elaborate known results? (b.3) conditional on some strong conjecture?

Note:

This is not asking whether twin primes are always of the form $6k\pm 1$, which is true and easy; it is also not asking if there are infinitely many primes in the arithmetic progression $6m-1$.

Thank you for any help !

Answer 1

Take $p=3$. Then $p$ and $p+2$ are both prime but $\frac{p+1}{6}=\frac{4}{6}$ is not even an integer.