can you give the validity or proof of the following statements of my observations on Primes?
$(1)$ For a positive integer $k$, there exists infinitely many primes of the form $29 + 72k$.
$(2)$ If the cited above $(1)$ is true, then the following is true. That is:
Let $ p = 29+72k$ (where $p$ is prime) and we get always or infinitely many primes $q$ by considering $q = (p+1)/6$
(1) is an instance of Dirichlet's theorem. At the elementary-number-theory level you aren't expected to prove this, just say that you're using it.
(2) is not known to be true, though it would follow from various unproven conjectures like Dickson's.