Arithmetic progression of primes question Is it known whether for all positive integers $k$ there is an integer $a$ such that $a+30n$ is a prime number for all $n\in \{1,\ldots,k\}$?

$k$ cannot be larger than $6$, since among any seven numbers of the form $b, b + 30, b + 60, \ldots, b + 180$ at least one of them is divisible by $7$, and hence only a prime if it is seven. But if $b = 7$, then $187 = 11\cdot 17$ is not a prime.
That being said, $b = 7, k = 6$ is one maximal example (w.r.t. $k$)as
$$
7, 37, 67, 97, 127, 157
$$
are all prime. There may be more examples.
Also note that it was proven in 2004 that for any $k$, there is a prime $p$ and a difference $d$ such that the numbers
$$
p, \,p + d,\, p + 2d, \ldots, p + (k-1)d 
$$
are all prime (the Green-Tao theorem).
