Arithmetic progression of primes question

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 $n\in \{1,\ldots,k\}$?

Answer 1

$k$ 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).

Answer 2

The numbers $a+30,a+60,\ldots,a+210$ are all distinct mod $7$, so one of them must be divisible by 7. So your conjecture fails.