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Problem:A positive integer $k$ greater than $1$ is given. Prove that there exist a prime $p$ and a strictly increasing sequence of positive integers $a_1, a_2, . . . , a_n, . . .$ such that the terms of the sequence $p + ka_1, p + ka_2, . . . , p + ka_n, . . .$ are all primes.
I have never encountered problems of this kind before and therefore I don't even know how to start. I am guessing that we need to provide some sort of construction for $a_n$, but I'm not sure how to proceed further.
This is another problem where Dirichlet's Theorem on Arithmetic Progressions applies, such that if $p$ and $k$ are coprime to each other, there are infinitely many primes of the form $\{p + ka \mid k \in \mathbb{N} \}$ Therefore, there exists a sequence $a_n$ such that each indexed $p + ka_i$ is prime.
