I have recently become interested in reading a little more on certain directions regarding primes in arithmetic progressions (AP). I would appreciate specific paper references (with the journal and the year, if possible) and suggested textbook/monograph materials that may explain any of the following questions I recently had.

Given an odd prime $p$, is there a $3$-AP in primes that it belongs to? Moreover, is there a $3$-AP whose first term is $p$? For instance, given $3$, the AP $3,5,7$ is a $3$-AP in primes, and for $5$, the AP $5,11,17$ is such an AP. I am trying to code for this right now on Python for small primes, to check it for primes until $1000$ at least, but I was wondering if there is a general proof or a counterexample. More generally, given an odd-prime $p$, is there an AP in primes of length $k$ (where $2 < k \leq p$), whose first term is $p$? If this is not true in general for every odd-prime, that is, $$\exists p_{0} \in \mathbb{P}_{\text{odd}}: \exists k_{0} \in \{3, \dotsc, p_{0}\}: \forall d \in 2\mathbb{N}: \{p_{0}, p_{0} + d, \dotsc, p_{0} + k_{0}d\} \cap (\mathbb{N} - \mathbb{P}) \neq \emptyset, $$ then one could ask whether there are infinitely many primes for which it is true (call the set of these primes $S_{\text{true}}$). An analogous question can be asked about ($\mathbb{P}_{\text{odd}} - S_{\text{true}}$). A trivial remark is that $3 \in S_{\text{true}}$, so, one at least knows that $S_{\text{true}} \neq \emptyset$. In fact, for $5$, the AP $5,11,17,23,29$ is an AP in primes, and so, a $3$-, $4$- and $5$-AP in primes exists, whose first term is $5$, so $5 \in S_{\text{true}}$. Therefore, if one can show that given a prime $p$, there is an AP in primes of length $p$, with the first term of the AP being $p$, then, one has shown that $S_{\text{true}} = \mathbb{P}$.

I am just a beginner, and so, this question might be too standard for MathOverflow (I don't know), so please excuse me if this is the case. Also, I hope my formulation of the question is non-trivial.

belongs toa 3-AP in primes, should really be whether every primebeginsa 3-AP in primes, right? $\endgroup$ – LSpice May 23 at 2:45