# Arithmetic progressions, given a prime

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.

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• Based on the rest of your question, the first thing you ask, whether every prime belongs to a 3-AP in primes, should really be whether every prime begins a 3-AP in primes, right? – LSpice May 23 at 2:45
• @LSpice Absolutely agree. However, if one can show that $\exists p \in \mathbb{P}_{odd}:$ $(p$ $is$ $not$ $in$ $any$ $3-A.P.$ $in$ $primes)$, then, the next question about whether every prime begins a 3-A.P. in primes becomes moot, although, I think the rest of the question is probably still valid, with some modification. I have added a sentence there to reflect your comment. Thank you. – Inershya May 23 at 2:59

The question, whether there are infinitely many three-term arithmetic progressions with first term $$3$$, is asking whether there are infinitely many primes $$p$$ such that $$2p-3$$ is also prime. In general, the question, given integers $$a,b$$ are there infinitely many primes $$p$$ such that $$ap+b$$ is prime, is wide open (except in the cases where the answer is trivially "no", such as when $$\gcd(a,b)>1$$, or when $$a\equiv b\equiv1\bmod2$$). It is expected that the answer is "yes", and it is worth consulting the literature on Dickson's conjecture.
• My original question asks only for one A.P. in primes, whose first term is $p$ (and in particular, $3$), but a natural extension would be to ask what you have stated- if there are infinitely many such 3-A.P.s, given a prime $p$. Of course, the Dickson's conjecture is far more general than that. Also, that is a neat reformulation in the first sentence- one is, after all, only really checking 'prime-ness' for the third term, since every prime greater than $3$ forms a 2-A.P. with $3$. Thanks! – Inershya May 23 at 12:38