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I was wondering if the following approach is being attempted to prove the twin-prime conjecture.

Tao and Green proved in their paper (2006), that there are arbitrarily long arithmetic progressions in primes.

I was wondering if there has been work done to show that given a prime p, there is an arithmetic progression iof length k it belongs to, where k is between 2 and p-1.

This is, of course, not intended to be anything more than a question to satisfy the curiosity of an undergraduate student about whether such problems are being worked on. I don't know anything at all about this area of research, and so, really am not too sure if my question is too basic for this site or not. If you can point me to any papers where I can find related work, be it using this method, or others, that would be very welcome.

Edit: Lucia's comment clarifies the question I initially asked regarding the Twin Prime Conjecture. I had made an error and a glaring one at that. I shall modify the question suitably to make it relevant, but shall record the error here, in order to ensure that the comment is still relevant. The error was that given a prime p, it can not be a part of an infinite arithmetic progression, since at some point, the term p + pd will arise, where d is the common difference. This is what Lucia points out in the comments.

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    $\begingroup$ If you start with a prime $p$, then the common difference of a progression that begins with $p$ must be coprime to $p$. But then you cannot have more than a $p$ term progression before hitting a multiple of $p$. $\endgroup$
    – Lucia
    Commented Feb 23, 2019 at 16:35
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    $\begingroup$ @Lucia Ah, of course. I see where I made an error. $\endgroup$
    – user136205
    Commented Feb 23, 2019 at 16:51
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    $\begingroup$ I suggest that the OP closes this question as obsolete. $\endgroup$
    – GH from MO
    Commented Feb 23, 2019 at 16:57
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    $\begingroup$ Also, every two numbers form an arithmetic progression of length $2$, so $k=2$ is a trivial case. $\endgroup$
    – Greg Martin
    Commented Feb 23, 2019 at 18:57

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