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 arbitrarily long A.P. it belongs to, or to disprove that statement in general. Even if that statement turns out to be untrue, one may be able to show that it holds for a pair of twin primes, p and p+2. If that is shown, then, one could consider all such APs containing p and all those containing p+2, and therefore, their corresponding sets of differences (that is, all those d's, such that p+kd forms an AP in primes, and similarly, all those b's, such that p + 2 + qb form an AP in primes). Then, if one shows that there exist an element in one of those sets that divides an element in another, wouldn't one have shown the conjecture?

This is, of course, not intended to be anything more than a question to satisfy the curiosity of an undergraduate student about whether such techniques are indeed being employed to prove the Twin Prime Conjecture. 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. Naturally, I have made an error in my understanding of the statement of the Green-Tao theorem.

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.

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

Tao and Green proved in their seminal 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 arbitrarily long A.P. it belongs to, or to disprove that statement in general. Even if that statement turns out to be untrue, one may be able to show that it holds for a pair of twin primes, p and p+2. If that is shown, then, one could consider all such APs containing p and all those containing p+2, and therefore, their corresponding sets of differences (that is, all those d's, such that p+kd forms an AP in primes, and similarly, all those b's, such that p + 2 + qb form an AP in primes). Then, if one shows that there exist an element in one of those sets that divides an element in another, wouldn't one have shown the conjecture?

This is, of course, not intended to be anything more than a question to satisfy the curiosity of an undergraduate student about whether such techniques are indeed being employed to prove the Twin Prime Conjecture. 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.

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 arbitrarily long A.P. it belongs to, or to disprove that statement in general. Even if that statement turns out to be untrue, one may be able to show that it holds for a pair of twin primes, p and p+2. If that is shown, then, one could consider all such APs containing p and all those containing p+2, and therefore, their corresponding sets of differences (that is, all those d's, such that p+kd forms an AP in primes, and similarly, all those b's, such that p + 2 + qb form an AP in primes). Then, if one shows that there exist an element in one of those sets that divides an element in another, wouldn't one have shown the conjecture?

This is, of course, not intended to be anything more than a question to satisfy the curiosity of an undergraduate student about whether such techniques are indeed being employed to prove the Twin Prime Conjecture. 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. Naturally, I have made an error in my understanding of the statement of the Green-Tao theorem.

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

Tao and Green proved in their seminal 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 arbitrarily long A.P. it belongs to, or to disprove that statement in general. Even if that statement turns out to be untrue, one may be able to show that it holds for a pair of twin primes, p and p+2. If that is shown, then, one could consider all such APs containing p and all those containing p+2, and therefore, their corresponding sets of differences (that is, all those d's, such that p+kd forms an AP in primes, and similarly, all those b's, such that p + 2 + qb form an AP in primes). Then, if one shows that there exist an element in one of those sets that divides an element in another, wouldn't one have shown the conjecture?

This is, of course, not intended to be a proof or a suggestion , merely the curiosity of an undergraduate student about whether such techniques are indeed being employed to prove the Twin Prime Conjecture. 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.

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

Tao and Green proved in their seminal 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 arbitrarily long A.P. it belongs to, or to disprove that statement in general. Even if that statement turns out to be untrue, one may be able to show that it holds for a pair of twin primes, p and p+2. If that is shown, then, one could consider all such APs containing p and all those containing p+2, and therefore, their corresponding sets of differences (that is, all those d's, such that p+kd forms an AP in primes, and similarly, all those b's, such that p + 2 + qb form an AP in primes). Then, if one shows that there exist an element in one of those sets that divides an element in another, wouldn't one have shown the conjecture?

This is, of course, not intended to be anything more than a question to satisfy the curiosity of an undergraduate student about whether such techniques are indeed being employed to prove the Twin Prime Conjecture. 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.