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The Green-Tao theorem states that if $A$ is an infinite subset of the the prime numbers such that

$\limsup_{n \to \infty} \frac{ |A \cap [1, n]| }{\pi(n)} > 0$ then for any integer $k$, $A$ contains an arithmetic progression of length $k$.

My question is the following: Suppose that $A = \{p_{n_k}\}$ such that $\sum_{n_k} \frac{1}{p_{n_k}} = \infty$. Does this set contain arbitrarily long arithmetic progressions?

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    $\begingroup$ In the paper Greene and Tao published, they said that your question is an unanswered hypothesis $\endgroup$ – D. Hershko Jan 7 '18 at 15:56
  • $\begingroup$ @D.Hershko if this were true then I have a simple argument that settles the general conjecture of Erdos via the Tao-Greene special case $\endgroup$ – Mustafa Said Jan 7 '18 at 16:03
  • $\begingroup$ on which Erdos conjecture are you talking? $\endgroup$ – D. Hershko Jan 7 '18 at 16:09
  • $\begingroup$ @D.Hershko Erdős-Turán probably, see here. $\endgroup$ – Daniel Fischer Jan 7 '18 at 16:19
  • $\begingroup$ what is your idea? $\endgroup$ – D. Hershko Jan 7 '18 at 18:15
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Yu-Chen Sun and Hao Pan have proved the green tao theorem for prime numbers of the form $x ^ 2 + y ^ 2 + 1$ in this paper https://arxiv.org/pdf/1708.08629.pdf .It is not known if the inverse sum of the prime numbers of the form $x^2+y^2+1$ is convergent or divergent. It is not known if the inverse sum of the prime numbers of the form is convergent or divergent. But the most certain that your preposition is true. Another thing, this theorem also fulfills for the chern prime numbers, and its sum of inverses is convergent.

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