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I am interested in the following question. I will be grateful for any reference, comment, or solution.

Let $p_1\geq 5$ be a given prime number. Does there exist an infinite sequence of prime numbers $q_k$ such that $p_{k+1}=2p_k+q_k$ is a prime number and $q_k<p_k$, for all positive integers $k$? Of course, one can relax the question by omitting the condition $q_k<p_k$. I am also interested in the related question with the recursion formula $p_{k+1}= N p_k+q_k$, where $N$ is a positive even integer.

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    $\begingroup$ Euristically yes. Suppose $p_k$ large enough. There are ~ $p_k/\log p_k$ primes $q$ with $q<p_k$ and the probability that $2p_k+q$ is prime is ~ $\log p_k$. So euristically there are ~ $p_k/(\log p_k)^2$ pairs of primes $(q,p)$, with $p=2p_k+q$ (or with $p=Np_k+q$). By central limit theorem, if $p_k$ is large enough, is very unlikely that the sequence cannot be extended to $p_{k+1}$ and indefinitely for infinitely many $k$. However, in a definitive solution perspective, additive problems involving two primes in an equation are generally considered as hopeless (as e.g. the Goldbach conjecture) $\endgroup$
    – G. Melfi
    Commented Jul 13 at 21:09
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    $\begingroup$ To add to @G.Melfi's comment, we can conjecture the answers to these kinds of problems in a relatively straightforward manner (i.e. using statistical models for the primes such as Cramer's model). However, research that makes substantial progress on these kinds of problems without relying on heavy conjectures (like GRH, Elliot-Halberstam, abc, etc.) is nonexistent, at least to my knowledge. $\endgroup$
    – Joshua Stucky
    Commented Jul 14 at 3:57
  • $\begingroup$ Thank you for your comments! Yes, it seemed to me that the problem was hard. It is equivalent to the question of whether, for a given prime number $p\geq 5$ there exist prime numbers $q, r$, such that $q<p<r$ and $2p=r-q$. Thus, "similar" to a variant of the Goldbach conjecture. $\endgroup$
    – Janko Bracic
    Commented Jul 14 at 8:57

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