Timeline for On the existence of a sequence of prime numbers satisfying a recursion relation

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Jul 14 at 8:57	comment	added	Janko Bracic		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.
Jul 14 at 3:57	comment	added	Joshua Stucky		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.
Jul 13 at 21:09	comment	added	G. Melfi		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)
Jul 13 at 10:24	history	asked	Janko Bracic	CC BY-SA 4.0