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I know about the importance in analytic number theory of the sutdy of series involving prime numbers or constellations of prime numbers, for example, if I am not wrong, major theorems are Mertens' theorems or Brun's theorem. On the other hand I know from an informative point of view that are in the literature other statements and/or conjectures more complicated about the convergence of series involving prime numbers, see for example the first paragraph of section E7 from [1].

Motivation (Edited, thanks for the comments). I am going to ask just for the following prime constellation with the goal that my question isn't too broad, but I think that to focus the attention about the sequence of strong primes is interesting itself, the definition is the definition in number theory of Wikipedia Strong prime. Also I would like to know your good intuition/reasoning to get statements or conjectures of this kind.

Question. What can be interesting series involving the sequences of strong primes? I am asking about statements or conjectures concerning convergent or divergent series involving strong primes. I am asking about series with a good mathematical content or significance in mathematics. Many thanks.

I don't know if some series involving such strong primes are in the literature. If this question is welcome, feel free to add the reference and I try to find and read those results from the literature, in other case add your genunine series for strong primes.

References:

[1] Richard K. Guy, Unsolved Problems in Number Theory, Unsolved problems in Intuitive Mathematics Volume I, Second Edition, Springer-Verlag (1994).

[2] I add also that the encyclopedia MarthWolrd has the article Prime Sums.

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    $\begingroup$ Wikipedia mentions several different definitions of strong primes. Which one do you mean? $\endgroup$ – Wojowu Aug 7 at 8:20
  • $\begingroup$ I'm sorry I am going to edit the post. Many thanks @Wojowu $\endgroup$ – user142929 Aug 7 at 8:25

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