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In the set of primes $\mathbb P \setminus \{2,3\}$ there is an infinite number of them of the form $6k-1$ and an infinite number of them of the form $6k-5$.

Denote by $q_n$ the $n$-th prime of the form $6k-1$ in the set $\mathbb P\setminus \{2,3\}$ and by $r_n$ the $n$-th prime of the form $6k-5$ in the set $\mathbb P\setminus \{2,3\}$.

Then $s_n=|q_n-r_n|$ is a well-defined sequence.

I would like to know is $$\sum_{n=1}^{+ \infty} \frac {1}{s_n} $$ convergent?

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All primes $\ge 3$ are of the form $6k \pm 1$ and Dirichlet theorem of primes in arithmetic progression says that asymptotic density of primes of the form $6k+1$ and $6k-1$ or $6k-5$ is equal. Hence

$$ \pi(x) = 2 + \pi_{6k+1}(x) + \pi_{6k-1}(x) \approx 2\pi_{6k+1}(x) \approx 2\pi_{6k-5}(x) $$

Hence by the prime number theorem $q_n \approx r_n \approx 2n\log n$. This implies $$ |q_n - r_n| < k_1 n\log n $$ for some positive constant $k_1$. Hence $$ \sum_{n \le x} \frac{1}{|q_n - r_n|} > \sum_{n \le x} \frac{k_2}{n\log n} > \sum_{n \le x} \frac{k_2}{p_n} > k_2\log\log x $$ where $p_n$ is the $n$-th prime and $k_2$ is some positive constant. This is clearly divergent.

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