# A question of convergence of a series closely related to two sequences of prime numbers

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?

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.