On the primes with \(p_{n+1}-p_n=8\) and the sum of their reciprocals
Let \(\mathbb P\) be the set of prime numbers, and let \(\pi_{2,8}(x):=\sum_{p\leq x: p\in {\mathbb P},p+8\in {\mathbb P}}1.\) Theorem 1. \(\pi_{2,8}(x)=O\left(x\left(\frac{\log\log(x)}{\log(x)}\right)^2\right) .\) Theorem 2. The series \(\sum_{p\in {\mathbb P}:p+8\in {\mathbb P}}\frac{1}{p}\) is convergent.
scientific article; zbMATH DE number 5076966