Erdős Problem #218

declared status 'open'. Formalized: yes. Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\leq d_n$. Furthermore, there are infinitely many $n$ such that $d_{n+1}=d_n$. Prize: no. OEIS: A064113, A333230, A333231. Tags: number theory, primes.

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