Problem Statement
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 $d_{n+1}=d_n$.
Categories:
Number Theory Primes
Progress
In [Er85c] Erdős also conjectures that $d_n=d_{n+1}=\cdots=d_{n+k}$ is solvable for every $k$ (which is equivalent to $k$ consecutive primes in arithmetic progression, see [141]).Source: erdosproblems.com/218 | Last verified: January 14, 2026