Problem Statement
Let $C \geq 0$. Is there an infinite sequence of $n_i$ such that $\lim_{i\to \infty}\frac{p_{n_i+1}-p_{n_i}}{\log n_i}=C$?
Categories:
Number Theory Primes
Progress
Progress on Limit Points
- Westzynthius (1931): $\infty \in S$
- Goldston-Pintz-Yildirim (2009): $0 \in S$
- Pintz (2016): $[0,c] \subset S$ for some $c > 0$
- Merikoski (2020): At least 1/3 of $[0,\infty)$ belongs to $S$
Source: erdosproblems.com/5 | Last verified: January 13, 2026