Problem 218

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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 $d_{n+1}=d_n$.

Status. Open.

Source. erdosproblems.com/218, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #218, https://www.erdosproblems.com/218.

References.

• [Ba23] Banks, William D., On ratios of consecutive prime gaps. Integers 23 (2023), Paper No. A50, 13 pp.
• [Er85c] Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.

Formalization. Statement in formal-conjectures.

Current assessment

Status gives the site's label. No literature search beyond the site record is recorded.

The problem has no claim page. The 2026 OpenAI mathematics release carries one paper near it, Positive lower density of large prime gaps (25 September 2026, with a Lean supplement; its card is openai_2026_positive_lower_density_large_prime_gaps), which concerns the frequency of single large gaps $d_n>C\log p_n$ and the indices at which $p_n/n$ increases. Nothing in it compares $d_{n+1}$ with $d_n$ or concerns equal consecutive gaps, so it bears on none of the three assertions and no claim page records it.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• banks_2023_ratios_consecutive_prime_gaps