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Problem 218 — Erdős Problems

Problem 218

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 $d_{n+1}=d_n$ .

Source: erdosproblems.com/218

Number theory
Primes

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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.

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