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Claim. If $q_1,q_2,\ldots$ are primes with $q_{n+1}-q_n\ge q_n-q_{n-1}>0$ for $n\ge2$, then $\liminf_n q_n/n^2\ge1/S$, where, with $p_1=2<p_2=3<\cdots$ the primes, $S=\sum_{r\ge1}(p_{r+1}-1)^2/(p_2\cdots p_{r+1})=2.84010\ldots$, a sum over the odd primes $p_{r+1}$; thus $\liminf_n q_n/n^2\ge0.3521\ldots$, the bound the site records as $>0.352$. The source is Richter, B., Über die Monotonie von Differenzenfolgen, Acta Arith. 30 (1976), 225--227, on the card richter_1976_uber_die_monotonie_von_differenzenfolgen. The proof compares the sequence with an extremal minorant. Since the gaps never decrease, equal gaps are consecutive, and a run of equal gaps $d$ is an arithmetic progression of primes. Such a progression has at most $P(d)-1$ terms, or at most $P(d)$ when it starts at $P(d)$, where $P(d)$ is the least prime not dividing $d$. So each gap value occurs at most $P(d)-1$ times, and the prime number theorem gives the minorant's growth $n^2/S$.

Covers. The lower bound $\liminf q_n/n^2\ge1/S=0.3521\ldots$. Whether $q_n/n^2\to\infty$, the question of Problem 455, is not addressed.

Acceptance. Refereed: Acta Arithmetica 30, no. 3 (1976), 225--227. The site's commentary credits the bound to Richter, but the site labels the problem OPEN, so the commentary is not reviewed evidence.

Date. The record gives the year 1976 and no finer date, so the page is dated to the first day of that year.

Depends on. Nothing in this wiki; the claim rests on the cited paper.