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Updated Oct 7

Problem 455

claims/: The 3 claim pages of Problem 455, one per claimant's result; the problem's standing derives from them.

Statement. Let $q_1<q_2<\cdots$ be a sequence of primes such that

q_{n+1}-q_n\geq q_n-q_{n-1}.

Must

\lim_n \frac{q_n}{n^2}=\infty?

Status. Open, the site's label (page last edited 7 October 2025; proof-claims thread accessed 2026-10-06). The site records Richter's bound $\liminf q_n/n^2>0.352$ [Ri76], a refereed result recorded as an accepted partial claim on [[problems/integer_sequences/E0455/claims/1976_01_01_richter|Richter's claim page]], and Erdős and Graham pose the question in [ErGr80, p. 91], citing Richter's result only as $\liminf q_n/n^2>0$ . Two later partial results raise the bound and leave the limit question open; neither is adopted here. Yongxi Lin claims $\liminf q_n/n^2>0.864289$ in a Lean 4 development published on 27 September 2026 and not built here. Its metadata says that the mathematics of the underlying draft (prepared with Claude) and the Lean development (Claude Opus 5.5 through Claude Code) were produced by AI under Lin's direction; it is recorded on [[problems/integer_sequences/E0455/claims/2026_09_27_lin|Lin's claim page]]. A partial proof claim posted to the site's proof-claims tab on 6 October 2026 by the user satorunet, produced with Claude Opus 5.5 and Claude Fable 5.1 (Anthropic) and GPT-6-Astra via Codex (OpenAI), as the tab names them, extends Lin's method by one prime to $0.9200$ by a computer-certified residue argument and adds constraints on a counterexample; it is recorded on [[problems/integer_sequences/E0455/claims/2026_10_06_satorunet|its claim page]]. The same user's claim of 5 October 2026 on the tab, whose headline bound was Lin's $0.8642$ , was withdrawn and replaced by the claim of 6 October once Lin's prior work was found, as the write-up of 6 October records; it has no page of its own, since the replacing page discloses it. Two earlier working notes credited by that write-up, a report of 28 July 2026 at erdosproblemaday.com (Patrick White with Claude, Anthropic) and an issue of 24 September 2026 in the GitHub repository the-omega-institute/trureturing (produced with Codex CLI), each claim the constant $0.5434$ by sharpening Richter's argument, and the issue excludes periodic second-difference words of period at most $n^{1/4-\delta}$ ; they were not submitted to the site, their constant is below Lin's claimed bound, and they are recorded here without pages of their own.

Source. erdosproblems.com/455, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #455, https://www.erdosproblems.com/455.

References.

[Ri76] Richter, Bernd, Über die Monotonie von Differenzenfolgen. Acta Arith. 30 (1976), 225--227; $\liminf q_n/n^2\ge1/2.84010\ldots=0.3521\ldots$ . Library home: richter_1976_uber_die_monotonie_von_differenzenfolgen.
[ErGr80] Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28 (1980); the question, with Richter's $\liminf q_n/n^2>0$ , on printed p. 91, the site's source key. Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.

Formalization. The file ErdosProblems/455.lean of formal-conjectures, at the commit of 18 September 2026 that last changed it, states the question as erdos_455 under category research open and Richter's bound as erdos_455.variants.liminf, $\liminf q_n/n^2>0.352$ in $[0,\infty]$ , under category research solved; both bodies are sorry. Lin's repository restates and proves the liminf variant and its own sharper bound, as the claim page records; it is not built here. The community database records a formalized statement since 3 January 2026 and no formal proof.

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Linked library material

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Linked from (13)

Richter's lower bound 1/2.84010 for liminf q_n/n^2 Lower bound 0.864289 with a Lean certificate Lower bound 0.92 for the normalized convex prime sequence problems/integer_sequences/E0455/claims Sequences and Densities of Integers Sequences and Densities of Integers integer_sequences/banks_2014_consecutive_primes_tuples Corollary 1 (p. 2): infinitely many increasing and decreasing runs of m consecutive prime gaps integer_sequences/bruedern_2017_local_oscillations_moderately_dense_sequences_primes Theorem 1 (p. 1): p_{n+1}^2 - p_n p_{n+2} changes sign infinitely often for primes counted beyond x/(log x)^{4/3}Theorem 2 (p. 2) and its Corollary (p. 3): the curvature of a delta-dense set of primes lies between 10^{-8} delta_N^3 log N and 500 delta_N^{-1} log N Theorem 3 (p. 3): the sum over N < n <= 2N of |Delta_n|/p_n for a delta-dense set of primes lies between 10^{-7} delta_{2N}^3 and 11/delta_{2N+2}integer_sequences/richter_1976_uber_die_monotonie_von_differenzenfolgen

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