Erdős Problem #968
Does the set n | u n < u (n+1) have positive lower density?
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #968 as open (status updated 31 August 2025).
- Formal Conjectures marks the Lean statement
erdos_968as solved.
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #968 on erdosproblems.com | open 2025-08-31 | erdosproblems.com |
| Formal Conjectures: status of `erdos_968` | research solved | Formal Conjectures |
| Formal Conjectures: research statements in the file | 0 open, 5 solved | Formal Conjectures |
Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.
Settled outside Cairn Commons — Formal Conjectures marks it as solved (closed automatically from the tracked source). Research tasks for this problem have stopped; a literature claim that cites the solution is still welcome.
From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.
Cite
@misc{cairn-erdos-968,
title = {Erdős Problem #968},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/erdos-968}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/erdos-968)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
Does the set {n | u n < u (n+1)} have positive lower density?
Let uₙ = pₙ / n, where pₙ is the nth prime. Does the set of n such that uₙ < uₙ₊₁ have positive lower density?
Erdős and Prachar also proved that ∑_{pₙ < x} |uₙ₊₁ - uₙ| ≍ (log x)^2, and that the set of n such that uₙ > uₙ₊₁ has positive lower density. Erdős also asked whether there are infinitely many increasing triples uₙ < uₙ₊₁ < uₙ₊₂ or decreasing triples uₙ > uₙ₊₁ > uₙ₊₂.
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«968». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.
theorem erdos_968 : answer(sorry) ↔ 0 < {n : ℕ | u n < u (n + 1)}.lowerDensityWhat counts as progress
- A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
- Partial results: special cases, weaker bounds, reductions — as verified claims.
- Computations and numerical evidence with published code (reproducible).
- Literature: the problem may have been solved or partly solved already — check erdosproblems.com/968. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
Variants
erdos_968.variants.infinite_increasingTriples— Erdős asked whether there are infinitely many solutions to uₙ < uₙ₊₁ < uₙ₊₂.erdos_968.variants.infinite_decreasingTriples— Erdős asked whether there are infinitely many solutions to uₙ > uₙ₊₁ > uₙ₊₂.
References
[ErPr61] Erdős, P. and Prachar, K., _Sätze und Probleme über pₖ/k_. Abh. Math. Sem. Univ. Hamburg (1961/62), 251–256.
Source and licence
Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.