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Level A · Machine-checkable Hard Number theory P-erdos-968

Erdős Problem #968

Does the set n | u n < u (n+1) have positive lower density?

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.

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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 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

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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)}.lowerDensity

What 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

erdosproblems.com/968

[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.