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Level A · Machine-checkable Hard Number theory P-erdos-5

Erdős Problem #5

Let C≥ 0. Is there an infinite sequence of n_i such that lim_i→ inftyp_n_i+1-p_n_i/log n_i=C? We formalise "an infinite sequence of n_i" as a strictly monotone sequence of indices n : ℕ → ℕ.

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-5,
  title        = {Erdős Problem #5},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-5}},
  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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The problem

The question

Let . Is there an infinite sequence of such that

We formalise "an infinite sequence of " as a strictly monotone sequence of indices n : ℕ → ℕ. Note that the numerator is the gap between the two consecutive primes and , which is primeGap (n i), and not the gap between the primes indexed by two consecutive members of the sequence.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«5». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_5 : answer(sorry) ↔ ∀ C : ℝ, 0 ≤ C →
    ∃ n : ℕ → ℕ, StrictMono n ∧ Tendsto (fun i => normalizedGap (n i)) atTop (𝓝 C)

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/5. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_5.variants.limit_point_set — Let S be the set of limit points of (p_n+1-p_n)/log n. This problem asks whether S=[0,∞].
  • erdos_5.variants.dense — In [Er65b], [Er85c], and [Er97c] Erdős asks whether S is everywhere dense (but Weisenberg notes that clearly S is closed so this is equivalent to asking…

References

  • erdosproblems.com/5
  • [BFM16] Banks, William D. and Freiberg, Tristan and Maynard, James, *On limit points of the sequence of normalized prime gaps*. Proc. Lond. Math. Soc. (3) (2016), 515-539.
  • [Er55] Erdős, Paul, Some remarks on number theory. Riveon Lematematika (1955), 45-48.
  • [Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.
  • [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.
  • [Er97c] Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67.
  • [GPY09] Goldston, Daniel A. and Pintz, János and Yıldırım, Cem Y., Primes in tuples. I. Ann. of Math. (2) (2009), 819-862.
  • [HiMa88] Hildebrand, Adolf and Maier, Helmut, Gaps between prime numbers. Proc. Amer. Math. Soc. (1988), 1-9.
  • [Me20] Merikoski, Jori, Limit points of normalized prime gaps. J. Lond. Math. Soc. (2) (2020), 99-124.
  • [Pi16] Pintz, János, *Polignac numbers, conjectures of Erdős on gaps between primes, arithmetic progressions in primes, and the bounded gap conjecture*. From arithmetic to zeta-functions (2016), 367-384.
  • [Ri56] Ricci, Giovanni, Recherches sur l'allure de la suite . Colloque sur la Théorie des Nombres, Bruxelles, 1955 (1956), 93-106.
  • [We31] Westzynthius, E., *Über die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind*. Commentat. Phys. Math. (1931), 1-37.

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.