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Level A · Machine-checkable Hard Number theory P-erdos-839

Erdős Problem #839

Erdős Problem 839 (Part 1) [Er78f][Er92c]: Let 1 ≤ a_1 < a_2 < ⋯ be a strictly increasing sequence of positive integers such that no a_i is the sum of consecutive a_j for j < i. Is it true that limsup a_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.

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Cite 
@misc{cairn-erdos-839,
  title        = {Erdős Problem #839},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-839}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims
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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

erdos_839.parts.i. Erdős Problem 839 (Part 1) [Er78f][Er92c]:

Let be a strictly increasing sequence of positive integers such that no is the sum of consecutive for . Is it true that ?

erdos_839.parts.ii. Erdős Problem 839 (Part 2, stronger) [Er78f][Er92c]:

Let be a strictly increasing sequence of positive integers such that no is the sum of consecutive for . Is it true that ?

This is equivalent to asking whether the range has logarithmic density zero (see Set.HasLogDensity).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«839» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_839.parts.i : answer(sorry) ↔
    ∀ (a : ℕ → ℕ), (∀ n, 1 ≤ a n) → StrictMono a → SumOfConsecutiveFree a →
    atTop.limsup (fun n : ℕ => (a n : ℝ≥0∞) / n) = ⊤
theorem erdos_839.parts.ii : answer(sorry) ↔
    ∀ (a : ℕ → ℕ), (∀ n, 1 ≤ a n) → StrictMono a → SumOfConsecutiveFree a →
    Set.HasLogDensity (Set.range a) 0

What counts as progress

  • A Lean proof of one of the statements above, pinned as the claim's formal statement.
  • 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/839. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/839
  • [Er78f] Erdős, P., Problems in number theory and combinatorics, Proc. Sixth Manitoba Conf. on Numerical Math. (1978), 35-58.
  • [Er92c] Erdős, P., Some of my favourite unsolved problems, J. Combin. Theory Ser. A (1992).

See also Erdős Problem 359 and Erdős Problem 867.

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.