Erdős (Erdos) Problem #839: status in 2026

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 = ∞?

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #359 as open (status updated 31 August 2025).
erdosproblems.com lists Erdős problem #839 as open (status updated 31 August 2025).
erdosproblems.com lists Erdős problem #867 as disproved, with a Lean proof (status updated 7 April 2026).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #359 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Status of Erdős problem #839 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Status of Erdős problem #867 on erdosproblems.com	disproved (Lean) 2026-04-07	erdosproblems.com
Formal Conjectures: research statements in the file	2 open, 0 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 — erdosproblems.com marks it as disproved (Lean) (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.

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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 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

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

Let $1 \leq a_{1} < a_{2} < \dots$ 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 $lim sup a_{n} / n = \infty$ ?

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

Let $1 \leq a_{1} < a_{2} < \dots$ 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 $lim_{x \to \infty} \frac{1}{l o g x} \sum_{a_{n} < x} \frac{1}{a _{n}} = 0$ ?

This is equivalent to asking whether the range ${a_{1}, a_{2}, \dots}$ has logarithmic density zero (see Set.HasLogDensity).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«839» (2 statements).

theorem erdos_839.parts.i :
    ∀ (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 :
    ∀ (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).

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.