Erdős (Erdos) Problem #876: status in 2026

Erdős Problem #876

Let A=a_1<a_2<⋯⊂ ℕ be an infinite sum-free set - that is, there are no solutions to a=b_1+⋯+b_r with b_1<⋯<b_r<a∈ A. How small can a_n+1-a_n be? Is it possible that a_n+1-a_n<n? The gap question is discussed by Erdős in [Er98]. We state the second question.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #790 as open (status updated 31 August 2025).
erdosproblems.com lists Erdős problem #876 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #790 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Status of Erdős problem #876 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: research statements in the file	2 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.

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.

Try it with your chatbot Submit a claim Follow (RSS)

Cite

@misc{cairn-erdos-876,
  title        = {Erdős Problem #876},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-876}},
  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):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Ferdos-876.json)](https://cairn-commons.com/problems/erdos-876)

Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

Get a task — one click

Your chatbot works on it — 5–10 minutes, in its own tab

Paste its answer here — check it and publish it under your name

What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

erdos_876.parts.i. Let $A = {a_{1} < a_{2} < \dots} \subset N$ be an infinite sum-free set - that is, there are no solutions to $a = b_{1} + \dots + b_{r}$ with $b_{1} < \dots < b_{r} < a \in A$ . How small can $a_{n + 1} - a_{n}$ be? Is it possible that $a_{n + 1} - a_{n} < n$ ?

The gap question is discussed by Erdős in [Er98]. We state the second question. The inequality $a_{n + 1} - a_{n} < n$ is required for all large $n$ , because it fails for $n = 1$ . The first question is vaguer and is not formalised.

erdos_876.parts.ii. In [Er75b] and [Er77c] Erdős asks to determine the maximum possible value of $\sum_{n \in A} \frac{1}{n}$ . Erdős had proved this is $< 100$ , and Sullivan had shown that this is $< 4$ , and Sullivan conjectured the maximum is slightly larger than $2$ [Er77c, p. 52].

Here $A$ ranges over infinite sum-free sets, and the maximum is read as the supremum maxRecipSum.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«876» (2 statements).

theorem erdos_876.parts.i :
    ∃ A : Set ℕ, A.Infinite ∧ IsSubsetSumFree A ∧ ∀ᶠ n : ℕ in atTop, gap A n < n

theorem erdos_876.parts.ii : maxRecipSum = erdos_876.parts.ii.answer

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/876. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/876
[Er62c] Erdős, P., Some remarks on number theory. III (in Hungarian). Mat. Lapok 13 (1962), 28-38. PDF
[Er75b] Erdős, P., Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, 1974), Astérisque 24-25 (1975), 295-310. PDF
[Er77c] Erdős, P., Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976), Lecture Notes in Math. 626 (1977), 43-72. PDF
[Er98] Erdős, P., *Some of my new and almost new problems and results in combinatorial number theory*. Number theory (Eger, 1996) (1998), 169-180.
[DEM99] Deshouillers, J.-M., Erdős, P. and Melfi, G., On a question about sum-free sequences. Discrete Math. 200 (1999), 49-54.
[LeOS77] Levine, E. and O'Sullivan, J., *An upper estimate for the reciprocal sum of a sum-free sequence*. Acta Arith. 34 (1977), 9-24.
[Ab87] Abbott, H. L., On sum-free sequences. Acta Arith. 48 (1987), 93-96.

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.