Erdős Problem #340
Let A = 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, … be the greedy Sidon sequence: we begin with 1 and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to a + b = c + d). What is the order of growth of A?
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #340 as open (status updated 31 August 2025).
- Formal Conjectures marks the Lean statement
erdos_340as open.
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #340 on erdosproblems.com | open 2025-08-31 | erdosproblems.com |
| Formal Conjectures: status of `erdos_340` | research open | Formal Conjectures |
| Formal Conjectures: research statements in the file | 6 open, 1 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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.
Cite
@misc{cairn-erdos-340,
title = {Erdős Problem #340},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/erdos-340}},
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):
[](https://cairn-commons.com/problems/erdos-340)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Nobody has worked on this problem here yet
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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.
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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
Let be the greedy Sidon sequence: we begin with and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to ). What is the order of growth of ? Is it true that for all and large ?
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«340».
theorem erdos_340 (ε : ℝ) (hε : ε > 0) :
(fun n : ℕ ↦ √n / n ^ ε) =O[atTop]
fun n : ℕ ↦ ((Set.range greedySidon ∩ Set.Icc 1 n).ncard : ℝ)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/340. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
Variants
erdos_340.variants.isTheta— Let A = 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, … be the greedy Sidon sequence: we begin with 1 and iteratively include the next smallest integer that…erdos_340.variants.sub_hasPosDensity— Erdős and Graham [ErGr80] also asked about the difference set A - A and whether this has positive density. [ErGr80] Erdős, P.erdos_340.variants._33_mem_sub— The smallest integer which is unknown to be in A - A is 33.erdos_340.variants.cofinite_sub— It may be true that all or almost all integers are in A - A.erdos_340.variants.co_density_zero_sub— It may be true that all or almost all integers are in A - A.
References
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.