Erdős Problem #340 — open problem in combinatorics

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?

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-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 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $A = {1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \dots}$ 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 $A$ ? Is it true that $∣ A \cap {1, \dots, N} ∣ ≫ N^{1/2 - ε}$ for all $ε > 0$ and large $N$ ?

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

erdosproblems.com/340

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.