Erdős (Erdos) Problem #338: status in 2026

Erdős Problem #338

The restricted order of a basis A is the least integer t (if it exists) such that every large integer is the sum of at most t distinct summands from A. What are necessary and sufficient conditions that this exists?

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 #338 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_338 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #338 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_338`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	5 open, 7 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.

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Cite

@misc{cairn-erdos-338,
  title        = {Erdős Problem #338},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-338}},
  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

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.

The problem

The question

The restricted order of a basis $A$ is the least integer $t$ (if it exists) such that every large integer is the sum of at most $t$ distinct summands from $A$ . What are necessary and sufficient conditions that this exists?

A solution must fill in Erdos338.erdos_338.answer with the condition and prove the equivalence. Whether a condition is a satisfactory answer is up to human judgement.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«338».

theorem erdos_338 :
    ∀ A : Set ℕ, (∃ h, HasOrder A h) → ((∃ t, HasRestrictedOrder A t) ↔ erdos_338.answer A)

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/338. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_338.variants.bounded_by_order — Can the restricted order of a basis be bounded (when it exists) in terms of the order of the basis? HasOrder A h only says that the order of A is at most h.
erdos_338.variants.eq_order — What are necessary and sufficient conditions that the restricted order of a basis A is equal to the order of A?
erdos_338.variants.basis_after_finite_removal — Is it true that if A ∖ F is a basis for all finite sets F then A must have a restricted order?
erdos_338.variants.same_order_after_finite_removal — Is it true that if the sets A ∖ F, for all finite sets F, are bases of the same order, then A must have a restricted order?

References

erdosproblems.com/338
[ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number theory*. Monographies de L'Enseignement Mathematique (1980).
[ErGr80b] Erdős, P. and Graham, R. L., On bases with an exact order. Acta Arith. (1980), 201-207.
[HHP07] Hegyvári, Norbert and Hennecart, François and Plagne, Alain, *Answer to a question by Burr and Erdős on restricted addition, and related results*. Combin. Probab. Comput. (2007), 747--756.
[He05] Hennecart, François, On the restricted order of asymptotic bases of order two. Ramanujan J. (2005), 123--130.
[Ke57] Kelly, John B., Restricted bases. Amer. J. Math. (1957), 258-264.
[Pa33] Pall, Gordon, On Sums of Squares. Amer. Math. Monthly (1933), 10-18.
[Sc54] Schinzel, A., *Sur la décomposition des nombres naturels en sommes de nombres triangulaires distincts*. Bull. Acad. Polon. Sci. Cl. III. (1954), 409-410.

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.