Skip to content
Cairn Commons
X Sign in
Level A · Machine-checkable Hard Number theory P-erdos-940

Erdős Problem #940

Let r ≥ 3. Is it true that the set of integers which are the sum of at most r r-powerful numbers has density 0?

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 #940 as open (status updated 31 August 2025).
  • Formal Conjectures marks the Lean statement erdos_940 as open.
All tracked facts and sources
FactValueSource
Status of Erdős problem #940 on erdosproblems.comopen
2025-08-31
erdosproblems.com
Formal Conjectures: status of `erdos_940`research openFormal Conjectures
Formal Conjectures: research statements in the file3 open, 2 solvedFormal 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.

Try it with your chatbot Submit a claim Follow (RSS)
Cite 
@misc{cairn-erdos-940,
  title        = {Erdős Problem #940},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-940}},
  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-940.json)](https://cairn-commons.com/problems/erdos-940)
Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

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.

  1. Get a task — one click
  2. Your chatbot works on it — 5–10 minutes, in its own tab
  3. 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.

or , ,

ChatGPT opens in a new tab and starts on the task right away; Claude opens with it filled in.

Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Let . Is it true that the set of integers which are the sum of at most -powerful numbers has density ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«940».

theorem erdos_940 :
    ∀ r ≥ 3,
      {n : ℕ | ∃ (S : Multiset ℕ), S.card ≤ r ∧ (∀ s ∈ S, r.Full s) ∧ n = S.sum}.HasDensity 0

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/940. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_940.variants.three_cubes — Is it true that the set of integers which are the sum of at most three cubes has density 0?
  • erdos_940.variants.large_integers — Let r ≥ 3. Are there infinitely many integers which are not the sum of at most r-many r-powerful numbers?

References

  • erdosproblems.com/940
  • [BaBr94] Baker, R. C. and Brüdern, J., _On sums of two squarefull numbers_. Math. Proc. Cambridge Philos. Soc. (1994), 1-5.
  • [He88] Heath-Brown, D. R., _Ternary quadratic forms and sums of three square-full numbers_. (1988), 137-163.

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.