Erdős Problem #323
Is it true that f_k,k(x) ≫_ε x^1-ε for all ε>0? This would have significant applications to Waring's problem. Erdős and Graham describe this as 'unattackable by the methods at our disposal'.
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #323 as open (status updated 31 August 2025).
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #323 on erdosproblems.com | open 2025-08-31 | erdosproblems.com |
| Formal Conjectures: research statements in the file | 3 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.
Cite
@misc{cairn-erdos-323,
title = {Erdős Problem #323},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/erdos-323}},
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-323)
- 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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Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
erdos_323.parts.i. Is it true that for all ?
This would have significant applications to Waring's problem. Erdős and Graham describe this as 'unattackable by the methods at our disposal'.
erdos_323.parts.ii. Is it true that if then for sufficiently large ?
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«323» (2 statements).
theorem erdos_323.parts.i :
∀ k ≥ 1, ∀ ε > (0 : ℝ),
(fun (x : ℕ) ↦ (x : ℝ) ^ (1 - ε)) =O[atTop] (fun (x : ℕ) ↦ (f k k x : ℝ))theorem erdos_323.parts.ii :
∀ k m : ℕ, 1 ≤ m → m < k →
(fun (x : ℕ) ↦ (x : ℝ) ^ ((m : ℝ) / (k : ℝ))) =O[atTop] (fun (x : ℕ) ↦ (f k m x : ℝ))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/323. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
Variants
erdos_323.variants.k_gt_2— For k>2 it is not known if f_k,k(x)=o(x).
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.