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Level A · Machine-checkable Hard Number theory P-erdos-323

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'.

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.

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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 2026-09-28}
}

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Claims
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Verified
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Disputed
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Refuted
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On the literature board
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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). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_323.parts.i :
    answer(sorry) ↔ ∀ 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 :
    answer(sorry) ↔ ∀ 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

erdosproblems.com/323

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.