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Level A · Machine-checkable Hard Number theory P-erdos-30

Erdős Problem #30

Is it true that, for every ε > 0, h(N) = sqrt N + O_ε(N^ε)

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-30,
  title        = {Erdős Problem #30},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-30}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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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

Is it true that, for every ,

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«30». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_30 : answer(sorry) ↔
    ∀ᵉ (ε > 0), (fun N => h N - (N : Real).sqrt) =O[atTop] fun N => (N : ℝ)^(ε : 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/30. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_30.variants.O_one — A stronger conjecture: is it true that h(N) = sqrt N + O(1)? Erdős thought this was perhaps too optimistic.

References

  • erdosproblems.com/30
  • [ErTu41] Erdős, P. and Turán, P., *On a problem of Sidon in additive number theory, and on some related problems*. J. London Math. Soc. 16 (1941), 212-215.
  • [Li69] Lindström, B., An inequality for -sequences. J. Combinatorial Theory 6 (1969), 211-212.
  • [Si38] Singer, J., *A theorem in finite projective geometry and some applications to number theory*. Trans. Amer. Math. Soc. 43 (1938), 377-385.
  • [BFR23] Balogh, J., Füredi, Z. and Roy, S., An upper bound on the size of Sidon sets. Amer. Math. Monthly 130 (2023), 437-445.
  • [OB22] O'Bryant, K., On the size of finite Sidon sets. arXiv:2207.07800 (2022).
  • [CHO25] Carter, D., Hunter, Z. and O'Bryant, K., On the diameter of finite Sidon sets. Acta Math. Hungar. 175 (2025), 108-126.

See also Ben Green's Open Problem 31 (formalised in FormalConjectures/GreensOpenProblems/31.lean).

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.