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Level A · Machine-checkable Hard Combinatorics P-erdos-241

Erdős Problem #241

Is it true that f(N)∼ N^1/3? Originally asked to Erdős by Bose. This is discussed in problem C11 of Guy's collection [Gu04].

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-241,
  title        = {Erdős Problem #241},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-241}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

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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 ?

Originally asked to Erdős by Bose.

This is discussed in problem C11 of Guy's collection [Gu04].

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«241». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_241 :
    answer(sorry) ↔ (fun N ↦ (f N 3 : ℝ)) ~[atTop] (fun N ↦ (N : ℝ) ^ ((1 : ℝ) / 3))

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/241. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_241.variants.generalization — More generally, Bose and Chowla [BoCh62] conjectured that the maximum size of A⊆ 1,…,N with all r-fold sums distinct (aside from the trivial coincidences) then…

References

  • erdosproblems.com/30
  • erdosproblems.com/241
  • [BoCh62] Bose, R. C. and Chowla, S., Theorems in the additive theory of numbers. Comment. Math. Helv. (1962/63), 141-147.
  • [Gr01] Green, Ben, The number of squares and {} sets. Acta Arith. (2001), 365-390.
  • [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
  • [Wh24] White, Ethan Patrick, An optimal autoconvolution inequality. Canad. Math. Bull. 67 (2024), 108-121. doi:10.4153/S0008439523000565

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.