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Level A · Machine-checkable Hard Number theory P-erdos-1063

Erdős Problem #1063

Estimate n_k by finding a better upper bound than Cambie's n_k ≤ k · lcm(1, dotsc, k-1). The comparator takes its least common multiple in ℕ and casts the result.

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-1063,
  title        = {Erdős Problem #1063},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1063}},
  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

Estimate by finding a better upper bound than Cambie's .

The comparator takes its least common multiple in ℕ and casts the result. Writing the ascription as ((… ).lcm (fun n : ℕ => n) : ℝ) instead puts it on the Finset.lcm application, so the coercion lands on n and the lcm is taken in ℝ, where lcm of non-zero elements is 1 and the whole comparator collapses to k.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1063». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1063.better_upper :
    let upper_bound : ℕ → ℝ := answer(sorry)
    (fun k => (n k : ℝ)) =O[atTop] upper_bound ∧
    upper_bound =o[atTop] fun k =>
      (k : ℝ) * (((Finset.Icc 1 (k - 1)).lcm id : ℕ) : 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/1063. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/1063
  • [ErSe83] Erdos, P. and Selfridge, J. L., Problem 6447. Amer. Math. Monthly (1983), 710.
  • [Gu04] Guy, Richard K., _Unsolved problems in number theory_. (2004), Problem B31.
  • [Mo85] Monier, Jean-Marie, _Problems and Solutions: Solutions of Advanced Problems: 6447_. Amer. Math. Monthly 92 (1985), 435-436.

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.