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Level A · Machine-checkable Hard Number theory P-erdos-878

Erdős Problem #878

Find an asymptotic formula for the number of n<x such that f(n)=F(n).

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

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-878,
  title        = {Erdős Problem #878},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-878}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

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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_878.parts.iv. Find an asymptotic formula for the number of such that .

erdos_878.parts.v. Find an asymptotic formula for .

erdos_878.parts.vi. Is it true that ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«878» (3 statements).

theorem erdos_878.parts.iv :
    (fun x : ℕ ↦
      (((Finset.Ico 1 x).filter fun n ↦ f n = F n).card : ℝ))
      ~[atTop] erdos_878.parts.iv.answer
theorem erdos_878.parts.v :
    H ~[atTop] erdos_878.parts.v.answer
theorem erdos_878.parts.vi :
    H =O[atTop] fun x : ℕ ↦
        (x : ℝ) * Real.log (Real.log (Real.log (Real.log (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/878. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/878
  • [Er84e] Erdős, P., _On two unconventional number theoretic functions and on some related problems_. (1984), 113--121.
  • [Ki26] Kitamura, K., _Erdős Problem #878 in Lean_, GitHub repository (2026).

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.