Skip to content
Cairn Commons
X Sign in
Level A · Machine-checkable Hard Number theory P-erdos-422

Erdős Problem #422

Let f(1) = f(2) = 1 and for n > 2 f(n) = f(n - f(n - 1)) + f(n - f(n - 2)). Does f(n) miss infinitely many integers?

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.

Start working on it Submit a claim Follow
Cite 
@misc{cairn-erdos-422,
  title        = {Erdős Problem #422},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-422}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Let and for Does miss infinitely many integers?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«422». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_422 : answer(sorry) ↔
    ∀ f : ℕ+ → ℕ+, IsHofstadterQ f → Set.Infinite {n | ∀ x, f x ≠ n}

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/422. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_422.variants.surjective — Is f surjective?
  • erdos_422.variants.growth_rate — How does f grow?
  • erdos_422.variants.eventually_const — Does f become stationary at some point?

References

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.