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Level A · Machine-checkable Hard Number theory P-erdos-1101

Erdős Problem #1101

1. There is NO good sequence with polynomial growth.

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-1101,
  title        = {Erdős Problem #1101},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1101}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims
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Verified
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Disputed
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Refuted
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On the literature board
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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_1101.parts.i. 1. There is NO good sequence with polynomial growth.

erdos_1101.parts.ii. 2. There is a good sequence with sub-exponential growth.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1101» (2 statements).

theorem erdos_1101.parts.i :
    ¬ ∃ u, IsGood u ∧ ∃ k : ℕ, (fun n => (u n : ℝ)) =O[atTop] (fun n => (n : ℝ) ^ k)
theorem erdos_1101.parts.ii :
    ∃ u, IsGood u ∧ (fun n => Real.log (u n : ℝ)) =o[atTop] (fun n => (n : 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/1101. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1101

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.