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Level A · Machine-checkable Hard Number theory P-erdos-821

Erdős Problem #821

Is it true that, for every ε>0, there exist infinitely many n such that g(n) > n^1-ε?

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-821,
  title        = {Erdős Problem #821},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-821}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

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

Is it true that, for every , there exist infinitely many such that ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«821». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_821 :
    answer(sorry) ↔ ∀ ε > (0 : ℝ), { n : ℕ | (g n : ℝ) > (n : ℝ) ^ (1 - ε) }.Infinite

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/821. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/821
  • [BaHa98] Baker, R. C. and Harman, G., Shifted primes without large prime factors. Acta Arith. (1998), 331--361.
  • [Er35b] Erdős, P., On the normal number of prime factors of and some related problems concerning Euler's -function. Quart. J. Math. (1935), 205-213.
  • [Er74b] Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.
  • [Li22] J. D. Lichtman, Primes in arithmetic progressions to large moduli and shifted primes without large prime factors. arXiv:2211.09641 (2022).
  • [LuPo11] Luca, Florian and Pollack, Paul, An arithmetic function arising from {C}armichael's conjecture. J. Théor. Nombres Bordeaux (2011), 697--714.

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.