Erdős Problem #821
Is it true that, for every ε>0, there exist infinitely many n such that g(n) > n^1-ε?
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #821 as open (status updated 31 August 2025).
- Formal Conjectures marks the Lean statement
erdos_821as solved.
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #821 on erdosproblems.com | open 2025-08-31 | erdosproblems.com |
| Formal Conjectures: status of `erdos_821` | research solved | Formal Conjectures |
| Formal Conjectures: research statements in the file | 0 open, 4 solved | Formal Conjectures |
Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.
Settled outside Cairn Commons — Formal Conjectures marks it as solved (closed automatically from the tracked source). Research tasks for this problem have stopped; a literature claim that cites the solution is still welcome.
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.
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 2026-11-12}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/erdos-821)
- 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
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) }.InfiniteWhat 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.