Erdős Problem #208
Let s_1 < s_2 < … be the sequence of squarefree numbers. Is it true that for any ε > 0 and large n, s_n+1 - s_n ≪_ε s_n^ε?
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #208 as open (status updated 31 August 2025).
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #208 on erdosproblems.com | open 2025-08-31 | erdosproblems.com |
| Formal Conjectures: research statements in the file | 3 open, 0 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.
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.
Cite
@misc{cairn-erdos-208,
title = {Erdős Problem #208},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/erdos-208}},
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-208)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Nobody has worked on this problem here yet
Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.
- Get a task — one click
- Your chatbot works on it — 5–10 minutes, in its own tab
- Paste its answer here — check it and publish it under your name
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.
ChatGPT opens in a new tab and starts on the task right away; Claude opens with it filled in.
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
erdos_208.parts.i. Let be the sequence of squarefree numbers. Is it true that for any and large , ?
erdos_208.parts.ii. Let be the sequence of squarefree numbers. Is it true that ?
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«208» (2 statements).
theorem erdos_208.parts.i :
∀ ε > (0 : ℝ), (fun n => (s (n + 1) - s n : ℝ)) =O[atTop] (fun n => (s n : ℝ)^ε)theorem erdos_208.parts.ii : ∃ (c : ℕ → ℝ), (c =o[atTop] (1 : ℕ → ℝ)) ∧ ∀ᶠ n in atTop,
s (n + 1) - s n ≤ (1 + (c n)) * (π^2 / 6) * log (s n) / log (log (s n))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/208. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
Variants
erdos_208.variants.log_bound— In [Er79] Erdős says perhaps s_n+1 - s_n ≪ log s_n, but he is 'very doubtful'. [Er79] Erdős, Paul, __Some unconventional problems in number theory__. Math.
References
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.