Erdős Problem #208 — open problem in number theory

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?

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.

Start working on it Submit a claim Follow

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 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

erdos_208.parts.i. Let $s_{1} < s_{2} < \dots$ be the sequence of squarefree numbers. Is it true that for any $ϵ > 0$ and large $n$ , $s_{n + 1} - s_{n} ≪_{ϵ} s_{n}^{ϵ}$ ?

erdos_208.parts.ii. Let $s_{1} < s_{2} < \dots$ be the sequence of squarefree numbers. Is it true that $s_{n + 1} - s_{n} \leq (1 + o (1)) \cdot (π^{2} /6) \cdot lo g (s_{n}) / lo g (lo g (s_{n}))$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«208» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_208.parts.i : answer(sorry) ↔
    ∀ ε > (0 : ℝ), (fun n => (s (n + 1) - s n : ℝ)) =O[atTop] (fun n => (s n : ℝ)^ε)

theorem erdos_208.parts.ii : answer(sorry) ↔ ∃ (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

erdosproblems.com/208

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.