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Level A · Machine-checkable Hard Number theory P-erdos-365

Erdős Problem #365

Is the number of positive n≤ x such that n and n+1 are both powerful bounded by (log x)^O(1)?

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-09-20).

Current state (tracked automatically)

  • erdosproblems.com lists Erdős problem #365 as open (status updated 20 September 2025).
  • Formal Conjectures marks the Lean statement erdos_365.parts.ii as open.
All tracked facts and sources
FactValueSource
Status of Erdős problem #365 on erdosproblems.comopen
2025-09-20
erdosproblems.com
Formal Conjectures: status of `erdos_365.parts.ii`research openFormal Conjectures
Formal Conjectures: research statements in the file1 open, 1 solvedFormal 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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

Try it with your chatbot Submit a claim Follow (RSS)
Cite 
@misc{cairn-erdos-365,
  title        = {Erdős Problem #365},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-365}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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On the literature board
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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.

  1. Get a task — one click
  2. Your chatbot works on it — 5–10 minutes, in its own tab
  3. 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.

or , ,

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

Is the number of positive such that and are both powerful bounded by ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«365».

theorem erdos_365.parts.ii :
    ∃ C : ℝ, 0 < C ∧ ∃ k : ℕ, ∀ᶠ x : ℕ in Filter.atTop,
      ({n : ℕ | 0 < n ∧ n ≤ x ∧ Nat.Powerful n ∧ Nat.Powerful (n + 1)}.ncard : ℝ) ≤
        C * (Real.log x) ^ k

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/365. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/365
  • [Go70] Golomb, S. W., Powerful numbers. Amer. Math. Monthly (1970), 848-855.
  • [Gu04] Guy, R. K., Unsolved problems in number theory (2004), Problem B16.
  • [Wa76] Walker, D. T., *Consecutive integer pairs of powerful numbers and related Diophantine equations*. Fibonacci Quart. (1976), 111-116.

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.