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Level A · Machine-checkable Hard Number theory P-brocard-conjecture

Brocard's Conjecture

Brocard's Conjecture For every n ≥ 2, between the squares of the n-th and (n+1)-th primes, there are at least four prime numbers.

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-brocard-conjecture,
  title        = {Brocard's Conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/brocard-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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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

Brocard's Conjecture For every n ≥ 2, between the squares of the n-th and (n+1)-th primes, there are at least four prime numbers.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.BrocardConjecture.

theorem brocard_conjecture (n : ℕ) (hn : 1 ≤ n) :
    letI prev := n.nth Nat.Prime;
    letI next := (n+1).nth Nat.Prime;
    4 ≤ ((Ioo (prev^2) (next^2)).filter Nat.Prime).card

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

References

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.