OP-6963Status unassessedNumber theory

Weak Grimm Conjecture

TheoremDB P9760 · Candidate awaiting source review

Inferred from the community field; finer classification has not been reviewed.

Source candidate · not reviewedAs of 2026-09-08Source & evidence ↓

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Problem statement

Let $n$ and $k$ be positive integers such that each of the $k$ consecutive integers $n, n+1, \dots, n+k-1$ is composite. Must the product $n(n+1)\cdots(n+k-1)$ have at least $k$ distinct prime divisors? Equivalently, do there exist $k$ distinct primes $p_0, p_1, \dots, p_{k-1}$ such that each $p_i$ divides at least one of the integers $n, n+1, \dots, n+k-1$ ?

Lean 4

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Formal statement

Lean 4

theorem grimm_conjecture_weak (n k : ℕ) (hn : 1 ≤ n) (hk : 1 ≤ k)
    (h : ∀ i : Fin k, (n + i).Composite) :
    ∃ ps : Fin k ↪ ℕ,  ∀ i : Fin k, (ps i).Prime ∧ ∃ j : Fin k, ps i ∣ (n + j)

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Progress & source assessment

Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.

Open. No signed Lean verification is attached to this problem or its selected resolution.

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References

[1]TheoremDB P9760 link
[2]Source formal statement (not locally verified) link

Recorded progress

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Machine attempts

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Discussion

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