OP-6967Status unassessedNumber theory

Weisenberg's variant of Erdős problem 891 on prime factor counts in short intervals

TheoremDB P8626 · 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 $p_1, p_2, \ldots$ denote the sequence of prime numbers in increasing order, so that $p_1 = 2$ . For an integer $m \geq 1$ , let $\omega(m)$ denote the number of distinct prime factors of $m$ . For an integer $k \geq 2$ , let $P_k = p_1 p_2 \cdots p_k = \prod_{i=0}^{k-1} p_{i+1}$ . Determine whether there exist infinitely many positive integers $n$ such that every integer $m$ in the interval $[n, n + P_k - 1)$ satisfies $\omega(m) \leq k$ .

Lean 4

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Formal statement

Lean 4

theorem erdos_891.variants.weisenberg (k : ℕ) (hk : k ≥ 2) :
    ∃ᶠ n in atTop,
      ∀ m ∈ Ico n (n + (∏ i ∈ range k, i.nth Nat.Prime) - 1),
      ω m ≤ k

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Progress & source assessment

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

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References

[1]TheoremDB P8626 link
[2]Source formal statement (not locally verified) link

Recorded progress

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Discussion

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