Source labels openChecked July 26, 2026

Erdős Problem 891

Let $2=p_1 < p_2 < \cdots$ be the primes and $k\geq 2$ . Is it true that, for all sufficiently large $n$ , there must exist an integer in $[n,n+p_1\cdots p_k)$ with $>k$ many prime factors?

Number theory AMS 11

Mathematical statement

Let $2=p_1 < p_2 < \cdots$ be the primes and $k\geq 2$ . Is it true that, for all sufficiently large $n$ , there must exist an integer in $[n,n+p_1\cdots p_k)$ with $>k$ many prime factors?

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

erdos_891

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_891 :    answer(sorry) ↔      ∀ k ≥ 2, ∀ᶠ n in atTop,      ∃ m ∈ Ico n (n + ∏ i ∈ range k, i.nth Nat.Prime), k < ω m := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization erdosproblems.com/891background