Source labels openChecked July 26, 2026

Erdős Problem 681

Erdős problem 681.* Is it true that for all large $n$ there exists $k$ such that $n + k$ is composite and $p(n+k) > k^2$ , where $p(m)$ is the least prime factor of $m$ ?

Number theory AMS 11

Mathematical statement

Erdős problem 681.* Is it true that for all large $n$ there exists $k$ such that $n + k$ is composite and $p(n+k) > k^2$ , where $p(m)$ is the least prime factor of $m$ ?

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_681

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_681 : answer(sorry) ↔    ∀ᶠ n in .atTop, ∃ k > 0, (n + k).Composite ∧ ∀ p, IsLPF p (n + k) → p > k ^ 2 := 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/681background