Source labels openChecked July 26, 2026

Erdős Problem 428

Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$ , all of $n-a$ are prime for all $a\in A$ with $0 < a < n$ and $\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?$

Number theory AMS 11

Mathematical statement

Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$ , all of $n-a$ are prime for all $a\in A$ with $0 < a < n$ and $\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?$

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_428

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_428 :    answer(sorry) ↔ ∃ A : Set ℕ,      (∃ᶠ n in atTop, ∀ a ∈ A, 0 < a → a < n → (n - a).Prime) ∧      liminf (fun n ↦ primeDensityRatio A n) atTop > 0 := 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/428background