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Erdős ProblemsNumber theory

Erdős Problem 486

For each n∈Nn \in \mathbb{N} choose some Xn⊆Z/nZX_n \subseteq \mathbb{Z}/n\mathbb{Z}. Let B={m∈N:∀n,m≢x(modn) for all x∈Xn}B = \{m \in \mathbb{N} : \forall n, m \not\equiv x \pmod{n} \text{ for all } x \in X_n\}. Must BB have a logarithmic density?

Number theoryAMS 11

Mathematical statement

For each n∈Nn \in \mathbb{N} choose some Xn⊆Z/nZX_n \subseteq \mathbb{Z}/n\mathbb{Z}. Let B={m∈N:∀n,m≢x(modn) for all x∈Xn}B = \{m \in \mathbb{N} : \forall n, m \not\equiv x \pmod{n} \text{ for all } x \in X_n\}. Must BB have a logarithmic density?

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_486

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem erdos_486 : answer(sorry) ↔    ∀ X : (n : ℕ) → Set (ZMod n), ∃ d, {m : ℕ | ∀ n, (m : ZMod n) ∉ X n}.HasLogDensity d := 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 sourceformalizationerdosproblems.com/486background