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Erdős ProblemsNumber theory

Erdős Problem 873

Let A={a1<a2<… }⊆NA = \{a_1 < a_2 < \dots\} \subseteq \mathbb{N} and let F(A,X,k)F(A,X,k) count the number of ii such that [ai,ai+1,…,ai+k−1]<X[a_i,a_{i+1}, \dots ,a_{i+k−1}] < X, where the left-hand side is the least common multiple. Is it true that, for every ϵ>0\epsilon > 0, there exists s...

Number theoryAMS 11

Mathematical statement

Let A={a1<a2<… }⊆NA = \{a_1 < a_2 < \dots\} \subseteq \mathbb{N} and let F(A,X,k)F(A,X,k) count the number of ii such that [ai,ai+1,…,ai+k−1]<X[a_i,a_{i+1}, \dots ,a_{i+k−1}] < X, where the left-hand side is the least common multiple. Is it true that, for every ϵ>0\epsilon > 0, there exists some kk such that F(A,X,k)<XϵF(A,X,k) < X^\epsilon?

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_873

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem erdos_873 : answer(sorry) ↔ ∀ᵉ (a : ℕ → ℕ) (ε > (0 : ℝ)), 0 < a 0 → StrictMono a →    ∃ k, ∀ X > 0, F a X k < (X^ε).toEReal := 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/873background