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Erdős ProblemsNumber theory

Erdős Problem 51

Is there an infinite set A⊂NA \subset \mathbb{N} such that for every a∈Aa \in A, there is an integer n such that ϕ(n)=a\phi(n)=a, and yet if nan_a is the smallest such integer, then naa→∞\frac{n_a}{a} → \infty as a→∞a → ∞?

Number theoryAMS 11

Mathematical statement

Is there an infinite set A⊂NA \subset \mathbb{N} such that for every a∈Aa \in A, there is an integer n such that ϕ(n)=a\phi(n)=a, and yet if nan_a is the smallest such integer, then naa→∞\frac{n_a}{a} → \infty as a→∞a → ∞?

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_51

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem erdos_51 : answer(sorry) ↔ ∃ A : Set ℕ, ∃ n : A → ℕ,      A.Infinite ∧      (∀ a : A, IsLeast (φ ⁻¹' {(a : ℕ)}) (n a)) ∧      Tendsto (fun a : A => (n a : ℝ) / (a : ℝ)) atTop atTop := 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/51background