Source labels openChecked July 26, 2026

Erdős Problem 897: I

Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$ ) such that $\limsup_{p,k} f(p^k) / \log(p^k) = ∞$ and $f(p^k) = f(p)$ or $f(p^k) = kf(p)$ . Is it true that $\limsup_n (f(n+1)−f(n))/ \log n = ∞$ ?

Number theory AMS 11

Mathematical statement

The known counterexample does not satisfy either of these extra hypotheses, so this variant remains open.

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_897.variants.parts.i

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_897.variants.parts.i : answer(sorry) ↔ ∀ (f : ℕ → ℝ),    (∀ᵉ (a > 0) (b > 0), a.Coprime b → f (a * b) = f a + f b) →    ((Filter.atTop ⊓ Filter.principal {(p, k) : ℕ × ℕ | p.Prime}).limsup      (fun (p, k) => (f (p^k) / (p^k : ℝ).log : EReal)) = ⊤) →    (∀ k p, p.Prime → f (p^k) = f p) ∨ (∀ (k p : ℕ), p.Prime → f (p^k) = k*f p) →    Filter.atTop.limsup (fun (n : ℕ) => ((f (n+1) - f n) / (n : ℝ).log : EReal)) = ⊤ := 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/897background An Additive Counterexample: Erdős Problem 897background Lean formalisation of Erdős problem 897background