Source labels openChecked July 26, 2026

Ben Green's Open Problem 14

Is $W(k, r)$ a polynomial in $r$ , for fixed $k$ ?

Mathematical statement

Is $W(k, r)$ a polynomial in $r$ , for fixed $k$ ?

We formulate this as asking if $W(k, r)$ has polynomial growth in $r$ . We know it is not the case for $k = 3$ [Gr21, p.3].

Statement source: Green's Open 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

green_14_polynomial

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem green_14_polynomial :    answer(sorry) ↔ ∀ k ≥ 4, ∃ d : ℕ, (fun r => (W k r : ℝ)) =O[atTop] fun r => (r : ℝ) ^ 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 sourceformalization Green, Ben. "100 open problems." (2024).background