Source labels openChecked July 26, 2026

Ben Green's Open Problem 31: Upper Eventually

Can we improve the upper bound $N^{1/2} + 0.98183 N^{1/4} + O(1)$ [CHO25], for all sufficiently large $N$ ?

Combinatorics AMS 5 AMS 11

Mathematical statement

Can we improve the upper bound $N^{1/2} + 0.98183 N^{1/4} + O(1)$ [CHO25], for all sufficiently large $N$ ?

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_31.variants.upper_eventually

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem green_31.variants.upper_eventually :    let ans := (answer(sorry) : ℕ → ℝ)    (∀ᶠ N in atTop, F N ≤ ans N) ∧    ∃ c < (0.98183 : ℝ), ∃ C : ℝ, ∀ᶠ N in atTop, ans N - Real.sqrt (N : ℝ) ≤ c * (N : ℝ) ^ (4⁻¹ : ℝ) + C := 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 Ben Green's Open Problem 31background