Source labels openChecked July 26, 2026

Green's Open Problem 44

Sieve $[N]$ by removing half the residue classes mod $p_i$ , for primes $2 \leqslant p_1 < p_2 < \dots < p_{1000} < N^{9/10}$ . Does the remaining set have size at most $\frac{1}{10} N$ ?

Number theory AMS 11

Mathematical statement

Sieve $[N]$ by removing half the residue classes mod $p_i$ , for primes $2 \leqslant p_1 < p_2 < \dots < p_{1000} < N^{9/10}$ . Does the remaining set have size at most $\frac{1}{10} N$ ?

We interpret "half the residue classes" as $\lfloor p_i / 2 \rfloor$ .

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_44

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem green_44 :    answer(sorry) ↔ ∀ (N : ℕ) (p : Fin 1000 → ℕ) (A : (i : Fin 1000) → Finset (ZMod (p i))),      let remaining := (Finset.Icc 1 N).filter (fun x => ∀ i, (x : ZMod (p i)) ∉ A i)      (∀ i, (p i).Prime) →      StrictMono p →      (p 999) ^ 10 < N ^ 9 →      (∀ i, (A i).card = (p i) / 2) →      10 * remaining.card ≤ N := 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 100 Open Problemsbackground