1. FormalConjectures
  2. GreensOpenProblems
  3. «58»
/- Copyright 2026 The Formal Conjectures Authors. Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at https://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. -/ import FormalConjecturesUtil

Ben Green's Open Problem 58

Reference: [Ben Green's Open Problem 58](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8 Problem 58)

open scoped Pointwisenamespace Green58

Suppose $A, B ⊆ {1, \dots, N}$ both have size at least $N^{0.49}$. Must the sumset $A + B$ contain a composite number?

@[category research open, AMS 5 11] theorem declaration uses 'sorry'green_58 : answer(sorry) ↔ ∀ᶠ (N : ℕ) in Filter.atTop, ∀ᵉ (A ⊆ Finset.Icc 1 N) (B ⊆ Finset.Icc 1 N), (N : ℝ) ^ (0.49 : ℝ) ≤ (A.card : ℝ) → (N : ℝ) ^ (0.49 : ℝ) ≤ (B.card : ℝ) → ∃ m ∈ (A + B), m.Composite := ⊢ True ↔ ∀ᶠ (N : ℕ) in Filter.atTop, ∀ A ⊆ Finset.Icc 1 N, ∀ B ⊆ Finset.Icc 1 N, ↑N ^ 0.49 ≤ ↑A.card → ↑N ^ 0.49 ≤ ↑B.card → ∃ m ∈ A + B, m.Composite All goals completed! 🐙 end Green58