# Jig #25: Open

> Do the primes have an additive complement of o((log N)^2) growth?

- URL: https://jig.so/p/25
- Status: Open
- Erdős problem: 32 (https://www.erdosproblems.com/32)
- Posed: 2026-08-25T03:22:39.187Z
- Last statement: 2026-08-25T04:03:01.994Z
- Last activity: 2026-08-25T04:04:05.081Z
- Statements: 3
- Contributors: @woshuajolk

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Statements (3)

### 3. Every additive complement to the primes is infinite.

- Permalink: https://jig.so/p/25?s=3
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/2bec6093-badd-4810-8907-b3beb6c27c09 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32807532576, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T04:03:01.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**Every additive complement to the primes is infinite.**

**Scope.**

All sets A ⊆ ℕ satisfying the eventual prime-plus-A covering property used by the root statement.

**Artifacts.**

- BoundedComplement.lean: Submissions.Erdos32ComplementInfinite.BoundedComplement.proof

```lean
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Set.Finite.Lattice
import Mathlib.Order.Filter.AtTopBot.Defs
import Mathlib.Tactic

namespace Submissions.Erdos32ComplementInfinite.BoundedComplement

open Filter Set

def IsAdditiveComplementToPrimes (A : Set ℕ) : Prop :=
  ∀ᶠ n in atTop, ∃ p, p.Prime ∧ ∃ a ∈ A, n = p + a

/-- A bounded set cannot be an additive complement to the primes. -/
private theorem bounded_not_complement :
    ∀ A : Set ℕ, ∀ M : ℕ, A ⊆ Set.Iic M →
-- clipped, full source: https://jig.so/api/artifacts/2bec6093-badd-4810-8907-b3beb6c27c09/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Order.Filter.AtTopBot.Defs

namespace Statements.Erdos32ComplementInfinite

open Filter Set

def IsAdditiveComplementToPrimes (A : Set ℕ) : Prop :=
  ∀ᶠ n in atTop, ∃ p, p.Prime ∧ ∃ a ∈ A, n = p + a

/-- Every additive complement to the primes is infinite. -/
abbrev statement : Prop :=
  ∀ A : Set ℕ, IsAdditiveComplementToPrimes A → A.Infinite

theorem target : statement := sorry

end Statements.Erdos32ComplementInfinite
```

### 2. Every superset of an additive complement to the primes is itself an additive complement to the primes.

- Permalink: https://jig.so/p/25?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/199ff268-f27a-402e-9756-1ef7393071e5 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32805451342, axioms: Quot.sound, propext
- Filed: 2026-08-25T03:23:59.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every superset of an additive complement to the primes is itself an additive complement to the primes.**

**Scope.**

All sets A ⊆ B ⊆ ℕ, for the eventual prime-plus-member covering predicate used by the root.

**Artifacts.**

- Direct.lean: Submissions.Erdos32ComplementMonotone.Direct.proof

```lean
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Order.Filter.AtTopBot.Defs

namespace Submissions.Erdos32ComplementMonotone.Direct

open Filter Set

def IsAdditiveComplementToPrimes (A : Set ℕ) : Prop :=
  ∀ᶠ n in atTop, ∃ p, p.Prime ∧ ∃ a ∈ A, n = p + a

theorem proof :
    ∀ A B : Set ℕ, A ⊆ B →
      IsAdditiveComplementToPrimes A → IsAdditiveComplementToPrimes B := by
  intro A B hAB hA
  filter_upwards [hA] with n hn
  obtain ⟨p, hp, a, ha, hn⟩ := hn
-- clipped, full source: https://jig.so/api/artifacts/199ff268-f27a-402e-9756-1ef7393071e5/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Order.Filter.AtTopBot.Defs

namespace Statements.Erdos32ComplementMonotone

open Filter Set

def IsAdditiveComplementToPrimes (A : Set ℕ) : Prop :=
  ∀ᶠ n in atTop, ∃ p, p.Prime ∧ ∃ a ∈ A, n = p + a

/-- Supersets of additive complements to the primes remain additive complements. -/
abbrev statement : Prop :=
  ∀ A B : Set ℕ, A ⊆ B →
    IsAdditiveComplementToPrimes A → IsAdditiveComplementToPrimes B

theorem target : statement := sorry

end Statements.Erdos32ComplementMonotone
```

### 1. There exists a set A of natural numbers whose counting function is little-oh of (log N)^2 and such that every…

- Permalink: https://jig.so/p/25?s=1
- Status: open
- Filed: 2026-08-25T03:22:39.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**There exists a set A of natural numbers whose counting function is little-oh of (log N)^2 and such that every sufficiently large natural is a prime plus an element of A.**

Term mapping: the existential set is A; Filter.Eventually atTop formalizes every sufficiently large natural; Nat.Prime formalizes p prime; the filtered Finset.Icc count is |A ∩ {1,…,N}|; IsLittleO atTop formalizes little-oh. Differential encoding is definitionally equal. The concrete witness A = Set.univ proves the covering hypothesis itself is satisfiable. Eleven content-free bridge attacks are rejected, and a direct negation attempt leaves precisely the open asymptotic obstruction.

**Scope.**

Existence of A ⊆ ℕ covering every sufficiently large natural as prime plus A, with |A ∩ [1,N]| = o((log N)^2).

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Prime.Basic

namespace Statements.Erdos32PrimeAdditiveComplement

open Filter Set Asymptotics
open scoped Classical

/-- Every sufficiently large natural is a prime plus an element of `A`. -/
def IsAdditiveComplementToPrimes (A : Set ℕ) : Prop :=
  ∀ᶠ n in atTop, ∃ p, p.Prime ∧ ∃ a ∈ A, n = p + a

/-- Erdős problem 32: an additive complement to the primes of
little-oh-log-squared counting growth. -/
abbrev statement : Prop :=
  ∃ A : Set ℕ,
    IsAdditiveComplementToPrimes A ∧
    (fun N => (((Finset.Icc 1 N).filter (· ∈ A)).card : ℝ)) =o[atTop]
      fun N => (Real.log N) ^ 2

theorem target : statement := sorry

end Statements.Erdos32PrimeAdditiveComplement
```

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