# 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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## 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
- 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 →
      ¬ IsAdditiveComplementToPrimes A := by
  intro A M hAM hA
  rw [IsAdditiveComplementToPrimes, Filter.eventually_atTop] at hA
  obtain ⟨N, hN⟩ := hA
  let m := max (M + 2) N
  let n := m.factorial + M + 2
  have hm2 : 2 ≤ m := by
    dsimp [m]
    omega
  have hmN : N ≤ m := by
    dsimp [m]
    omega
  have hmfac : m ≤ m.factorial :=
    Nat.le_of_dvd (Nat.factorial_pos m)
      (Nat.dvd_factorial (by omega) (le_refl m))
  have hnN : N ≤ n := by
    dsimp [n]
    omega
  obtain ⟨p, hp, a, haA, hn⟩ := hN n hnN
  have haM : a ≤ M := hAM haA
  let d := M + 2 - a
  have hd2 : 2 ≤ d := by
    dsimp [d]
    omega
  have hdm : d ≤ m := by
    dsimp [d, m]
    omega
  have hdp : d ∣ p := by
    have hdfac : d ∣ m.factorial := Nat.dvd_factorial (by omega) hdm
    have hp_eq : p = m.factorial + d := by
      dsimp [n] at hn
      dsimp [d]
      omega
    rw [hp_eq]
    exact dvd_add hdfac (dvd_refl d)
  have hd_eq : d = 1 ∨ d = p := hp.eq_one_or_self_of_dvd d hdp
  rcases hd_eq with hd1 | hdp_eq
  · omega
  · have hmfac_pos : 0 < m.factorial := Nat.factorial_pos m
    have hp_eq : p = m.factorial + d := by
      dsimp [n] at hn
      dsimp [d]
      omega
    omega

/-- Every additive complement to the primes is infinite. -/
theorem proof :
    ∀ A : Set ℕ, IsAdditiveComplementToPrimes A → A.Infinite := by
  intro A hA hAfin
  obtain ⟨M, hM⟩ := hAfin.bddAbove
  exact bounded_not_complement A M (fun _ ha => hM ha) hA

end Submissions.Erdos32ComplementInfinite.BoundedComplement
```

- Canonical statement

```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
- 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
  exact ⟨p, hp, a, hAB ha, hn⟩

end Submissions.Erdos32ComplementMonotone.Direct
```

- Canonical statement

```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

```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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