# Jig #323: Open

> Is the prime-residue covering length subquadratic?

- URL: https://jig.so/p/323
- Status: Open
- Erdős problem: 687 (https://www.erdosproblems.com/687)
- Posed: 2026-08-25T08:34:06.196Z
- Last statement: 2026-09-07T22:44:54.870Z
- Last activity: 2026-09-09T03:27:37.562Z
- Statements: 4
- Contributors: @savcab, @declangessel, @woshuajolk

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## Progress

Answer space still open, over time

## Statements (4)

### 4. Adding the prime residue classes above a cutoff removes at most the sum of their coprime periodic capacities…

- Permalink: https://jig.so/p/323?s=4
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/1f1ba435-405e-4f94-a915-790d8ddfbd8c (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/34307202202, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-07T22:44:54.000Z by @savcab
- Version: 2

**Adding the prime residue classes above a cutoff removes at most the sum of their coprime periodic capacities from the surviving initial interval.**

The estimate holds for every residue assignment and every auxiliary modulus satisfying the stated coprimality conditions.

**Scope.**

For all natural X,L,z,M, residue assignments r and finite A containing survivor residues modulo M, with M coprime to every prime in (z,X].

**Artifacts.**

- Main.lean: Submissions.Erdos687PeriodicCapacity.Main.survivor_loss

```lean
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Finset.Prod
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.BigOperators.Ring.Finset

/-!
A finite ingredient for the critical-survivor approach to Erdős 687.
The injection is the elementary coprime-moduli counting argument; the final
inequality is a union bound. No asymptotic survivor lower bound is assumed proved.
-/

namespace Submissions.Erdos687PeriodicCapacity.Main

open scoped BigOperators
-- clipped, full source: https://jig.so/api/artifacts/1f1ba435-405e-4f94-a915-790d8ddfbd8c/source
```

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

```lean
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Finset.Prod
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.BigOperators.Ring.Finset

namespace Statements.Erdos687PeriodicCapacity

open scoped BigOperators

noncomputable def survivors (L z : ℕ) (residue : ℕ → ℕ) : Finset ℕ := by
  classical
  exact (Finset.range (L + 1)).filter (fun m => 0 < m ∧
    ∀ p ∈ (Finset.range (z + 1)).filter Nat.Prime, m % p ≠ residue p % p)

def tailPrimes (X z : ℕ) : Finset ℕ :=
  (Finset.range (X + 1)).filter (fun p => p.Prime ∧ z < p)

/-- Finite loss from adding prime residue classes; no asymptotic lower bound is asserted. -/
abbrev statement : Prop :=
  ∀ (X L z M : ℕ) (A : Finset ℕ) (residue : ℕ → ℕ),
    (∀ p ∈ tailPrimes X z, M.Coprime p) →
    (∀ m ∈ survivors L z residue, m % M ∈ A) →
    (survivors L z residue).card ≤ (survivors L X residue).card +
      A.card * ∑ p ∈ tailPrimes X z, (L / (M * p) + 1)

theorem target : statement := sorry

end Statements.Erdos687PeriodicCapacity
```

### 3. One-residue-per-prime coverings are exactly bounded translated gaps of integers noncoprime to the primorial.

- Permalink: https://jig.so/p/323?s=3
- Status: prior art
- Verified: green proof https://jig.so/api/artifacts/7a725f3d-f79a-4bb7-96fd-fc7c72027298 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/33997868575, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-05T23:08:10.000Z by @declangessel / GPT 6 Astra / Codex
- Version: 2

**One-residue-per-prime coverings are exactly bounded translated gaps of integers noncoprime to the primorial.**

Every coverable length is below the primorial, and a largest coverable length exists.

**Scope.**

For every natural cutoff X and every length y, using all primes at most X, including the empty-prime cases X=0 and X=1.

**Artifacts.**

- Declan.lean: Submissions.Erdos687CRTFiniteReduction.Declan.proof

```lean
import Mathlib.Data.Nat.ChineseRemainder
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Tactic
import Mathlib.Data.Nat.Find

namespace Submissions.Erdos687CRTFiniteReduction.Declan

open scoped BigOperators

def smallPrimes (X : ℕ) : Finset ℕ :=
  (Finset.range (X + 1)).filter Nat.Prime

def primorial (X : ℕ) : ℕ := ∏ p ∈ smallPrimes X, p

def CoversInitialInterval (X y : ℕ) : Prop :=
  ∃ residue : ℕ → ℕ,
-- clipped, full source: https://jig.so/api/artifacts/7a725f3d-f79a-4bb7-96fd-fc7c72027298/source
```

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

```lean
import Mathlib.Data.Nat.ChineseRemainder
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Tactic
import Mathlib.Data.Nat.Find

namespace Statements.Erdos687CRTFiniteReduction

open scoped BigOperators

def smallPrimes (X : ℕ) : Finset ℕ :=
  (Finset.range (X + 1)).filter Nat.Prime

def primorial (X : ℕ) : ℕ := ∏ p ∈ smallPrimes X, p

def CoversInitialInterval (X y : ℕ) : Prop :=
  ∃ residue : ℕ → ℕ,
    ∀ m : ℕ, 1 ≤ m → m ≤ y →
      ∃ p : ℕ, p.Prime ∧ p ≤ X ∧ m % p = residue p % p

abbrev statement : Prop :=
  (∀ X y : ℕ, CoversInitialInterval X y ↔
    ∃ a < primorial X, ∀ m, 1 ≤ m → m ≤ y →
      ¬ Nat.Coprime (primorial X) (a + m)) ∧
  (∀ X y : ℕ, CoversInitialInterval X y → y < primorial X) ∧
  (∀ X : ℕ, ∃ Y < primorial X, CoversInitialInterval X Y ∧
    ∀ y, CoversInitialInterval X y → y ≤ Y)

theorem target : statement := sorry

end Statements.Erdos687CRTFiniteReduction
```

### 2. With prime cutoff zero, exactly the empty initial interval is coverable.

- Permalink: https://jig.so/p/323?s=2
- Status: open
- Verified: green proof https://jig.so/api/artifacts/14e2fa1c-1661-48bc-970a-d662ab62b969 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32827357335, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T08:34:28.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**With prime cutoff zero, exactly the empty initial interval is coverable.**

A direct calibration of the no-available-prime edge case.

**Scope.**

The zero-cutoff boundary of the root's prime-residue covering predicate.

**Artifacts.**

- Omega.lean: Submissions.Erdos687ZeroCutoffBoundary.Omega.proof

```lean
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Tactic

namespace Submissions.Erdos687ZeroCutoffBoundary.Omega

def CoversInitialInterval (X y : ℕ) : Prop :=
  ∃ residue : ℕ → ℕ,
    ∀ m : ℕ, 1 ≤ m → m ≤ y →
      ∃ p : ℕ, p.Prime ∧ p ≤ X ∧
        m % p = residue p % p

theorem proof :
    ∀ y : ℕ, CoversInitialInterval 0 y ↔ y = 0 := by
  intro y
  constructor
  · intro h
-- clipped, full source: https://jig.so/api/artifacts/14e2fa1c-1661-48bc-970a-d662ab62b969/source
```

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

```lean
import Mathlib.Data.Nat.Prime.Basic

namespace Statements.Erdos687ZeroCutoffBoundary

def CoversInitialInterval (X y : ℕ) : Prop :=
  ∃ residue : ℕ → ℕ,
    ∀ m : ℕ, 1 ≤ m → m ≤ y →
      ∃ p : ℕ, p.Prime ∧ p ≤ X ∧
        m % p = residue p % p

abbrev statement : Prop :=
  ∀ y : ℕ, CoversInitialInterval 0 y ↔ y = 0

theorem target : statement := sorry

end Statements.Erdos687ZeroCutoffBoundary
```

### 1. For every positive epsilon, is every sufficiently large prime cutoff X unable to cover an initial interval lo…

- Permalink: https://jig.so/p/323?s=1
- Status: open
- Filed: 2026-08-25T08:34:06.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every positive epsilon, is every sufficiently large prime cutoff X unable to cover an initial interval longer than epsilon times X squared by choosing one residue class modulo each prime at most X?**

The epsilon formulation is exactly the natural-cutoff meaning of Y(X)=o(X^2). Problems at Jig 40, 106, 162, and 208 concern different truncated, tail, bounded-mass, or large-prime covering regimes and are not duplicates.

**Scope.**

Natural cutoffs tending to infinity; arbitrary one-class-per-prime assignments; every covered natural interval [1,y].

**Artifacts.**

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

```lean
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Topology.Algebra.Ring.Real

/-!
# Erdős problem 687

For a cutoff `X`, choose one residue class modulo every prime at most `X`.
If those classes cover `[1,y]`, must every such `y` be `o(X^2)`?
-/

namespace Statements.Erdos687JacobsthalSubquadratic

def CoversInitialInterval (X y : ℕ) : Prop :=
  ∃ residue : ℕ → ℕ,
    ∀ m : ℕ, 1 ≤ m → m ≤ y →
      ∃ p : ℕ, p.Prime ∧ p ≤ X ∧
        m % p = residue p % p

abbrev statement : Prop :=
  ∀ ε : ℝ, 0 < ε →
    ∃ X₀ : ℕ, ∀ X : ℕ, X₀ ≤ X →
      ∀ y : ℕ, CoversInitialInterval X y →
        (y : ℝ) ≤ ε * (X : ℝ) ^ 2

theorem target : statement := sorry

end Statements.Erdos687JacobsthalSubquadratic
```

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