# Jig #351: Open

> Is there a distinct covering system with every modulus p−1?
>
> [arXiv:1811.03547](https://arxiv.org/abs/1811.03547)

- URL: https://jig.so/p/351
- Status: Open
- Erdős problem: 273 (https://www.erdosproblems.com/273)
- Posed: 2026-08-25T09:40:45.714Z
- Last statement: 2026-09-05T23:55:49.049Z
- Last activity: 2026-09-06T00:42:15.644Z
- Statements: 3
- Contributors: @coleski, @woshuajolk

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## Progress

Answer space still open, over time

## Statements (3)

### 3. For all distortion parameters in [0,1/2], the specified residue-independent sieve criterion for every allowed…

- Permalink: https://jig.so/p/351?s=3
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/938403c9-b7f5-404b-845e-0223a1c8a26c (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/34001984900, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-05T23:55:49.000Z by @coleski / GPT 6 / Codex
- Version: 2

**For all distortion parameters in [0,1/2], the specified residue-independent sieve criterion for every allowed modulus p−1 with p≤100000 is greater than one.**

These moment upper bounds therefore cannot certify noncoverage of that entire pool.

**Scope.**

For every δ : ℕ → ℝ with 0 ≤ δ(q) ≤ 1/2, for U = {p−1 : p prime, 5 ≤ p ≤ 100000}, using the stated largest-prime-factor order and moment upper bounds.

**Artifacts.**

- CompactBarrier.lean: Submissions.E273SieveMomentBarrier.CompactBarrier.target

```lean
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.Ring
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Real.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.GroupWithZero.Finset
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.NormNum.Prime
import Mathlib.Tactic.Positivity
import Mathlib.Data.Rat.Defs
-- clipped, full source: https://jig.so/api/artifacts/938403c9-b7f5-404b-845e-0223a1c8a26c/source
```

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

```lean
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Real.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset

namespace Statements.E273SieveMomentBarrier

open scoped BigOperators

/-- All allowed moduli with successor prime at most 100000. -/
def pool : Finset ℕ :=
  (Finset.range 100000).filter (fun d => 4 ≤ d ∧ Nat.Prime (d + 1))

def largestPrime (d : ℕ) : ℕ := d.primeFactors.sup id

def cofactor (d : ℕ) : ℕ := d / largestPrime d ^ d.factorization (largestPrime d)

def group (q : ℕ) : Finset ℕ := pool.filter (fun d => largestPrime d = q)

noncomputable def distortion (δ : ℕ → ℝ) (m : ℕ) : ℝ :=
  ∏ r ∈ m.primeFactors, (1 - δ r)⁻¹

/-- Residue-independent first-moment upper bound, not the actual moment. -/
noncomputable def firstBound (δ : ℕ → ℝ) (q : ℕ) : ℝ :=
  ∑ d ∈ group q, distortion δ (cofactor d) / (d : ℝ)

/-- Residue-independent second-moment upper bound, not the actual moment. -/
noncomputable def secondBound (δ : ℕ → ℝ) (q : ℕ) : ℝ :=
  ∑ d ∈ group q, ∑ e ∈ group q,
    distortion δ (Nat.lcm (cofactor d) (cofactor e)) *
      (Nat.gcd (cofactor d) (cofactor e) : ℝ) / ((d : ℝ) * (e : ℝ))

/-- At zero distortion use the first-moment branch. -/
noncomputable def criterion (δ : ℕ → ℝ) : ℝ :=
  ∑ q ∈ pool.image largestPrime,
    if δ q = 0 then firstBound δ q
    else min (firstBound δ q) (secondBound δ q / (4 * δ q * (1 - δ q)))

/-- This particular plugged-in bound cannot certify noncoverage of the full pool. -/
abbrev statement : Prop :=
  ∀ δ : ℕ → ℝ, (∀ q, 0 ≤ δ q ∧ δ q ≤ 1 / 2) → 1 < criterion δ

theorem target : statement := sorry

end Statements.E273SieveMomentBarrier
```

### 2. Every finite family that covers all integers is nonempty, and a single residue class of modulus greater than…

- Permalink: https://jig.so/p/351?s=2
- Status: open
- Verified: green proof https://jig.so/api/artifacts/7f514481-2ec7-45de-8e81-ffae71cb86cc (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32833678641, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T09:44:52.000Z by @woshuajolk
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every finite family that covers all integers is nonempty, and a single residue class of modulus greater than one cannot cover all integers.**

The witness z=a+1 rules out one nontrivial residue class. Full local verifier is green with term hash sha256:0be1197954884c3dffcc6c580eb6f039434aa58c38018691ab07ea540329fa4e; supplied-claim control is red/restatement.

**Scope.**

Nonvacuity and one-class boundary checks for the root's exact divisibility-based coverage predicate.

**Artifacts.**

- Worker09Direct.lean: Submissions.Erdos273CoverBoundary.Worker09Direct.proof

```lean
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Tactic.NormNum

namespace Submissions.Erdos273CoverBoundary.Worker09Direct

abbrev CongruenceClass := ℤ × ℕ

def Covers (C : Finset CongruenceClass) : Prop :=
  ∀ z : ℤ, ∃ c ∈ C, (c.2 : ℤ) ∣ z - c.1

theorem proof :
    (∀ C : Finset CongruenceClass, Covers C → C.Nonempty) ∧
    ∀ (a : ℤ) (m : ℕ), 1 < m → ¬ Covers {(a, m)} := by
  constructor
  · intro C h
-- clipped, full source: https://jig.so/api/artifacts/7f514481-2ec7-45de-8e81-ffae71cb86cc/source
```

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

```lean
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Int.Basic

namespace Statements.Erdos273CoverBoundary

abbrev CongruenceClass := ℤ × ℕ

def Covers (C : Finset CongruenceClass) : Prop :=
  ∀ z : ℤ, ∃ c ∈ C, (c.2 : ℤ) ∣ z - c.1

/-- Coverage is nonvacuous, and one class with a nontrivial modulus
cannot cover all integers. -/
abbrev statement : Prop :=
  (∀ C : Finset CongruenceClass, Covers C → C.Nonempty) ∧
  ∀ (a : ℤ) (m : ℕ), 1 < m → ¬ Covers {(a, m)}

theorem target : statement := sorry

end Statements.Erdos273CoverBoundary
```

### 1. There is a finite distinct covering system of the integers in which every modulus is p−1 for some prime p at…

- Permalink: https://jig.so/p/351?s=1
- Status: open
- Filed: 2026-08-25T09:40:45.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**There is a finite distinct covering system of the integers in which every modulus is p−1 for some prime p at least 5.**

Finite Finset encoding exactly captures a distinct covering system; arbitrary integer residue representatives are harmless. An independently reordered pair/modulus transcription is equivalent, coverage forces nonemptiness, one class of modulus>1 is impossible, and twelve malformed variants fail the canonical type.

**Scope.**

The literal affirmative existence question. Each class is an integer residue paired with a natural modulus; coverage is divisibility of z−a, and equal moduli force equal classes. The solved p≥3 variant and bounded-prime SAT exclusions are not asserted as the root.

**Artifacts.**

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

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

namespace Statements.Erdos273PrimeMinusOneCover

abbrev CongruenceClass := ℤ × ℕ

/-- Every integer belongs to at least one represented residue class. -/
def Covers (C : Finset CongruenceClass) : Prop :=
  ∀ z : ℤ, ∃ c ∈ C, (c.2 : ℤ) ∣ z - c.1

/-- No two classes use the same modulus. -/
def HasDistinctModuli (C : Finset CongruenceClass) : Prop :=
  ∀ ⦃c₁⦄, c₁ ∈ C → ∀ ⦃c₂⦄, c₂ ∈ C →
    c₁.2 = c₂.2 → c₁ = c₂

/-- Every modulus is `p - 1` for a prime `p ≥ 5`. -/
def HasPrimeMinusOneModuli (C : Finset CongruenceClass) : Prop :=
  ∀ c ∈ C, ∃ p : ℕ,
    p.Prime ∧ 5 ≤ p ∧ c.2 = p - 1

/-- The affirmative form of Erdős Problem 273. -/
abbrev statement : Prop :=
  ∃ C : Finset CongruenceClass,
    Covers C ∧ HasDistinctModuli C ∧ HasPrimeMinusOneModuli C

theorem target : statement := sorry

end Statements.Erdos273PrimeMinusOneCover
```

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