# 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
- 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

namespace Submissions.E273SieveMomentBarrier.CompactBarrier

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

lemma weight_one_le (δ : ℕ → ℝ) (hδ : ∀ q, 0 ≤ δ q ∧ δ q ≤ 1/2) (q : ℕ) :
    1 ≤ (1 - δ q)⁻¹ := by
  have hpos : 0 < 1 - δ q := by linarith [(hδ q).2]
  rw [one_le_inv₀ hpos]
  linarith [(hδ q).1]

lemma distortion_one_le (δ : ℕ → ℝ) (hδ : ∀ q, 0 ≤ δ q ∧ δ q ≤ 1/2) (m : ℕ) :
    1 ≤ distortion δ m := by
  exact Finset.one_le_prod (fun q _ => weight_one_le δ hδ q)

lemma distortion_subset (δ : ℕ → ℝ) (hδ : ∀ q, 0 ≤ δ q ∧ δ q ≤ 1/2)
    (m : ℕ) (s : Finset ℕ) (hs : s ⊆ m.primeFactors) :
    (∏ q ∈ s, (1 - δ q)⁻¹) ≤ distortion δ m := by
  exact Finset.prod_le_prod_of_subset_of_one_le hs
    (fun q _ => le_trans zero_le_one (weight_one_le δ hδ q))
    (fun q _ _ => weight_one_le δ hδ q)

lemma firstBound_nonneg (δ : ℕ → ℝ) (hδ : ∀ q, 0 ≤ δ q ∧ δ q ≤ 1/2) (q : ℕ) :
    0 ≤ firstBound δ q := by
  apply Finset.sum_nonneg
  intro d hd
  exact div_nonneg (le_trans zero_le_one (distortion_one_le δ hδ _)) (Nat.cast_nonneg _)

lemma secondBound_nonneg (δ : ℕ → ℝ) (hδ : ∀ q, 0 ≤ δ q ∧ δ q ≤ 1/2) (q : ℕ) :
    0 ≤ secondBound δ q := by
  apply Finset.sum_nonneg
  intro d hd
  apply Finset.sum_nonneg
  intro e he
  exact div_nonneg (mul_nonneg (le_trans zero_le_one (distortion_one_le δ hδ _))
    (Nat.cast_nonneg _)) (mul_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))

lemma first_subset (δ : ℕ → ℝ) (hδ : ∀ q, 0 ≤ δ q ∧ δ q ≤ 1/2)
    (q : ℕ) (s : Finset ℕ) (hs : s ⊆ group q) :
    (∑ d ∈ s, distortion δ (cofactor d) / (d : ℝ)) ≤ firstBound δ q := by
  apply Finset.sum_le_sum_of_subset_of_nonneg hs
  intro d _ _
  exact div_nonneg (le_trans zero_le_one (distortion_one_le δ hδ _)) (Nat.cast_nonneg _)

lemma second_subset (δ : ℕ → ℝ) (hδ : ∀ q, 0 ≤ δ q ∧ δ q ≤ 1/2)
    (q : ℕ) (s : Finset ℕ) (hs : s ⊆ group q) :
    (∑ d ∈ s, ∑ e ∈ s, distortion δ (Nat.lcm (cofactor d) (cofactor e)) *
      (Nat.gcd (cofactor d) (cofactor e) : ℝ) / ((d : ℝ) * (e : ℝ))) ≤ secondBound δ q := by
  have hn (d e : ℕ) : 0 ≤ distortion δ (Nat.lcm (cofactor d) (cofactor e)) *
      (Nat.gcd (cofactor d) (cofactor e) : ℝ) / ((d : ℝ) * (e : ℝ)) :=
    div_nonneg (mul_nonneg (le_trans zero_le_one (distortion_one_le δ hδ _))
      (Nat.cast_nonneg _)) (mul_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))
  apply le_trans (Finset.sum_le_sum (fun d _ =>
    Finset.sum_le_sum_of_subset_of_nonneg hs (fun e _ _ => hn d e)))
  exact Finset.sum_le_sum_of_subset_of_nonneg hs
    (fun d _ _ => Finset.sum_nonneg (fun e _ => hn d e))

lemma criterion_term_nonneg (δ : ℕ → ℝ) (hδ : ∀ q, 0 ≤ δ q ∧ δ q ≤ 1/2) (q : ℕ) :
    0 ≤ (if δ q = 0 then firstBound δ q else
      min (firstBound δ q) (secondBound δ q / (4 * δ q * (1 - δ q)))) := by
  have hd : 0 ≤ 4 * δ q * (1 - δ q) := by
    apply mul_nonneg (mul_nonneg (by norm_num) (hδ q).1)
    linarith [(hδ q).2]
  split_ifs
  · exact firstBound_nonneg δ hδ q
  · exact le_min (firstBound_nonneg δ hδ q) (div_nonneg (secondBound_nonneg δ hδ q) hd)

lemma criterion_lower_bound (δ : ℕ → ℝ) (hδ : ∀ q, 0 ≤ δ q ∧ δ q ≤ 1/2)
    (Q : Finset ℕ) (hQ : Q ⊆ pool.image largestPrime)
    (a b : ℕ → ℝ) (ha : ∀ q ∈ Q, a q ≤ firstBound δ q)
    (hb : ∀ q ∈ Q, b q ≤ secondBound δ q) :
-- 2644 more lines, see https://jig.so/p/
```

- Canonical statement

```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
- 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
    obtain ⟨c, hc, -⟩ := h 0
    exact ⟨c, hc⟩
  · intro a m hm h
    obtain ⟨c, hc, hdvd⟩ := h (a + 1)
    simp only [Finset.mem_singleton] at hc
    subst c
    norm_num at hdvd
    have hnat : m ∣ 1 := Int.natCast_dvd_natCast.mp (by simpa using hdvd)
    exact (Nat.not_lt_of_ge (Nat.le_of_dvd (by decide) hnat)) hm

end Submissions.Erdos273CoverBoundary.Worker09Direct
```

- Canonical statement

```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

```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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