# 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
- 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

theorem class_capacity (L M p a : ℕ) (A S : Finset ℕ)
    (hS : ∀ m ∈ S, m ≤ L ∧ m % M ∈ A) (hcop : M.Coprime p) :
    (S.filter (fun m => m % p = a % p)).card ≤
      A.card * (L / (M * p) + 1) := by
  let T := S.filter (fun m => m % p = a % p)
  have hcard : T.card ≤ (A ×ˢ Finset.range (L / (M * p) + 1)).card := by
    apply Finset.card_le_card_of_injOn (fun m : ℕ => (m % M, m / (M * p)))
    · intro m hm
      have hmS : m ∈ S := (Finset.mem_filter.mp hm).1
      apply Finset.mem_product.mpr
      exact ⟨(hS m hmS).2, Finset.mem_range.mpr
        (Nat.lt_succ_of_le (Nat.div_le_div_right (hS m hmS).1))⟩
    · intro m hm n hn heq
      have hmS := Finset.mem_filter.mp hm
      have hnS := Finset.mem_filter.mp hn
      have hM : m % M = n % M := congrArg Prod.fst heq
      have hdiv : m / (M * p) = n / (M * p) := congrArg Prod.snd heq
      have hp : m % p = n % p := hmS.2.trans hnS.2.symm
      have hmod : m % (M * p) = n % (M * p) :=
        (Nat.modEq_and_modEq_iff_modEq_mul hcop).mp ⟨hM, hp⟩
      calc
        m = m % (M * p) + M * p * (m / (M * p)) :=
          (Nat.mod_add_div m (M * p)).symm
        _ = n % (M * p) + M * p * (n / (M * p)) := by rw [hmod, hdiv]
        _ = n := Nat.mod_add_div n (M * p)
  simpa [T, Finset.card_product] using hcard

theorem proof (L M : ℕ) (A S P : Finset ℕ) (residue : ℕ → ℕ)
    (hS : ∀ m ∈ S, m ≤ L ∧ m % M ∈ A)
    (hcop : ∀ p ∈ P, M.Coprime p)
    (hcover : ∀ m ∈ S, ∃ p ∈ P, m % p = residue p % p) :
    S.card ≤ A.card * ∑ p ∈ P, (L / (M * p) + 1) := by
  have hsub : S ⊆ P.biUnion (fun p => S.filter (fun m => m % p = residue p % p)) := by
    intro m hm
    obtain ⟨p, hp, hmp⟩ := hcover m hm
    exact Finset.mem_biUnion.mpr ⟨p, hp, Finset.mem_filter.mpr ⟨hm, hmp⟩⟩
  calc
    S.card ≤ (P.biUnion (fun p => S.filter (fun m => m % p = residue p % p))).card :=
      Finset.card_le_card hsub
    _ ≤ ∑ p ∈ P, (S.filter (fun m => m % p = residue p % p)).card :=
      Finset.card_biUnion_le
    _ ≤ ∑ p ∈ P, A.card * (L / (M * p) + 1) :=
      Finset.sum_le_sum (fun p hp => class_capacity L M p (residue p) A S hS (hcop p hp))
    _ = A.card * ∑ p ∈ P, (L / (M * p) + 1) := (Finset.mul_sum P _ _).symm

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)

theorem mem_survivors {L z m : ℕ} {residue : ℕ → ℕ} :
    m ∈ survivors L z residue ↔ m ∈ Finset.range (L + 1) ∧ 0 < m ∧
      ∀ p ∈ (Finset.range (z + 1)).filter Nat.Prime, m % p ≠ residue p % p := by
  simp only [survivors, Finset.mem_filter]

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

/-- Apply the finite estimate to the root's exact one-class-per-prime cover. -/
theorem covering_capacity (X L z M : ℕ) (A : Finset ℕ) (residue : ℕ → ℕ)
    (hcover : ∀ m : ℕ, 1 ≤ m → m ≤ L →
      ∃ p : ℕ, p.Prime ∧ p ≤ X ∧ m % p = residue p % p)
    (hcop : ∀ p ∈ tailPrimes X z, M.Coprime p)
    (hA : ∀ m ∈ survivors L z residue, m % M ∈ A) :
    (survivors L z residue).card ≤
      A.card * ∑ p ∈ tailPrimes X z, (L / (M * p) + 1) := by
  apply proof L M A (survivors L z residue) (tailPrimes X z) residue
  · intro m hm
    exact ⟨Nat.le_of_lt_succ (Finset.mem_range.mp (mem_survivors.mp hm).1), hA m hm⟩
  · exact hcop
  · intro m hm
    have hm' := mem_survivors.mp hm
    have hmL : m ≤ L := Nat.le_of_lt_succ (Finset.mem_range.mp hm'.1)
    obtain ⟨p, hp, hpX, hmp⟩ := hcover m hm'.2.1 hmL
    have hzp : z < p := Nat.lt_of_not_ge (fun hpz =>
      hm'.2.2 p (Finset.mem_filter.mpr
        ⟨Finset.mem_range.mpr (Nat.lt_succ_of_le hpz), hp⟩) hmp)
    exact ⟨p, Finset.mem_filter.mpr
      ⟨Finset.mem_range.mpr (Nat.lt_succ_of_le hpX), hp, hzp⟩, hmp⟩

/-- Uniform loss from adding the remaining prime classes, without a cover assumption. -/
theorem survivor_loss (X L z M : ℕ) (A : Finset ℕ) (residue : ℕ → ℕ)
    (hcop : ∀ p ∈ tailPrimes X z, M.Coprime p)
    (hA : ∀ 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) := by
  classical
  let T := (survivors L z residue).filter
    (fun m => ∃ p ∈ tailPrimes X z, m % p = residue p % p)
  have hT : T.card ≤ A.card * ∑ p ∈ tailPrimes X z, (L / (M * p) + 1) := by
    apply proof L M A T (tailPrimes X z) residue
    · intro m hm
      have hmS := (Finset.mem_filter.mp hm).1
      exact ⟨Nat.le_of_lt_succ (Finset.mem_range.mp (mem_survivors.mp hmS).1),
        hA m hmS⟩
    · exact hcop
    · intro m hm
      exact (Finset.mem_filter.mp hm).2
  have hsub : survivors L z residue ⊆ survivors L X residue ∪ T := by
    intro m hm
    by_cases hmT : m ∈ T
    · exact Finset.mem_union.mpr (Or.inr hmT)
    · apply Finset.mem_union.mpr
      left
-- 36 more lines, see https://jig.so/p/
```

- Canonical statement

```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
- 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,
    ∀ m : ℕ, 1 ≤ m → m ≤ y →
      ∃ p : ℕ, p.Prime ∧ p ≤ X ∧ m % p = residue p % p

@[simp] theorem mem_smallPrimes {X p : ℕ} :
    p ∈ smallPrimes X ↔ p.Prime ∧ p ≤ X := by
  simp [smallPrimes, and_comm]

theorem primorial_pos (X : ℕ) : 0 < primorial X := by
  apply Finset.prod_pos
  intro p hp
  exact (mem_smallPrimes.mp hp).1.pos

theorem primes_pairwise (X : ℕ) :
    Set.Pairwise (smallPrimes X : Set ℕ) Nat.Coprime := by
  intro p hp q hq hpq
  exact (Nat.coprime_primes (mem_smallPrimes.mp hp).1
    (mem_smallPrimes.mp hq).1).mpr hpq

theorem crt (X : ℕ) (r : ℕ → ℕ) :
    ∃ a < primorial X, ∀ p ∈ smallPrimes X, Nat.ModEq p a (r p) := by
  have hp : ∀ p ∈ smallPrimes X, p ≠ 0 := by
    intro p hp
    exact (mem_smallPrimes.mp hp).1.ne_zero
  let a := Nat.chineseRemainderOfFinset r id (smallPrimes X) hp (primes_pairwise X)
  exact ⟨a, Nat.chineseRemainderOfFinset_lt_prod r id hp (primes_pairwise X), a.property⟩

theorem complement_zero {p r : ℕ} (hp : 0 < p) :
    Nat.ModEq p ((p - r % p) + r) 0 := by
  change ((p - r % p) + r) % p = 0 % p
  have hh : ((p - r % p) + r) % p = ((p - r % p) + r % p) % p := by
    simp only [Nat.add_mod, Nat.mod_mod]
  rw [hh, Nat.sub_add_cancel (Nat.le_of_lt (Nat.mod_lt r hp))]
  simp

theorem div_iff_residue {p a m r : ℕ} (ha : Nat.ModEq p (a + r) 0) :
    p ∣ a + m ↔ m % p = r % p := by
  change p ∣ a + m ↔ Nat.ModEq p m r
  rw [← Nat.modEq_zero_iff_dvd]
  constructor
  · intro hm
    exact Nat.ModEq.add_left_cancel' a (hm.trans ha.symm)
  · intro hm
    exact (hm.add_left a).trans ha

theorem coprime_primorial_iff (X n : ℕ) :
    Nat.Coprime (primorial X) n ↔
      ∀ p, p.Prime → p ≤ X → ¬ p ∣ n := by
  simp only [primorial, Nat.coprime_prod_left_iff, mem_smallPrimes]
  constructor
  · intro h p hp hpx
    exact hp.coprime_iff_not_dvd.mp (h p ⟨hp,hpx⟩)
  · intro h p hp
    exact hp.1.coprime_iff_not_dvd.mpr (h p hp.1 hp.2)

theorem not_coprime_primorial_iff (X n : ℕ) :
    ¬ Nat.Coprime (primorial X) n ↔
      ∃ p, p.Prime ∧ p ≤ X ∧ p ∣ n := by
  rw [coprime_primorial_iff]
  push Not
  rfl

theorem covers_iff_shift (X y : ℕ) :
    CoversInitialInterval X y ↔
      ∃ a < primorial X, ∀ m, 1 ≤ m → m ≤ y →
        ¬ Nat.Coprime (primorial X) (a + m) := by
  constructor
  · rintro ⟨r, hr⟩
    obtain ⟨a, ha, hamod⟩ := crt X (fun p => p - r p % p)
    refine ⟨a, ha, ?_⟩
    intro m hm hmy
    obtain ⟨p, hp, hpX, hmr⟩ := hr m hm hmy
    apply (not_coprime_primorial_iff X (a + m)).mpr
    refine ⟨p, hp, hpX, ?_⟩
    apply (div_iff_residue ?_).mpr hmr
    exact ((hamod p (mem_smallPrimes.mpr ⟨hp,hpX⟩)).add_right (r p)).trans
      (complement_zero hp.pos)
  · rintro ⟨a, ha, hr⟩
    refine ⟨fun p => p - a % p, ?_⟩
    intro m hm hmy
    obtain ⟨p,hp,hpX,hpdiv⟩ := (not_coprime_primorial_iff X (a+m)).mp (hr m hm hmy)
    refine ⟨p,hp,hpX,?_⟩
    apply (div_iff_residue ?_).mp hpdiv
    simpa [Nat.add_comm] using (complement_zero (r:=a) hp.pos)

theorem covers_lt_primorial {X y : ℕ} (h : CoversInitialInterval X y) :
    y < primorial X := by
  obtain ⟨a, ha, hgap⟩ := (covers_iff_shift X y).mp h
  by_contra! hyp
  by_cases ha0 : a = 0
  · have hh := hgap 1 (by omega) (by have := primorial_pos X; omega)
    subst a
    simpa using hh
  · have ham : a + (primorial X - a + 1) = primorial X + 1 := by omega
    have hh := hgap (primorial X - a + 1) (by omega) (by omega)
    rw [ham] at hh
    simpa using hh

theorem covers_zero (X : ℕ) : CoversInitialInterval X 0 := by
  refine ⟨fun _ => 0, ?_⟩
  intro m hm hm0
  omega

theorem exists_maximum (X : ℕ) :
    ∃ Y < primorial X, CoversInitialInterval X Y ∧
-- 23 more lines, see https://jig.so/p/
```

- Canonical statement

```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
- 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
    by_contra hy
    obtain ⟨residue, hresidue⟩ := h
    obtain ⟨p, hp, hp0, hmod⟩ :=
      hresidue 1 (by omega) (by omega)
    have hp2 : 2 ≤ p := hp.two_le
    omega
  · intro hy
    subst y
    refine ⟨fun _ => 0, ?_⟩
    omega

end Submissions.Erdos687ZeroCutoffBoundary.Omega
```

- Canonical statement

```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

```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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