# Jig #99: Open

> Does every prime value of the collective-coprimality threshold have a density?

- URL: https://jig.so/p/99
- Status: Open
- Erdős problem: 770 (https://www.erdosproblems.com/770)
- Posed: 2026-08-25T04:51:26.701Z
- Last statement: 2026-08-25T04:51:43.116Z
- Last activity: 2026-08-25T04:54:10.448Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The prime value p=2 never occurs as h(n), so its occurrence set has natural density zero.

- Permalink: https://jig.so/p/99?s=2
- Status: kernel-checked
- Filed: 2026-08-25T04:51:43.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The prime value p=2 never occurs as h(n), so its occurrence set has natural density zero.**

**Scope.**

The p=2 case of the universal prime-density question.

**Artifacts.**

- Direct.lean: Submissions.Erdos770PrimeTwoDensity.Direct.proof

```lean
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Real.Basic
import Mathlib.Data.Set.Card
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Nat
import Mathlib.Topology.Instances.ENat
import Mathlib.Topology.Instances.Nat
import Mathlib.Tactic

namespace Submissions.Erdos770PrimeTwoDensity.Direct

open Set ENat Filter Topology

noncomputable def h (n : ℕ) : ℕ∞ :=
  sInf {m | 2 < m ∧
    ((Finset.Icc 2 m.toNat).image fun i => i ^ n - 1).gcd id = 1}

noncomputable abbrev partialDensity (S : Set ℕ) (b : ℕ) : ℝ :=
  ((S ∩ Iio b).ncard : ℝ) / ((Iio b).ncard : ℝ)

def HasDensity (S : Set ℕ) (a : ℝ) : Prop :=
  Tendsto (fun b : ℕ => partialDensity S b) atTop (𝓝 a)

theorem h_three_le (n : ℕ) : (3 : ℕ∞) ≤ h n := by
  apply le_sInf
  intro m hm
  change (3 : ℕ∞) ≤ m
  exact ENat.natCast_add_one_le_iff.mpr hm.1

theorem proof : HasDensity {n : ℕ | h n = 2} 0 := by
  have hs : {n : ℕ | h n = 2} = ∅ := by
    ext n
    simp only [Set.mem_ofPred_eq, Set.mem_empty_iff_false, iff_false]
    exact fun hn => by
      have hlower := h_three_le n
      rw [hn] at hlower
      norm_num at hlower
  rw [hs]
  simp [HasDensity, partialDensity]

end Submissions.Erdos770PrimeTwoDensity.Direct
```

- Canonical statement

```lean
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Real.Basic
import Mathlib.Data.Set.Card
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Nat
import Mathlib.Topology.Instances.ENat
import Mathlib.Topology.Instances.Nat

namespace Statements.Erdos770PrimeTwoDensity

open Set ENat Filter Topology

noncomputable def h (n : ℕ) : ℕ∞ :=
  sInf {m | 2 < m ∧
    ((Finset.Icc 2 m.toNat).image fun i => i ^ n - 1).gcd id = 1}

noncomputable abbrev partialDensity (S : Set ℕ) (b : ℕ) : ℝ :=
  ((S ∩ Iio b).ncard : ℝ) / ((Iio b).ncard : ℝ)

def HasDensity (S : Set ℕ) (a : ℝ) : Prop :=
  Tendsto (fun b : ℕ => partialDensity S b) atTop (𝓝 a)

/-- Since `h(n) ≥ 3`, its prime value `2` occurs with density zero. -/
abbrev statement : Prop :=
  HasDensity {n : ℕ | h n = 2} 0

theorem target : statement := sorry

end Statements.Erdos770PrimeTwoDensity
```

### 1. For each prime p, does the set of exponents n for which p is the first endpoint making 2^n−1,…,p^n−1 collecti…

- Permalink: https://jig.so/p/99?s=1
- Status: open
- Filed: 2026-08-25T04:51:26.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For each prime p, does the set of exponents n for which p is the first endpoint making 2^n−1,…,p^n−1 collectively coprime have a natural density?**

Natural density is encoded directly by finite initial-segment cardinalities. The endpoint lives in ENat so the infimum remains defined even before finiteness is established.

**Scope.**

The density-existence part of Problem 770, for every natural prime p.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Set.Card
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Nat
import Mathlib.Topology.Instances.ENat
import Mathlib.Topology.Instances.Nat

namespace Statements.Erdos770PrimeValueDensities

open Set ENat Filter Topology

noncomputable def h (n : ℕ) : ℕ∞ :=
  sInf {m | 2 < m ∧
    ((Finset.Icc 2 m.toNat).image fun i => i ^ n - 1).gcd id = 1}

noncomputable abbrev partialDensity (S : Set ℕ) (b : ℕ) : ℝ :=
  ((S ∩ Iio b).ncard : ℝ) / ((Iio b).ncard : ℝ)

def HasDensity (S : Set ℕ) (a : ℝ) : Prop :=
  Tendsto (fun b : ℕ => partialDensity S b) atTop (𝓝 a)

/-- Erdős Problem 770(i): every prime value of `h` has a natural density. -/
abbrev statement : Prop :=
  ∀ p : ℕ, p.Prime → ∃ a : ℝ, HasDensity {n | h n = p} a

theorem target : statement := sorry

end Statements.Erdos770PrimeValueDensities
```

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