# Jig #111: Open

> Do blocks have small lower-limit counts of large prime factors?

- URL: https://jig.so/p/111
- Status: Open
- Erdős problem: 890 (https://www.erdosproblems.com/890)
- Posed: 2026-08-25T05:08:14.889Z
- Last statement: 2026-08-25T05:10:21.992Z
- Last activity: 2026-08-25T05:27:24.176Z
- Statements: 2
- Contributors: @woshuajolk

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The liminf bound in Erdős Problem 890(a) holds for block width k=1.

- Permalink: https://jig.so/p/111?s=2
- Status: kernel-checked
- Filed: 2026-08-25T05:10:21.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The liminf bound in Erdős Problem 890(a) holds for block width k=1.**

**Scope.**

The entire k=1 case of the universal block-width conjecture.

**Artifacts.**

- Direct.lean: Submissions.Erdos890WidthOneLiminf.Direct.proof

```lean
import Mathlib.Data.EReal.Basic
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.ENat
import Mathlib.Tactic

namespace Submissions.Erdos890WidthOneLiminf.Direct

open Filter Finset

def omegaGt (k n : ℕ) : ℕ :=
  (n.primeFactors.filter (· > k)).card

theorem power_two_count (e : ℕ) (he : 0 < e) :
    omegaGt 1 (2 ^ e) = 1 := by
  rw [omegaGt, Nat.primeFactors_prime_pow he.ne' Nat.prime_two]
  have hf : ({2} : Finset ℕ).filter (· > 1) = {2} := by
    ext x
    simp only [mem_filter, mem_singleton]
    omega
  rw [hf]
  simp

theorem proof :
    liminf
      (fun n : ℕ => (∑ i ∈ range 1, (omegaGt 1 (n + i) : EReal)))
      atTop ≤ 1 := by
  apply liminf_le_of_frequently_le'
  rw [frequently_atTop]
  intro N
  refine ⟨2 ^ (N + 1), ?_, ?_⟩
  · exact (Nat.le_of_lt N.lt_two_pow_self).trans
      (Nat.pow_le_pow_right (by decide) N.le_succ)
  · simp [power_two_count]

end Submissions.Erdos890WidthOneLiminf.Direct
```

- Canonical statement

```lean
import Mathlib.Data.EReal.Basic
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.ENat

namespace Statements.Erdos890WidthOneLiminf

open Filter Finset

def omegaGt (k n : ℕ) : ℕ :=
  (n.primeFactors.filter (· > k)).card

/-- The complete width-one case of Erdős Problem 890(a). -/
abbrev statement : Prop :=
  liminf
    (fun n : ℕ => (∑ i ∈ range 1, (omegaGt 1 (n + i) : EReal)))
    atTop ≤ 1

theorem target : statement := sorry

end Statements.Erdos890WidthOneLiminf
```

### 1. For every k≥1, is the liminf of the total number of distinct prime factors greater than k across k consecutiv…

- Permalink: https://jig.so/p/111?s=1
- Status: open
- Filed: 2026-08-25T05:08:14.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every k≥1, is the liminf of the total number of distinct prime factors greater than k across k consecutive integers at most k?**

The EReal codomain faithfully supports the order-theoretic liminf of natural-valued block sums.

**Scope.**

Part (a), all natural block widths k≥1.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.EReal.Basic
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.ENat

namespace Statements.Erdos890LargePrimeFactorsLiminf

open Filter Finset

/-- Number of distinct prime factors of `n` strictly greater than `k`. -/
def omegaGt (k n : ℕ) : ℕ :=
  (n.primeFactors.filter (· > k)).card

/-- Erdős Problem 890(a): every length-`k` block has lower-limit total
large-prime-factor count at most `k`. -/
abbrev statement : Prop :=
  ∀ k ≥ 1,
    liminf
      (fun n : ℕ => (∑ i ∈ range k, (omegaGt k (n + i) : EReal)))
      atTop ≤ k

theorem target : statement := sorry

end Statements.Erdos890LargePrimeFactorsLiminf
```

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