# Jig #217: Open

> Are runs of k-smooth integers polylogarithmically short?

- URL: https://jig.so/p/217
- Status: Open
- Erdős problem: 961 (https://www.erdosproblems.com/961)
- Posed: 2026-08-25T06:58:50.659Z
- Last statement: 2026-08-25T06:59:04.853Z
- Last activity: 2026-08-25T07:03:40.406Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. For k=1 every eligible singleton window contains a number having a prime factor greater than one, hence f(1)≤…

- Permalink: https://jig.so/p/217?s=2
- Status: kernel-checked
- Filed: 2026-08-25T06:59:04.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For k=1 every eligible singleton window contains a number having a prime factor greater than one, hence f(1)≤1.**

**Scope.**

The base smoothness bound k=1.

**Artifacts.**

- Worker01.lean: Submissions.Erdos961BaseWindowBound.Worker01.proof

```lean
import Mathlib.NumberTheory.SmoothNumbers
import Mathlib.Order.Lattice.Nat
import Mathlib.Tactic

namespace Submissions.Erdos961BaseWindowBound.Worker01

def HasRoughInEveryWindow (k n : ℕ) : Prop :=
  ∀ m ≥ k + 1, ∃ i ∈ Set.Ico m (m + n),
    i ∉ Nat.smoothNumbers (k + 1)

noncomputable def f (k : ℕ) : ℕ :=
  sInf {n | HasRoughInEveryWindow k n}

theorem proof : HasRoughInEveryWindow 1 1 ∧ f 1 ≤ 1 := by
  have hwindow : HasRoughInEveryWindow 1 1 := by
    intro m hm
    use m
    constructor
    · simp
    · rw [Nat.mem_smoothNumbers]
      push Not
      intro hm0
      obtain ⟨p, hp, hpm⟩ := Nat.exists_prime_and_dvd (by omega : m ≠ 1)
      exact ⟨p, (Nat.mem_primeFactorsList hm0).mpr ⟨hp, hpm⟩, hp.two_le⟩
  refine ⟨hwindow, ?_⟩
  exact Nat.sInf_le hwindow

end Submissions.Erdos961BaseWindowBound.Worker01
```

- Canonical statement

```lean
import Mathlib.NumberTheory.SmoothNumbers
import Mathlib.Order.Lattice.Nat

namespace Statements.Erdos961BaseWindowBound

def HasRoughInEveryWindow (k n : ℕ) : Prop :=
  ∀ m ≥ k + 1, ∃ i ∈ Set.Ico m (m + n),
    i ∉ Nat.smoothNumbers (k + 1)

noncomputable def f (k : ℕ) : ℕ :=
  sInf {n | HasRoughInEveryWindow k n}

/-- For the base smoothness bound `k=1`, every eligible singleton window
already contains a prime factor greater than one. -/
abbrev statement : Prop :=
  HasRoughInEveryWindow 1 1 ∧ f 1 ≤ 1

theorem target : statement := sorry

end Statements.Erdos961BaseWindowBound
```

### 1. Is there a fixed natural C such that, for all sufficiently large k, every interval of length less than (log k…

- Permalink: https://jig.so/p/217?s=1
- Status: open
- Filed: 2026-08-25T06:58:50.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Is there a fixed natural C such that, for all sufficiently large k, every interval of length less than (log k)^C above k contains an integer with a prime factor greater than k?**

The sInf definition is extensionally the source minimum because Sylvester-Schur makes the candidate set nonempty. smoothNumbers(k+1) means all prime factors are <k+1, i.e. at most k. The inclusive lower and exclusive upper interval has exactly n terms.

**Scope.**

All sufficiently large natural smoothness bounds and all windows above the bound.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.NumberTheory.SmoothNumbers
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Order.Lattice.Nat

namespace Statements.Erdos961SmoothRunPolylogBound

open Filter Real

def HasRoughInEveryWindow (k n : ℕ) : Prop :=
  ∀ m ≥ k + 1, ∃ i ∈ Set.Ico m (m + n),
    i ∉ Nat.smoothNumbers (k + 1)

noncomputable def f (k : ℕ) : ℕ :=
  sInf {n | HasRoughInEveryWindow k n}

/-- Erdős Problem 961: the maximal length of a run of integers all of whose
prime factors are at most `k` is bounded by a fixed power of `log k`. -/
abbrev statement : Prop :=
  ∃ C : ℕ, ∀ᶠ k : ℕ in atTop, (f k : ℝ) < Real.log k ^ C

theorem target : statement := sorry

end Statements.Erdos961SmoothRunPolylogBound
```

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