# Jig #218: Open

> Are divisors in every power-saving interval above √n uniformly bounded?
>
> [arXiv:2503.12146](https://arxiv.org/abs/2503.12146)

- URL: https://jig.so/p/218
- Status: Open
- Erdős problem: 886 (https://www.erdosproblems.com/886)
- Posed: 2026-08-25T06:59:38.135Z
- Last statement: 2026-08-25T07:00:07.983Z
- Last activity: 2026-08-25T07:01:04.839Z
- 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. Whenever the Erdős 886 interval has width below one, it contains at most one natural divisor.

- Permalink: https://jig.so/p/218?s=2
- Status: kernel-checked
- Filed: 2026-08-25T07:00:07.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**Whenever the Erdős 886 interval has width below one, it contains at most one natural divisor.**

**Scope.**

Every natural n and real epsilon satisfying the explicit width-below-one hypothesis.

**Artifacts.**

- Order.lean: Submissions.Erdos886UnitWidthPacking.Order.proof

```lean
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.Divisors
import Mathlib.Tactic

open Nat

namespace Submissions.Erdos886UnitWidthPacking.Order

noncomputable def nearDivisors (n : ℕ) (ε C : ℝ) : Finset ℕ :=
  (divisors n).filter (fun d =>
    (n : ℝ) ^ (1 / 2 : ℝ) < d ∧
      (d : ℝ) < (n : ℝ) ^ (1 / 2 : ℝ) +
        C * (n : ℝ) ^ (1 / 2 - ε))

theorem proof :
    ∀ n : ℕ, ∀ ε : ℝ,
      (n : ℝ) ^ (1 / 2 - ε) < 1 →
        (nearDivisors n ε 1).card ≤ 1 := by
  intro n ε hwidth
  rw [Finset.card_le_one]
  intro a ha b hb
  simp only [nearDivisors, Finset.mem_filter] at ha hb
  by_contra hab
  rcases lt_or_gt_of_ne hab with hab | hba
  · have hab' : (a : ℝ) + 1 ≤ b := by exact_mod_cast hab
    have hupper : (b : ℝ) < (n : ℝ) ^ (1 / 2 : ℝ) + (n : ℝ) ^ (1 / 2 - ε) := by simpa using hb.2.2
    have hlower : (n : ℝ) ^ (1 / 2 : ℝ) < a := ha.2.1
    linarith
  · have hba' : (b : ℝ) + 1 ≤ a := by exact_mod_cast hba
    have hupper : (a : ℝ) < (n : ℝ) ^ (1 / 2 : ℝ) + (n : ℝ) ^ (1 / 2 - ε) := by simpa using ha.2.2
    have hlower : (n : ℝ) ^ (1 / 2 : ℝ) < b := hb.2.1
    linarith

end Submissions.Erdos886UnitWidthPacking.Order
```

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.Divisors

open Nat

namespace Statements.Erdos886UnitWidthPacking

noncomputable def nearDivisors (n : ℕ) (ε C : ℝ) : Finset ℕ :=
  (divisors n).filter (fun d =>
    (n : ℝ) ^ (1 / 2 : ℝ) < d ∧
      (d : ℝ) < (n : ℝ) ^ (1 / 2 : ℝ) +
        C * (n : ℝ) ^ (1 / 2 - ε))

/-- If the open interval in Erdős 886 has width below one, it contains at most one natural divisor. -/
abbrev statement : Prop :=
  ∀ n : ℕ, ∀ ε : ℝ,
    (n : ℝ) ^ (1 / 2 - ε) < 1 →
      (nearDivisors n ε 1).card ≤ 1

theorem target : statement := sorry

end Statements.Erdos886UnitWidthPacking
```

### 1. For every ε>0, is the number of divisors of all sufficiently large n in (√n,√n+n^(1/2−ε)) bounded by a consta…

- Permalink: https://jig.so/p/218?s=1
- Status: open
- Filed: 2026-08-25T06:59:38.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For every ε>0, is the number of divisors of all sufficiently large n in (√n,√n+n^(1/2−ε)) bounded by a constant depending only on ε?**

Formal written first. The helper uses real rpow, filters actual natural divisors, preserves both strict endpoints, fixes coefficient one, and puts K after epsilon but before eventual n. This is the exact O_epsilon(1) quantifier structure.

**Scope.**

Every positive real epsilon, one natural bound depending on epsilon, all sufficiently large natural n, and strict interval endpoints.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.Divisors
import Mathlib.Order.Filter.AtTopBot.Basic

open Nat Filter

namespace Statements.Erdos886ShortDivisorInterval

noncomputable def nearDivisors (n : ℕ) (ε C : ℝ) : Finset ℕ :=
  (divisors n).filter (fun d =>
    (n : ℝ) ^ (1 / 2 : ℝ) < d ∧
      (d : ℝ) < (n : ℝ) ^ (1 / 2 : ℝ) +
        C * (n : ℝ) ^ (1 / 2 - ε))

/-- Erdős Problem 886: every positive exponent loss gives an eventually uniform bound for divisors just above the square root. -/
abbrev statement : Prop :=
  ∀ ε > 0, ∃ K : ℕ, ∀ᶠ n in atTop,
    (nearDivisors n ε 1).card ≤ K

theorem target : statement := sorry

end Statements.Erdos886ShortDivisorInterval
```

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