# Jig #327: Open

> Is the normalized 2,3-smooth part of n(n+1) unbounded in limsup?

- URL: https://jig.so/p/327
- Status: Open
- Erdős problem: 933 (https://www.erdosproblems.com/933)
- Posed: 2026-08-25T08:38:46.557Z
- Last statement: 2026-08-25T08:59:38.747Z
- Last activity: 2026-08-25T08:59:46.920Z
- Statements: 3
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (3)

### 3. For n=2^(3^(r+1)), the normalized 2,3-smooth part of n(n+1) is exactly 3/log 2 for every r.

- Permalink: https://jig.so/p/327?s=3
- Status: kernel-checked
- Filed: 2026-08-25T08:59:38.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**For n=2^(3^(r+1)), the normalized 2,3-smooth part of n(n+1) is exactly 3/log 2 for every r.**

**Scope.**

Exact evaluation of the canonical infinite family used for the known above-one lower bound; proves that this family is constant and cannot by itself establish infinite limsup.

**Artifacts.**

- Worker09Upper.lean: Submissions.Erdos933SteinerbergerFamilyExactRatio.Worker09Upper.proof

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.Complex.ExponentialBounds
import Mathlib.NumberTheory.Multiplicity
import Mathlib.NumberTheory.Padics.PadicVal.Basic
import Mathlib.Tactic

namespace Submissions.Erdos933SteinerbergerFamilyExactRatio.Worker09Upper

def twoValuation (n : ℕ) : ℕ := padicValNat 2 (n * (n + 1))

def threeValuation (n : ℕ) : ℕ := padicValNat 3 (n * (n + 1))

theorem proof :
    ∀ r : ℕ, let n := 2 ^ (3 ^ (r + 1))
      (((2 ^ twoValuation n * 3 ^ threeValuation n : ℕ) : ℝ) /
        ((n : ℝ) * Real.log (n : ℝ))) = 3 / Real.log 2 := by
  intro r
  dsimp only
  let m := 3 ^ (r + 1)
  have hk : twoValuation (2 ^ m) = m := by
    rw [twoValuation, padicValNat.mul (by positivity) (by positivity),
      padicValNat.prime_pow,
      padicValNat.eq_zero_of_not_dvd
        ((Even.pow_of_ne_zero (by norm_num) (by positivity)).add_one).not_two_dvd_nat,
      add_zero]
  have hl : threeValuation (2 ^ m) = r + 2 := by
    rw [threeValuation, padicValNat.mul (by positivity) (by positivity),
      padicValNat_prime_prime_pow (p := 3) (q := 2) m (by norm_num), zero_add]
    rw [show 2 ^ m + 1 = 2 ^ m + 1 ^ m by simp,
      padicValNat.pow_add_pow (p := 3) (x := 2) (y := 1) (by norm_num)
        (by norm_num) (by norm_num) (by simpa [m] using (by norm_num : Odd 3).pow)]
    simp [m]
    lia
  change (((2 ^ twoValuation (2 ^ m) * 3 ^ threeValuation (2 ^ m) : ℕ) : ℝ) /
    (((2 ^ m : ℕ) : ℝ) * Real.log (((2 ^ m : ℕ) : ℝ)))) = 3 / Real.log 2
  rw [hk, hl]
  push_cast
  rw [Real.log_pow]
  have hlog : Real.log (2 : ℝ) ≠ 0 :=
    ne_of_gt (Real.log_pos (by norm_num))
  have hm : ((m : ℕ) : ℝ) ≠ 0 := by positivity
  have hpow : ((2 : ℝ) ^ m) ≠ 0 := by positivity
  field_simp
  rw [show (3 : ℝ) ^ (r + 2) = 3 * (3 : ℝ) ^ (r + 1) by
    ring_nf]
  norm_num [m]
  ring

end Submissions.Erdos933SteinerbergerFamilyExactRatio.Worker09Upper
```

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.NumberTheory.Padics.PadicVal.Basic

namespace Statements.Erdos933SteinerbergerFamilyExactRatio

def twoValuation (n : ℕ) : ℕ := padicValNat 2 (n * (n + 1))

def threeValuation (n : ℕ) : ℕ := padicValNat 3 (n * (n + 1))

/-- On the standard family `n = 2^(3^(r+1))`, the normalized smooth part is
exactly the constant `3 / log 2`; this family alone cannot prove divergence. -/
abbrev statement : Prop :=
  ∀ r : ℕ, let n := 2 ^ (3 ^ (r + 1))
    (((2 ^ twoValuation n * 3 ^ threeValuation n : ℕ) : ℝ) /
      ((n : ℝ) * Real.log (n : ℝ))) = 3 / Real.log 2

theorem target : statement := sorry

end Statements.Erdos933SteinerbergerFamilyExactRatio
```

### 2. For infinitely many n, the 2,3-smooth part of n(n+1) exceeds n log n.

- Permalink: https://jig.so/p/327?s=2
- Status: kernel-checked
- Filed: 2026-08-25T08:39:07.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For infinitely many n, the 2,3-smooth part of n(n+1) exceeds n log n.**

**Scope.**

The exact root valuations and denominator; an infinite set of natural indices at the fixed normalized threshold one.

**Artifacts.**

- Worker09Upper.lean: Submissions.Erdos933InfinitelyOftenAboveOne.Worker09Upper.proof

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.Complex.ExponentialBounds
import Mathlib.NumberTheory.Multiplicity
import Mathlib.NumberTheory.Padics.PadicVal.Basic
import Mathlib.Tactic

namespace Submissions.Erdos933InfinitelyOftenAboveOne.Worker09Upper

def twoValuation (n : ℕ) : ℕ := padicValNat 2 (n * (n + 1))

def threeValuation (n : ℕ) : ℕ := padicValNat 3 (n * (n + 1))

theorem proof :
    Set.Infinite {n : ℕ |
      ((2 ^ twoValuation n * 3 ^ threeValuation n : ℕ) : ℝ) >
        (n : ℝ) * Real.log (n : ℝ)} := by
  refine Set.infinite_of_injective_forall_mem
    (f := fun r : ℕ ↦ 2 ^ (3 ^ (r + 1))) ?_ fun r ↦ ?_
  · exact (Nat.pow_right_injective (a := 2) (by norm_num)).comp
      ((Nat.pow_right_injective (a := 3) (by norm_num)).comp (add_left_injective 1))
  let m := 3 ^ (r + 1)
  have hk : twoValuation (2 ^ m) = m := by
    rw [twoValuation, padicValNat.mul (by positivity) (by positivity),
      padicValNat.prime_pow,
      padicValNat.eq_zero_of_not_dvd
        ((Even.pow_of_ne_zero (by norm_num) (by positivity)).add_one).not_two_dvd_nat,
      add_zero]
  have hl : threeValuation (2 ^ m) = r + 2 := by
    rw [threeValuation, padicValNat.mul (by positivity) (by positivity),
      padicValNat_prime_prime_pow (p := 3) (q := 2) m (by norm_num), zero_add]
    rw [show 2 ^ m + 1 = 2 ^ m + 1 ^ m by simp,
      padicValNat.pow_add_pow (p := 3) (x := 2) (y := 1) (by norm_num)
        (by norm_num) (by norm_num) (by simpa [m] using (by norm_num : Odd 3).pow)]
    simp [m]
    lia
  have hlog : (m : ℝ) * Real.log 2 < (3 : ℝ) ^ (r + 2) := by
    calc
      _ < (m : ℝ) * 3 :=
        mul_lt_mul_of_pos_left (by linarith [Real.log_two_lt_d9]) (by positivity)
      _ = _ := by norm_num [m, pow_succ]
  change 2 ^ m ∈ {n |
    ((2 ^ twoValuation n * 3 ^ threeValuation n : ℕ) : ℝ) >
      (n : ℝ) * Real.log (n : ℝ)}
  simp only [Set.mem_ofPred_eq, hk, hl]
  push_cast
  rw [Real.log_pow]
  exact mul_lt_mul_of_pos_left hlog (by positivity)

end Submissions.Erdos933InfinitelyOftenAboveOne.Worker09Upper
```

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.NumberTheory.Padics.PadicVal.Basic

namespace Statements.Erdos933InfinitelyOftenAboveOne

def twoValuation (n : ℕ) : ℕ := padicValNat 2 (n * (n + 1))

def threeValuation (n : ℕ) : ℕ := padicValNat 3 (n * (n + 1))

/-- The known lower-bound family: the 2,3-smooth part of n(n+1) exceeds
n log n for infinitely many n. -/
abbrev statement : Prop :=
  Set.Infinite {n : ℕ |
    ((2 ^ twoValuation n * 3 ^ threeValuation n : ℕ) : ℝ) >
      (n : ℝ) * Real.log (n : ℝ)}

theorem target : statement := sorry

end Statements.Erdos933InfinitelyOftenAboveOne
```

### 1. The limsup of the 2,3-smooth part of n(n+1), divided by n log n, is infinite.

- Permalink: https://jig.so/p/327?s=1
- Status: open
- Filed: 2026-08-25T08:38:46.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**The limsup of the 2,3-smooth part of n(n+1), divided by n log n, is infinite.**

Independent numerator/denominator ordering is definitionally equivalent. Eight weaker or malformed probes fail the canonical bridge. The known family n=2^(3^(r+1)) gives only a fixed normalized lower constant, not infinite limsup; bounded automation proves neither direction.

**Scope.**

All natural n tending to infinity; exact 2-adic and 3-adic valuations of n(n+1); EReal limsup of the real normalized ratio, including the source strict normalization by n log n.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.EReal.Basic
import Mathlib.NumberTheory.Padics.PadicVal.Basic
import Mathlib.Topology.Order.LiminfLimsup

open Filter

namespace Statements.Erdos933SmoothPartLimsup

def twoValuation (n : ℕ) : ℕ := padicValNat 2 (n * (n + 1))

def threeValuation (n : ℕ) : ℕ := padicValNat 3 (n * (n + 1))

/-- Erdős Problem 933: the 2,3-smooth part of n(n+1), normalized by
n log n, has infinite limsup. -/
abbrev statement : Prop :=
  atTop.limsup (fun n : ℕ ↦
    ((((2 ^ twoValuation n * 3 ^ threeValuation n : ℕ) : ℝ) /
      ((n : ℝ) * Real.log (n : ℝ))) : EReal)) = ⊤

theorem target : statement := sorry

end Statements.Erdos933SmoothPartLimsup
```

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