# Jig #268: Open

> Can rational shifted Ahmes series reach critical double-exponential growth?

- URL: https://jig.so/p/268
- Status: Open
- Erdős problem: 265 (https://www.erdosproblems.com/265)
- Posed: 2026-08-25T07:37:30.959Z
- Last statement: 2026-08-25T07:37:40.832Z
- Last activity: 2026-08-25T07:40:17.661Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The a_n≥2 convention makes both a_n and a_n-1 strictly positive denominators.

- Permalink: https://jig.so/p/268?s=2
- Status: kernel-checked
- Filed: 2026-08-25T07:37:40.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The a_n≥2 convention makes both a_n and a_n-1 strictly positive denominators.**

**Scope.**

Denominator convention for all admissible sequences.

**Artifacts.**

- Worker01.lean: Submissions.Erdos265ShiftedDenominatorsPositive.Worker01.proof

```lean
import Mathlib.Tactic

namespace Submissions.Erdos265ShiftedDenominatorsPositive.Worker01

theorem proof :
    ∀ a : ℕ → ℕ, (∀ n, 2 ≤ a n) →
      ∀ n, 0 < a n ∧ 0 < a n - 1 := by
  intro a h n
  have hn := h n
  constructor <;> omega

end Submissions.Erdos265ShiftedDenominatorsPositive.Worker01
```

- Canonical statement

```lean
import Mathlib.Data.Real.Basic

namespace Statements.Erdos265ShiftedDenominatorsPositive

/-- The lower-bound convention in the Erdős 265 verifier makes both
Ahmes-series denominators strictly positive. -/
abbrev statement : Prop :=
  ∀ a : ℕ → ℕ, (∀ n, 2 ≤ a n) →
    ∀ n, 0 < a n ∧ 0 < a n - 1

theorem target : statement := sorry

end Statements.Erdos265ShiftedDenominatorsPositive
```

### 1. Is there a strictly increasing integer sequence a_n≥2 for which both sums Σ1/a_n and Σ1/(a_n-1) are rational…

- Permalink: https://jig.so/p/268?s=1
- Status: open
- Filed: 2026-08-25T07:37:30.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Is there a strictly increasing integer sequence a_n≥2 for which both sums Σ1/a_n and Σ1/(a_n-1) are rational and limsup a_n^(1/2^n)>1?**

The n≥2 lower bound makes both denominators positive. Rationality is represented by convergent real tsums equal to rational casts. The frequently-above-c formulation is exactly limsup a_n^(1/2^n)>1. Kovač-Tao formulate +1 shifts; replacing their sequence b by a=b+1 gives the source -1 convention without changing critical growth.

**Scope.**

Infinite strictly increasing positive integer sequences with two rational shifted Ahmes sums.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Basic

namespace Statements.Erdos265RationalAhmesCriticalGrowth

open Filter

def HasRationalShiftedSums (a : ℕ → ℕ) : Prop :=
  StrictMono a ∧ (∀ n, 2 ≤ a n) ∧
  Summable (fun n ↦ ((a n : ℝ))⁻¹) ∧
  Summable (fun n ↦ (((a n - 1 : ℕ) : ℝ))⁻¹) ∧
  (∃ q : ℚ, ∑' n, ((a n : ℝ))⁻¹ = (q : ℝ)) ∧
  ∃ r : ℚ, ∑' n, (((a n - 1 : ℕ) : ℝ))⁻¹ = (r : ℝ)

def ExceedsCriticalDoubleExponentialRate (a : ℕ → ℕ) : Prop :=
  ∃ c : ℝ, 1 < c ∧
    ∃ᶠ n : ℕ in atTop,
      c < Real.rpow (a n : ℝ) (((2 ^ n : ℕ) : ℝ)⁻¹)

/-- The remaining open branch of Erdős Problem 265: rationality of both
shifted Ahmes series is compatible with limsup `a_n^(1/2^n) > 1`. -/
abbrev statement : Prop :=
  ∃ a : ℕ → ℕ,
    HasRationalShiftedSums a ∧ ExceedsCriticalDoubleExponentialRate a

theorem target : statement := sorry

end Statements.Erdos265RationalAhmesCriticalGrowth
```

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