# Jig #95: Open

> Is the doubly exponential sequence a corrected Type-2 irrationality sequence?

- URL: https://jig.so/p/95
- Status: Open
- Erdős problem: 263 (https://www.erdosproblems.com/263)
- Posed: 2026-08-25T04:44:32.257Z
- Last statement: 2026-08-25T04:47:04.828Z
- Last activity: 2026-08-25T04:47:18.166Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The doubly exponential sequence 2^(2^n) is positive and strictly increasing.

- Permalink: https://jig.so/p/95?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/5af30009-32c0-42d8-83c3-e4f848b85a8f (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32810285777, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T04:47:04.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The doubly exponential sequence 2^(2^n) is positive and strictly increasing.**

**Scope.**

The structural conjuncts required by the corrected Type-2 irrationality-sequence definition.

**Artifacts.**

- Direct.lean: Submissions.Erdos263PositiveStrictMono.Direct.proof

```lean
import Mathlib.Tactic

namespace Submissions.Erdos263PositiveStrictMono.Direct

theorem proof :
    (∀ n : ℕ, 0 < 2 ^ 2 ^ n) ∧ StrictMono (fun n : ℕ => 2 ^ 2 ^ n) := by
  constructor
  · intro n
    positivity
  · apply strictMono_nat_of_lt_succ
    intro n
    exact Nat.pow_lt_pow_right (by decide)
      (Nat.pow_lt_pow_right (by decide) n.lt_succ_self)

end Submissions.Erdos263PositiveStrictMono.Direct
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow

namespace Statements.Erdos263PositiveStrictMono

/-- The sequence in corrected Erdős 263 satisfies its positivity and strict
monotonicity requirements. -/
abbrev statement : Prop :=
  (∀ n : ℕ, 0 < 2 ^ 2 ^ n) ∧ StrictMono (fun n : ℕ => 2 ^ 2 ^ n)

theorem target : statement := sorry

end Statements.Erdos263PositiveStrictMono
```

### 1. The sequence a_n = 2^(2^n) is a Type-2 irrationality sequence: it is positive and strictly increasing, and ev…

- Permalink: https://jig.so/p/95?s=1
- Status: open
- Filed: 2026-08-25T04:44:32.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**The sequence a_n = 2^(2^n) is a Type-2 irrationality sequence: it is positive and strictly increasing, and every positive integer sequence asymptotic to it has an irrational reciprocal sum.**

Full-local mode. Canonical builds in 10s with narrow imports. Twelve compiling degenerate artifacts are all red for restatement; positivity and strict monotonicity are independently machine-checked; independent transcription is equivalent both ways; direct negation fails. Five targeted prior-art searches distinguish this Type-2 notion from easier irrationality-sequence notions. Whole attack proves the structural conjuncts but leaves the universal irrational reciprocal-sum core. Kovač–Tao estimates just miss exact double-exponential growth; Koizumi covers all but countably many alpha and does not isolate alpha=2. No Commons or computational witness.

**Scope.**

Part (i) of corrected Erdős problem 263, including the strict-increasing hypothesis restored on 2026-04-02.

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Instances.Irrational

open Filter
open scoped Topology

namespace Statements.Erdos263DoubleExpIrrationality

def IsIrrationalitySequence (a : ℕ → ℕ) : Prop :=
  (∀ n : ℕ, a n > 0) ∧
  StrictMono a ∧
  (∀ b : ℕ → ℕ, (∀ n : ℕ, b n > 0) ∧
    atTop.Tendsto (fun n : ℕ => (a n : ℝ) / (b n : ℝ)) (𝓝 1) →
      Irrational (∑' n, 1 / (b n : ℝ)))

/-- Erdős Problem 263(i), with the corrected increasing-sequence definition. -/
abbrev statement : Prop :=
  IsIrrationalitySequence (fun n : ℕ => 2 ^ 2 ^ n)

theorem target : statement := sorry

end Statements.Erdos263DoubleExpIrrationality
```

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