https://github.com/jjaassoonn/transcendental

https://github.com/jjaassoonn/transcendental

leanprover

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• Host: GitHub
• URL: https://github.com/jjaassoonn/transcendental
• Owner: jjaassoonn
• Created: 2020-01-17T01:23:52.000Z (over 6 years ago)
• Default Branch: master
• Last Pushed: 2020-09-09T13:54:55.000Z (almost 6 years ago)
• Last Synced: 2025-02-01T03:42:23.985Z (over 1 year ago)
• Topics: leanprover
• Language: HTML
• Homepage: https://jjaassoonn.github.io
• Size: 4.72 MB
• Stars: 6
• Watchers: 2
• Forks: 1
• Open Issues: 0
• Metadata Files:
  • Readme: README.md

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README

          

# Transcendental Numbers

This project is to prove several theorems in transcendental number theory:

1. [x] Countability argument: abstract existence of transcendental number;

2. [x] Liouvielle theorem and hence ![liouville number](https://latex.codecogs.com/gif.latex?\sum_{i=0}^\infty&space;\frac{1}{10^{n!}}) is transcendental;

3. [x] ![e](https://latex.codecogs.com/gif.latex?e) is transcendental;

4. [ ] ![pi](https://latex.codecogs.com/gif.latex?\pi) is transcendental.

## Part 1, countability argument

The main theorem is in [algebraic_countable_over_Z.lean](https://github.com/jjaassoonn/transcendental/blob/master/src/algebraic_countable_over_Z.lean#L731)

```lean

theorem transcendental_number_exists : ∃ x : real, ¬ (is_algebraic ℤ x)

```

The other version is in [algebraic_countable_over_Q.lean](https://github.com/jjaassoonn/transcendental/blob/master/src/algebraic_countable_over_Q.lean#L897)

```lean

theorem transcendental_number_exists : ∃ x : real, ¬ (is_algebraic ℚ x)

```

## Part 2, Liouville theorem and an explicit Liouville number

Definition of the explicit Liouville number is in [liouville_theorem.lean](https://github.com/jjaassoonn/transcendental/blob/master/src/liouville_theorem.lean#L1136)

```lean

def α := ∑' n, ten_pow_n_fact_inverse n

```

The main theorem is in [liouville_theorem.lean](https://github.com/jjaassoonn/transcendental/blob/master/src/liouville_theorem.lean#L863):

```lean

theorem liouville_numbers_transcendental : ∀ x : real, liouville_number x -> ¬(is_algebraic ℤ x)

theorem transcendental_α : transcendental α := liouville_numbers_transcendental α liouville_α

```

## Part 3, the transcendence of e

We defined e in [e_transcendental.lean](https://github.com/jjaassoonn/transcendental/blob/699e50a6d262ee73ab20bfa6362ed637d4e88c77/src/e_transcendental.lean#L15) as :

``` lean

def e : ℝ := real.exp 1

```

The main theorem is at [e_transcendental.lean](https://github.com/jjaassoonn/transcendental/blob/699e50a6d262ee73ab20bfa6362ed637d4e88c77/src/e_transcendental.lean#L1798):

```lean

theorem e_transcendental : ¬ is_algebraic ℤ e :=

```

Almost immediately, we can prove

```lean

theorem e_irrational : irrational e

theorem e_pow_transcendental (n : ℕ) (hn : n ≥ 1) : transcendental (e^n)

theorem e_pow_n_irrational (n : ℕ) (hn : n ≥ 1) : irrational (e ^ n)

```

Please see [this](https://jjaassoonn.github.io/e_transcendence_doc.html) for an explanation of the proof of transcendence of $e$ with reference to Lean code.

I haven't finished documentation (not even close), but you can click around the proves I documented so far

at [e_trans_helpers2.lean](https://jjaassoonn.github.io/transcendental/html/e_trans_helpers2.html) and[e_transcendental.lean](https://jjaassoonn.github.io/transcendental/html/e_transcendental.html).