Datasets:
| /- Copyright (c) Heather Macbeth, 2023. All rights reserved. -/ | |
| import Library.Theory.Parity | |
| /-! # Assorted number theory lemmas from earlier needed in section 7.3 (square root of 2) -/ | |
| -- from Section 2.3 | |
| theorem sq_ne_two (n : ℤ) : n ^ 2 ≠ 2 := by | |
| intro hn | |
| obtain ⟨hn1, hn2⟩ : -2 < n ∧ n < 2 | |
| · apply abs_lt_of_sq_lt_sq' | |
| · linarith | |
| · norm_num | |
| interval_cases n <;> norm_num at hn | |
| -- from Section 6.1 | |
| theorem Nat.Odd.pow {a : ℕ} (ha : Nat.Odd a) (n : ℕ) : Nat.Odd (a ^ n) := by | |
| induction' n with k IH | |
| · use 0 | |
| change a ^ 0 = _ | |
| ring | |
| · obtain ⟨x, hx⟩ := ha | |
| obtain ⟨y, hy⟩ := IH | |
| use 2 * x * y + x + y | |
| rw [pow_succ, hy, hx] | |
| ring | |
| -- from Section 6.1 | |
| theorem Nat.even_of_pow_even {a n : ℕ} (ha : Nat.Even (a ^ n)) : Nat.Even a := by | |
| rw [even_iff_not_odd] at * | |
| intro h | |
| have : Odd (a ^ n) := Odd.pow h n | |
| contradiction | |