Documentation

Mathlib.Algebra.IsPrimePow

Prime powers #

This file deals with prime powers: numbers which are positive integer powers of a single prime.

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def IsPrimePow {R : Type u_1} [CommMonoidWithZero R] (n : R) :
Prop

n is a prime power if there is a prime p and a positive natural k such that n can be written as p^k.

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    theorem isPrimePow_def {R : Type u_1} [CommMonoidWithZero R] (n : R) :
    IsPrimePow n ↔ ∃ (p : R) (k : ℕ), Prime p ∧ 0 < k ∧ p ^ k = n
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    theorem isPrimePow_iff_pow_succ {R : Type u_1} [CommMonoidWithZero R] (n : R) :
    IsPrimePow n ↔ ∃ (p : R) (k : ℕ), Prime p ∧ p ^ (k + 1) = n

    An equivalent definition for prime powers: n is a prime power iff there is a prime p and a natural k such that n can be written as p^(k+1).

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    theorem not_isPrimePow_zero {R : Type u_1} [CommMonoidWithZero R] [NoZeroDivisors R] :
    ¬IsPrimePow 0
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    theorem IsPrimePow.not_unit {R : Type u_1} [CommMonoidWithZero R] {n : R} (h : IsPrimePow n) :
    ¬IsUnit n
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    theorem IsUnit.not_isPrimePow {R : Type u_1} [CommMonoidWithZero R] {n : R} (h : IsUnit n) :
    ¬IsPrimePow n
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    theorem not_isPrimePow_one {R : Type u_1} [CommMonoidWithZero R] :
    ¬IsPrimePow 1
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    theorem Prime.isPrimePow {R : Type u_1} [CommMonoidWithZero R] {p : R} (hp : Prime p) :
    IsPrimePow p
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    theorem IsPrimePow.pow {R : Type u_1} [CommMonoidWithZero R] {n : R} (hn : IsPrimePow n) {k : ℕ} (hk : k ≠ 0) :
    IsPrimePow (n ^ k)
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    theorem IsPrimePow.ne_zero {R : Type u_1} [CommMonoidWithZero R] [NoZeroDivisors R] {n : R} (h : IsPrimePow n) :
    n ≠ 0
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    theorem IsPrimePow.ne_one {R : Type u_1} [CommMonoidWithZero R] {n : R} (h : IsPrimePow n) :
    n ≠ 1
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    theorem isPrimePow_nat_iff (n : ℕ) :
    IsPrimePow n ↔ ∃ (p : ℕ) (k : ℕ), Nat.Prime p ∧ 0 < k ∧ p ^ k = n
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    theorem Nat.Prime.isPrimePow {p : ℕ} (hp : Prime p) :
    IsPrimePow p
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    theorem isPrimePow_nat_iff_bounded (n : ℕ) :
    IsPrimePow n ↔ ∃ p ≤ n, ∃ k ≤ n, Nat.Prime p ∧ 0 < k ∧ p ^ k = n
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    theorem isPrimePow_nat_iff_bounded_log (n : ℕ) :
    IsPrimePow n ↔ ∃ k ≤ Nat.log 2 n, 0 < k ∧ ∃ p ≤ n, n = p ^ k ∧ Nat.Prime p
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    theorem isPrimePow_nat_iff_bounded_log_minFac (n : ℕ) :
    IsPrimePow n ↔ ∃ k ≤ Nat.log 2 n, 0 < k ∧ n = n.minFac ^ k
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    @[implicit_reducible]
    instance instDecidableIsPrimePowNat {n : ℕ} :
    Decidable (IsPrimePow n)
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    theorem IsPrimePow.dvd {n m : ℕ} (hn : IsPrimePow n) (hm : m ∣ n) (hm₁ : m ≠ 1) :
    IsPrimePow m
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    theorem IsPrimePow.two_le {n : ℕ} :
    IsPrimePow n → 2 ≤ n
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    theorem IsPrimePow.pos {n : ℕ} (hn : IsPrimePow n) :
    0 < n
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    theorem IsPrimePow.one_lt {n : ℕ} (h : IsPrimePow n) :
    1 < n