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number_theory.​primorial

number_theory.​primorial

source
dvd_choose_of_middling_prime
primorial
primorial_le_4_pow
primorial_succ
prod_primes_dvd

Primorial

This file defines the primorial function (the product of primes less than or equal to some bound), and proves that primorial n ≤ 4 ^ n.

Notations

We use the local notation n# for the primorial of n: that is, the product of the primes less than or equal to n.

source
def primorial  :
ℕ → ℕ

The primorial n# of n is the product of the primes less than or equal to n.

Equations
source
theorem primorial_succ {n : ℕ} :
1 < n → n % 2 = 1 → primorial (n + 1) = primorial n

source
theorem dvd_choose_of_middling_prime (p : ℕ) (is_prime : nat.prime p) (m : ℕ) :
m + 1 < p → p ≤ 2 * m + 1 → p ∣ (2 * m + 1).choose (m + 1)

source
theorem prod_primes_dvd {s : finset ℕ} (n : ℕ) :
(∀ (a : ℕ), a ∈ s → nat.prime a) → (∀ (a : ℕ), a ∈ s → a ∣ n) → s.prod (λ (p : ℕ), p) ∣ n

source
theorem primorial_le_4_pow (n : ℕ) :
primorial n ≤ 4 ^ n

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