File: Mathlib/Data/Nat/Choose/Factorization.lean

full_name: Nat.factorization_choose_le_log
code: /-- A logarithmic upper bound on the multiplicity of a prime in a binomial coefficient. -/
theorem factorization_choose_le_log : (choose n k).factorization p ≤ log p n 
goal: (n.choose k).factorization p ≤ Nat.log p n

full_name: Nat.pow_factorization_choose_le
code: /-- A `pow` form of `Nat.factorization_choose_le` -/
theorem pow_factorization_choose_le (hn : 0 < n) : p ^ (choose n k).factorization p ≤ n 
goal: p ^ (n.choose k).factorization p ≤ n

full_name: Nat.factorization_choose_le_one
code: /-- Primes greater than about `sqrt n` appear only to multiplicity 0 or 1
in the binomial coefficient. -/
theorem factorization_choose_le_one (p_large : n < p ^ 2) : (choose n k).factorization p ≤ 1 
goal: (n.choose k).factorization p ≤ 1

full_name: Nat.factorization_choose_of_lt_three_mul
code: theorem factorization_choose_of_lt_three_mul (hp' : p ≠ 2) (hk : p ≤ k) (hk' : p ≤ n - k)
    (hn : n < 3 * p) : (choose n k).factorization p = 0 
goal: (n.choose k).factorization p = 0

full_name: Nat.factorization_centralBinom_of_two_mul_self_lt_three_mul
code: /-- Primes greater than about `2 * n / 3` and less than `n` do not appear in the factorization of
`centralBinom n`. -/
theorem factorization_centralBinom_of_two_mul_self_lt_three_mul (n_big : 2 < n) (p_le_n : p ≤ n)
    (big : 2 * n < 3 * p) : (centralBinom n).factorization p = 0 
goal: n.centralBinom.factorization p = 0

full_name: Nat.factorization_factorial_eq_zero_of_lt
code: theorem factorization_factorial_eq_zero_of_lt (h : n < p) : (factorial n).factorization p = 0 
goal: n.factorial.factorization p = 0

full_name: Nat.factorization_choose_eq_zero_of_lt
code: theorem factorization_choose_eq_zero_of_lt (h : n < p) : (choose n k).factorization p = 0 
goal: (n.choose k).factorization p = 0

full_name: Nat.factorization_centralBinom_eq_zero_of_two_mul_lt
code: /-- If a prime `p` has positive multiplicity in the `n`th central binomial coefficient,
`p` is no more than `2 * n` -/
theorem factorization_centralBinom_eq_zero_of_two_mul_lt (h : 2 * n < p) :
    (centralBinom n).factorization p = 0 
goal: n.centralBinom.factorization p = 0

full_name: Nat.le_two_mul_of_factorization_centralBinom_pos
code: /-- Contrapositive form of `Nat.factorization_centralBinom_eq_zero_of_two_mul_lt` -/
theorem le_two_mul_of_factorization_centralBinom_pos
    (h_pos : 0 < (centralBinom n).factorization p) : p ≤ 2 * n 
goal: p ≤ 2 * n

full_name: Nat.prod_pow_factorization_choose
code: /-- A binomial coefficient is the product of its prime factors, which are at most `n`. -/
theorem prod_pow_factorization_choose (n k : ℕ) (hkn : k ≤ n) :
    (∏ p ∈ Finset.range (n + 1), p ^ (Nat.choose n k).factorization p) = choose n k 
goal: ∏ p ∈ Finset.range (n + 1), p ^ (n.choose k).factorization p = n.choose k

full_name: Nat.prod_pow_factorization_centralBinom
code: /-- The `n`th central binomial coefficient is the product of its prime factors, which are
at most `2n`. -/
theorem prod_pow_factorization_centralBinom (n : ℕ) :
    (∏ p ∈ Finset.range (2 * n + 1), p ^ (centralBinom n).factorization p) = centralBinom n 
goal: ∏ p ∈ Finset.range (2 * n + 1), p ^ n.centralBinom.factorization p = n.centralBinom