# Documentation

Mathlib.Data.Nat.Prime

# Prime numbers #

This file deals with prime numbers: natural numbers p ≥ 2 whose only divisors are p and 1.

## Important declarations #

• Nat.Prime: the predicate that expresses that a natural number p is prime
• Nat.Primes: the subtype of natural numbers that are prime
• Nat.minFac n: the minimal prime factor of a natural number n ≠ 1
• Nat.exists_infinite_primes: Euclid's theorem that there exist infinitely many prime numbers. This also appears as Nat.not_bddAbove_setOf_prime and Nat.infinite_setOf_prime (the latter in Data.Nat.PrimeFin).
• Nat.prime_iff: Nat.Prime coincides with the general definition of Prime
• Nat.irreducible_iff_nat_prime: a non-unit natural number is only divisible by 1 iff it is prime
def Nat.Prime (p : ) :

Nat.Prime p means that p is a prime number, that is, a natural number at least 2 whose only divisors are p and 1.

Instances For
theorem Nat.Prime.ne_zero {n : } (h : ) :
n 0
theorem Nat.Prime.pos {p : } (pp : ) :
0 < p
theorem Nat.Prime.two_le {p : } :
2 p
theorem Nat.Prime.one_lt {p : } :
1 < p
instance Nat.Prime.one_lt' (p : ) [hp : Fact ()] :
Fact (1 < p)
theorem Nat.Prime.ne_one {p : } (hp : ) :
p 1
theorem Nat.Prime.eq_one_or_self_of_dvd {p : } (pp : ) (m : ) (hm : m p) :
m = 1 m = p
theorem Nat.prime_def_lt'' {p : } :
2 p ∀ (m : ), m pm = 1 m = p
theorem Nat.prime_def_lt {p : } :
2 p ∀ (m : ), m < pm pm = 1
theorem Nat.prime_def_lt' {p : } :
2 p ∀ (m : ), 2 mm < p¬m p
theorem Nat.prime_def_le_sqrt {p : } :
2 p ∀ (m : ), 2 mm ¬m p
theorem Nat.prime_of_coprime (n : ) (h1 : 1 < n) (h : ∀ (m : ), m < nm 0) :

This instance is slower than the instance decidablePrime defined below, but has the advantage that it works in the kernel for small values.

If you need to prove that a particular number is prime, in any case you should not use by decide, but rather by norm_num, which is much faster.

Instances For
theorem Nat.Prime.five_le_of_ne_two_of_ne_three {p : } (hp : ) (h_two : p 2) (h_three : p 3) :
5 p
theorem Nat.Prime.pred_pos {p : } (pp : ) :
0 <
theorem Nat.succ_pred_prime {p : } (pp : ) :
Nat.succ () = p
theorem Nat.dvd_prime {p : } {m : } (pp : ) :
m p m = 1 m = p
theorem Nat.dvd_prime_two_le {p : } {m : } (pp : ) (H : 2 m) :
m p m = p
theorem Nat.prime_dvd_prime_iff_eq {p : } {q : } (pp : ) (qp : ) :
p q p = q
theorem Nat.Prime.not_dvd_one {p : } (pp : ) :
¬p 1
theorem Nat.not_prime_mul {a : } {b : } (a1 : 1 < a) (b1 : 1 < b) :
theorem Nat.not_prime_mul' {a : } {b : } {n : } (h : a * b = n) (h₁ : 1 < a) (h₂ : 1 < b) :
theorem Nat.prime_mul_iff {a : } {b : } :
Nat.Prime (a * b) b = 1 a = 1
theorem Nat.Prime.dvd_iff_eq {p : } {a : } (hp : ) (a1 : a 1) :
a p p = a
theorem Nat.minFac_lemma (n : ) (k : ) (h : ¬n < k * k) :
- k < + 2 - k
def Nat.minFacAux (n : ) :

If n < k * k, then minFacAux n k = n, if k | n, then minFacAux n k = k. Otherwise, minFacAux n k = minFacAux n (k+2) using well-founded recursion. If n is odd and 1 < n, then minFacAux n 3 is the smallest prime factor of n.

Equations
Instances For
def Nat.minFac (n : ) :

Returns the smallest prime factor of n ≠ 1.

Instances For
@[simp]
theorem Nat.minFac_zero :
= 2
@[simp]
theorem Nat.minFac_one :
= 1
theorem Nat.minFac_eq (n : ) :
= if 2 n then 2 else
theorem Nat.minFacAux_has_prop {n : } (n2 : 2 n) (k : ) (i : ) :
k = 2 * i + 3(∀ (m : ), 2 mm nk m) → Nat.minFacProp n ()
theorem Nat.minFac_has_prop {n : } (n1 : n 1) :
theorem Nat.minFac_dvd (n : ) :
n
theorem Nat.minFac_prime {n : } (n1 : n 1) :
theorem Nat.minFac_le_of_dvd {n : } {m : } :
2 mm n m
theorem Nat.minFac_pos (n : ) :
0 <
theorem Nat.minFac_le {n : } (H : 0 < n) :
n
theorem Nat.le_minFac {m : } {n : } :
n = 1 m ∀ (p : ), p nm p
theorem Nat.le_minFac' {m : } {n : } :
n = 1 m ∀ (p : ), 2 pp nm p
theorem Nat.prime_def_minFac {p : } :
2 p = p
@[simp]
theorem Nat.Prime.minFac_eq {p : } (hp : ) :
= p
instance Nat.decidablePrime (p : ) :

This instance is faster in the virtual machine than decidablePrime1, but slower in the kernel.

If you need to prove that a particular number is prime, in any case you should not use by decide, but rather by norm_num, which is much faster.

theorem Nat.not_prime_iff_minFac_lt {n : } (n2 : 2 n) :
< n
theorem Nat.minFac_le_div {n : } (pos : 0 < n) (np : ) :
n /
theorem Nat.minFac_sq_le_self {n : } (w : 0 < n) (h : ) :
^ 2 n

The square of the smallest prime factor of a composite number n is at most n.

@[simp]
theorem Nat.minFac_eq_one_iff {n : } :
= 1 n = 1
@[simp]
theorem Nat.minFac_eq_two_iff (n : ) :
= 2 2 n
theorem Nat.exists_dvd_of_not_prime {n : } (n2 : 2 n) (np : ) :
m, m n m 1 m n
theorem Nat.exists_dvd_of_not_prime2 {n : } (n2 : 2 n) (np : ) :
m, m n 2 m m < n
theorem Nat.exists_prime_and_dvd {n : } (hn : n 1) :
p, p n
theorem Nat.dvd_of_forall_prime_mul_dvd {a : } {b : } (hdvd : ∀ (p : ), p ap * a b) :
a b
theorem Nat.exists_infinite_primes (n : ) :
p, n p

Euclid's theorem on the infinitude of primes. Here given in the form: for every n, there exists a prime number p ≥ n.

A version of Nat.exists_infinite_primes using the BddAbove predicate.

theorem Nat.Prime.eq_two_or_odd {p : } (hp : ) :
p = 2 p % 2 = 1
theorem Nat.Prime.eq_two_or_odd' {p : } (hp : ) :
p = 2 Odd p
theorem Nat.Prime.even_iff {p : } (hp : ) :
Even p p = 2
theorem Nat.Prime.odd_of_ne_two {p : } (hp : ) (h_two : p 2) :
Odd p
theorem Nat.Prime.even_sub_one {p : } (hp : ) (h2 : p 2) :
Even (p - 1)
theorem Nat.Prime.mod_two_eq_one_iff_ne_two {p : } [Fact ()] :
p % 2 = 1 p 2

A prime p satisfies p % 2 = 1 if and only if p ≠ 2.

theorem Nat.coprime_of_dvd {m : } {n : } (H : ∀ (k : ), k m¬k n) :
theorem Nat.coprime_of_dvd' {m : } {n : } (H : ∀ (k : ), k mk nk 1) :
theorem Nat.factors_lemma {k : } :
(k + 2) / Nat.minFac (k + 2) < k + 2
theorem Nat.Prime.coprime_iff_not_dvd {p : } {n : } (pp : ) :
¬p n
theorem Nat.Prime.dvd_iff_not_coprime {p : } {n : } (pp : ) :
p n ¬
theorem Nat.Prime.not_coprime_iff_dvd {m : } {n : } :
¬ p, p m p n
theorem Nat.Prime.dvd_mul {p : } {m : } {n : } (pp : ) :
p m * n p m p n
theorem Nat.Prime.not_dvd_mul {p : } {m : } {n : } (pp : ) (Hm : ¬p m) (Hn : ¬p n) :
¬p m * n
theorem Nat.prime_iff {p : } :
theorem Nat.Prime.prime {p : } :

Alias of the forward direction of Nat.prime_iff.

theorem Prime.nat_prime {p : } :

Alias of the reverse direction of Nat.prime_iff.

theorem Nat.Prime.dvd_of_dvd_pow {p : } {m : } {n : } (pp : ) (h : p m ^ n) :
p m
theorem Nat.Prime.pow_not_prime' {x : } {n : } (hn : n 1) :
theorem Nat.Prime.pow_not_prime {x : } {n : } (hn : 2 n) :
theorem Nat.Prime.eq_one_of_pow {x : } {n : } (h : Nat.Prime (x ^ n)) :
n = 1
theorem Nat.Prime.pow_eq_iff {p : } {a : } {k : } (hp : ) :
a ^ k = p a = p k = 1
theorem Nat.pow_minFac {n : } {k : } (hk : k 0) :
Nat.minFac (n ^ k) =
theorem Nat.Prime.pow_minFac {p : } {k : } (hp : ) (hk : k 0) :
Nat.minFac (p ^ k) = p
theorem Nat.Prime.mul_eq_prime_sq_iff {x : } {y : } {p : } (hp : ) (hx : x 1) (hy : y 1) :
x * y = p ^ 2 x = p y = p
theorem Nat.Prime.dvd_factorial {n : } {p : } :
→ ( p n)
theorem Nat.Prime.coprime_pow_of_not_dvd {p : } {m : } {a : } (pp : ) (h : ¬p a) :
Nat.Coprime a (p ^ m)
theorem Nat.coprime_primes {p : } {q : } (pp : ) (pq : ) :
p q
theorem Nat.coprime_pow_primes {p : } {q : } (n : ) (m : ) (pp : ) (pq : ) (h : p q) :
Nat.Coprime (p ^ n) (q ^ m)
theorem Nat.coprime_or_dvd_of_prime {p : } (pp : ) (i : ) :
p i
theorem Nat.coprime_of_lt_prime {n : } {p : } (n_pos : 0 < n) (hlt : n < p) (pp : ) :
theorem Nat.eq_or_coprime_of_le_prime {n : } {p : } (n_pos : 0 < n) (hle : n p) (pp : ) :
p = n
theorem Nat.dvd_prime_pow {p : } (pp : ) {m : } {i : } :
i p ^ m k, k m i = p ^ k
theorem Nat.Prime.dvd_mul_of_dvd_ne {p1 : } {p2 : } {n : } (h_neq : p1 p2) (pp1 : ) (pp2 : ) (h1 : p1 n) (h2 : p2 n) :
p1 * p2 n
theorem Nat.eq_prime_pow_of_dvd_least_prime_pow {a : } {p : } {k : } (pp : ) (h₁ : ¬a p ^ k) (h₂ : a p ^ (k + 1)) :
a = p ^ (k + 1)

If p is prime, and a doesn't divide p^k, but a does divide p^(k+1) then a = p^(k+1).

theorem Nat.ne_one_iff_exists_prime_dvd {n : } :
n 1 p, p n
theorem Nat.eq_one_iff_not_exists_prime_dvd {n : } :
n = 1 ∀ (p : ), ¬p n
theorem Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul {p : } (p_prime : ) {m : } {n : } {k : } {l : } (hpm : p ^ k m) (hpn : p ^ l n) (hpmn : p ^ (k + l + 1) m * n) :
p ^ (k + 1) m p ^ (l + 1) n

The type of prime numbers

Instances For
theorem Nat.Primes.coe_nat_inj (p : Nat.Primes) (q : Nat.Primes) :
p = q p = q
instance Nat.monoid.primePow {α : Type u_1} [] :