# Euler's totient function #

This file defines Euler's totient function Nat.totient n which counts the number of naturals less than n that are coprime with n. We prove the divisor sum formula, namely that n equals φ summed over the divisors of n. See sum_totient. We also prove two lemmas to help compute totients, namely totient_mul and totient_prime_pow.

def Nat.totient (n : ) :

Euler's totient function. This counts the number of naturals strictly less than n which are coprime with n.

Equations
Instances For

Euler's totient function. This counts the number of naturals strictly less than n which are coprime with n.

Equations
Instances For
@[simp]
theorem Nat.totient_zero :
= 0
@[simp]
theorem Nat.totient_one :
= 1
theorem Nat.totient_eq_card_coprime (n : ) :
n.totient = (Finset.filter n.Coprime ()).card
theorem Nat.totient_eq_card_lt_and_coprime (n : ) :
n.totient = Nat.card {m : | m < n n.Coprime m}

A characterisation of Nat.totient that avoids Finset.

theorem Nat.totient_le (n : ) :
n.totient n
theorem Nat.totient_lt (n : ) (hn : 1 < n) :
n.totient < n
@[simp]
theorem Nat.totient_eq_zero {n : } :
n.totient = 0 n = 0
@[simp]
theorem Nat.totient_pos {n : } :
0 < n.totient 0 < n
theorem Nat.filter_coprime_Ico_eq_totient (a : ) (n : ) :
(Finset.filter a.Coprime (Finset.Ico n (n + a))).card = a.totient
theorem Nat.Ico_filter_coprime_le {a : } (k : ) (n : ) (a_pos : 0 < a) :
(Finset.filter a.Coprime (Finset.Ico k (k + n))).card a.totient * (n / a + 1)
@[simp]
theorem ZMod.card_units_eq_totient (n : ) [] [Fintype (ZMod n)ˣ] :
= n.totient

Note this takes an explicit Fintype ((ZMod n)ˣ) argument to avoid trouble with instance diamonds.

theorem Nat.totient_even {n : } (hn : 2 < n) :
Even n.totient
theorem Nat.totient_mul {m : } {n : } (h : m.Coprime n) :
(m * n).totient = m.totient * n.totient
theorem Nat.totient_div_of_dvd {n : } {d : } (hnd : d n) :
(n / d).totient = (Finset.filter (fun (k : ) => n.gcd k = d) ()).card

For d ∣ n, the totient of n/d equals the number of values k < n such that gcd n k = d

theorem Nat.sum_totient (n : ) :
n.divisors.sum Nat.totient = n
theorem Nat.sum_totient' (n : ) :
mFinset.filter (fun (x : ) => x n) (Finset.range n.succ), m.totient = n
theorem Nat.totient_prime_pow_succ {p : } (hp : ) (n : ) :
(p ^ (n + 1)).totient = p ^ n * (p - 1)

When p is prime, then the totient of p ^ (n + 1) is p ^ n * (p - 1)

theorem Nat.totient_prime_pow {p : } (hp : ) {n : } (hn : 0 < n) :
(p ^ n).totient = p ^ (n - 1) * (p - 1)

When p is prime, then the totient of p ^ n is p ^ (n - 1) * (p - 1)

theorem Nat.totient_prime {p : } (hp : ) :
p.totient = p - 1
theorem Nat.totient_eq_iff_prime {p : } (hp : 0 < p) :
p.totient = p - 1
theorem Nat.card_units_zmod_lt_sub_one {p : } (hp : 1 < p) [Fintype (ZMod p)ˣ] :
p - 1
@[simp]
theorem Nat.totient_two :
= 1
theorem Nat.totient_eq_one_iff {n : } :
n.totient = 1 n = 1 n = 2
theorem Nat.dvd_two_of_totient_le_one {a : } (han : 0 < a) (ha : a.totient 1) :
a 2

### Euler's product formula for the totient function #

We prove several different statements of this formula.

theorem Nat.totient_eq_prod_factorization {n : } (hn : n 0) :
n.totient = n.factorization.prod fun (p k : ) => p ^ (k - 1) * (p - 1)

Euler's product formula for the totient function.

theorem Nat.totient_mul_prod_primeFactors (n : ) :
n.totient * pn.primeFactors, p = n * pn.primeFactors, (p - 1)

Euler's product formula for the totient function.

theorem Nat.totient_eq_div_primeFactors_mul (n : ) :
n.totient = (n / pn.primeFactors, p) * pn.primeFactors, (p - 1)

Euler's product formula for the totient function.

theorem Nat.totient_eq_mul_prod_factors (n : ) :
n.totient = n * pn.primeFactors, (1 - (p)⁻¹)

Euler's product formula for the totient function.

theorem Nat.totient_gcd_mul_totient_mul (a : ) (b : ) :
(a.gcd b).totient * (a * b).totient = a.totient * b.totient * a.gcd b
theorem Nat.totient_super_multiplicative (a : ) (b : ) :
a.totient * b.totient (a * b).totient
theorem Nat.totient_dvd_of_dvd {a : } {b : } (h : a b) :
a.totient b.totient
theorem Nat.totient_mul_of_prime_of_dvd {p : } {n : } (hp : ) (h : p n) :
(p * n).totient = p * n.totient
theorem Nat.totient_mul_of_prime_of_not_dvd {p : } {n : } (hp : ) (h : ¬p n) :
(p * n).totient = (p - 1) * n.totient