Campus Livre
eulerPhi
Euler's totient function: how many integers up to n are coprime to n.
Category: Mathematics > Number Theory
Syntax
p = eulerPhi(n)Returns the number of integers k with 1 <= k <= n and gcd(k, n) = 1.
Description
Euler's totient phi(n) is computed from the prime factorization: if n = p1^a1 * ... * pk^ak, then phi(n) = n*(1 - 1/p1)*...*(1 - 1/pk). For a prime p the value is p - 1, and phi(1) = 1.
It gives the order of the group of units modulo n, which is why a^phi(n) = 1 (mod n) when gcd(a, n) = 1 (Euler's theorem). Non-integer input is an error.
Input arguments
n- Integer scalar or array, double or symbolic.
Output arguments
p- Array of the same size as
nwith the totient values.
Example
Totients of 1 to 6 and of 36
eulerPhi(1:6)
eulerPhi(36)
eulerPhi(sym(97))Output
ans = 1×6
1 1 2 2 4 2
ans = 12
ans = 96