在数论，对正整数n，欧拉函数是少于或等于n的数中与n互质的数的数目。

φ函数的值 通式：φ(x)=x(1-1/p1)(1-1/p2)(1-1/p3)(1-1/p4)…..(1-1/pn) 其中p1, p2……pn为x的所有质因数，x是不为0的整数。φ(1)=1（唯一和1互质的数就是1本身）。 (注意：每种质因数只一个。比如12=2*2*3　　那么φ（12）=12*（1-1/2）*(1-1/3)=4)　　若n是质数p的k次幂，φ(n)=p^k-p^(k-1)=(p-1)p^(k-1)，因为除了p的倍数外，其他数都跟n互质。



欧拉函数是积性函数——若m,n互质，φ(mn)=φ(m)φ(n)。

特殊性质：当n为奇数时，φ(2n)=φ(n), 证明于上述类似。-Positive integer n, Euler function is less than or equal to n number coprime with n number of number of number theory. the the the φ function value formula: φ (x) = x (1-1/p1) (1-1/p2) (1-1/p3) (1-1/p4) ..... (1-1/pn ) wherein p1, p2 ...... pn all the prime factors of the x, x is not an integer of 0. φ (1) = 1 (the sole and a relatively prime number is 1 itself). (Note: The each germplasm factor only one example, 12 = 2* 2* 3 then φ of (12) = 12* (1-1/2)* (1-1/3) = 4) when n is a prime number p The k-th power, φ (n) = p ^ kp ^ (k-1) = (p-1) p ^ (k-1), because, in addition to a multiple of p, the number of other related n coprime. The Eulerian function is a multiplicative function- if the m, n are coprime, and φ (Mn) = φ (m) φ (n). The special properties: When n is odd, and φ (2n) = φ (n), proved similar to the above.