Euler nature and function templates

Euler function

Euler function, symbolically written as [Phi] ( n- ) [Phi] (n-), which is smaller than the n- n-and the n- n-number of prime numbers

nature

①

For the prime number n- n-

z ( n ) = n - 1 z (n) = n-1

 

②

For n- = P K n-PK =

z ( n ) = ( p - 1 ) * p k - 1 z (n) = (p-1 ) * pk-1

 

③

Multiplicative function []
for G C D ( n- , m ) = . 1 GCD (n-, m) =. 1

z ( n * m ) = z ( n ) * f ( m ) f (n * m) = z (n) * f (m)

 

④

[Formula] is calculated
for the n- = [pi P K I I n-= Πpiki

z ( n ) = n * W ( 1 - 1 p i ) z (n) = n * P (1-1pi)

 

⑤

Euler's theorem []
for coprime A , m A, m

aφ(m)≡1(modm)aφ(m)≡1(modm)

 

⑥

Less than n- n-and the n- n-prime number and:

S=n∗φ(n)2S=n∗φ(n)2

 

⑦

For the prime number P P
when the n- MOD P = 0 nmodp = 0

z ( n * p ) = z ( n ) * p z (n * p) = z (n) * p

If the n- MOD P ≠ 0 nmodp ≠ 0

z ( n * p ) = z ( n ) * ( p - 1 ) z (n * p) = z (n) * (p-1 )

 

⑧

 

∑d|nφ(d)=n∑d|nφ(d)=n

φ ( n ) = Σ d | n μ ( d ) * n d