Euler's theorem:
When a p-prime when there \ (a ^ {\ varphi (
p)} \ equiv 1 \ (mod \ p) \) Term Formula and Proof:
If \ (n = p ^ k, p \) is a prime number, the \ (\ varphi (p ^ k
) = p ^ k - p ^ {k - 1} \) when a number is not included prime factors \ (P \) when the can with \ (n-\) prime ,
less \ (n-\) number included in the quality factor \ (P \) only \ (p ^ {k-1 } \) th, they are:
\ (P, 2 * P,. 3 * P,. .., {P ^ K -. 1} * P \) , to remove them.
It can be obtained by a unique decomposition theorem: \ (n-A_1} = {P_2 P_1 ^ ^ ^ {P_3 A_2}} ... {A_3 a_k P_K ^ {} \)
then \ (\ varphi (n) = \ varphi (p_1 ^ {a_1}) \ varphi (p_2 ^ {a_2}) \ varphi (p_3 ^ {a_3}) ... \ varphi (p_k ^ {a_k}) \)
According to the \ (\ varphi (p ^ k ) = p ^ k - p ^ {k - 1} \) can be obtained:
$ \ varphi (P) = P ^ K (. 1 - $ \ ({. 1} \ over { K} ^ P \) )
则 \(\varphi (n) = \varphi(p_1^{a_1})\varphi(p_2^{a_2})\varphi(p_3^{a_3})...\varphi(p_k^{a_k})\)可化为
\(\ \ \ \ \varphi (n) = p_1 ^{a_1}(1 - \frac {1} {p_1}) p_2 ^{a_2}(1 - \frac {1} {p_2})p_3 ^{a_3}(1 - \frac {1} {p_3})...p_k ^{a_k}(1 - \frac {1} {p_k})\)
\(\ \ \ \ \ \ \ \ \ \ \ = n (1 - \frac {1} {p_1})(1 - \frac {1} {p_2})(1 - \frac {1} {p_3})...(1 - \frac {1} {p_k})\)
Euler function
\ (\ varphi (n) or
\ phi (n) \) represents the number of the number of positive integers less than n and prime to n.
apparently:
when n is prime \ (\ varphi (n) \)
When n is odd \ (\ varphi (2n) =
\ varphi (n) \) demonstrated:
\ (\ Because \) Euler function multiplicative function.
\ (\ THEREFORE \ varphi (2N) = \ varphi (2 ) \ AST \ varphi (n-) \)
\ (\ Because \ varphi (2) =. 1 \)
\ (\ THEREFORE \ varphi (2N) = \ varphi (n-) \)
Intergrable of evidence Ming Oula function.
With the proviso that m and n are coprime
can be \ (\ phi (mn) = \ phi (m) \ ast \ phi (n) \)
demonstrated:
\ (P_1 ^ m = {P_2 A_1} ^ {} ... A_2 } ^ {a_k P_K \)
\ (\ Phi (m) = m (l- \ FRAC. 1 {{}} P_1) (l- \ FRAC. 1} {} {P_2) ... (l- \ {FRAC. 1 P_K {}}) \)
\ (n-P_1 = 'A_1 ^ {'} P_2 'A_2 ^ {'} ... P_K 'a_k ^ {'} \)
\ (\ Phi (n-) = n-(l- \ {} {P_1. 1 FRAC '}) (l- \ FRAC {} {P_2. 1'}) ... (l- \ FRAC. 1} {{P_K '}) \)
\ (\ Because m coprime with n \ )
\ (\ THEREFORE P_1, P_2 ... P_K with p_1'p_2 '... p_k' \) twenty-two different from each other
\ (\ therefore \ phi (mn ) = mn (1- \ frac {1} {p_1 }) (1- \ frac {1 } {p_2}) ... (1- \ frac {1} {p_k}) (1- \ frac {1} {p_1 '}) (1- \ frac {1} P_2 { '}) ... (l- \ FRAC. 1} {{P_K'}) \)
\ (\ THEREFORE \ Phi (Mn) = \ Phi (m) \ AST \ Phi (n-) \)