欧拉定理:
当a 与 p互质的时候则有 \(a^{\varphi(p)} \equiv 1 \ (mod \ p)\)
通项公式及证明：
若\(n = p ^ k ,p\)为质数,则\(\varphi (p^k) = p ^k - p ^{k - 1}\)
当一个数不包含质因子\(p\)时就能与\(n\)互质，
小于等于\(n\)的数中包含质因子\(p\)的只有\(p^{k-1}\) 个，他们是:
\(p, 2*p, 3* p, ...,p ^{k - 1} ∗p\)，把他们去除即可.
由唯一分解定理可得: \(n = p_1 ^{a_1} p_2 ^{a_2}p_3 ^{a_3}...p_k ^{a_k}\)
则 \(\varphi (n) = \varphi(p_1^{a_1})\varphi(p_2^{a_2})\varphi(p_3^{a_3})...\varphi(p_k^{a_k})\)
根据上述\(\varphi (p^k) = p ^k - p ^{k - 1}\)可得:
$ \varphi (p) = p^k(1 - $\({1}\over {p^k}\))
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
\(\ \ \ \ \varphi (n) = p_1 ^{a_1}(1 - \frac {1} {p_1}) p_2 ^{a_2}(1 - \frac {1} {p_2})p_3 ^{a_3}(1 - \frac {1} {p_3})...p_k ^{a_k}(1 - \frac {1} {p_k})\)
\(\ \ \ \ \ \ \ \ \ \ \ = n (1 - \frac {1} {p_1})(1 - \frac {1} {p_2})(1 - \frac {1} {p_3})...(1 - \frac {1} {p_k})\)
欧拉函数
\(\varphi(n) or \phi(n)\)
表示小于n的正整数与n互质的数的个数.
显然有:
当n为质数时 \(\varphi(n)\)
当n为奇数时 \(\varphi(2n) = \varphi(n)\)
证明：
\(\because\)欧拉函数为积性函数.
\(\therefore \varphi(2n) = \varphi(2) \ast \varphi(n)\)
\(\because \varphi(2)=1\)
\(\therefore \varphi(2n) = \varphi(n)\)
证明欧拉函数的积性证明.
条件是m与n互质
可以得到\(\phi(mn) = \phi(m) \ast \phi(n)\)
证明：
\(m = p_1^{a_1}p_2^{a_2}...p_k^{a_k}\)
\(\phi (m) = m(1- \frac {1}{p_1})(1- \frac {1}{p_2})...(1- \frac {1}{p_k})\)
\(n = p_1'^{a_1'}p_2'^{a_2'}...p_k'^{a_k'}\)
\(\phi(n) = n(1- \frac {1}{p_1'})(1- \frac {1}{p_2'})...(1- \frac {1}{p_k'})\)
\(\because m与n互质\)
\(\therefore p_1,p_2...p_k与p_1'p_2'...p_k'\)两两互不相同
\(\therefore \phi(mn) = mn(1- \frac {1}{p_1})(1- \frac {1}{p_2})...(1- \frac {1}{p_k})(1- \frac {1}{p_1'})(1- \frac {1}{p_2'})...(1- \frac {1}{p_k'})\)
\(\therefore \phi(mn) = \phi(m) \ast \phi(n)\)