基础数论公式集合（无推导）

1.整除的性质

\[\text{if }(a|b) \text{ } \text{ ,} \text{ } b \% a =0 \text{ }(a \in factor \ b) \]

\[\text{if }(a!|b) \text{ } \text{ ,} \text{ } b \% a !=0 \text{ }(a \in factor \ b) \]

2.质因数分解

\[x=p_1^{q_1} \times p_2^{q_2} \times ...... \times p_n^{q_n} \text{ } (x,y \in composite,p \in prime) \]

\[y=p_1^{w_1} \times p_2^{w_2} \times ...... \times p_n^{w_n} \text{ } (x,y \in composite,p \in prime) \]

3.算术基本定理

\[gcd(x,y)=p_1^{min(q_1,w_1)} \times p_2^{min(q_2,w_2)} \times ...... \times p_n^{min(q_n,w_n)} \]

\[lcm(x,y)=p_1^{max(q_1,w_1)} \times p_2^{max(q_2,w_2)} \times ...... \times p_n^{max(q_n,w_n)} \]

4.欧拉函数

\[\varphi (n) = \sum_{i = 1}^{n} [ gcd(i,n)=1 ] \text{ }(n=p_1^{k_1} \times p_2^{k_2} \times ...... \times p_m^{q_m})\text{ ,cost }O(n) \text{ times} \]

\[= n \times [ \prod_{i=1}^{m} (1- \frac {1}{p_i})] \text{ }(n=p_1^{k_1} \times p_2^{k_2} \times ...... \times p_m^{q_m})\text{ ,cost }O(\sqrt {n}) \text{ times} \]

5.扩展欧拉函数

\[\varphi(n)=n-1 \text{ } (n \in prime) \]

\[\sum_{i = 1}^{n} i \times [ gcd(i,n)=1 ] \]

\[= n \times \frac{\varphi(n)}{2} \]

\[\varphi(a*b)=\varphi(a)*\varphi(b) \text{ }(gcd(a,b)=1) \]

\[\left \{\varphi(i \times p) \right \} \left\{\begin{matrix} \varphi(i) \times p ，p | i \\ \varphi(i) \times (p-1)，p !|i\end{matrix}\right. \text{ }(p \in prime) \]

\[\sum_{i|n} \varphi(i)=n \]

6.同余性质(线性运算)

\[ \left\{\begin{matrix} a \equiv b \pmod m \\ c \equiv d \pmod m \end{matrix}\right. \Rightarrow \left\{\begin{matrix} a\pm c \equiv b \pm d \\ ac \equiv bd \end{matrix}\right. \]

\[ \left\{\begin{matrix} a \equiv b \pmod m \\ a=a_1 \times d \\ b=b_1 \times d \\ gcd(d,m)=1\end{matrix}\right. \Rightarrow a_1 \equiv b_1 \pmod m \]

\[\left.\begin{matrix} a \equiv b \pmod {m_1} \\ a = a_1 \times d \\ b = b_1 \times d \\ m=m_1 \times d\end{matrix}\right\}\Rightarrow a_1 \equiv b_1 \pmod {m_1} \]

7.费马小定理

\[a^{p-1} \equiv 1\pmod p \text{ } (p \in prime,gcd(a,p)=1) \]

\[=a^{p-2} \equiv a^{-1} \pmod p \text{ } (p \in prime,gcd(a,p)=1) \]

8.裴蜀定理

\[a|b \text{^} a|c \Longrightarrow a|(xb+yc)\text{ }(\forall x,y \in N) \]

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