欧拉降幂公式与证明

欧拉降幂公式与证明
转载自D-Tesla

欧拉降幂公式
$$A^K\equiv A^{K\%\varphi (m)+\varphi (m)}(modm)K>\varphi (m)$$
证明
今天在牛客多校的群里看一个数学大佬写的证明，不过是拍照，我决定动手自己写一下

$$A^K\equiv A^{K\%\varphi (m)+\varphi (m)}(modm)K>\varphi (m)(1)$$
证明如下
1 若 $(A,m)=1$，根据欧拉定理 $A^{\varphi (m)}\equiv 1(modm)$,即可轻易得证
2 若 $(A,m)\ne 1$,证明如下
设$K=a\ast \varphi (m)+c$ $a\ge 1,0\le c<\varphi (m)$
那么欧拉降幂公式就是
$$A^K\equiv A^{a\ast \varphi (m)+c}\equiv A^{\varphi (m)+c}(modm)(2)$$

即 证
$$A^{a\ast \varphi (m)}\equiv A^{\varphi (m)}(modm)$$

即 证
$$A^{2\ast \varphi (m)}\equiv A^{\varphi (m)}(modm)$$

移项
$$A^{\varphi (m)}(A^{\varphi (m)}-1)\equiv 0(modm)$$

即证
$$m|A^{\varphi (m)}(A^{\varphi (m)}-1)(3)$$
若有
$$(\frac{m}{(m,A^{\varphi (m)})},A)=1(4)$$
根据欧拉定理
$$A^{\varphi (m)}\equiv A^{k\ast \varphi (\frac{m}{(m,A^{\varphi (m)})})}\equiv {(A^{\varphi (\frac{m}{(m,A^{\varphi (m)})})})}^k\equiv 1(mod(\frac{m}{(m,A^{\varphi (m)})})$$
其中$k\ge 1$
移项即得 $\frac{m}{(m,A^{\varphi (m)})}|(A^{\varphi (m)}-1)$
同时乘 $(m,A^{\varphi (m)})$
即 $m|(m,A^{\varphi (m)})\ast (A^{\varphi (m)}-1)$
即 $m|A^{\varphi (m)}(A^{\varphi (m)}-1)$
就是 式 3

所以证明 式子 4
$$(\frac{m}{(m,A^{\varphi (m)})},A)=1$$
就好了
进行素因子分解
$$A=p_1^{a_1}\ast p_2^{a_2}\ast ....\ast p_{t1}^{a_{t1}}\ast q_1^{b_1}\ast q_2^{b_2}\ast ...\ast q_{t2}^{b_{t2}}$$

$$m=p_1^{c_1}\ast p_2^{c_2}\ast ....\ast p_{t1}^{c_{t1}}\ast r_1^{d_1}\ast r_2^{d_2}\ast ...\ast r_{t3}^{d_{t3}}3$$

$$(A,m)=p_1^{min(a_1,c_1)}\ast p_2^{min(a_2,c_2)}\ast ....\ast p_{t1}^{min(a_{t1},c_{t1})}$$

$$(A^{\varphi (m)},m)=p_1^{min(a_1\ast \varphi (m),c_1)}\ast p_2^{min(a_2\ast \varphi (m),c_2)}\ast ....\ast p_{t1}^{min(a_{t1}\ast \varphi (m),c_{t1})}$$

欧拉函数 $\varphi (m)=p_1^{c_1-1}\ast p_2^{c_2-1}\ast ....\ast p_{t1}^{c_{t1}-1}(p_1-1)\ast (p_2-1)\ast ....\ast (p_{t1}-1)$
$a_i\ast \varphi (m)\ge a_i\ast p_i^{c_i-1}\ast (p_i-1)\ge p_i^{c_i-1}\ast (p_i-1)\ge p_i^{c_i-1}$
证明
$$p_i^{c_i-1}\ge c_i(6)$$
若 $c_i=1$ ,成立

令 $f(x)=\frac{ln(x)}{x-1}$
在 $[3,\infty ]$ 单调减\
又有
$$f(2)=ln2<2\le p_i$$

$$f(3)=ln3/2<2\le p_i$$

于是有 式子6 成立
于是有
$$(A^{\varphi (m)},m)=p_1^{min(a_1\ast \varphi (m),c_1)}\ast p_2^{min(a_2\ast \varphi (m),c_2)}\ast ....\ast p_{t1}^{min(a_{t1}\ast \varphi (m),c_{t1})}=p_1^{c_1}\ast p_2^{c_2}\ast ....\ast p_{t1}^{c_{t1}}$$

$$\frac{m}{(A^{\varphi (m)},m)}=q_1^{b_1}\ast q_2^{b_2}\ast ...\ast q_{t2}^{b_{t2}}$$

式子 4
$$(\frac{m}{(m,A^{\varphi (m)})},A)=1$$

得证
式子 3
$$m|A^{\varphi (m)}(A^{\varphi (m)}-1)(3)$$
得证
式子 2
$$A^K\equiv A^{a\ast \varphi (m)+c}\equiv A^{\varphi (m)+c}(modm)(2)$$
得证
欧拉降幂公式得证
$$A^K\equiv A^{K\%\varphi (m)+\varphi (m)}(modm)K>\varphi (m)(1)$$