codeforce 17D Notepad
这题实际求的是\(a^b\ mod\ n\),但是\(gcd(a,n)\)[下面记作(a,n)]不一定等于1.......
(a,n)等于1时直接费马小定理能推倒...
(a,n)不为1时,有如下事实(当然互素时显然成立):
\[a^{2\phi (n)}\ mod\ n = a^{\phi (n)}\ mod\ n\]
即
\[a^{\phi (n)}(a^{\phi(n)}-1)\ mod \ n =0\]
简单证明如下:
1.如果\(a^{\phi (n)}\ mod \ n =0\),等式显然成立.
2.否则,命题等价于
\[\frac{a^{\phi (n)} }{(a^{\phi (n)},n)} (a^{\phi (n)}-1) \ mod \ \frac{n}{(a^{\phi (n)} ,n)} = 0\]
即:
\[a^{\phi (n)}\ mod \ \frac{n}{(a^{\phi (n)} ,n)} = 1\]
到这里就能使用费马小定理了~
回到题目上:\(a^b\ mod\ n\)$实际上分两种情况:
1.\(b<\phi (n)\),此时直接算
2.else,\(a^b\ mod\ n=a^{b\ mod\ \phi (n)+\phi (n)}\ mod\ n\)
对任意(?)a,n成立
2022年9月28日 13:24
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