费马小定理—— HDU4704 Sum

费马小定理：

假如p是素数，且a与p互质，那么a^(p-1) = 1 (mod p)。

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应用:

说明了模素数的指数是有循环节的，且循环节长度为p-2，减少运算量2^n%m == ( 2^(n%(m-1))*    2^(n/(m-1)*(m-1))    )%m == (2^(n%(m-1)))%m * ((2^k)^(m-1))%m == (2^(n%(m-1)))%m * 1 == (2^(n%(m-1)))%m;

HDU4704 Sum

//求2^(n-1); n很大#include<iostream>#include<string>#include<vector>#include<algorithm>#include<queue>#include<cstdio>#include<cstring>#include<cmath>#include<map>using namespace std;typedef long long ll;const int INF_INT  = 0x3f3f3f3f;const ll INF_LL  = 1e18;const int MOD=1e9 + 7;string s;ll pow_mod(ll a,ll n,ll m){    ll re = 1;    while(n>0)    {        if(n%2)        {            re = re*a%m;        }        a = a*a%m;        n/=2;    }    return re;}int main(){    ios_base::sync_with_stdio(false);//    freopen("data.txt","r",stdin);    while(cin>>s)    {        ll sum = 0;        for(int i=0;i<s.size();i++)        {            sum = (sum*10+s[i]-'0')%(MOD-1);        }        cout << pow_mod(2,sum-1,MOD)<<endl;    }}