HDU 6608：Fansblog（威尔逊定理）

Fansblog

Time Limit: 2000/2000 MS (Java/Others) Memory Limit: 65536/65536 K (Java/Others)

Total Submission(s): 3170 Accepted Submission(s): 671

Problem Description

Farmer John keeps a website called ‘FansBlog’ .Everyday , there are many people visited this blog.One day, he find the visits has reached P , which is a prime number.He thinks it is a interesting fact.And he remembers that the visits had reached another prime number.He try to find out the largest prime number Q ( Q < P ) ,and get the answer of Q! Module P.But he is too busy to find out the answer. So he ask you for help. ( Q! is the product of all positive integers less than or equal to n: n! = n * (n-1) * (n-2) * (n-3) *… * 3 * 2 * 1 . For example, 4! = 4 * 3 * 2 * 1 = 24 )

Input

First line contains an number T(1<=T<=10) indicating the number of testcases.
Then T line follows, each contains a positive prime number P (1e9≤p≤1e14)

Output

For each testcase, output an integer representing the factorial of Q modulo P.

Sample Input

1
1000000007 

Sample Output

328400734

题意

给出一个素数P，找出小于P的最小素数Q，并计算Q的阶乘对P取模的结果

解决

从P-1开始进行素数检测，因为素数的分布是比较密集的，所以可以用试除法来判断素数。

在找到Q之后，由威尔逊定理可知：当P是素数的情况下，(P-1)! Ξ -1(mod P)

因为求的是Q! mod P，所以我们可以先将(P-1)! mod P求出来的，然后利用除法来计算Q! mod P

Code

#include <bits/stdc++.h>
#define ll long long
#define ull unsigned long long
#define ms(a,b) memset(a,b,sizeof(a))
const int inf=0x3f3f3f3f;
const ll INF=0x3f3f3f3f3f3f3f3f;
const int maxn=1e6+10;
const int mod=1e9+7;
const int maxm=1e3+10;
using namespace std;
bool check(ll n)
{
    for(ll i=2;i*i<=n;i++)
        if(n%i==0)
            return false;
    return true;
}
ll modmul(ll A,ll B,ll Mod)
{
    return (A*B-(ll)((long double)A*B/Mod)*Mod+Mod)%Mod;
}
ll Pow(ll a,ll b,ll c)
{
    ll ans=1;
    while(b)
    {
        if(b&1)
            ans=modmul(ans,a,c);
        b>>=1;
        a=modmul(a,a,c);
    }
    return ans;
}
ll inv(ll a,ll b)
{
    return Pow(a,b-2,b);
}
int main(int argc, char const *argv[])
{
    #ifndef ONLINE_JUDGE
        freopen("in.txt", "r", stdin);
        freopen("out.txt", "w", stdout);
        srand((unsigned int)time(NULL));
    #endif
    ios::sync_with_stdio(false);
    cin.tie(0);
    int t;
    ll p;
    cin>>t;
    while(t--)
    {
        cin>>p;
        ll num;
        for(ll i=p-1;;i--)
            if(check(i))
            {
                num=i;
                break;
            }
        ll ans=p-1;
        for(ll i=num+1;i<=p-1;i++)
            ans=modmul(ans,inv(i,p),p);
        cout<<ans<<endl;
    }
    #ifndef ONLINE_JUDGE
        cerr<<"Time elapsed: "<<1.0*clock()/CLOCKS_PER_SEC<<" s."<<endl;
    #endif
    return 0;
}