SPOJ PON

题目链接：http://www.spoj.com/problems/PON/

题目大意：判断N是不是素数，N<2⁶⁴-1.

解题思路：需要用到拉宾-米勒素性判定。

（选自数论书籍）合数的拉宾-米勒测试：设n是奇素数，记n-1=2^kq，q为奇数。对不被n整除的某个a，如果下述两个条件都成立，则n是合数。

　　a): a^q !≡ 1 (mod n), 　　即a的q次方不与1模n同余

　　b): 对所有i = 0,1,2，……，k-1， a^{2^i * q} !≡ -1 (mod n) 

所以可以打出100个素数表然后迭代判定。其中需要用的：快速幂取模，大数乘取模。

代码：

const int maxn = 1e5 + 10;
VI primes;
bool vis[maxn];

ll mul_mod(ll a, ll b, ll m){
    ll ans = 0, base = a;
    while(b){
        if(b & 1) ans = (ans + base) % m;
        base %= m;
        base = base * 2 % m;
        b >>= 1;
    }
    return ans;
}
ll pow_mod(ll a, ll b, ll m){
    ll ans = 1, base = a % m;
    while(b){
        if(b & 1) ans = mul_mod(ans, base, m);
        base %= m;
        base = mul_mod(base, base, m);
        b >>= 1;
    }
    return ans % m;
}
void dowork(){
    memset(vis, 0, sizeof(vis));
    for(int i = 2; i < maxn / 2; i++)
        for(int j = i * 2; j > 0 && j < maxn; j += i)
            vis[j] = 1;
    for(int i = 2; i < 1e5; i++)
        if(!vis[i]) primes.push_back(i);
}
bool check(ll x){
    if(x == 2) return true;
    if(x % 2 == 0) return false;
    ll q = x - 1;
    int k = 0;
    while(!(q & 1)){
        k++; q >>= 1;
    } 
    bool flag = true;
    for(int i = 0; i < 100; i++){
        ll a = primes[i];
        if(a == x) return true;
        if(pow_mod(a, q, x) == 1) 
            continue;
        bool none = true;
        for(int j = 0; j < k; j++){
            ll qi = 1 << j;
            ll tmp = pow_mod(a, qi * q, x);
            if(tmp == x - 1){
                none = false;
                break;
            }
            else if(tmp == 1) 
                break;
        }
        if(none) {
            flag = false;
            break;
        }
    }
    return flag;
}
int main(){
    dowork();
    int t;
    scanf("%d", &t);
    while(t--){
        ll n;
        scanf("%lld", &n);
        if(check(n)) puts("YES");
        else puts("NO");
    }
}

题目：

PON - Prime or Not

#cryptography #prime-numbers-1 #primality-test

Given the number, you are to answer the question: "Is it prime?"

Solutions to this problem can be submitted in C, C++, Pascal, Perl, Python, Ruby, Lisp, Hask, Ocaml, Prolog, Whitespace, Brainf**k and Intercal only.

Input

t – the number of test cases, then t test cases follows. [t <= 500]
Each line contains one integer: N [2 <= N <= 2^63-1]

Output

For each test case output string "YES" if given number is prime and "NO" otherwise.

Example

Input:
5
2
3
4
5
6

Output:
YES
YES
NO
YES
NO

 Submit solution!