ACM数论模板

首先是最经典的素数判定问题，先贴上基本的素数判定算法和米勒拉宾测试算法。

//1：基本素数判定算法
//判断n是否为素数，只需要将i从2枚举到sqrt(n)。
//理论支持： 若n是合数,则必然可以分解成两数之积,而
//             这两个数不可能同时大于sqrt(n),则n必有
//           一个小于等于sqrt(n)的约数。
bool isPrime_fundamental( int n )
{
    int r = sqrt(n) + 1e-6;                            
    for ( int i = 2; i <= r; i++ )                    /*这样最快，1e-6避免误差
    //for ( int i = 2; i <= sqrt(n) + 1e-6; i++ )    /*由于每次sqrt(n)都会计算，所以比较慢*/
    //for ( int i = 2; i * i <= n; i++ )            /*由于每次都会计算i*i，所以最慢*/
    {
        if ( n % i == 0 ) return false;
    }
    return true;
}

//2：米勒拉宾测试算法
//快速幂求a^b%mod
int pow_mod( int a, int b, int mod )
{
    int ans = 1, w = a % mod;
    while ( b )
    {
        if ( b & 1 )
        {
            ans = ( long long ) ans * w % mod;
        }
        b = b >> 1;
        w = ( long long ) w * w % mod;
    }
    return ans;
}

//米勒拉宾大法
bool mr( int n, int a )
{
    if ( n % a == 0 ) return false;
    int r = 0, s = n - 1;
    while ( ( s & 1 ) == 0 )
    {
        s = s >> 1;
        r++;
    }
    int k = pow_mod( a, s, n );
    if ( k == 1 ) return true;
    for ( int j = 0; j < r; j++, k = ( long long ) k * k % n )
    {
        if ( k == n - 1 ) return true;    //I don't understand!
    }
    return false;
}

//判断素数
bool isPrime( int n )
{
    if ( n == 2 || n == 3 || n == 5 || n == 7 ) return true;
    if ( n & 1 == 0 ) return false;
    int tab[] = { 2, 3, 5, 7 };
    for ( int i = 0; i < 4; i++ )
    {
        if ( !mr( n, tab[i] ) ) return false;
    }
    return true;
}

常用的模板集合：

//update date: 2015.7.27
#include <algorithm>
#include <iostream>
#include <cstring>
#include <cstdio>
#include <cmath>
using namespace std;

typedef long long ll;
const int N = 10000001;
const int M = 1000001;
bool visit[N];
int prime[M];
int pn;

void sieve( int n )
{
    int r = sqrt( n + 0.5 );
    memset( visit, 0, sizeof(visit) );
    visit[0] = visit[1] = 1;
    for ( int i = 2; i <= r; i++ )
    {
        if ( !visit[i] )
        {
            for ( int j = i * i; j <= n; j += i )
            {
                visit[j] = 1;
            }
        }
    }
}

void get_prime( int n )
{
    sieve(n);
    pn = 0;
    for ( int i = 0; i <= n; i++ )
    {
        if ( !visit[i] )
        {
            prime[pn++] = i;
        }
    }
}

void better_get_prime()
{
    pn = 0;
    memset( visit, 0, sizeof(visit) );
    visit[0] = visit[1] = 1;
    for ( int i = 2; i < N; i++ )
    {
        if ( !visit[i] ) prime[pn++] = i;
        for ( int j = 0; j < pn && ( ll ) i * prime[j] < N; j++ )
        {
            visit[i * prime[j]] = 1;
            if ( i % prime[j] == 0 ) break;
        }
    }
}

ll gcd( ll a, ll b )
{
    return b ? gcd( b, a % b ) : a;
}

ll extgcd( ll a, ll b, ll &d, ll &x, ll &y )
{
    if ( !b )
    {
        d = a, x = 1, y = 0;
    }
    else
    {
        extgcd( b, a % b, d, y, x );
        y -= x * ( a / b );
    }
}

ll inv( ll a, ll mod )
{
    ll d, x, y;
    extgcd( a, mod, d, x, y );
    return d == 1 ? ( x + mod ) % mod : -1;
}

ll pow_mod( ll a, ll n, ll mod )
{
    ll ans = 1, w = a % mod;
    while ( n )
    {
        if ( n & 1 )
        {
            ans = ans * w % mod;
        }
        w = w * w % mod;
        n >>= 1;
    }
    return ans;
}

int euler_phi( int n )
{
    int ans = n;
    for ( int i = 2; i * i <= n; i++ )
    {
        if ( n % i == 0 )
        {
            ans = ans / i * ( i - 1 );
            do
            {
                n = n / i;
            }
            while ( n % i == 0 );
        }
    }
    if ( n > 1 )
    {
        ans = ans / n * ( n - 1 );
    }
    return ans;
}

int phi[N];

void phi_table( int n )
{
    for ( int i = 1; i <= n; i++ )
    {
        phi[i] = i;
    }
    for ( int i = 2; i <= n; i++ )
    {
        if ( phi[i] == i )
        {
            phi[i] = i - 1;
            for ( int j = i + i; j <= n; j += i )
            {
                phi[j] = phi[j] / i * ( i - 1 );
            }
        }
    }
}