关于莫比乌斯函数的应用1

include

include

include

include

include

using namespace std;

// 快速幂算法：计算 (a^b) % mod
long long fast_power(long long a, long long b, long long mod) {
long long result = 1;
a = a % mod;

while (b > 0) {
    if (b & 1) {
        result = (result * a) % mod;
    }
    a = (a * a) % mod;
    b = b >> 1;
}
return result;

}

// 判断一个数是否为素数
bool is_prime(long long n) {
if (n <= 1) return false;
if (n <= 3) return true;
if (n % 2 == 0 || n % 3 == 0) return false;

for (long long i = 5; i * i <= n; i += 6) {
    if (n % i == 0 || n % (i + 2) == 0) {
        return false;
    }
}
return true;

}

// 计算欧拉函数 φ(n)
long long euler_phi(long long n) {
long long result = n;

for (long long p = 2; p * p <= n; ++p) {
    if (n % p == 0) {
        while (n % p == 0) {
            n /= p;
        }
        result -= result / p;
    }
}

if (n > 1) {
    result -= result / n;
}

return result;

}

// 获取n的所有质因数
vector get_prime_factors(long long n) {
vector factors;

for (long long p = 2; p * p <= n; ++p) {
    if (n % p == 0) {
        factors.push_back(p);
        while (n % p == 0) {
            n /= p;
        }
    }
}

if (n > 1) {
    factors.push_back(n);
}

return factors;

}

// 判断g是否是模n的原根
bool is_primitive_root(long long g, long long n) {
if (gcd(g, n) != 1) {
return false;
}

long long phi_n = euler_phi(n);
vector<long long> prime_factors = get_prime_factors(phi_n);

for (long long factor : prime_factors) {
    if (fast_power(g, phi_n / factor, n) == 1) {
        return false;
    }
}

return true;

}

// 找到模n的所有原根
vector find_primitive_roots(long long n) {
vector primitive_roots;

// 只有特定的数有原根：1, 2, 4, p^k, 2p^k，其中p是奇素数
if (n == 1) {
    primitive_roots.push_back(1);
    return primitive_roots;
}
if (n == 2) {
    primitive_roots.push_back(1);
    return primitive_roots;
}
if (n == 4) {
    primitive_roots.push_back(3);
    return primitive_roots;
}

// 检查n是否是p^k或2p^k的形式
bool has_primitive_root = false;

if (is_prime(n)) {
    has_primitive_root = true;
} else {
    // 检查是否是p^k
    long long temp = n;
    long long p = -1;
    for (long long i = 2; i * i <= temp; ++i) {
        if (temp % i == 0) {
            p = i;
            while (temp % i == 0) {
                temp /= i;
            }
            break;
        }
    }
    
    if (temp > 1 && p == -1) {
        p = temp;
        temp = 1;
    }
    
    if (temp == 1 && is_prime(p) && p % 2 == 1) {
        has_primitive_root = true;
    } else if (n % 2 == 0) {
        // 检查是否是2p^k
        long long half_n = n / 2;
        temp = half_n;
        p = -1;
        for (long long i = 2; i * i <= temp; ++i) {
            if (temp % i == 0) {
                p = i;
                while (temp % i == 0) {
                    temp /= i;
                }
                break;
            }
        }
        
        if (temp > 1 && p == -1) {
            p = temp;
            temp = 1;
        }
        
        if (temp == 1 && is_prime(p) && p % 2 == 1) {
            has_primitive_root = true;
        }
    }
}

if (!has_primitive_root) {
    return primitive_roots;
}

long long phi_n = euler_phi(n);
vector<long long> prime_factors = get_prime_factors(phi_n);

// 寻找最小的原根
long long min_primitive_root = -1;
for (long long g = 2; g < n; ++g) {
    if (gcd(g, n) != 1) {
        continue;
    }
    
    bool is_root = true;
    for (long long factor : prime_factors) {
        if (fast_power(g, phi_n / factor, n) == 1) {
            is_root = false;
            break;
        }
    }
    
    if (is_root) {
        min_primitive_root = g;
        break;
    }
}

if (min_primitive_root == -1) {
    return primitive_roots;
}

// 找到所有原根
primitive_roots.push_back(min_primitive_root);

// 其他原根是 g^k，其中gcd(k, φ(n)) = 1
for (long long k = 2; k < phi_n; ++k) {
    if (gcd(k, phi_n) == 1) {
        primitive_roots.push_back(fast_power(min_primitive_root, k, n));
    }
}

sort(primitive_roots.begin(), primitive_roots.end());

return primitive_roots;

}

// 测试函数
int main() {
cout << "求原根的程序" << endl;
cout << "============" << endl;

vector<long long> test_cases = {7, 11, 17, 19, 23, 29, 31};

for (long long n : test_cases) {
    cout << "\n模 " << n << " 的原根：" << endl;
    
    vector<long long> primitive_roots = find_primitive_roots(n);
    
    if (primitive_roots.empty()) {
        cout << "  模 " << n << " 没有原根" << endl;
    } else {
        cout << "  原根数量: " << primitive_roots.size() << endl;
        cout << "  原根列表: ";
        for (size_t i = 0; i < primitive_roots.size(); ++i) {
            cout << primitive_roots[i];
            if (i != primitive_roots.size() - 1) {
                cout << ", ";
            }
        }
        cout << endl;
        
        // 验证最小的原根
        if (!primitive_roots.empty()) {
            long long g = primitive_roots[0];
            long long phi_n = euler_phi(n);
            cout << "  验证最小原根 " << g << ":" << endl;
            cout << "  ";
            for (long long i = 1; i <= phi_n; ++i) {
                cout << fast_power(g, i, n);
                if (i != phi_n) cout << ", ";
            }
            cout << endl;
        }
    }
}

// 用户输入测试
cout << "\n请输入一个数来求其原根（输入0退出）: ";
long long n;
while (cin >> n && n != 0) {
    vector<long long> primitive_roots = find_primitive_roots(n);
    
    if (primitive_roots.empty()) {
        cout << "模 " << n << " 没有原根" << endl;
    } else {
        cout << "模 " << n << " 的原根数量: " << primitive_roots.size() << endl;
        cout << "原根列表: ";
        for (size_t i = 0; i < primitive_roots.size(); ++i) {
            cout << primitive_roots[i];
            if (i != primitive_roots.size() - 1) {
                cout << ", ";
            }
        }
        cout << endl;
    }
    
    cout << "\n请输入一个数来求其原根（输入0退出）: ";
}

return 0;

}