數論的東西太多了,決定把一些遇到的定理在這裏整理一下
L(x) 求的是 LCM(1,2,3.....x)
LCM 求最小公倍數
L(1) = 1
L(x+1) = { L(x) * p if x+1 is a perfect power of prime p
{ L(x) otherwise
PI的定義方式
const double PI = acos(-1);
歐拉函數(求1~n與n互質的數的個數)
Φ(n) = n*(1 - 1/p1)*(1 - 1/p2)....(1 - 1/pn) pn is prime
關於 ax + by = 1 的必要條件是 gcd(a,b) = 1的簡單證明
假設a,b的最大公約數爲m,如果方程ax+by=1有解,那麼方程(a/m)x+(b/m)y=1/m.有整數解。
顯然不成立(a/m, b/m 是整數,整數加整數不等於小於1的數),所以假設不成立
關於取整的兩個函數
#include <math.h> double floor(double x);
float floorf(float x);
long double floorl(long double x);
double floor(double x);
double ceil(double x);
使用floor函數。floor(x)返回的是小於或等於x的最大整數。
如: floor(10.5) == 10 floor(-10.5) == -11
使用ceil函數。ceil(x)返回的是大於x的最小整數。
如: ceil(10.5) == 11 ceil(-10.5) ==-10
floor()是向負無窮大舍入,floor(-10.5) == -11;
ceil()是向正無窮大舍入,ceil(-10.5) == -10
不得不說點:/ % 四捨五入 向上取整(ceil()) 向下取整(floor)
1. /
//Test "/"
cout << "Test \"/\"!" << endl;
cout << "7 / 2 = " << 7/2 << endl; ///3
cout << "7 / 2.0 = " << 7/2.0 << endl; ///3.5
cout << "7.0 / 2 = " << 7.0/2 << endl; ///3.5
cout << "7.0 / 2.0 = " << 7.0/2.0 << endl; ///3.5
cout << "7 / 3 = " << 7/3 << endl; ///2
cout << endl;
2. %
///Test "%"
cout << "Test \"%\"!" << endl;
cout << "9 % 3 = " << 9%3 << endl; ///0
cout << "9 % 4 = " << 9%4 << endl; ///1
///cout << "9.0 % 3 = " << 9.0%3 << endl;
///cout << "9 % 3.0 = " << 9%3.0 << endl;
cout << endl;
3. 四捨五入
///Test round()
cout << "Test \"四捨五入\"!" << endl;
double dRoundA = 1.4;
double dRoundB = 1.6;
double dRoundLowA = -1.4;
double dRoundLowB = -1.6;
double dRoundLowC = 0.0;
cout << dRoundA << " = " << RoundEx(dRoundA) << endl; ///1
cout << dRoundB << " = " << RoundEx(dRoundB) << endl; ///2
cout << dRoundLowA << " = " << RoundEx(dRoundLowA) << endl; ///-1
cout << dRoundLowB << " = " << RoundEx(dRoundLowB) << endl; ///-2
cout << dRoundLowC << " = " << RoundEx(dRoundLowC) << endl; ///0
cout << endl;
double RoundEx(const double& dInput)
{
double dIn = dInput;
if (dInput >= 0.0) ///???
{
return int(dIn + 0.5);
}
else
{
return int(dIn - 0.5);
}
}
4. ceil() 向上取整
///Test ceil() 向上取整
cout << "Test ceil() 向上取整!" << endl;
cout << "ceil 1.2 = " << ceil(1.2) << endl; ///2
cout << "ceil 1.8 = " << ceil(1.8) << endl; ///2
cout << "ceil -1.2 = " << ceil(-1.2) << endl; ///-1
cout << "ceil -1.8 = " << ceil(-1.8) << endl; ///-1
cout << "ceil 0.0 = " << ceil(0.0) << endl; ///0
cout << endl;
5. floor() 向下取整
///Test floor() 向下取整
cout << "Test floor() 向下取整!" << endl;
cout << "floor 1.2 = " << floor(1.2) << endl; ///1
cout << "floor 1.8 = " << floor(1.8) << endl; ///1
cout << "floor -1.2 = " << floor(-1.2) << endl; ///-2
cout << "floor -1.8 = " << floor(-1.8) << endl; ///-2
cout << "floor 0.0 = " << floor(0.0) << endl; ///0
cout << endl;
拓撲排序
如果in[](入度數組)中存在兩個以上的in[x] == 0,該拓撲排序的順序無法判斷
如果in[]判定完畢後,還存在in[x] != 0,該拓撲排序中存在衝突,即存在環