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法律研究C++ 中的欧拉四平方恒等式
在这个问题中，我们得到两个数字，我们需要使用欧拉四平方恒等式来找到这两个数字的乘积。
欧拉四平方恒等式是一种寻找两个数字乘积的方法，如果这两个数字可以表示为四个平方数之和，则可以使用该方法。
如果 a * b 可以表示为四个平方数之和，则：
a = x12 + x22 + x32 + x42
b = y12 + y22 + y32 + y42
a * b = z12 + z22 + z32 + z42
让我们举个例子来理解这个问题：
输入：
a = 54 = 22 + 32 + 42 + 52
b = 10 = 12 + 22 + 12 + 22
输出：1*1 + 1*1 + 3*3 + 23*23
解释：
a 和 b 的乘积 = 540，
可以有多种表示方式。这里我们将考虑一种解法：
540 = 1*1 + 1*1 + 3*3 + 23*23 = 1 + 1 + 9 + 529
解决方案方法：
解决这个问题的一个简单方法是使用试错法，尝试每四个平方的组合来解决问题。为此，我们将使用嵌套循环，每个平方值一个嵌套循环。然后找到计算出的值并打印所有可能的和表示组合。
这种解决方案有点复杂，时间复杂度为
O( (a*b)4 )。
程序说明了我们解决方案的工作原理：
示例
#include <bits/stdc++.h>
using namespace std;
void findEulerSquareNumberValue(int a, int b) {
int prod = a * b;
int sumSquare = 0;
for (int x = 0; x <= sqrt(prod); x++) {
for (int y = x; y <= sqrt(prod); y++) {
for (int z = y; z <= sqrt(prod); z++) {
for (int k = z; k <= sqrt(prod); k++) {
sumSquare = (x*x) + (y*y) + (z*z) + (k*k);
if (sumSquare == prod) {
cout<<"The product "<<a<<"*"<<b<<" = "<<prod<<" represented as ";
cout<<x<<"*"<<x<<" + ";
cout<<y<<"*"<<y<<" + ";
cout<<z<<"*"<<z<<" + ";
cout<<k<<"*"<<k<<endl;
cout<<endl;
}
}
}
}
}
}
int main() {
int a = (2*2) + (3*3) + (4*4) + (5*5);
int b = (1*1) + (2*2) + (1*1) + (2*2);
cout<<"a = (2*2) + (3*3) + (4*4) + (5*5) = "<<a<<endl;
cout<<"b = (1*1) + (2*2) + (1*1) + (2*2) = "<<b<<endl;
findEulerSquareNumberValue(a, b);
return 0;
}输出：
a = (2*2) + (3*3) + (4*4) + (5*5) = 54 b = (1*1) + (2*2) + (1*1) + (2*2) = 10 The product 54*10 = 540 represented as 1*1 + 1*1 + 3*3 + 23*23 The product 54*10 = 540 represented as 1*1 + 3*3 + 13*13 + 19*19 The product 54*10 = 540 represented as 1*1 + 5*5 + 15*15 + 17*17 The product 54*10 = 540 represented as 1*1 + 7*7 + 7*7 + 21*21 The product 54*10 = 540 represented as 1*1 + 9*9 + 13*13 + 17*17 The product 54*10 = 540 represented as 2*2 + 4*4 + 6*6 + 22*22 The product 54*10 = 540 represented as 2*2 + 4*4 + 14*14 + 18*18 The product 54*10 = 540 represented as 2*2 + 6*6 + 10*10 + 20*20 The product 54*10 = 540 represented as 2*2 + 12*12 + 14*14 + 14*14 The product 54*10 = 540 represented as 3*3 + 3*3 + 9*9 + 21*21 The product 54*10 = 540 represented as 3*3 + 7*7 + 11*11 + 19*19 The product 54*10 = 540 represented as 3*3 + 9*9 + 15*15 + 15*15 The product 54*10 = 540 represented as 3*3 + 11*11 + 11*11 + 17*17 The product 54*10 = 540 represented as 4*4 + 10*10 + 10*10 + 18*18 The product 54*10 = 540 represented as 5*5 + 5*5 + 7*7 + 21*21 The product 54*10 = 540 represented as 5*5 + 11*11 + 13*13 + 15*15 The product 54*10 = 540 represented as 6*6 + 6*6 + 12*12 + 18*18 The product 54*10 = 540 represented as 7*7 + 7*7 + 9*9 + 19*19 The product 54*10 = 540 represented as 7*7 + 9*9 + 11*11 + 17*17 The product 54*10 = 540 represented as 9*9 + 11*11 + 13*13 + 13*13 The product 54*10 = 540 represented as 10*10 + 10*10 + 12*12 + 14*14
另一种解决此问题的方法是使用三个嵌套循环并检查第四个值，如果剩余数字是完全平方数，则其平方根就是第四个数，否则该解不存在。这种方法可能会去掉第四个循环，使解决方案更有效。
程序说明了我们解决方案的工作原理：
示例
#include <bits/stdc++.h>
using namespace std;
void findEulerSquareNumberValue(int a, int b) {
int prod = a * b;
int sumSquare = 0;
for (int x = 0; x <= sqrt(prod); x++) {
for (int y = x; y <= sqrt(prod); y++) {
for (int z = y; z <= sqrt(prod); z++) {
sumSquare = (x*x) + (y*y) + (z*z);
float k = sqrt(prod - sumSquare);
if ( floor(k) == ceil(k)) {
cout<<"The product "<<a<<"*"<<b<<" = "<<prod<<" represented as ";
cout<<x<<"*"<<x<<" + ";
cout<<y<<"*"<<y<<" + ";
cout<<z<<"*"<<z<<" + ";
cout<<k<<"*"<<k<<endl;
cout<<endl;
}
}
}
}
}
int main() {
int a = (2*2) + (3*3) + (4*4) + (5*5);
int b = (1*1) + (2*2) + (1*1) + (2*2);
cout<<"a = (2*2) + (3*3) + (4*4) + (5*5) = "<<a<<endl;
cout<<"b = (1*1) + (2*2) + (1*1) + (2*2) = "<<b<<endl;
findEulerSquareNumberValue(a, b);
return 0;
}输出：
a = (2*2) + (3*3) + (4*4) + (5*5) = 54 b = (1*1) + (2*2) + (1*1) + (2*2) = 10 The product 54*10 = 540 represented as 1*1 + 1*1 + 3*3 + 23*23 The product 54*10 = 540 represented as 1*1 + 1*1 + 23*23 + 3*3 The product 54*10 = 540 represented as 1*1 + 3*3 + 13*13 + 19*19 The product 54*10 = 540 represented as 1*1 + 3*3 + 19*19 + 13*13 The product 54*10 = 540 represented as 1*1 + 3*3 + 23*23 + 1*1 The product 54*10 = 540 represented as 1*1 + 5*5 + 15*15 + 17*17 The product 54*10 = 540 represented as 1*1 + 5*5 + 17*17 + 15*15 The product 54*10 = 540 represented as 1*1 + 7*7 + 7*7 + 21*21 The product 54*10 = 540 represented as 1*1 + 7*7 + 21*21 + 7*7 The product 54*10 = 540 represented as 1*1 + 9*9 + 13*13 + 17*17 The product 54*10 = 540 represented as 1*1 + 9*9 + 17*17 + 13*13 The product 54*10 = 540 represented as 1*1 + 13*13 + 17*17 + 9*9 The product 54*10 = 540 represented as 1*1 + 13*13 + 19*19 + 3*3 The product 54*10 = 540 represented as 1*1 + 15*15 + 17*17 + 5*5 The product 54*10 = 540 represented as 2*2 + 4*4 + 6*6 + 22*22 The product 54*10 = 540 represented as 2*2 + 4*4 + 14*14 + 18*18 The product 54*10 = 540 represented as 2*2 + 4*4 + 18*18 + 14*14 The product 54*10 = 540 represented as 2*2 + 4*4 + 22*22 + 6*6 The product 54*10 = 540 represented as 2*2 + 6*6 + 10*10 + 20*20 The product 54*10 = 540 represented as 2*2 + 6*6 + 20*20 + 10*10 The product 54*10 = 540 represented as 2*2 + 6*6 + 22*22 + 4*4 The product 54*10 = 540 represented as 2*2 + 10*10 + 20*20 + 6*6 The product 54*10 = 540 represented as 2*2 + 12*12 + 14*14 + 14*14 The product 54*10 = 540 represented as 2*2 + 14*14 + 14*14 + 12*12 The product 54*10 = 540 represented as 2*2 + 14*14 + 18*18 + 4*4 The product 54*10 = 540 represented as 3*3 + 3*3 + 9*9 + 21*21 The product 54*10 = 540 represented as 3*3 + 3*3 + 21*21 + 9*9 The product 54*10 = 540 represented as 3*3 + 7*7 + 11*11 + 19*19 The product 54*10 = 540 represented as 3*3 + 7*7 + 19*19 + 11*11 The product 54*10 = 540 represented as 3*3 + 9*9 + 15*15 + 15*15 The product 54*10 = 540 represented as 3*3 + 9*9 + 21*21 + 3*3 The product 54*10 = 540 represented as 3*3 + 11*11 + 11*11 + 17*17 The product 54*10 = 540 represented as 3*3 + 11*11 + 17*17 + 11*11 The product 54*10 = 540 represented as 3*3 + 11*11 + 19*19 + 7*7 The product 54*10 = 540 represented as 3*3 + 13*13 + 19*19 + 1*1 The product 54*10 = 540 represented as 3*3 + 15*15 + 15*15 + 9*9 The product 54*10 = 540 represented as 4*4 + 6*6 + 22*22 + 2*2 The product 54*10 = 540 represented as 4*4 + 10*10 + 10*10 + 18*18 The product 54*10 = 540 represented as 4*4 + 10*10 + 18*18 + 10*10 The product 54*10 = 540 represented as 4*4 + 14*14 + 18*18 + 2*2 The product 54*10 = 540 represented as 5*5 + 5*5 + 7*7 + 21*21 The product 54*10 = 540 represented as 5*5 + 5*5 + 21*21 + 7*7 The product 54*10 = 540 represented as 5*5 + 7*7 + 21*21 + 5*5 The product 54*10 = 540 represented as 5*5 + 11*11 + 13*13 + 15*15 The product 54*10 = 540 represented as 5*5 + 11*11 + 15*15 + 13*13 The product 54*10 = 540 represented as 5*5 + 13*13 + 15*15 + 11*11 The product 54*10 = 540 represented as 5*5 + 15*15 + 17*17 + 1*1 The product 54*10 = 540 represented as 6*6 + 6*6 + 12*12 + 18*18 The product 54*10 = 540 represented as 6*6 + 6*6 + 18*18 + 12*12 The product 54*10 = 540 represented as 6*6 + 10*10 + 20*20 + 2*2 The product 54*10 = 540 represented as 6*6 + 12*12 + 18*18 + 6*6 The product 54*10 = 540 represented as 7*7 + 7*7 + 9*9 + 19*19 The product 54*10 = 540 represented as 7*7 + 7*7 + 19*19 + 9*9 The product 54*10 = 540 represented as 7*7 + 7*7 + 21*21 + 1*1 The product 54*10 = 540 represented as 7*7 + 9*9 + 11*11 + 17*17 The product 54*10 = 540 represented as 7*7 + 9*9 + 17*17 + 11*11 The product 54*10 = 540 represented as 7*7 + 9*9 + 19*19 + 7*7 The product 54*10 = 540 represented as 7*7 + 11*11 + 17*17 + 9*9 The product 54*10 = 540 represented as 7*7 + 11*11 + 19*19 + 3*3 The product 54*10 = 540 represented as 9*9 + 11*11 + 13*13 + 13*13 The product 54*10 = 540 represented as 9*9 + 11*11 + 17*17 + 7*7 The product 54*10 = 540 represented as 9*9 + 13*13 + 13*13 + 11*11 The product 54*10 = 540 represented as 9*9 + 13*13 + 17*17 + 1*1 The product 54*10 = 540 represented as 9*9 + 15*15 + 15*15 + 3*3 The product 54*10 = 540 represented as 10*10 + 10*10 + 12*12 + 14*14 The product 54*10 = 540 represented as 10*10 + 10*10 + 14*14 + 12*12 The product 54*10 = 540 represented as 10*10 + 10*10 + 18*18 + 4*4 The product 54*10 = 540 represented as 10*10 + 12*12 + 14*14 + 10*10 The product 54*10 = 540 represented as 11*11 + 11*11 + 17*17 + 3*3 The product 54*10 = 540 represented as 11*11 + 13*13 + 13*13 + 9*9 The product 54*10 = 540 represented as 11*11 + 13*13 + 15*15 + 5*5 The product 54*10 = 540 represented as 12*12 + 14*14 + 14*14 + 2*2
更新于：2021年1月22日
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