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C++でのオイラーの四平方アイデンティティ

この問題では、2つの数値が与えられ、オイラーの4平方アイデンティティを使用して数値の積を見つける必要があります。

オイラーの四平方アイデンティティ は、4つの正方形の合計を使用して表すことができる2つの数値の積を見つける方法です。 数が4つの正方形の合計として表すことができる場合は数の。

積a*bは、次の場合に4つの二乗の合計として表すことができます

a =x1 2 + x2 2 + x3 2 + x4 2
b =y1 2 + y2 2 + y3 2 + y4 2
a * b =z1 2 + z2 2 + z3 2 + z4 2

問題を理解するために例を見てみましょう

入力：

a =54 =2 * 2 + 3 * 3 + 4 * 4 + 5 * 5

b =10 =1 * 1 + 2 * 2 + 1 * 1 + 2 * 2


出力： 1 * 1 + 1 * 1 + 3 * 3 + 23 * 23

説明：

aとbの積=540、

これは複数の方法で表すことができます。ここで1つの解決策を検討します

540 =1 * 1 + 1 * 1 + 3 * 3 + 23 * 23 =1 + 1 + 9 + 529

ソリューションアプローチ-

問題の簡単な解決策は、問題を解決するために4つの正方形の組み合わせのそれぞれを試すための試行方法を使用することです。このために、ネストされたループを使用します。正方形の値ごとに1つのネストされたループです。次に、計算された値を見つけて、合計表現のすべての可能な組み合わせを出力します。

このソリューションは少し複雑であり、時間の複雑さはオーダーです
O（（a * b） 4 ）。

ソリューションの動作を説明するプログラム

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;
}

出力-

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

別の 時間計算量を減らして問題を解決する方法は、3つのネストされたループを使用し、4番目の値をチェックすることです。残りの数が完全な平方である場合、ルート値は4番目の数であり、そうでない場合、解決は不可能です。この方法では、4番目のループが削除され、ソリューションが効果的になる可能性があります。

ソリューションの動作を説明するプログラム

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;
}

出力-

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
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