欧拉的四方形身份

根据欧拉的四个平方恒等式，如果a和b两者都可以分别表示为四个平方的和，则任何两个数字a和b的乘积可以表示为四个平方的和。

数学上，如果a = $c1^2 + c2^2 + c3^2 + c4^2$和b = $d1^2 + d2^2 + d3^2 + d4^2$
然后，a * b = $e1^2 + e2^2 + e3^2 + e4^2$
其中c1，c2，c3，c4，d1，d2，d3，d4，e1，e2，e3，e4是任何整数。

一些示例是，a = $1^2 + 2^2 + 3^2 + 4^2$ = 30头= $1^2 + 1^2 + 1^2 + 1^2$ = 4 ab = a * b = 120 = $2^2 + 4^2 + 6^2 + 8^2$ a = $1^2 + 2^2 + 3^2 + 1^2$ = 15羽= $2^2 + 3^2 + 4^2 + 5^2$ = 24 ab = a * b = 810 = $1^2 + 4^2 + 8^2 + 27^2$ a = $1^2 + 2^2 + 3^2 + 1^2$ = 15羽= $2^2 + 3^2 + 2^2 + 3^2$ = 26 ab = a * b = 390 = $4^2 + 7^2 + 10^2 + 15^2$

例子：

Input: a = 1 * 1 + 2 * 2 + 3 * 3 + 4 * 4
            b = 1 * 1 + 1 * 1 + 1 * 1 + 1 * 1
  
Output: i = 0
j = 2
k = 4
l = 10
Product of 30 and 4 can be written as sum of squares of i, j, k, l
120 = 0 * 0 + 2 * 2 + 4 * 4 + 10 * 10

i = 2
j = 4
k = 6
l = 8
Product of 30 and 4 can be written as sum of squares of i, j, k, l
120 = 2 * 2 + 4 * 4 + 6 * 6 + 8 * 8

解释 ：
2个数字a(30)和b(4)的乘积可以表示为4个平方的和，如欧拉的四个平方相同。上面是乘积a * b的2个表示，以4个平方的形式表示。显示了乘积a * b的所有可能表示，并以四个平方的形式表示。

Input: a = 1*1 + 2*2 + 3*3 + 1*1
       b = 1*1 + 2*2 + 1*1 + 1*1

Output: i = 0
j = 1
k = 2
l = 10
Product of 15 and 7 can be written as sum of squares of i, j, k, l
105 = 0*0 + 1*1 + 2*2 + 10*10

i = 0
j = 4
k = 5
l = 8
Product of 15 and 7 can be written as sum of squares of i, j, k, l
105 = 0*0 + 4*4 + 5*5 + 8*8

i = 1
j = 2
k = 6
l = 8
Product of 15 and 7 can be written as sum of squares of i, j, k, l
105 = 1*1 + 2*2 + 6*6 + 8*8

i = 2
j = 2
k = 4
l = 9
Product of 15 and 7 can be written as sum of squares of i, j, k, l
105 = 2*2 + 2*2 + 4*4 + 9*9

i = 2
j = 4
k = 6
l = 7
Product of 15 and 7 can be written as sum of squares of i, j, k, l
105 = 2*2 + 4*4 + 6*6 + 7*7

i = 3
j = 4
k = 4
l = 8
Product of 15 and 7 can be written as sum of squares of i, j, k, l
105 = 3*3 + 4*4 + 4*4 + 8*8

方法 ：

蛮力：
通过使用4个循环i，j，k，l查找四个平方中的每个平方，可以以4个平方的形式表示给定的数字(a * b)。这给出了所有可能的组合，以形成a * b作为四个平方的总和。在最内层循环(l循环)的每次迭代中，用乘积a * b检查总和。如果存在匹配项，则打印平方和等于a * b的4个数字(i，j，k和l)。

C++

// CPP code to verify euler's four square identity
#include 
  
using namespace std;
  
#define show(x) cout << #x << " = " << x << "\n";
  
// function to check euler four square identity
void check_euler_four_square_identity(int a, int b,
                                      int ab)
{
    int s = 0;
      
    // loops checking the sum of squares
    for (int i = 0;i * i <= ab;i ++)
    {
        s = i * i;
        for (int j = i;j * j <= ab;j ++)
        {
            // sum of 2 squares
            s = j * j + i * i;
              
            for (int k = j;k * k <= ab;k ++)
            {
                // sum of 3 squares
                s = k * k + j * j + i * i;
                  
                for (int l = k;l * l <= ab;l ++)
                {
                    // sum of 4 squares
                    s = l * l + k * k + j * j + i * i;
  
                    // product of 2 numbers represented
                    // as sum of four squares i, j, k, l 
                    if (s == ab)
                    {
                        // product of 2 numbers a and b
                        // represented as sum of four 
                        // squares i, j, k, l 
                        show(i);
                        show(j);
                        show(k);
                        show(l);
                        cout <<"" 
                        << "Product of " << a
                        << " and " << b;
                        cout << " can be written"<<
                        " as sum of squares of i, "<<
                         "j, k, l\n";
                        cout << ab << " = ";
                        cout << i << "*" << i << " + ";
                        cout << j << "*" << j << " + ";
                        cout << k << "*" << k << " + ";
                        cout << l << "*" << l << "\n";
                        cout << "\n";
                    }
                }
            }
        }
    }
}
  
// Driver code
int main() 
{
    // a and b such that they can be expressed 
    // as sum of squares of numbers
    int a = 30; // 1*1 + 2*2 + 3*3 + 4*4;
    int b = 4;  // 1*1 + 1*1 + 1*1 + 1*1;
  
    // given numbers can be represented as
    // sum of 4 squares By euler's four
    // square identity product also can be 
    // represented as sum of 4 squares
    int ab = a * b;
      
    check_euler_four_square_identity(a, b, ab);
      
    return 0;
}

Java

// Java code to verify euler's 
// four square identity
import java.io.*;
  
class GFG 
{
      
// function to check euler
// four square identity
static void check_euler_four_square_identity(int a, 
                                             int b,
                                             int ab)
{
    int s = 0;
      
    // loops checking the
    // sum of squares
    for (int i = 0; 
             i * i <= ab; i ++)
    {
        s = i * i;
        for (int j = i; 
                 j * j <= ab; j ++)
        {
            // sum of 2 squares
            s = j * j + i * i;
              
            for (int k = j;
                     k * k <= ab; k ++)
            {
                // sum of 3 squares
                s = k * k + j * 
                    j + i * i;
                  
                for (int l = k; 
                         l * l <= ab; l ++)
                {
                    // sum of 4 squares
                    s = l * l + k * k +
                        j * j + i * i;
  
                    // product of 2 numbers 
                    // represented as sum of
                    // four squares i, j, k, l 
                    if (s == ab)
                    {
                        // product of 2 numbers 
                        // a and b represented 
                        // as sum of four squares
                        // i, j, k, l 
                        System.out.print("i = " + 
                                          i + "\n");
                        System.out.print("j = " + 
                                          j + "\n");
                        System.out.print("k = " + 
                                          k + "\n");
                        System.out.print("l = " + 
                                          l + "\n");
                        System.out.print("Product of " + 
                                         a + " and " + b);
                        System.out.print(" can be written"+
                               " as sum of squares of i, "+
                                              "j, k, l\n");
                        System.out.print(ab + " = ");
                        System.out.print(i + "*" +
                                         i + " + ");
                        System.out.print(j + "*" +
                                         j + " + ");
                        System.out.print(k + "*" +
                                         k + " + ");
                        System.out.print(l + "*" + 
                                         l + "\n");
                        System.out.println();
                    }
                }
            }
        }
    }
}
  
// Driver code
public static void main (String[] args)
{
    // a and b such that 
    // they can be expressed 
    // as sum of squares 
    // of numbers
    int a = 30; // 1*1 + 2*2 + 
                // 3*3 + 4*4;
    int b = 4;  // 1*1 + 1*1 + 
                // 1*1 + 1*1;
  
    // given numbers can be 
    // represented as sum of 
    // 4 squares By euler's 
    // four square identity 
    // product also can be 
    // represented as sum 
    // of 4 squares
    int ab = a * b;
      
    check_euler_four_square_identity(a, b, ab);
}
}
  
// This code is contributed by ajit

Python3

# Python3 code to verify euler's 
# four square identity
  
# function to check euler
# four square identity
def check_euler_four_square_identity(a, b, ab):
  
    s = 0;
      
    # loops checking the sum of squares
    i = 0;
    while (i * i <= ab):
      
        s = i * i;
        j = i;
        while (j * j <= ab):
              
            # sum of 2 squares
            s = j * j + i * i;
            k = j;
            while (k * k <= ab):
                  
                # sum of 3 squares
                s = k * k + j * j + i * i;
                l = k;
                while (l * l <= ab):
                      
                    # sum of 4 squares
                    s = l * l + k * k + j * j + i * i;
  
                    # product of 2 numbers represented
                    # as sum of four squares i, j, k, l 
                    if (s == ab):
                          
                        # product of 2 numbers a and b
                        # represented as sum of four 
                        # squares i, j, k, l 
                        print("i =", i);
                        print("j =", j);
                        print("k =", k);
                        print("l =", l);
                        print("Product of ", a, 
                              "and", b, end = "");
                        print(" can be written as sum of", 
                                  "squares of i, j, k, l");
                        print(ab, "= ", end = "");
                        print(i, "*", i, "+ ", end = "");
                        print(j, "*", j, "+ ", end = "");
                        print(k, "*", k, "+ ", end = "");
                        print(l, "*", l);
                        print("");
                    l += 1;
                k += 1;
            j += 1;
        i += 1;
  
# Driver code
  
# a and b such that they can be expressed 
# as sum of squares of numbers
a = 30; # 1*1 + 2*2 + 3*3 + 4*4;
b = 4; # 1*1 + 1*1 + 1*1 + 1*1;
  
# given numbers can be represented as
# sum of 4 squares By euler's four
# square identity product also can be 
# represented as sum of 4 squares
ab = a * b;
  
check_euler_four_square_identity(a, b, ab);
  
# This code is contributed
# by mits

C#

// C# code to verify euler's 
// four square identity
using System;
  
class GFG
{
    // function to check euler
    // four square identity
    static void check_euler_four_square_identity(int a, 
                                                 int b,
                                                 int ab)
    {
        int s = 0;
          
        // loops checking the
        // sum of squares
        for (int i = 0; i * i <= ab; i ++)
        {
            s = i * i;
            for (int j = i; j * j <= ab; j ++)
            {
                // sum of 2 squares
                s = j * j + i * i;
                  
                for (int k = j; k * k <= ab; k ++)
                {
                    // sum of 3 squares
                    s = k * k + j * 
                        j + i * i;
                      
                    for (int l = k; l * l <= ab; l ++)
                    {
                        // sum of 4 squares
                        s = l * l + k * k +
                            j * j + i * i;
      
                        // product of 2 numbers 
                        // represented as sum of
                        // four squares i, j, k, l 
                        if (s == ab)
                        {
                            // product of 2 numbers a 
                            // and b represented as  
                            // sum of four squares i, j, k, l 
                            Console.Write("i = " + i + "\n");
                            Console.Write("j = " + j + "\n");
                            Console.Write("k = " + k + "\n");
                            Console.Write("l = " + l + "\n");
                            Console.Write("Product of " + a + 
                                                " and " + b);
                            Console.Write(" can be written"+
                                " as sum of squares of i, "+
                                               "j, k, l\n");
                            Console.Write(ab + " = ");
                            Console.Write(i + "*" + i + " + ");
                            Console.Write(j + "*" + j + " + ");
                            Console.Write(k + "*" + k + " + ");
                            Console.Write(l + "*" + l + "\n");
                            Console.Write("\n");
                        }
                    }
                }
            }
        }
    }
      
    // Driver code
    static void Main()
    {
        // a and b such that 
        // they can be expressed 
        // as sum of squares of numbers
        int a = 30; // 1*1 + 2*2 + 3*3 + 4*4;
        int b = 4; // 1*1 + 1*1 + 1*1 + 1*1;
      
        // given numbers can be 
        // represented as sum of 
        // 4 squares By euler's 
        // four square identity 
        // product also can be 
        // represented as sum 
        // of 4 squares
        int ab = a * b;
          
        check_euler_four_square_identity(a, b, ab);
    }
}
  
// This code is contributed by 
// Manish Shaw(manishshaw1)

---

输出：

i = 0
j = 2
k = 4
l = 10
Product of 30 and 4 can be written as sum of squares of i, j, k, l
120 = 0*0 + 2*2 + 4*4 + 10*10

i = 2
j = 4
k = 6
l = 8
Product of 30 and 4 can be written as sum of squares of i, j, k, l
120 = 2*2 + 4*4 + 6*6 + 8*8

改进算法：

上述算法的时间复杂度为$O((a*b)^4)$在最坏的情况下。可以减少到$O((a*b)^3)$通过从乘积a * b中减去所有(i，j，k)的i，j和k的平方，并检查该值是否为理想平方。如果这是一个完美的正方形，那么我们已经找到了解决方案。

C++

// CPP code to verify Euler's four-square identity 
#include
using namespace std;
  
// This function prints the four numbers 
// if a solution is found Else prints 
// solution doesn't exist 
void checkEulerFourSquareIdentity(int a, int b) 
{ 
    // Number for which we want to 
    // find a solution 
    int ab = a * b; 
    bool flag = false; 
      
    int i = 0; 
    while(i * i <= ab) // loop for first number 
    { 
        int j = i; 
        while (i * i + j * j <= ab) // loop for second number 
        { 
            int k = j; 
            while(i * i + j * j + 
                k * k <= ab) // loop for third number 
            { 
                // Calculate the fourth number 
                // and apply square root 
                double l = sqrt(ab - (i * i + j * 
                                        j + k * k)); 
                  
                // Check if the fourthNum is Integer or 
                // not. If yes, then solution is found 
                if (floor(l) == ceil(l) && l >= k) 
                { 
                    flag = true; 
                    cout<<"i = " << i << "\n"; 
                    cout<<"j = " << j << "\n"; 
                    cout<<"k = " << k << "\n"; 
                    cout<<"l = " << (int)l << "\n"; 
                    cout<<"Product of " << a << " and "<< b << 
                                " can be written as sum of squares"<< 
                                                " of i, j, k, l \n"; 
                                                  
                    cout<

Java

// Java code to verify Euler's four-square identity
class GFG
{
      
// This function prints the four numbers 
// if a solution is found Else prints 
// solution doesn't exist
public static void checkEulerFourSquareIdentity(int a, 
                                                int b)
{
    // Number for which we want to
    // find a solution
    int ab = a * b;
    boolean flag = false;
      
    int i = 0;
    while(i * i <= ab) // loop for first number
    {
        int j = i;
        while (i * i + j * j <= ab) // loop for second number
        {
            int k = j;
            while(i * i + j * j + 
                  k * k <= ab) // loop for third number
            {
                // Calculate the fourth number 
                // and apply square root
                double l = Math.sqrt(ab - (i * i + j * 
                                           j + k * k));
                  
                // Check if the fourthNum is Integer or 
                // not. If yes, then solution is found
                if (Math.floor(l) == Math.ceil(l) && l >= k)
                {
                    flag = true;
                    System.out.print("i = "  + i + "\n");
                    System.out.print("j = " + j + "\n");
                    System.out.print("k = " + k + "\n");
                    System.out.print("l = " + (int)l + "\n");
                    System.out.print("Product of " + a + " and "+ b + 
                                 " can be written as sum of squares"+
                                                " of i, j, k, l \n");
                                                  
                    System.out.print(ab + " = " + i + "*" + i + " + " + 
                                        j + "*" + j + " + " + k + "*" + 
                                             k + " + " + (int)l + "*" + 
                                                        (int)l + "\n");
                      
                }
                k += 1;
            }
            j += 1;
        }
        i += 1;
    }
      
    // Solution cannot be found
    if (flag == false)
    {
        System.out.println("Solution doesn't exist!");
        return ;
    }
}
  
// Driver Code
public static void main(String[] args)
{
    int a = 30;
    int b = 4;
    checkEulerFourSquareIdentity(a, b);
}
}
  
// This code is contributed by mits

Python3

# Python3 code to verify Euler's four-square identity
# This function prints the four numbers if a solution is found
# Else prints solution doesn't exist
def checkEulerFourSquareIdentity(a, b):
  
    # Number for which we want to find a solution
    ab = a*b
    flag = False
      
    i = 0
    while i*i <= ab: # loop for first number
          
        j = i
        while i*i + j*j <= ab: # loop for second number
          
            k = j
            while i*i + j*j + k*k <= ab: # loop for third number
                  
                # Calculate the fourth number and apply square root
                l = (ab - (i*i + j*j + k*k))**(0.5)
                  
                # Check if the fourthNum is Integer or not
                # If yes, then solution is found
                if l == int(l) and l >= k:
                    flag = True
                    print("i = ",i)
                    print("j = ",j)
                    print("k = ",k)
                    print("l = ",l)
                    print("Product of", a , "and" , b , 
                          "can be written as sum of squares of i, j, k, l" ) 
                    print(ab," = ",i,"*",i,"+",j,"*",j,"+",
                          k,"*",k,"+",l,"*",l)
                      
                      
                k += 1
              
            j += 1
          
        i += 1
          
    # Solution cannot be found
    if flag == False:
        print("Solution doesn't exist!")
        return
  
a, b = 30, 4
checkEulerFourSquareIdentity(a,b)

C＃

// C# code to verify Euler's four-square identity
using System;
  
class GFG
{
      
// This function prints the four numbers 
// if a solution is found Else prints 
// solution doesn't exist
public static void checkEulerFourSquareIdentity(int a, 
                                                int b)
{
    // Number for which we want to
    // find a solution
    int ab = a * b;
    bool flag = false;
      
    int i = 0;
    while(i * i <= ab) // loop for first number
    {
        int j = i;
        while (i * i + j * j <= ab) // loop for second number
        {
            int k = j;
            while(i * i + j * j + 
                  k * k <= ab) // loop for third number
            {
                // Calculate the fourth number 
                // and apply square root
                double l = Math.Sqrt(ab - (i * i + j * 
                                           j + k * k));
                  
                // Check if the fourthNum is Integer or 
                // not. If yes, then solution is found
                if (Math.Floor(l) == Math.Ceiling(l) && l >= k)
                {
                    flag = true;
                    Console.Write("i = " + i + "\n");
                    Console.Write("j = " + j + "\n");
                    Console.Write("k = " + k + "\n");
                    Console.Write("l = " + (int)l + "\n");
                    Console.Write("Product of " + a + " and "+ b + 
                              " can be written as sum of squares"+
                                             " of i, j, k, l \n");
                                                  
                    Console.Write(ab + " = " + i + "*" + i + " + " + 
                                     j + "*" + j + " + " + k + "*" + 
                                          k + " + " + (int)l + "*" + 
                                                      (int)l + "\n");
                      
                }
                k += 1;
            }
            j += 1;
        }
        i += 1;
    }
      
    // Solution cannot be found
    if (flag == false)
    {
        Console.WriteLine("Solution doesn't exist!");
        return ;
    }
}
  
// Driver Code
public static void Main()
{
    int a = 30;
    int b = 4;
    checkEulerFourSquareIdentity(a, b);
}
}
  
// This code is contributed by mits

的PHP

= $k)
                {
                    $flag = true;
                    print("i = " . $i . "\n");
                    print("j = " . $j . "\n");
                    print("k = " . $k . "\n");
                    print("l = " . $l . "\n");
                    print("Product of " . $a . " and " . $b . 
                          " can be written as sum of squares" .
                                          " of i, j, k, l \n");
                    print($ab . " = " . $i . "*" . $i . " + " . 
                          $j . "*" . $j . " + " . $k . "*" .
                          $k . " + " . $l . "*" . $l . "\n");
                      
                }
                $k += 1;
            }
            $j += 1;
        }
        $i += 1;
    }
    // Solution cannot be found
    if ($flag == false)
    {
        print("Solution doesn't exist!");
        return 0;
    }
}
  
// Driver Code
$a = 30;
$b = 4;
checkEulerFourSquareIdentity($a, $b);
  
// This code is contributed by mits
?>

输出：

i = 0
j = 2
k = 4
l = 10
Product of 30 and 4 can be written as sum of squares of i, j, k, l
120 = 0*0 + 2*2 + 4*4 + 10*10
i = 2
j = 4
k = 6
l = 8
Product of 30 and 4 can be written as sum of squares of i, j, k, l
120 = 2*2 + 4*4 + 6*6 + 8*8