According to Euler’s four square identity, the product of any two numbers a and b can be expressed as a sum of four squares if a and b both can individually be expressed as the sum of four squares.
Mathematically, if a =
Then, a * b =
where c1, c2, c3, c4, d1, d2, d3, d4, e1, e2, e3, e4 are any integer.
Some examples are, a == 30 b = = 4 ab = a * b = 120 = a = = 15 b = = 24 ab = a * b = 810 = a = = 15 b = = 26 ab = a * b = 390 =
Example:
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
Explanation :
The product of the 2 numbers a(30) and b(4) can be represented as the sum of 4 squares as stated by Euler’s four square identity. The above are the 2 representations of the product a * b in the sum of 4 squares form. All possible representations of the product a*b in the sum of four squares form are shown.
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
Approach :
Brute Force :
A given number(a*b) can be represented in a sum of 4 squares form by using 4 loops i, j, k, l to find each of the four squares. This gives all possible combinations to form a*b as a sum of four squares. At each iteration of the innermost loop(l loop), check the sum with the product a*b. If there is a match, then print the 4 numbers(i, j, k, and l) whose sum of squares equals a*b.
// CPP code to verify euler's four square identity #include <bits/stdc++.h> 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 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 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# 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) |
<?php // PHP code to verify euler's // four square identity // function to check euler // four square identity function check_euler_four_square_identity( $a , $b , $ab )
{ $s = 0;
// loops checking the sum of squares
for ( $i = 0; $i * $i <= $ab ; $i ++)
{
$s = $i * $i ;
for ( $j = $i ; $j * $j <= $ab ; $j ++)
{
// sum of 2 squares
$s = $j * $j + $i * $i ;
for ( $k = $j ; $k * $k <= $ab ; $k ++)
{
// sum of 3 squares
$s = $k * $k + $j * $j + $i * $i ;
for ( $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
echo ( "i = " . $i . "\n" );
echo ( "j = " . $j . "\n" );
echo ( "k = " . $k . "\n" );
echo ( "l = " . $l . "\n" );
echo "" . "Product of " .
$a . " and " . $b ;
echo " can be written" .
" as sum of squares of i, " .
"j, k, l\n" ;
echo $ab . " = " ;
echo $i . "*" . $i . " + " ;
echo $j . "*" . $j . " + " ;
echo $k . "*" . $k . " + " ;
echo $l . "*" . $l . "\n" ;
echo "\n" ;
}
}
}
}
}
} // 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 Abby_akku ?> |
<script> // Javascript code to verify euler's
// four square identity
// function to check euler
// four square identity
function check_euler_four_square_identity(a, b, ab)
{
let s = 0;
// loops checking the
// sum of squares
for (let i = 0; i * i <= ab; i ++)
{
s = i * i;
for (let j = i; j * j <= ab; j ++)
{
// sum of 2 squares
s = j * j + i * i;
for (let k = j; k * k <= ab; k ++)
{
// sum of 3 squares
s = k * k + j *
j + i * i;
for (let 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
document.write( "i = " + i +
"</br>" );
document.write( "j = " + j +
"</br>" );
document.write( "k = " + k +
"</br>" );
document.write( "l = " + l +
"</br>" );
document.write( "Product of " + a +
" and " + b);
document.write( " can be written" +
" as sum of squares of i, " +
"j, k, l" +
"</br>" );
document.write(ab + " = " );
document.write(i + "*" + i +
" + " );
document.write(j + "*" + j +
" + " );
document.write(k + "*" + k +
" + " );
document.write(l + "*" + l +
"</br>" );
document.write( "</br>" );
}
}
}
}
}
}
// a and b such that
// they can be expressed
// as sum of squares of numbers
let a = 30; // 1*1 + 2*2 + 3*3 + 4*4;
let 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
let ab = a * b;
check_euler_four_square_identity(a, b, ab);
</script> |
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
Improved Algorithm:
The time complexity of the above algorithm is
// CPP code to verify Euler's four-square identity #include<bits/stdc++.h> 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<<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 )
{
cout<< "Solution doesn't exist!\n" ;
return ;
}
} // Driver Code int main()
{ int a = 30;
int b = 4;
checkEulerFourSquareIdentity(a, b);
return 0;
} // This code is contributed by mits |
// 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 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# 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 |
<?php // PHP 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 function 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 = 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;
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 ?> |
<script> // Javascript 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
function checkEulerFourSquareIdentity(a, b)
{
// Number for which we want to
// find a solution
let ab = a * b;
let flag = false ;
let i = 0;
while (i * i <= ab) // loop for first number
{
let j = i;
while (i * i + j * j <= ab) // loop for second number
{
let k = j;
while (i * i + j * j +
k * k <= ab) // loop for third number
{
// Calculate the fourth number
// and apply square root
let 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 ;
document.write( "i = " + i + "</br>" );
document.write( "j = " + j + "</br>" );
document.write( "k = " + k + "</br>" );
document.write( "l = " + l + "</br>" );
document.write( "Product of " + a + " and " + b +
" can be written as sum of squares" +
" of i, j, k, l " + "</br>" );
document.write(ab + " = " + i + "*" + i + " + " +
j + "*" + j + " + " + k + "*" +
k + " + " + l + "*" +
l + "</br>" );
}
k += 1;
}
j += 1;
}
i += 1;
}
// Solution cannot be found
if (flag == false )
{
document.write( "Solution doesn't exist!" + "</br>" );
return ;
}
}
let a = 30;
let b = 4;
checkEulerFourSquareIdentity(a, b);
</script> |
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