Lagrange’s Four Square Theorem states that every natural number can be written as sum of squares of four non negative integers.
For eg.
Similarly
Similarly for any
The above identity may be derived from Euler’s four square identity: which says we can write a product of 2 numbers (which can be written as sum of 4 squares) as the sum of 4 squares.
C++
// CPP program for Lagrange's four square identity#include <bits/stdc++.h>using namespace std;// Prints all the possible combinations 4 numbers // whose sum of squares is equal to the given no.void printFourSquares(int a){ // loops checking the sum of squares for (int i = 0; i * i <= a; i++) { for (int j = i; j * j <= a; j++) { for (int k = j; k * k <= a; k++) { for (int l = k; l * l <= a; l++) { // if sum of four squares equals // the given no. if (i * i + j * j + k * k + l * l == a) { // printing the numbers cout << a << " = " << i << "*" << i << " + " << j << "*" << j << " + "; cout << k << "*" << k << " + " << l << "*" << l << "\n"; } } } } }}// Driver Codeint main(){ int a = 74; // 74 = 0*0 + 0*0 + 5*5 + 7*7 // 74 = 0*0 + 1*1 + 3*3 + 8*8 // 74 = 0*0 + 3*3 + 4*4 + 7*7 // 74 = 1*1 + 1*1 + 6*6 + 6*6 // 74 = 2*2 + 3*3 + 5*5 + 6*6 printFourSquares(a); return 0;} |
Java
// Java program for Lagrange's four square identityclass GFG{// Prints all the possible combinations 4 numbers // whose sum of squares is equal to the given no.static void printFourSquares(int a){ // loops checking the sum of squares for (int i = 0; i * i <= a; i++) { for (int j = i; j * j <= a; j++) { for (int k = j; k * k <= a; k++) { for (int l = k; l * l <= a; l++) { // if sum of four squares equals // the given no. if (i * i + j * j + k * k + l * l == a) { // printing the numbers System.out.print( a + " = " + i + "*" + i + " + " + j + "*" + j + " + "); System.out.println( k + "*" + k + " + " + l + "*" + l); } } } } }}// Driver Codepublic static void main(String[] args){ int a = 74; // 74 = 0*0 + 0*0 + 5*5 + 7*7 // 74 = 0*0 + 1*1 + 3*3 + 8*8 // 74 = 0*0 + 3*3 + 4*4 + 7*7 // 74 = 1*1 + 1*1 + 6*6 + 6*6 // 74 = 2*2 + 3*3 + 5*5 + 6*6 printFourSquares(a);}}// This code is contributed by smitha |
Python3
# Python program for Lagrange's four square identity# Prints all the possible combinations 4 numbers # whose sum of squares is equal to the given no.def printFourSquares(a) : # loops checking the sum of squares i = 0 while (i * i <= a) : j = i while (j * j <= a) : k = j while (k * k <= a) : l = k while (l * l <= a) : # if sum of four squares equals # the given no. if (i * i + j * j + k * k + l * l == a) : # printing the numbers print ("{} = {}*{} + {}*{} +". format(a,i,i,j,j), end = " ") print ("{}*{} + {}*{}". format(k,k,l,l), end="\n") l = l + 1 k = k + 1 j = j + 1 i = i + 1 # Driver Codea = 74# 74 = 0*0 + 0*0 + 5*5 + 7*7# 74 = 0*0 + 1*1 + 3*3 + 8*8# 74 = 0*0 + 3*3 + 4*4 + 7*7# 74 = 1*1 + 1*1 + 6*6 + 6*6# 74 = 2*2 + 3*3 + 5*5 + 6*6printFourSquares(a)# This code is contributed by Manish Shaw# (manishshaw1) |
C#
// C# program for Lagrange's four square identityusing System;class GFG{// Prints all the possible combinations 4 numbers // whose sum of squares is equal to the given no.static void printFourSquares(int a){ // loops checking the sum of squares for (int i = 0; i * i <= a; i++) { for (int j = i; j * j <= a; j++) { for (int k = j; k * k <= a; k++) { for (int l = k; l * l <= a; l++) { // if sum of four squares equals // the given no. if (i * i + j * j + k * k + l * l == a) { // printing the numbers Console.Write( a + " = " + i + "*" + i + " + " + j + "*" + j + " + "); Console.Write( k + "*" + k + " + " + l + "*" + l + "\n"); } } } } }}// Driver codepublic static void Main(){ int a = 74; // 74 = 0*0 + 0*0 + 5*5 + 7*7 // 74 = 0*0 + 1*1 + 3*3 + 8*8 // 74 = 0*0 + 3*3 + 4*4 + 7*7 // 74 = 1*1 + 1*1 + 6*6 + 6*6 // 74 = 2*2 + 3*3 + 5*5 + 6*6 printFourSquares(a);}}// This code is contributed by Smitha |
PHP
<?php//PHP program for Lagrange's four square identity// Prints all the possible combinations 4 numbers // whose sum of squares is equal to the given no.function printFourSquares($a){ // loops checking the sum of squares for ($i = 0; $i * $i <= $a; $i++) { for ( $j = $i; $j * $j <= $a; $j++) { for ($k = $j; $k * $k <= $a; $k++) { for ($l = $k; $l * $l <= $a; $l++) { // if sum of four squares equals // the given no. if ($i * $i + $j * $j + $k * $k + $l * $l == $a) { // printing the numbers echo $a , " = " , $i , "*" ,$i, " + " , $j , "*" , $j , " + "; echo $k , "*" , $k , " + ", $l , "*" , $l , "\n"; } } } } }}// Driver Code $a = 74; // 74 = 0*0 + 0*0 + 5*5 + 7*7 // 74 = 0*0 + 1*1 + 3*3 + 8*8 // 74 = 0*0 + 3*3 + 4*4 + 7*7 // 74 = 1*1 + 1*1 + 6*6 + 6*6 // 74 = 2*2 + 3*3 + 5*5 + 6*6 printFourSquares($a);// This code is contributed by ajit.?> |
Javascript
<script>// Javascript program for Lagrange's four // square identity// Prints all the possible combinations// 4 numbers whose sum of squares is // equal to the given no.function printFourSquares(a){ // Loops checking the sum of squares for(let i = 0; i * i <= a; i++) { for(let j = i; j * j <= a; j++) { for(let k = j; k * k <= a; k++) { for(let l = k; l * l <= a; l++) { // If sum of four squares equals // the given no. if (i * i + j * j + k * k + l * l == a) { // Printing the numbers document.write(a + " = " + i + "*" + i + " + " + j + "*" + j + " + "); document.write(k + "*" + k + " + " + l + "*" + l + "</br>"); } } } } }}// Driver codelet a = 74;// 74 = 0*0 + 0*0 + 5*5 + 7*7// 74 = 0*0 + 1*1 + 3*3 + 8*8// 74 = 0*0 + 3*3 + 4*4 + 7*7// 74 = 1*1 + 1*1 + 6*6 + 6*6// 74 = 2*2 + 3*3 + 5*5 + 6*6printFourSquares(a);// This code is contributed by decode2207</script> |
Output:
74 = 0*0 + 0*0 + 5*5 + 7*7 74 = 0*0 + 1*1 + 3*3 + 8*8 74 = 0*0 + 3*3 + 4*4 + 7*7 74 = 1*1 + 1*1 + 6*6 + 6*6 74 = 2*2 + 3*3 + 5*5 + 6*6
Time complexity : O(a2)
Auxiliary Space : O(1)
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