El teorema de los cuatro cuadrados de Lagrange establece que todo número natural se puede escribir como suma de cuadrados de cuatro números enteros no negativos.
Por ej.
De manera similar
De manera similar para cualquier
La identidad anterior puede derivarse de la identidad de cuatro cuadrados de Euler: que dice que podemos escribir un producto de 2 números (que se puede escribir como suma de 4 cuadrados) como la suma de 4 cuadrados.
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 Code
int 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 identity
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
System.out.print( a + " = " + i + "*" + i
+ " + " + j + "*" + j + " + ");
System.out.println( k + "*" + k + " + "
+ l + "*" + l);
}
}
}
}
}
}
// Driver Code
public 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 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 Manish Shaw
# (manishshaw1)
C#
// C# program for Lagrange's four square identity
using 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 code
public 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 code
let 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 decode2207
</script>
Producción:
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
Publicación traducida automáticamente
Artículo escrito por VaradaDesikan y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA