# Number of ways of writing N as a sum of 4 squares

Given a number N, the task is to find the number of ways of writing N as a sum of 4 squares. Two representations are considered different if their terms are in a different order or if the integer being squared (not just the square) is different.

Examples:

Input : n=1
Output :8
12 + 02 + 02 + 02
02 + 12 + 02 + 02
02 + 02 + 12 + 02
02 + 02 + 02 + 12
Similarly there are 4 other possible perumutations by replacing 1 with -1
Hence there are 8 possible ways.

Input :n=5
Output :48

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:
Jacobi’s four-square theorem states that the number of ways of writing n as a sum of 4 squares is 8 times the sum of divisor of n if n is odd and is 24 times the sum of odd divisor of n if n is even.Find the sum of odd and even divisor of n by running a loop from 1 to sqrt(n) .

## C++

 `// C++ implementation of above approach ` `#include ` `using` `namespace` `std; ` ` `  `// Number of ways of writing n ` `// as a sum of 4 squares ` `int` `sum_of_4_squares(``int` `n) ` `{ ` `    ``// sum of odd and even factor ` `    ``int` `i, odd = 0, even = 0; ` ` `  `    ``// iterate from 1 to the number ` `    ``for` `(i = 1; i <= ``sqrt``(n); i++) { ` `        ``// if i is the factor of n ` `        ``if` `(n % i == 0) { ` `            ``// if factor is even ` `            ``if` `(i % 2 == 0) ` `                ``even += i; ` ` `  `            ``// if factor is odd ` `            ``else` `                ``odd += i; ` ` `  `            ``// n/i is also a factor ` `            ``if` `((n / i) != i) { ` `                ``// if factor is even ` `                ``if` `((n / i) % 2 == 0) ` `                    ``even += (n / i); ` ` `  `                ``// if factor is odd ` `                ``else` `                    ``odd += (n / i); ` `            ``} ` `        ``} ` `    ``} ` `    ``// if n is odd ` `    ``if` `(n % 2 == 1) ` `        ``return` `8 * (odd + even); ` ` `  `    ``// if n is even ` `    ``else` `        ``return` `24 * (odd); ` `} ` `// Driver code ` `int` `main() ` `{ ` `    ``int` `n = 4; ` ` `  `    ``cout << sum_of_4_squares(n); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java implementation of above approach ` `import` `java.io.*; ` ` `  `class` `GFG ` `{ ` `     `  `// Number of ways of writing n ` `// as a sum of 4 squares ` `static` `int` `sum_of_4_squares(``int` `n) ` `{ ` `    ``// sum of odd and even factor ` `    ``int` `i, odd = ``0``, even = ``0``; ` ` `  `    ``// iterate from 1 to the number ` `    ``for` `(i = ``1``; i <= Math.sqrt(n); i++)  ` `    ``{ ` `        ``// if i is the factor of n ` `        ``if` `(n % i == ``0``)  ` `        ``{ ` `            ``// if factor is even ` `            ``if` `(i % ``2` `== ``0``) ` `                ``even += i; ` ` `  `            ``// if factor is odd ` `            ``else` `                ``odd += i; ` ` `  `            ``// n/i is also a factor ` `            ``if` `((n / i) != i)  ` `            ``{ ` `                ``// if factor is even ` `                ``if` `((n / i) % ``2` `== ``0``) ` `                    ``even += (n / i); ` ` `  `                ``// if factor is odd ` `                ``else` `                    ``odd += (n / i); ` `            ``} ` `        ``} ` `    ``} ` `    ``// if n is odd ` `    ``if` `(n % ``2` `== ``1``) ` `        ``return` `8` `* (odd + even); ` ` `  `    ``// if n is even ` `    ``else` `        ``return` `24` `* (odd); ` `} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `main (String[] args)  ` `    ``{ ` `            ``int` `n = ``4``; ` `        ``System.out.println (sum_of_4_squares(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by tushil.  `

## Python

 `# Python3 implementation of above approach ` ` `  `# Number of ways of writing n ` `# as a sum of 4 squares ` `def` `sum_of_4_squares(n): ` ` `  `    ``# sum of odd and even factor ` `    ``i, odd, even ``=` `0``,``0``,``0` ` `  `    ``# iterate from 1 to the number ` `    ``for` `i ``in` `range``(``1``,``int``(n``*``*``(.``5``))``+``1``): ` `        ``# if i is the factor of n ` `        ``if` `(n ``%` `i ``=``=` `0``): ` `             `  `            ``# if factor is even ` `            ``if` `(i ``%` `2` `=``=` `0``): ` `                ``even ``+``=` `i ` ` `  `            ``# if factor is odd ` `            ``else``: ` `                ``odd ``+``=` `i ` ` `  `            ``# n/i is also a factor ` `            ``if` `((n ``/``/` `i) !``=` `i): ` `                 `  `                ``# if factor is even ` `                ``if` `((n ``/``/` `i) ``%` `2` `=``=` `0``): ` `                    ``even ``+``=` `(n ``/``/` `i) ` ` `  `                ``# if factor is odd ` `                ``else``: ` `                    ``odd ``+``=` `(n ``/``/` `i) ` `             `  `         `  `     `  `    ``# if n is odd ` `    ``if` `(n ``%` `2` `=``=` `1``): ` `        ``return` `8` `*` `(odd ``+` `even) ` ` `  `    ``# if n is even ` `    ``else` `: ` `        ``return` `24` `*` `(odd) ` ` `  `# Driver code ` ` `  `n ``=` `4` ` `  `print``(sum_of_4_squares(n)) ` ` `  `# This code is contributed by mohit kumar 29 `

## C#

 `// C# implementation of above approach ` `using` `System; ` ` `  `class` `GFG ` `{ ` `         `  `// Number of ways of writing n ` `// as a sum of 4 squares ` `static` `int` `sum_of_4_squares(``int` `n) ` `{ ` `    ``// sum of odd and even factor ` `    ``int` `i, odd = 0, even = 0; ` ` `  `    ``// iterate from 1 to the number ` `    ``for` `(i = 1; i <= Math.Sqrt(n); i++)  ` `    ``{ ` `        ``// if i is the factor of n ` `        ``if` `(n % i == 0)  ` `        ``{ ` `            ``// if factor is even ` `            ``if` `(i % 2 == 0) ` `                ``even += i; ` ` `  `            ``// if factor is odd ` `            ``else` `                ``odd += i; ` ` `  `            ``// n/i is also a factor ` `            ``if` `((n / i) != i)  ` `            ``{ ` `                ``// if factor is even ` `                ``if` `((n / i) % 2 == 0) ` `                    ``even += (n / i); ` ` `  `                ``// if factor is odd ` `                ``else` `                    ``odd += (n / i); ` `            ``} ` `        ``} ` `    ``} ` `    ``// if n is odd ` `    ``if` `(n % 2 == 1) ` `        ``return` `8 * (odd + even); ` ` `  `    ``// if n is even ` `    ``else` `        ``return` `24 * (odd); ` `} ` ` `  `// Driver code ` `static` `public` `void` `Main () ` `{ ` `         `  `    ``int` `n = 4; ` `    ``Console.WriteLine(sum_of_4_squares(n)); ` `} ` `} ` ` `  `// This code is contributed by ajit.  `

Output:

```24
```

Time Complexity : O(sqrt(N))

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Second year Department of Information Technology Jadavpur University

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Improved By : mohit kumar 29, jit_t