The sum-of-squares function (denoted $r_{2}(n)$) gives the number of ways in which a given number $n$ is expressible as the sum of two squares. The following is from the article on this function from Wolfram Mathworld.

To find the the number of representations of $n$ as the sum of two squares (ignoring order and signs), factor $n$ as

$$n= 2^{a_0}p_{1}^{2a_{1}}...p_{r}^{2a_{r}}q_{1}^{b_{1}}...q_{s}^{b_{s}}$$ where $p_{i}$s are primes of the form $4k+3$ and $q_{i}$s are primes of the form $4k+1$. If $n$ does not have such a representation with integer $a_{i}$ because one or more of the powers of $p_{i}$ is odd, then there are no representations. Otherwise, define $$B=\prod_{i=1}^{r}(b_{i}+1)$$ Then, the required number will be $0$ if any $a_{i}$ is a half-integer,$\frac{B}{2}$ if all $a_{i}$ are integers and $B$ is even, and $\frac{1}{2}(B-(-1)^{a_0})$ if all $a_{i}$ are integers and $B$ is odd.

Now the questions I want to ask are two.

If I want to write an algorithm to find out the value of this function at a large integer, then using the above definition wouldn't give the best result since it is required to

*factor*the integer first, and Integer Factorization is known to be computationally difficult. So, is there any other formula for this function which allows us to "put in" the value of $n$ to get the required result without having to resort to finding out the prime factorization of $n$ first?If there

*was*such a formula for the sum-of-squares function, would that be preferred over the above? Are there any reasons why such a formula would be more advantageous?Thanks in advance!