Originally Posted by

**wonderboy1953** Hello Defunkt.

To answer your question about a natural number having two or more non-trivial sum of squares representations, I believe I read that such representations don't exist.

Dude, what are you talking about? Why do you keep bringing up "multigrades" when they are not in any obvious way related to Defunkt's question, all while feeding him false statements. Read Soroban's post.

Anyways. Let $\displaystyle U(n)$ be the number of representations of $\displaystyle n$ as a sum of two squares. (We count $\displaystyle a^2+b^2$ and $\displaystyle b^2+a^2$ as different representations if $\displaystyle a \neq b$.) Then we have the beautiful theorem

$\displaystyle U(n) = 4\sum_{d|n,\ d\equiv 1 \mod 2}(-1)^\frac{d-1}{2}.$