Can 2017 be written uniquely as a sum of 2 squares?

Question

I know that $2017 = 9^2+44^2$, but how can I prove that this is the only possible way you can write 2017 as a sum of 2 squares?

Answer 1

$2017$ is a prime number of the form $4k+1$, and for such a numbers it is well known that the decomposition as a sum of two squares is essentially unique.

The point is that the equality $2017=9^2+44^2$ can be translated in the ring of Gaussian integers $\mathbb{Z}[i]$ as $$2017 = (9+44i)(9-44i).$$

Since $\mathbb{Z}[i]$ is a unique factorization domain, any other factorization of $2017$ coincides with the one above up to units. But the units of Gaussian integers are $\{\pm1, \, \pm i\}$, so the representation $2017= a^2+b^2$ is unique up to order and up to sign of $a$ and $b$.