Consider, . This is a principal ideal domain. Now the equation is solvable if . Therefore, for some .Ifwas irreducible then would also be prime (since the integral domain is a PID and so "irreducible" is equivalent to "prime") and so . But this is impossible. Therefore, is reducible and where and are non-units. This means (norm). However, since they are not units and so . Therefore, if then it means .

Additional problem: Prove that if and where are odd and are even and then . In other words, we have a certain level of uniqueness in the decomposition of into sum of squares.