Prove that $S = \{a^2 + b^2: a,b \in \Bbb N\}$ is closed under multiplication. Is it true? 
Can you prove or disprove this?
$S = \{a^2 + b^2: a,b \in \Bbb N\}$ is closed under multiplication.

Suppose that $x=a^2+b^2$ and $y=c^2+d^2$. Then
$$xy=\det
\begin{pmatrix}
a&b\\-b&a
\end{pmatrix}
\det\begin{pmatrix}
c&d\\-d&c
\end{pmatrix}
=\det
\begin{pmatrix}
a&b\\-b&a
\end{pmatrix}
\begin{pmatrix}
c&d\\-d&c
\end{pmatrix}
=\det\left(
\begin{array}{cc}
 a c-b d & b c+a d \\
 -b c-a d & a c-b d \\
\end{array}
\right)=(ac-bd)^2+(bc+ad)^2$$
