I need to prove " If m and n are each sum of two squares, then their product mn is also a sum of two squares. THANKS!!
Write $\displaystyle \displaystyle \begin{align*} m = p^2 + q^2 \end{align*}$ and $\displaystyle \displaystyle \begin{align*} n = r^2 + s^2 \end{align*}$. Then
$\displaystyle \displaystyle \begin{align*} mn &= \left( p^2 + q^2 \right) \left( r^2 + s^2 \right) \\ &= p^2 r^2 + p^2 s^2 + q^2 r^2 + q^2 s^2 \\ &= \left( p\, r \right)^2 + \left( q \, s \right)^2 + \left( p\, s \right)^2 + \left( q\, r \right)^2 \\ &= \left( p\, r \right)^2 + 2\, p\, q\, r\, s + \left( q \, s \right)^2 + \left( p\, s \right)^2 - 2\,p\, q\, r\, s + \left( q\, r \right)^2 \\ &= \left( p\, r + q\, s \right)^2 + \left( p\, s - q\, r \right)^2 \end{align*}$
which is also the sum of two squares