I need help in proving " If m and n are each sum of two squares, then their p....

**erockdontstop** · February 5th 2013, 09:13 PM

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!!

**Prove It** · February 5th 2013, 09:48 PM

Write $\displaystyle \begin{align*} m = p^2 + q^2 \end{align*}$ and $\displaystyle \begin{align*} n = r^2 + s^2 \end{align*}$ . Then

$\displaystyle \begin{align*} mn &= \left( p^2 + q^2 \right) \left( r^2 + s^2 \right) \\ &= p^2 r^2 + p^2 s^2 + q^2 r^2 + q^2 s^2 \\ &= \left( p\, r \right)^2 + \left( q \, s \right)^2 + \left( p\, s \right)^2 + \left( q\, r \right)^2 \\ &= \left( p\, r \right)^2 + 2\, p\, q\, r\, s + \left( q \, s \right)^2 + \left( p\, s \right)^2 - 2\,p\, q\, r\, s + \left( q\, r \right)^2 \\ &= \left( p\, r + q\, s \right)^2 + \left( p\, s - q\, r \right)^2 \end{align*}$

which is also the sum of two squares

**erockdontstop** · February 5th 2013, 10:01 PM

Thank you! I have more questions!

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