Writing a Product as a Sum of Two Squares
\[x, \;\]
can each be written as the sum of two squares, \[x=a^2+b^2, \; y=c^2+d^2\]
then so can their product, using the identity\[(a^2+b^2)(c^2+d^2)=(ab+cd)^2+(bc-ad)^2\]
.We can write each of 245, 260 as the sum of two squares.
\[245=14^2+7^2\]
\[260=16^2+2^2\]
Then
\[\begin{equation} \begin{aligned}245 \times 260 &= (14^2+7^2)(16^2+2^2) \\ &= (14 \times 16+7 \times 2)^2+(7 \times 16-14 \times 2)^2 \\ &= 238^2+84^2 \end{aligned} \end{equation}\]
This result is not unique We can also write for example
\[\begin{equation} \begin{aligned}245 \times 260 &= (7^2+14^2)(16^2+2^2) \\ &= (7 \times 16+14 \times 2)^2+(14 \times 16-7 \times 2)^2 \\ &= 140^2+210^2 \end{aligned} \end{equation}\]
