## Writing a Product as a Sum of Two Squares

If two numbers
$x, \;$
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{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}

This result is not unique We can also write for example
\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}