# How do you find a number that is the sum of two squares and whose square is also the sum of two squares?

Jul 31, 2017

If you take any two numbers, square them and add the squares, then the resulting number will satisfy your criteria.

#### Explanation:

Choose two numbers $a$ and $b$.

Then:

${\left({a}^{2} - {b}^{2}\right)}^{2} + {\left(2 a b\right)}^{2} = \left({a}^{4} - 2 {a}^{2} {b}^{2} + {b}^{4}\right) + 4 {a}^{2} {b}^{2}$

$\textcolor{w h i t e}{{\left({a}^{2} - {b}^{2}\right)}^{2} + {\left(2 a b\right)}^{2}} = {a}^{4} + 2 {a}^{2} {b}^{2} + {b}^{4}$

$\textcolor{w h i t e}{{\left({a}^{2} - {b}^{2}\right)}^{2} + {\left(2 a b\right)}^{2}} = {\left({a}^{2} + {b}^{2}\right)}^{2}$

That is ${\left({a}^{2} + {b}^{2}\right)}^{2}$ is the sum of two squares too.