How do you find a number that is the sum of two squares and whose square is also the sum of two squares?
If you take any two numbers, square them and add the squares, then the resulting number will satisfy your criteria.
Choose two numbers
#(a^2-b^2)^2 + (2ab)^2 = (a^4-2a^2b^2+b^4) + 4a^2b^2#
#color(white)((a^2-b^2)^2 + (2ab)^2) = a^4+2a^2b^2+b^4#
#color(white)((a^2-b^2)^2 + (2ab)^2) = (a^2+b^2)^2#