2022/049) Prove that if a number is sum of 3 squares then square of the number is sum of 3 squares

 Let n be sum of 3 squares so

$n= a^2 + b ^2 + c^2$

we need to show that $n^2$ is sum of 3 squares

from given condition

$n^2 = (a^2+b^2+c^2)^2 = a^4 + b^4 + c^4 + 2 a^2b^2 +2 b^2 c^2 + 2 c^2 a^2$

$=  a^4 + b^4 + c^4 + 2 a^2b^2 -2 b^2 c^2 - 2 c^2 a^2 + 4 b^2c^2 + 4 c^2 a^2$

$=(a^2+ b^2 -c^2)^2 + (2bc)^2 + (2ca)^2$

sum of 3 sqaures and hence proved