Direct proof if $n = a^2 + b^2$

Question

I need to use a direct proof to show this:

"A positive integer $n$ is said to be enriched if there are integers $a$ and $b$ such that $n = a^2 + b^2$. Use direct proof to show that if $n$ is an enriched number, then $2n$ is also an enriched number."

all I have is if $n = a^2 + b^2$ then $2n = 2(a^2+b^2) = 2a^2 + 2b^2$

I don't really know what to do next, if anyone could direct me to the next step, that would be great. :)

Answer 1

You can see that $$2n=2a^2+2b^2=a^2+2ab +b^2+a^2-2ab +b^2=(a+b)^2+(a-b) ^2$$ And here's your proof

Answer 2

If you know a bit about complex numbers, it's easy to see that $a^2+b^2=|a+ib|^2$ and $2=1^2+1^2=|1+i|^2$ and therefore

$$2(a^2+b^2)=|1+i|^2|a+ib|^2=|(1+i)(a+ib)|^2=|(a-b)+i(a+b)|^2=(a-b)^2+(a+b)^2$$