"If x is a difference of squares, prove that 3x is a difference of squares as well."

Question

I'm banging my head against the wall with this task:

Prove that if $x$ is a difference of integer squares, then $3x$ is a difference of integer squares as well.

What strategies could I utilise in order to prove this?

Answer 1

Write $x=u^2-v^2$ with $u,v\in \Bbb Z$, then $3x=3(u^2-v^2)=3(u+v)(u-v)=(3u+3v)(u-v)=((2u+v)+(u+2v))((2u+v)-(u+2v))=(2u+v)^2-(u+2v)^2$