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?

  • $\begingroup$ When is an odd number a difference of squares? What about an even number? $\endgroup$
    – saulspatz
    Dec 6 '20 at 16:43

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

  • $\begingroup$ Thank you very much for answering! I wouldn't have thought of this. $\endgroup$
    – aachh
    Dec 6 '20 at 16:53
  • 1
    $\begingroup$ +1 elegance.... $\endgroup$ Dec 6 '20 at 16:54

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