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?
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$\begingroup$ When is an odd number a difference of squares? What about an even number? $\endgroup$– saulspatzDec 6, 2020 at 16:43
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1 Answer
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Write $x=u^2-v^2$ with $u,v\in \Bbb Z$, then $3x=3(u^2-v^2)=3(u+v)(u-v)=(3u+3v)(u-v)=((2u+v)+(u+2v))((2u+v)-(u+2v))=(2u+v)^2-(u+2v)^2$
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$\begingroup$ Thank you very much for answering! I wouldn't have thought of this. $\endgroup$– aachhDec 6, 2020 at 16:53
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