Show that if the integers $x, y,$ and $z$ satisfy $x^3 + 3y^3 = 9z^3$
then $x = y = z = 0.$
How should I interpret this question and how to proceed?
I am thinking about the Euclidean algorithm but it becomes confusing when $x,y,z$ comes like variables?
Answer
First notice that if $d=\mbox{gcd}(x,y,z)$ then $d^3$ can be factored out of the equation. So we can assume that $d=1$. Then $x^3 = 9z^3-3y^3$, so $3$ divides $x$, say $ x=3k$. So we have $3^3k^3 = 9z^3-3y^3$ and we can divide everything by $3$ to get $9k^3= 3 z^3-y^3$. A similar argument shows $3$ divides $y$. Repeat to show $3$ divides $z$. This contradicts that $d=1$.
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