<s>[INST] Proof: For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$ I need help proving the following statement:
For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$
The statement is true, I just need to know the thought process, or a lead in the right direction. I think I might have to use a contradiction, but I don't know where to begin.
Any help would be much appreciated.
 [/INST] We have
\begin{eqnarray*}
x^3+x=y^3+y&\Longleftrightarrow& (x^3-y^3)+(x-y)=0\\
&\Longleftrightarrow& (x-y)(x^2+y^2+xy+1)=0.
\end{eqnarray*}
Since $x^2+y^2+xy+1=(x+\frac{y}{2})^2+\frac{3}{4}y^2+1>0$, we get $x=y$.
The hypothesis $x,y$ are integer numbers is redundant.
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