<s>[INST] If $n$ is a positive integer such that $n^3 − n − 6 = 0$, then, for every positive integer $m$ with $m \not = n$, $m^3 − m − 6 \not = 0$. Prove, by contradiction, that, if $n$ is a positive integer such that $n^3 − n − 6 = 0$, then, for every positive integer $m$ with $m \not = n$, $m^3  − m − 6 \not = 0$.

Proposition: If $n$ is a positive integer such that $n^3 − n − 6 = 0$, then, for every positive integer $m$ with $m \not = n$, $m^3  − m − 6 \not = 0$.
Hypothesis: $n$ is a positive integer such that $n^3 − n − 6 = 0$.
Conclusion: For every positive integer $m$ with $m \not = n$, $m^3  − m − 6 \not = 0$.

My workings
A (Hypothesis): $n$ is a positive integer such that $n^3 − n − 6 = 0$.
A1 ($\neg B$): There exists a positive integer $m$ with $m \not = n$ such that $m^3  − m − 6 = 0$.
A2: $n^3 - n = 6$
$\implies n(n^2 - 1) = 6$ where $n \in \mathbb{Z}^+$
A3: $m^3 - m = 6$
$\implies m(m^2 - 1) = 6$ where $m \in \mathbb{Z}^+$ and $m \not = n$.
A4: $n(n^2 - 1) = m(m^2 - 1)$ where $m \not = n$.
$\implies \dfrac{n(n^2 - 1)}{m(m^2 - 1)} = 1$ where $m \not = n$.
Since $m,n \in \mathbb{Z}^+$, $(n^2 - 1) \ge 0$ and $(m^2 - 1) \ge 0$. Therefore, $n(n^2 - 1) = m(m^2 - 1)$ iff $n = m$. 
Contradiction.
$Q.E.D.$

I would greatly appreciate it if people could please take the time to provide feedback on the correctness of my proof. Given that $m \not = n$, did I construct my proof correctly? Or is it incorrect to structure it this way if $m \not = n$?
 [/INST] Let $f(x)=x^3-x-6.$ Let $n$ be any real number such that $f(n)=0$.  Then $f(x)=(x-n)(x^2+Ax+B)$ for all $x,$ for some constants $A, B.$ 
$$\text {So }\quad  x^3-x-6=x^3+x^2(A-n)+x(B-nA)+(-nB)$$ for all $x.$ This requires $0=A-n$ and $-1=B-nA ,$ so $A=n$ and $B=nA-1=n^2-1.$ 
Now if $f(m)=0$ with $m\ne n$ then $0=m^2+Am+B=m^2+nm+(n^2-1),$ implying $m=(-n\pm \sqrt {4-3n^2}\;)/2.\;$ In order for $m$ to be a real number we must have  $4-3n^2\geq 0,$ which  implies $|n|\leq 2/\sqrt 3.$
But  if $|n|\leq 2/\sqrt 3$ then $|n^3-n|\leq |n|^3+|n|\leq 8/(3\sqrt 3\;)+2/\sqrt 3<6.$
So $f(n)=0$ has at most one real-number solution.   
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