If $x$ is rational then $\,x+\frac{1}x \,$ is an integer $\iff x = \pm 1$

elementary-number-theory

Show that $X+\dfrac{1}{X}$ is not an integer number for any rational $X$ and $X \neq 1, X \neq -1$

I think we can substitue $X=\dfrac{P}{Q}$ but I don't know if I can now assume that $\gcd(P,Q)=1$


Hint $\ n = x+1/x \Rightarrow x^2\!-nx+1 = 0\,$ so $\,x\,$ is an integer dividing $1$, by the Rational Root Test.


$$x + \frac{1}{x} = n =>x^2-nx+1=0$$ The equation $x^2-nx+1=0$ has rational root $=> \delta =n^2-4=k^2$, where $ k$ is integer. $=>n^2-k^2=4=>(n+k)(n-k)=4=>n+k=+-2$ and $ n-k=+-2$ $=> n=+-2 => x=+-1$ false!

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