$r=\pm1$ are the only rationals with $\,r+1/r\in \Bbb Z$ (sum with its reciprocal is an integer) 
Can sum of a rational number and its reciprocal be an integer?

My brother asked me this question and I was unable to answer it. 
The only trivial solutions which I could think of are $1$ and $-1$.
As to what I tried, I am afraid not much. I have never tried to solve such a question, and if someone could point me in the right direction, maybe I could complete it on my own.
Please don't misunderstand my question. 
I am looking for a rational number $r$ where $r + \frac{1}{r}$ is an integer.

Suppose $\frac pq+\frac qp =n$ then $p^2+q^2=pqn$ for integers $p,q,n$. As a quadratic in $p$ this is $p^2-qnp+q^2=0$ so that $$p=\frac {qn\pm \sqrt {q^2n^2-4q^2}}{2}$$ so that for the square root to yield an integer we require $n^2-4=m^2$ for some integer  $m$. The only two integer squares which differ by $4$ are $0$ and $4$, so $n=\pm 2$ and the only solutions are $p=\pm q$.
