Proving there exist no positive integers $m,n$ such that $ m/n +(n+1)/m = 4$

Question

Prove that there exist no positive integers $m,n$ such that

$$ \frac{m}{n} + \frac{n+1}{m} = 4.$$

I worked on cases considering $m$ and $n$ are even or odd, but I couldn't get anything.

Answer 1

Times $\,mn\,\Rightarrow\, m^2 -4n\ m + n^2\!+\!n = 0$ $\!\iff\!$ $ m = 2n\pm \sqrt{3n^2-n}\in \Bbb Z$ $\!\iff\!$ $\,n(3n\!-\!1) = k^2,\ $ for $\, k\in\Bbb Z.\, $ $\ n,\, 3n\!-\!1 > 0\,$ are coprime factors of a square so, by unique factorization, are squares, say $\ n = i^2,\,\ j^2 = 3n\!-\!1 = 3i^2-1\,\Rightarrow\, j^2\equiv -1\pmod 3,\,$ contradiction.