Show that there do not exist any integer $m,n$ such that $\frac{m}{n}+\frac{n+1}{m}=4$

Question

The question is Show that there do not exist positive integers $$m,n$$ such that $\frac{m}{n}+\frac{n+1}{m}=4$ . It is the problem.I tried to make a quadratic equation of m. And I found that we have to show that there do not exist positive integer $n$ such that $\sqrt{3n^2-n}$ . But how?? Anyone have any idea!

Answer 1

Continuing your proof on showing that the discriminant $D$ is not a perfect square:

$D= 3n^2-n = n(3n-1)$

$\gcd(n, 3n-1) =1$ by elementary methods.

So we need both $3n-1$ and $n$ to be perfect squares. But $3n-1$ cannot be a perfect square taking mod 3.