show that there is no a positive integer $n$ for which $\sqrt{n+1} + \sqrt{n-1}$ is rational
If $n = 1 \implies a =\sqrt{2} $ is irrational. If $n \ge 2$, then if $a = \sqrt{n+1} + \sqrt{n-1}$ is a rational number, then $a^2 = 2n+2\sqrt{n^2-1}$ is also a rational number. But this shows that $\sqrt{n^2-1}$ is a rational as well. And because $n$ is an integer, this shows that $n^2-1$ must be a perfect square, and this is not possible. Thus there is no $n$ such that $\sqrt{n+1}+\sqrt{n-1}$ is a rational number.