Is this problem irrational or simply ab-surd?

Let's say that $\sqrt{n-1}+\sqrt{n+2014}={x}_{n}$ is rational for some n. So, $1/{x}_{n}$ should also be rational. $\Rightarrow\frac{\sqrt{n+2014}-\sqrt{n-1}}{2015}$ is also rational. Therefore, $\sqrt{n+2014}-\sqrt{n-1}$ is also rational. As the sum of two rationals cannot be irrational, we find out that $\sqrt{n-1}$ and $\sqrt{n+2014}$ are rational. As $n\epsilon N,$ let for some $x,y\epsilon N$ , $n+2014={x}^{2} \quad and \quad n-1 = {y}^{2}$ . $\Rightarrow {x}^{2}-{y}^{2}=(x-y)(x+y)=2015=5\times 13\times 31.$ 2015 has only 8 factors. This means that x+y=65, 155, 403, 2015.and x-y=31, 13, 5, 1. Solving these equations, we can get that minimum value of y=17. $\\ \therefore n=289+1=\boxed{290}$