How do you find the greatest $n$ such that the difference of its square root from some other integer is greater than or equal to one? For example : $$2011^{1/2} - n^{1/2} \ge1$$
What should be the greatest integer value of $n$ such that the difference is more than one? Is there a specific method to find this?
\begin{align*} \sqrt{2011} - \sqrt{n} & \geq 1 \\ \sqrt{n} & \leq \sqrt{2011} - 1 \\ n & \leq (\sqrt{2011} - 1)^{2} \\ &= 2011+1-2\sqrt{2011} \\ &= 1922.31165 \ldots \\ \end{align*}
The greatest value of $n$ is $1922$.