2008 AIME II Problem 15

Find the largest integer $n$ satisfying the following conditions:
$(i)$ $n^2$ can be expressed as the difference of two consecutive cubes;
$(ii)$ $2n+79$ is a perfect square.

Leading zeroes must be inputted, so if your answer is 34, then input 034. Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.