Find all pairs of positive integers m and n such that (n^{3}+1) / (mn-1) is an integer, and prove you've identified them all.

This was a problem I solved back in high school with a proof that could barely fit on one page, written front and back, with small handwriting.

Even looking at it again now, the most elegant proof I can come up with is still pretty complex, but nevertheless sort of neat.

In part, I'm sharing a tricky problem. In part, I'm wondering if there's a more elegant way of solving it than the one I came up with.