Conjecture: The only integer solutions to the function are , , and . No others exist. Furthermore, for n odd and for n even, when .
I agree, it is an interesting problem, and looking at it numerically, I believe it to be true. But why these three solutions should exist uniquely, I do not know.
UPDATE: http://www.mathhelpforum.com/math-he...duction-3.html Proof for general case found a posteriori.