I guess I won't even need to give this link, Polynomial representing all nonnegative integers, since this is perhaps the most famous question of MO until now.

Yes, I agree that this is a very interesting and natural curiosity seeing the proof of Gauss and Lagrange. But may someone explain me why this question particularly interesting? (recall the exclamation What a nice problem! by Gil Kalai). Is there something deeper here? The same query applies to Fermat polygonal number theorem. Is there anything that these theorems reveal us about the deeper structure of integers?