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.

I found this question interesting and natural as a curiosity. But from the interest of people, there seems to be more in it. (recall the exclamation What a nice problem! by Gil Kalai). May someone explain me why this question particularly interesting from, perhap, a number theorist perspective ? For example,is there some deeper linkage to other results ?

The same query applies to Fermat polygonal number theorem. Is there anything that these theorems reveal us about the deeper structure of integers?