The following question is in particular reference to the previous question by Bjorn Poonen. I guess I won't even need to give this link, https://mathoverflow.net/questions/9731/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? 

More generally, I hope we can address the question: What is it that makes some diophantine equations interesting, while others are less so?

(The change was suggested by Kevin O'Bryant)