Do you happen to know the proof? I had the opinion Diophantine equations usually have only undecidability results.

Good explicit example. Thankyou.

Hi Gerry: I don't have the book and I can't see it in google book preview. Could you provide some example problems if you have the book? It would greatly help.

@Franz lemmermeyer Your book does not have the Elliptic curves chapter and the appendix.

and d=2 I believe.

@Jyrki Lahtonen The Frobenius ring alphabet technique looks interesting. Let me look at it. Do you know a connection of Lee-metric to codes on torus. They look very similar. But at q=5 and n=2, the number of code words are different. I am not sure why this is??

@Jyrki Lahtonen I have seen only $q=4$ case or only even cases. I have not seen that much of odd cases. Do you know of any asymptotic results?