It is quite clear why certain differential equations, among the jungle of possible diff equations that is possible to conceive, are studied: some come from physical problems, or from "spontaneous" mathematical generalizations thereof, others come from geometry in a variety of ways. 

For diophantine equations there seem not to be such a direct link to other areas.
I would like to roughly understand why the attention of number theorists concentrates on some kinds of diophantine equations and not on others. 

Why an equation such as 

$x^2-ny^2=1$

or

$x^3+y^3=z^3$

is (or have been) considered worth studying, and not, say, any other random variant such as (if that specific example is not enough nontrivial for you or if it actually happens to have been studied, feel free to substitute it with your favourite "random" diophantine equation):

$x^3+y^5=z^2$  ?  So:

> Are there any reasons why certain diophantine equations are worth attention besides the mere approachability (i.e. being neither trivial nor hopelessly difficult to analyze)?