Yes.

We take a fixed diophantine equation in variables $y,x_1, x_2, \ldots, x_k$.
Task: for an input $n \in \mathbb{Z}$, output either

 1. Five values of $y$ for which solutions exist to the equation.
 2. A solution to the equation for which $y=n$. 
 3. "No", in which case there must be no solution with $y=n$.

It is clear that for any diophantine equation, there is a program which performs this task. If there are five values of $y$ for which solutions exist, you need only have the program output these for all inputs. If there are fewer than four values of $y$ for which solutions exist, once you know the solutions, then writing the program is trivial. 

However, telling whether a solution exists to a diophantine equation is undecidable.