Do you mean something similar to
In[4]:= With[{n = 100},
FindInstance[y == (5 n + x)/3 x && x > 1 && y > 1 && y < 2000 && z^2 == x y && z > 0, {x, y, z}, Integers]]
Out[4]= {{x -> 7, y -> 1183, z -> 91}}
We can also use Reduce to get an exhaustive list of solutions, for example
Reduce[y == (5 n + x)/3 x && x > 1 && y > 1 && y < 2000 && z^2 == x y && z > 0, {x, y, z}, Integers]
