Consider rational functions in two positive integer variables $a,b$ with integer coëfficiënts : $f_i(a,b)$.
Some of them have special properties.
For instance
$$ \frac{a^2 + b^2}{1 + ab} = f_2(a,b) = c $$
It is obvious that $c$ is a positive fraction.
But the special things here are :
Property A2 : whenever $c$ is an integer , it is a square integer.
Property B2 : every integer square has at least one representation of the form $f_2(a,b)$.
We can easily prove property A2 with the famous vièta jumping technique.
Now im looking for such a rational function $f_3(a,b)$ such that
$$ f_3(a,b) = d $$
where $ d $ is always positive and we have the following properties :
Property A3 : whenever $d$ is an integer , it is a cube integer.
Property B3 : every positive integer cube has at least one representation of the form $f_3(a,b)$.
Does such a rational function $f_3(a,b) $ even exist ?
How to decide existance ?
Assuming existence , is there uniqueness up to substitions $a,b$ to $x(a),y(b)$ where $x,y $ are integer polynomials ? ( replacing $f_3(a,b)$ with $ f_3(x(a),(y(b))$ is also a solution in many cases )
Can we construct such a solution ?