Consider solution of 
$$x^3-y^2=1728 \text{ unit} \qquad (1)$$ 
in a number field.


This is related to the discriminant of elliptic curve
in terms of $c\_4,c\_6$.


Via elliptic curves it might have infinitely solutions
for fixed unit.


I am interested if $x,y$ are in the ring of integers
of the number field.


If one solution exists, scaling by powers of units
gives infinitely, so define a set of solutions
to be primitive if the norms of $x$ are distinct.



> 
> Q1 Does (1) have infinitely many primitively solutions
>  when $x,y$ are in the ring of integers?
> 
> 
> Q2 Does positive solution to Q1 contradicts the
>  abc conjecture over number fields?
> 
> 
> Q3 What is the expected abc quality of triples
>  resulting from (1)? Say for $w=\sqrt{2}$, `A= (w + 1) * 3^3 ;B= (-70*w - 99) * w; C= (-2*w - 1)^4, A+B+C=0`
> 
> 
> 


I suppose this is impossible since it might solve $x^3-y^2=u^n$
for many $n$.



On $x^3-y^2=1728 \text{ unit}$ in number fields
You can always absorb 6th powers into $x$ and $y$, so there is a finite set of units $u\_1,\ldots,u\_n$ (covering the congruence classes of units mod 6th powers), so that every solution to $x^3-y^2=1728u$ with $x,y\in\mathcal{O}\_K$ and $u\in\mathcal{O}\_K^\*$ yields a solution $(X,Y)\in\mathcal{O}\_K\times\mathcal{O}\_K$ to one of the finitely many equations $x^3-y^2=1728u\_i$ with $1\le i\le n$. Each of these finitely many equations has finitely many solutions (by Siegel). From this it is easy to answer your question, Start from the finitely many solutions $(X\_j,Y\_j)$ with $1\le j\le N$, 
then the full set of solutions to your equation is 
$$
 \{ (v^2X\_j,v^3Y\_j) : 1\le j\le N\quad\hbox{and}\quad v\in\mathcal{O}\_K^\*\}.
$$
Since the units have norm 1, there are only finitely many solutions with distinct $x$-norms.


