Is there any implementation available of an algorithm which solves in full generality the $S$-unit equation $x+y=1$ in a number field? It seems that Magma solves $ax+by=c$ but only in the algebraic integers, while Sage solves $x+y=1$ with $x,y$ $S$-integers, but only for $S$ the set of primes over a fixed rational primes. Is that true? Is there any other implementation available?



Software for $S$-unit equation
This SageMath implementation promises the full generality you are seeking:


[A robust implementation for solving the S-unit equation and several applications](https://arxiv.org/abs/1903.00977)


See also this [Phys.Org](https://phys.org/news/2019-04-centuries-old-mathematical-puzzles.html) announcement.


