Let $n$ be a positive integer.


Does there exist a number field $K$ such that the number of solutions of the unit equation $$a+b =1, \quad a,b\in O\_{K}^\ast$$ is at least $n$? Can we write down such a number field explicitly?


I know that the number of solutions is always finite in a fixed number field. 



For any $n$, does there exist a number field with at least $n$ solutions to the unit equation
A slightly more general form of the above mentioned lemma states: whenever $m$ has at least two distinct prime factors and $\zeta\_m$ is a primitive $m$-th root of unity, $1-\zeta\_m$ is a unit in $\mathbf Z[\zeta\_m]$.


Choosing $a=1-\zeta\_m$ and $b=\zeta\_m$ for the various primitive roots of unity, we get $\varphi(m)$ solutions for $K=\mathbf Q(\zeta\_m)$. So any such $m$ satisfying $\varphi(m)\geq n$ will do.


