TITLE: For any $n$, does there exist a number field with at least $n$ solutions to the unit equation
QUESTION [5 upvotes]: 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.

REPLY [6 votes]: The unit equation is discussed in detail in Chapter 5 of Bombieri and Gubler's book Heights in diophantine geometry.  In particular, they note (see Example 5.2.6) that in the cyclotomic field ${\Bbb Q}(e^{2\pi i/p})$ there are exactly $(p-1)(p-2)$ non-real solutions to the unit equation $x+y=1$.  This observation is attributed to H.W. Lenstra.  
In Example 5.2.7 it is observed that there seem to be many more real solutions to the unit equation in cyclotomic fields.  For example the maximal real subfield of ${\Bbb Q}(e^{2\pi i/19})$ contains $28 398$ solutions to the unit equation.  
Another intriguing example is 5.2.8.  Let $\alpha$ be the real root $\alpha>1$ of the Lehmer polynomial 
$$ 
x^{10}+x^{9}-x^7-x^6-x^5-x^4-x^3+x+1. 
$$ 
Then the field ${\Bbb Q}(\alpha)$ (with unit group of rank $5$) contains $2532$ solutions to the unit equation.