Two $S$-unit equations with many solutions

Abstract

We show that there exist arbitrarily large sets $S$ of $s$ prime numbers such that the equation $a+b=c$ has more than $\exp (s^{2-\sqrt{2}-\epsilon })$ solutions in coprime integers $a$, $b$, $c$ all of whose prime factors lie in the set $S$. We also show that there exist sets $S$ for which the equation $a+1=c$ has more than $\exp (s^{\frac{1}{16}})$ solutions with all prime factors of $a$ and $c$ lying in $S$.

6 pages