Many solutions to the $S$-unit equation $a+1=c$

TL;DR

This paper demonstrates that for large sets of primes, the number of solutions to the equation a+1=c with prime factors in the set can grow extremely rapidly, exceeding exponential bounds.

Contribution

It establishes the existence of arbitrarily large prime sets where the solution count to a+1=c is exponentially large in the size of the set.

Findings

Number of solutions grows faster than exponential in the size of prime set

Existence of large prime sets with many solutions to a+1=c

Solution count exceeds  \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, solutions for large s

Abstract

We show that there are arbitrarily large sets $S$ of $s$ primes for which the number of solutions to $a+1=c$ where all prime factors of $ac$ lie in $S$ has $\gg \exp( s^{1/4}/\log s)$ solutions.