Effective $S$-unit Equations Beyond 3 Terms : Newman's Conjecture

TL;DR

This paper develops methods to effectively solve 5-term $S$-unit equations with small prime sets and applies these techniques to answer a longstanding question about representing integers as sums of $S$-units.

Contribution

It introduces an effective approach for solving 5-term $S$-unit equations with small prime sets and addresses Newman's conjecture on integer representations as sums of $S$-units.

Findings

Successfully solved 5-term $S$-unit equations for |S| ≤ 3

Provided explicit solutions to Newman's conjecture

Extended understanding of $S$-unit equations beyond 3 terms

Abstract

We show how to effectively solve 5-term $S$-unit equations when the set of primes $S$ has cardinality at most 3, and use this to provide an explicit answer to an old question of D.J. Newman on representations of integers as sums of $S$-units.