Improved lower bounds for strong $n$-conjectures

TL;DR

This paper improves lower bounds for the qualities of solutions to a generalized $n$-conjecture related to the $abc$-conjecture, showing Ramaekers's conjecture is false for all $n \

Contribution

It provides new, higher lower bounds for the limit superior of qualities in a restricted solution set, disproving Ramaekers's conjecture for all $n \\geq 5$.

Findings

Lower bound of 5/4 for $n \\geq 6$

Lower bound of 5/3 for odd $n \\geq 5$

Ramaekers's conjecture is false for all $n \\geq 5$

Abstract

The well-known $abc$-conjecture concerns triples $(a,b,c)$ of non-zero integers that are coprime and satisfy ${a+b+c=0}$. The strong $n$-conjecture is a generalisation to $n$ summands where integer solutions of the equation ${a_1 + \ldots + a_n = 0}$ are considered such that the $a_i$ are pairwise coprime and satisfy a certain subsum condition. Ramaekers studied a variant of this conjecture with a slightly different set of conditions. He conjectured that in this setting the limit superior of the so-called qualities of the admissible solutions equals $1$ for any $n$. In this article, we follow results of Konyagin and Browkin. We restrict to a smaller, and thus more demanding, set of solutions, and improve the known lower bounds on the limit superior: for ${n \geq 6}$ we achieve a lower bound of $\frac54$; for odd $n \geq 5$ we even achieve $\frac53$. In particular, Ramaekers's conjecture is false for every ${n \ge 5}$.