Improved lower bounds for strong $n$-conjectures

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 $a_1+\dots +a_n=0$ 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 $n\ge 6$ we achieve a lower bound of $\frac{5}{4}$; for odd $n\ge 5$ we even achieve $\frac{5}{3}$. In particular, Ramaekers's conjecture is false for every $n\ge 5$.