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IMPROVED LOWER BOUNDS FOR STRONG n-CONJECTURES
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 119 / Issue 1 / August 2025
- Published online by Cambridge University Press:
- 02 June 2025, pp. 61-81
- Print publication:
- August 2025
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- Article
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The well-known
$abc$-conjecture concerns triples 
$(a,b,c)$ of nonzero 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 
${a_1 + \cdots + a_n = 0}$ 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 paper, 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 
$\frac 54$; for odd 
$n \geq 5$ we even achieve 
$\frac 53$. In particular, Ramaekers’ conjecture is false for every 
${n \ge 5}$.
