We prove that there exist infinitely many coprime numbers $a$, $b$, $c$ with $a+b=c$ and $c>\operatorname{rad}(abc)\exp(6.563\sqrt{\log c}/\log\log c)$. These are the most extremal examples currently known in the $abc$ conjecture, thereby providing a new lower bound on the tightest possible form of the conjecture. This builds on work of van Frankenhuysen (1999) whom proved the existence of examples satisfying the above bound with the constant $6.068$ in place of $6.563$. We show that the constant $6.563$ may be replaced by $4\sqrt{2\delta/e}$ where $\delta$ is a constant such that all full-rank unimodular lattices of sufficiently large dimension $n$ contain a nonzero vector with $\ell_1$ norm at most $n/\delta$.

翻译：我们证明了存在无穷多互素的三元组$(a,b,c)$满足$a+b=c$且$c>\operatorname{rad}(abc)\exp(6.563\sqrt{\log c}/\log\log c)$。这些是目前$abc$猜想中已知的最极端例子，从而为该猜想最紧致形式提供了一个新的下界。该结果基于van Frankenhuysen (1999)的工作，他证明了存在满足上述不等式的例子，其中常数$6.563$被替换为$6.068$。我们指出常数$6.563$可替换为$4\sqrt{2\delta/e}$，其中$\delta$是使得所有充分大维数$n$的满秩幺模格均包含一个$\ell_1$范数不超过$n/\delta$的非零向量的常数。