A New Lower Bound in the $abc$ Conjecture

We prove that there exist infinitely many coprime numbers $a$, $b$, $c$ with $a+b=c$ and $c>\mathrm{rad}(abc)\exp (6.563\sqrt{\log c}\mathrm{/}\log \log c)$. 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) who 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 $4\sqrt{2\delta \mathrm{/}e}$ where $\delta$ is a constant such that all full-rank unimodular lattices of sufficiently large dimension $n$ contain a nonzero vector with ${\mathrm{\ell }}_1$ norm at most $n\mathrm{/}\delta$.