Abstract
Abstract
We prove that there exist infinitely many coprime numbers a, b, c with
$a+b=c$
and
$c>\operatorname {\mathrm {rad}}(abc)\exp (6.563\sqrt {\log c}/\log \log c)$
. 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. Our work builds on that of van Frankenhuysen (J. Number Theory 82(2000), 91–95) 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
$4\sqrt {2\delta /e}$
where
$\delta $
is a constant such that all unimodular lattices of sufficiently large dimension n contain a nonzero vector with
$\ell _1$
-norm at most
$n/\delta $
.
Publisher
Canadian Mathematical Society
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