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 > rad(abc) exp(6.563 root 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 root 2 delta/e where delta is a constant such that all unimodular lattices of sufficiently large dimension n contain a nonzero vector with l(1)-norm at most n/delta.