A New Lower Bound in the abc Conjecture
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We prove that there exist infinitely many coprime numbers a, b, c with a+b=c and c>rad(abc)exp(6.563√(log c)/loglog 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√(2δ/e) where δ is a constant such that all full-rank unimodular lattices of sufficiently large dimension n contain a nonzero vector with ℓ_1 norm at most n/δ.
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