A lower bound in the \(abc\) conjecture
The \(abc\) conjecture is the following claim: Let \(a\), \(b\) and \(c\) be relatively prime rational integers such that \(a+b=c\), then, for all fixed \(c>0\), there exists a constant \(K\) such that \[ h(a,b,c) \leq r(a,b,c)+c h(a,b,c)+K, \] where \(h(a,b,c)=\max\{\log |a|, \log |b|, \log |c|\}\) and \(r(a,b,c) =\sum_{p \mid abc} \log p\), \(p\) prime. The author shows that there exist infinitely many \((a,b,c)\) as above such that \[ h(a,b,c) \geq r(a,b,c) + 4K_1 {\sqrt{h(a,b,c)}\over \log h(a,b,c)}, \] where \(K_1=1{.}517\). The proof uses sphere packing.
scientific article; zbMATH DE number 1538255
scientific article