A new lower bound in the \(abc\) conjecture (Q6634708)

The natural numbers \(a\), \(b\), \(c\) are said to be an \(abc\) triple if they are relatively prime and satisfy the equation \(a+b=c\). The radical \(\mathrm{rad}(abc)\) of \(abc\) triple is defined to be the product of the primes in the prime factorization of \(abc\). The abc conjecture states that for every \(\epsilon > 0\), the \(abc\) triples satisfy \[ c = O_{\epsilon}(\mathrm{rad}(abc)^{1+\epsilon}).\] In this paper, it is proved that there are infinitely many \(abc\) triples satisfying \[\exp \left( \frac{4\sqrt{2(\delta/\epsilon)\log c}}{\log\log c} \right) \mathrm{rad}(abc) \leq c,\] 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 \(n/\delta\). A permissible value for \(\delta\) is 3.65931. In this case the constant in the exponent becomes approximately 6.56338. This improves the constant 6.068 of the lower bound of the same form provided in [\textit{M. van Frankenhuysen}, J. Number Theory 82, No. 1, 91--95 (2000; Zbl 0998.11033)].