The author introduces a new definition: the \textit{radical index} \(\nu(n)\) of an integer \(n\) with \(|n|>1\) which is given by the formula \(|\text{rad} (n)|^{\nu(n)}=|n|\) where, as usual, \(\text{rad} (n)=\prod_{p \mid n}p\). Then this concept is used in relation to the \((ABC)\) conjecture. In particular, the author considers the equation \(Ax+By+Cz=0\) with some conditions on the radicals of the numbers \(A\), \dots, \(z\), and the links with \((ABC)\) conjecture and some older conjectures of Erdős and Pillai. The other sections deal with Langevin's conjecture on values of polynomials and arithmetical properties of binary recurrences.