The authors first prove the following result on the radical of a perfect number.
(1) If \(x\) is perfect, then \(\text{rad}(x) < 2x^{17/26}\), where \(\text{rad}(x)\) is the radical or the squarefree kernel of \(x\), that is, \(\text{rad}(x)=\prod_{p\mid x} p\).
With the aid of (1) the authors then consider the question of whether large perfect numbers have the tendency to become far apart from each other and prove the following results towards this under the ABC conjecture.
(2) If the ABC conjecture holds, then for every odd integer \(k\), the equation \(x-y=k\) has only finitely many solutions in perfect numbers \(x\) and \(y\).
(3) If the ABC conjecture holds, then for every nonzero integer \(k\), the equation \(x-y=k\) has only finitely many solutions in squarefull perfect numbers \(x\) and \(y\).
(4) Let \((a_n)_{n\geq 1}\) be the increasing sequence of perfect numbers. If the ABC conjecture holds, then for every positive integer \(k\), the inequality \(a_{n+2}-a_n\leq k\) has only finitely many solutions in \(n\).