Radical of perfect numbers and perfect numbers among polynomial values (Q2800143)

Let \(\mathrm{rad}(n)=\prod_{p|n}p\). In the paper under review, the author proves that \(\mathrm{rad}(m)\ll m^{9/14}\) for each perfect number \(m\). As some corollaries, assuming that the ABC-conjecture is true, the author implies that:NEWLINENEWLINE-- Assume that \(P(x)\in\mathbb{Z}[x]\) with \(\mathrm{deg}(P)\geq 3\) has no repeated factors. Then there are only finitely many integer \(n\) such that \(P(n)\) is perfect number.NEWLINENEWLINE-- Assume that \(f(x,y)\in\mathbb{Z}[x,y]\) be a homogeneous form with \(\mathrm{deg}(f)\geq 6\) without repeated linear factors. Then there are only finitely many perfect numbers of the form \(f(m,n)\) for \(m,n\in\mathbb{Z}\).