On the Diophantine equation σ2(X¯n) = σn(X¯n)

Abstract

In this paper, we investigate the set [Formula: see text] of positive integer solutions of the title Diophantine equation. In particular, for a given [Formula: see text] we prove boundedness of the number of solutions, give precise upper bound on the common value of [Formula: see text] and [Formula: see text] together with the biggest value of the variable [Formula: see text] appearing in the solution. Moreover, we enumerate all solutions for [Formula: see text] and discuss the set of values of [Formula: see text] over elements of [Formula: see text].