On the Diophantine equation $\sigma_{2}(\overline{X}_{n})=\sigma_{n}(\overline{X}_{n})$

TL;DR

This paper studies positive integer solutions to a specific Diophantine equation involving sum-of-divisors functions, establishing bounds, enumerating solutions for small n, and analyzing solution ratios.

Contribution

It proves boundedness of solutions, provides explicit upper bounds, and enumerates all solutions for n up to 16, advancing understanding of this Diophantine problem.

Findings

Bounded number of solutions for each n

Explicit upper bounds on the sum-of-divisors values

Complete enumeration of solutions for n ≤ 16

Abstract

In this note we investigate the set $S(n)$ of positive integer solutions of the title Diophantine equation. In particular, for a given $n$ we prove boundedness of the number of solutions, give precise upper bound on the common value of $\sigma_{2}(\overline{X}_{n})$ and $\sigma_{n}(\overline{X}_{n})$ together with the biggest value of the variable $x_{n}$ appearing in the solution. Moreover, we enumerate all solutions for $n\leq 16$ and discuss the set of values of $x_{n}/x_{n-1}$ over elements of $S(n)$.