# On the Radical of Multiperfect Numbers and Applications

## TL;DR

This paper investigates bounds on the radical of multiperfect numbers, explores their distribution and representation, and establishes finiteness results for certain multiperfect multirepdigit numbers, assuming the ABC conjecture.

## Contribution

It provides new bounds on the radical of multiperfect numbers based on their abundancy index and proves finiteness of multiperfect multirepdigit numbers in any base with digit counts as powers of two.

## Key findings

- Bounds on the radical of multiperfect numbers depending on their abundancy index.
- Conditional results on gaps and polynomial representations of multiperfect numbers assuming ABC conjecture.
- Finiteness of multiperfect multirepdigit numbers in any base with digit counts as powers of two.

## Abstract

It is conjectured that for a perfect number $m,$ $\rm{rad}(m)\ll m^{\frac{1}{2}}.$ We prove bounds on the radical of multiperfect number $m$ depending on its abundancy index. Assuming the ABC conjecture, we apply this result to study gaps between multiperfect numbers, multiperfect numbers represented by polynomials. Finally, we prove that there are only finitely many multiperfect multirepdigit numbers in any base $g$ where the number of digits in the repdigit is a power of $2.$ This generalizes previous works of several authors including O. Klurman, F. Luca, P. Polack, C. Pomerance and others.