# Perfect polynomials over <strong>F</strong><sub>4</sub> with less than five prime factors

### Luis H. Gallardo

Université de Brest, France### Olivier Rahavandrainy

Université de Brest, France

## Abstract

A perfect polynomial $A\in\mathbb{F}_4[x]$ is a monic polynomial that equals the sum of its monic divisors. There are no perfect polynomials $A\in\mathbb{F}_4[x]$ with exactly $3$ prime divisors, i.e., of the form $A=P^aQ^bR^c$ where $P,Q,R\in\mathbb{F}_4[x]$ are irreducible and $a,b,c$ are positive integers. We characterize the perfect polynomials $A$ with $4$ prime divisors such that one of them has degree $1$. Assume that $A$ has an arbitrary number of distinct prime divisors, we discuss some simple congruence obstructions that arise and we propose three conjectures.

## Cite this article

Luis H. Gallardo, Olivier Rahavandrainy, Perfect polynomials over <strong>F</strong><sub>4</sub> with less than five prime factors. Port. Math. 64 (2007), no. 1, pp. 21–38

DOI 10.4171/PM/1774