# There is no odd perfect polynomial over $F_{2}$ with four prime factors

### Luis H. Gallardo

Université de Brest, France### Olivier Rahavandrainy

Université de Brest, France

## Abstract

A perfect polynomial over the binary field $F_{2}$ is a polynomial $A∈F_{2}[x]$ that equals the sum of all its divisors. If $gcd(A,x_{2} +x)=1$ then we say that $A$ is odd. It is believed that odd perfect polynomials do not exist. In this article we prove this for odd perfect polynomials $A$ with four prime divisors, i.e., polynomials of the form $A=P_{a}Q_{b}R_{c}S_{d}$ where $P,Q,R,S$ are distinct irreducible polynomials of degree $>$1 over $F_{2}$ and $a,b,c,d$ are positive integers.

## Cite this article

Luis H. Gallardo, Olivier Rahavandrainy, There is no odd perfect polynomial over $F_{2}$ with four prime factors. Port. Math. 66 (2009), no. 2, pp. 131–145

DOI 10.4171/PM/1836