There is no odd perfect polynomial over ${\mathbb{F}}_2$ with four prime factors

Abstract

A perfect polynomial over the binary field ${\mathbb{F}}_2$ is a polynomial $A\in {\mathbb{F}}_2[x]$ 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 ${\mathbb{F}}_2$ and $a,b,c,d$ are positive integers.