There is no odd perfect polynomial over $\mathbb F_2$ with four prime factors

Abstract

A perfect polynomial over the binary field F2 is a polynomial A ∈ F2\[x] that equals the sum of all its divisors. If gcd(A,x2 + 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 = PaQbRcSd where P, Q, R, S are distinct irreducible polynomials of degree > 1 over F2 and a, b, c, d are positive integers.