There is no odd perfect polynomial over \(\mathbb F_{2}\) with four prime factors (Q1018206)

A polynomial \(A\in\mathbb F_2[x]\) is called perfect if it is the sum of all its divisors, and it is called odd if \(\gcd(A,x^2+x)=1\). It is conjectured that there is no polynomial over \(\mathbb F_2\) which is perfect and odd. By \textit{E. F. Canaday} [Duke Math. J. 8, 721--737, (1941; Zbl 0061.06605)], there is no odd perfect polynomial over \(\mathbb F_2\) with less than three prime factors. In [J. Théor. Nombres Bordx. 19, 165--174 (2007; Zbl 1145.11081)], the authors proved the nonexistence of odd perfect polynomials in \(\mathbb F_2[x]\) composed of three prime divisors. In this paper, they show that an odd perfect polynomial over \(\mathbb F_2\) needs to have at least five prime factors.