Perfect polynomials over $\mathbb F_4$ with less than five prime factors
Luis H. Gallardo, Olivier Rahavandrainy · Portugaliae Mathematica · 2007
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.