On Even 2n-Unitary Perfect Polynomials over F2

Let k be a positive integer. A polynomial A∈F2[x] is called k-unitary perfectif the sum of the k-th powers of its distinct unitary divisors equals Ak. In this paper, we study the case k=2n and prove that every 2n-unitary perfect polynomial over F2 is even. Moreover, we completely classify all even 2n-unitary perfect polynomials having at most three distinct irreducible factors. In particular, we characterize all such polynomials of the form A=xa(x+1)bPh, where P is a Mersenne prime over F2 and a,b,h∈N. As a consequence, several explicit infinite families of k-unitary perfect polynomials over F2 are obtained.