On k−Unitary Perfect Polynomials over F2

Let k be a positive integer. A polynomial A∈F_2[x] is called k-unitary perfect if the sum of the k-th powers of its distinct unitary divisors equals A^k. In this paper, we focus on the case k=2^n and prove that every 2^n-unitary perfect polynomial over F_2 is necessarily even. Moreover, we obtain a complete classification of all even 2^n-unitary perfect polynomials having at most three distinct irreducible factors. In particular, we characterize all such polynomials of the formA=x^a (x+1)^b P^h,
where P is a Mersenne prime over F_2 and a, b, and h are positive integers.
As a consequence, several explicit infinite families of k-unitary perfect polynomials over F_2 are obtained.