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A perfect polynomial is a monic polynomial that equals the sum of its monic divisors. There are no perfect polynomials with exactly prime divisors, i.e., of the form where are irreducible and are positive integers. We characterize the perfect polynomials with prime divisors such that one of them has degree . Assume that has an arbitrary number of distinct prime divisors, we discuss some simple congruence obstructions that arise and we propose three conjectures.