EXISTENCE OF EVEN PERFECT POLYNOMIALS

Perfect polynomials are a natural analogue (in the ring F-p[x]) of multiperfect numbers (in the ring of integers). The latter numbers are classical objects that are poorly understood, since only their definition is simple. We describe, by elementary methods, the most basic objects in the polynomial case of the general problem. We display, for every prime number p not equivalent to 1mod 12 (resp. p not equivalent to 1mod 24) many new even non-splitting perfect (resp. unitary perfect) polynomials over F-p. Moreover, for any prime number p not equivalent to 1mod 24, new bi-unitary perfect polynomials are also given. These examples substantially improve our knowledge about these kinds of polynomials.