On the index of special perfect polynomials - Gallardo, L.H. (Carpathian Mathematical Publications, 2023)
We give a lower bound of the degree and the number of distinct prime divisors of the index of special perfect polynomials. More precisely, we prove that ω(d) ≥ 9, and (d) ≥ 258, where d := (Q^2,σ(Q^2)) is the index of the special perfect polynomial A := p_1^2 Q^2, in which p_1 is irreducible and has minimal degree. This means that σ(A)=A in the polynomial ring F _2[x]. The function σ is a natural analogue of the usual sums of divisors function over the integers. The index considered is an analogue of the index of an odd perfect number, for which a lower bound of 135 is known. Our work use elementary properties of the polynomials as well as results of the paper [J. Théor. Nombres Bordeaux 2007, 19 (1), 165-174].