On Mersenne polynomials over \(\mathbb{F}_2\) (Q2326698)

Let \(A\in \mathbb{F}_2[x]\) be a nonzero polynomial. \(A\) is called \textit{even} if it has a linear factor and it is called \textit{odd} otherwise. \textit{Mersenne polynomial} over \(\mathbb{F}_2\) is defined as a polynomial of the form \(1+x^a(x+1)^b\), for some \(a, b\in \mathbb{N}^*\). If such a polynomial is irreducible, it is called a Mersenne prime. Let \(\omega(A)\) denote the number of distinct irreducible (or prime) factors of \(A\) over \(\mathbb{F}_2\), \(\sigma(A)\) the sum of all divisors of \(A\) and ord\((A)\) the order or period of \(A\). Note that \(\sigma\) is a multiplicative function. If \(\sigma(A)=A\), then \(A\) is called a \textit{perfect} polynomial. The polynomial \(1+x^a(x+1)^b\) is obviously reducible when \(\mathrm{gcd}(a,b)\neq 1\) and it is square-free if and only if \(a\) and \(b\) are not both even. So, the authors only consider the case in which \(\mathrm{gcd}(a,b)=1\). They discover a new relation between Mersenne polynomials \(M_{ab}:= x^a(x+1)^b+1\) with \(\mathrm{gcd}(a,b)=1\) and trinomials \(T_{ab}:= x^{a+b}+x^b+1\), that is: \(r:= \omega(M_{ab})=\omega(T_{ab})\). This is important since most known results about trinomials (and more general polynomials) are about the parity of \(r\). Testing irreducibility for polynomials (in particular, trinomials) over a finite field remains a difficult problem. The following conjecture, which is about the factorization of \(\sigma(M^{2h})\) (and hence about that of \(M^{2h+1}+1=(M+1)\sigma(M^{2h}))\), appeared in [\textit{L. H. Gallardo} and \textit{O. Rahavandrainy}, Finite Fields Appl. 18, No. 5, 920--932 (2012; Zbl 1273.11175)]. \textbf{Conjecture} Let \(M\) be a Mersenne prime over \(\mathbb{F}_2\). Then the polynomial \(\sigma(M^{2h})\) is divisible by a non Mersenne prime if and only if \((h\geq 2\,\,\mathrm{ or}\,\,M\not \in \{1+x+x^3, 1+x^2+x^3\})\). The conjecture implies that all even perfect polynomials over \(\mathbb{F}_2\), which are products of Mersenne primes, are indeed the nine of the eleven non-trivial known even perfect polynomials. In this paper, the authors show that for any \(h\), \(\sigma(M^{2h})\) is square-free and it is not a Mersenne prime (even if it may be irreducible), and then consider the special case \(\omega(\sigma(M^{2h})=2\).