A Swan-like note for a family of binary pentanomials

Abstract

In this note, we employ the techniques of Swan (Pacific J. Math. 12(3): 1099-1106, 1962) with the purpose of studying the parity of the number of the irreducible factors of the penatomial $X^n+X^{3s}+X^{2s}+X^s+1\in {\mathbb{F}}_2[X]$, where $s$ is even and $n>3s$. Our results imply that if $n\not\equiv \pm 1(\mathrm{mod}8)$, then the polynomial in question is reducible.