Primitive Polynomials of the Form $g(x)+λ$ over Finite Fields: Non-Existence Results and Conjectures

In this paper, we investigate the existence of primitive polynomials over $\mathbb{F}_{q^n}$ whose constant term is a primitive element of $\mathbb{F}_{q^n}$. We prove that such polynomials do not exist if $q$ is odd, $q^n\equiv3\pmod{4}$, and the degree $m$ of the polynomial is odd. In particular,