9 problems

Let $p\geq 5$ be a prime, and let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind. A complete permutation polynomial (CPP) is a polynomial $f$ such that both…

Let $p>7$ be a prime, and let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind. A permutation polynomial (PP) is a polynomial that permutes $\mathbb{F}_p$. The…

Let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind, and let a permutation polynomial (PP) be a polynomial that permutes the field. The $p=5$ reversed Dickson…

Let $p>3$ be a prime, let $n_1,n_2>1$, and let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind over $\mathbb{Z}_p$. Reversed Dickson second-kind period conject…

Let $p>3$ be a non-Mersenne prime, and let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind. Consider the sequence of values $E_n(1,\frac14)$ modulo $p$. Non-Me…

Let $p$ be an odd prime, let $1\leq n\leq p^2-1$, and let $D_n(1,x)$ be the reversed Dickson polynomial of the first kind. A polynomial $f$ is a complete permutation polynomial (CP…

Let $p>3$ be a prime, and let $n$ satisfy $1leq nleq p^2-1$. Write $D_n(1,x)$ for the reversed Dickson polynomial of the first kind. A polynomial that permutes $\mathbb{F}_p$ is ca…

Existence conjecture. There exists $a\in\mathbb{F}_{p^{nk}}^*$ such that $h_a(x)$ is a Dickson polynomial of degree $n+1$ over $\mathbb{F}_{p^k}$. This conjecture was subsequently…

Dickson-polynomial conjecture. There exist $a\in\mathbb{F}_{p^{rk}}^*$ such that $h_a(x)$ are Dickson polynomials of degree $r+1$ over $\mathbb{F}_{p^k}$.