6 problems

Let $p\geq 5$ be prime and let $n\geq 3$ be an integer. An element of $\mathbb{F}_{p^n}$ is $k$-normal over $\mathbb{F}_p$ when the greatest common divisor of its associated polyno…

Arnold's conjecture. If $|g|>1$, then

Exponential-sum and small-order conjecture. There exists a non-increasing mapping $f:\mathbb{N}\to\mathbb{R}_+$ such that:

Small-order scarcity conjecture. For every such $b$ and $f$,

Let $q$ be a prime power, let $n$ be a positive integer, and write $\bar{n}$ for the least power of $q$ that is at least $n$. For a polynomial $g\in\mathbb{F}_q[X]$, consider the p…

For an integer $a$ and a positive integer $r$, define … where $\operatorname{ord}_p(a)$ is the multiplicative order of $a$ modulo the prime $p$. Erdős's conjecture. For every…