The $3$-sparsity of $X^n-1$ over finite fields

Abstract: Let $q$ be a prime power and $\mathbb{F}_q$ the finite field with $q$ elements. For a positive integer $n$, the binomial $f_n(X):= X^n-1\in\mathbb{F}_q[X]$ is called $3$-sparse over $\mathbb{F}_q$ if each irreducible factor of $f_n(X)$ over $\mathbb{F}_q$ is either a binomial or a trinomial. In 2021, Oliveira and Reis characterized all positive integers $n$ for which $f_n(X)$ is $3$-sparse over $\mathbb{F}_q$ when $q = 2$ and $q = 4$, and posed the open problem of whether, for any given $q$, there are only finitely many primes $p$ such that $f_p(X)$ is $3$-sparse over $\mathbb{F}_q$. In this paper, we prove that for any given odd prime power $q$, any prime $p$ for which $f_p(X)$ is $3$-sparse over $\mathbb{F}_q$ must divides $q^2-1$, thus resolving the problem of Oliveira and Reis for odd characteristic. Furthermore, we extend the results of Oliveira and Reis by determining all such integers $n$ for $q = 3$ and $q = 9$. More precisely, for any positive integer $n$ not divisible by $3$, we establish that: (i) $f_n(X)$ is $3$-sparse over $\mathbb{F}_3$ if and only if $n = 2^k$ for some nonnegative integer $k$; and (ii) $f_n(X)$ is $3$-sparse over $\mathbb{F}_9$ if and only if $n = 2^{k_1}5^{k_2}$ for some nonnegative integers $k_1$ and $k_2$.