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

Let $q$ be a prime power and $\mathbb{F}_q$ the finite field with $q$ elements. For a positive integer $n$, the binomial $X^n - 1 \in \mathbb{F}_q[X]$ is said to be $3$-sparse over $\mathbb{F}_q$ if every irreducible factor of $X^n-1$ in $\mathbb{F}_q[X]$ is either a binomial or a trinomial. In 2021, Oliveira and Reis characterized all positive integers $n$ for which $X^n-1$ is $3$-sparse over $\mathbb{F}_q$ when $q = 2$ and $q = 4$, and raised the open problem of whether, for any given $q$, there are only finitely many primes $p$ such that $X^p-1$ is $3$-sparse over $\mathbb{F}_q$. In this paper, if $q$ is a power of an odd prime $r$, we then establish that for any positive integer not divisible by $r$, $X^n-1$ is $3$-sparse over $\mathbb{F}_q$ if and only if $n =p_1^{e_1} \cdots p_s^{e_s}$ for some nonnegative integers $e_1, \dots, e_s$, where $p_1, \dots, p_s$ are distinct prime divisors of $q^2 - 1$. This resolves the problem posed by Oliveira and Reis for odd characteristic.