2025/07/09 by Cheng, Kaimin
#FOS: Mathematics #Number Theory (math.NT) #T1T06
paper · doi:10.48550/arxiv.2507.06655
Let q be a prime power and \mathbbFq the finite field with q elements. For a positive integer n, the binomial Xn - 1 ∈ \mathbbFq[X] is said to be 3-sparse over \mathbbFq if every irreducible factor of Xn-1 in \mathbbFq[X] is either a binomial or a trinomial. In 2021, Oliveira and Reis characterized all positive integers n for which Xn-1 is 3-sparse over \mathbbFq 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 Xp-1 is 3-sparse over \mathbbFq. 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, Xn-1 is 3-sparse over \mathbbFq if and only if n =p1e1 ⋯ pses for some nonnegative integers e1, …, es, where p1, …, ps are distinct prime divisors of q2 - 1. This resolves the problem posed by Oliveira and Reis for odd characteristic.