On consecutive primitive nth roots of unity modulo q

Abstract

Given n ∈ N, we study the conditions under which a finite field of prime order q will have adjacent elements of multiplicative order n. In particular, we analyze the resultant of the cyclotomic polynomial Φ n (x) with Φ n (x + 1), and exhibit Lucas and Mersenne divisors of this quantity. For each n = 1, 2, 3, 6, we prove the existence of a prime q n for which there is an element α ∈ Z qn where α and α + 1 both have multiplicative order n. Additionally, we use algebraic norms to set analytic upper bounds on the size and quantity of these primes.