On consecutive primitive nth roots of unity modulo q

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 Φn(x) with Φn(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 qn for which there is an element α∈Zqn 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.