Article ID Journal Published Year Pages File Type
5772720 Journal of Number Theory 2017 12 Pages PDF
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 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.
Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory
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