Zigzag polynomials, Artin's conjecture and trinomials

Abstract

Let Fp denote the finite field of prime order p. We introduce a new natural family of polynomials in Fp[X]. These polynomials first arose from considering faithful representations of the elementary abelian group of order p2 in characteristic p. The polynomials exhibit a number of interesting special properties. For example they satisfy a three term recursion, are closely related to zigzag zero-one sequences and form strong divisibility sequences. These polynomials are shown to be closely connected to the order of appearance of prime numbers in the Fibonacci sequence, Artin's Primitive Root Conjecture, and the factorization of trinomials over finite fields.