Finite Planes, Zigzag Sequences, Fibonacci Numbers, Artin's Conjecture and Trinomials

TL;DR

This paper explores the properties of a new family of polynomials derived from matrix representations of elementary abelian groups, revealing deep links with Fibonacci numbers, Artin's conjecture, and polynomial factorizations over finite fields.

Contribution

It introduces a novel family of polynomials characterized by a three-term recursion, connecting group representations, Fibonacci numbers, and number theory conjectures.

Findings

Polynomials satisfy a three-term recursion.

Connections established with Fibonacci numbers and prime distributions.

Classical Morgan-Voyce polynomials are special cases.

Abstract

We begin by considering faithful matrix representations of elementary abelian groups in prime characteristic. The representations considered are seen to be determined up to change of bases by a single number. Studying this number leads to a new family of polynomials which exhibit a number of special properties. These polynomials satisfy a three term recursion and are closely related to zigzag zero-one sequences. Interpreting the polynomials for the "prime" 1 yields the classical Morgan-Voyce polynomials, which form twoorthogonal families of polynomials and which have applications in the study of electrical resistance. Study of the general polynomials reveals deep connections with the Fibonacci series, the order of appearance of prime numbers in the Fibonacci sequence, the order of elements in cyclic groups, Artin's conjecture on primitive roots and the factorization of trinomials over finite fields.