Infinitely many twin prime polynomials of odd degree

TL;DR

This paper explores the infinite existence of twin prime polynomial pairs of odd degree over finite fields, extending classical twin prime conjecture ideas into algebraic polynomial settings.

Contribution

It introduces a new infinite family of twin prime polynomial tuples of odd degree and links their existence to Wieferich primes and Bell number properties.

Findings

Existence of infinitely many twin prime polynomial pairs over finite fields.

Connection between twin prime polynomials and Wieferich primes.

Relation to arithmetic properties of Bell numbers.

Abstract

While the twin prime conjecture is still famously open, it holds true in the setting of finite fields: There are infinitely many pairs of monic irreducible polynomials over $\mathbb{F}_q$ that differ by a fixed constant, for each $q \geq 3$. Elementary, constructive proofs were given for different cases by Hall and Pollack. In the same spirit, we discuss the construction of a further infinite family of twin prime tuples of odd degree, and its relations to the existence of certain Wieferich primes and to arithmetic properties of the combinatorial Bell numbers.