Algebraic divisibility sequences over function fields

TL;DR

This paper investigates the properties of divisibility sequences over function fields, demonstrating the infinite occurrence of irreducible elements and the finiteness of terms lacking primitive divisors in elliptic sequences.

Contribution

It extends classical divisibility sequence results to function fields, proving infinite irreducible elements and finite exceptions for primitive divisors in elliptic sequences.

Findings

Lucas sequences over function fields have infinitely many irreducible elements

Elliptic divisibility sequences over function fields contain finitely many terms without primitive divisors

Results depend on various hypotheses about the function fields

Abstract

We study the existence of primes and of primitive divisors in classical divisibility sequences defined over function fields. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.