On the use of the least common multiple to build a prime-generating recurrence

TL;DR

This paper investigates a prime-generating recurrence based on the least common multiple, linking its properties to an unproven conjecture and exploring how initial conditions affect the appearance of composite numbers.

Contribution

It establishes a connection between the prime-generating recurrence and a strong form of Linnik's Theorem, and generalizes the sequence to include composite numbers.

Findings

Sequence terms are conjectured to be only 1 or prime.

The prime conjecture relates to an unproven version of Linnik's Theorem.

Generalizations show conditions under which composite numbers may appear.

Abstract

We study a recursively defined sequence which is constructed using the least common multiple. It has been conjectured that every term of that sequence is $1$ or a prime. In this paper we show that this claim is connected to a strong version of Linnik's Theorem, which is yet unproved. We also study a generalization on which composite numbers may appear depending on the initial term.