On the Gap sequence and the Gilbreath conjecture

TL;DR

This paper explores the gap sequence concept inspired by the Gilbreath conjecture, introducing new notions of paths, circuits, traces, and path lengths to analyze the conjecture.

Contribution

It develops a novel framework involving paths and circuits to study the Gilbreath conjecture through the concepts of trace and path length.

Findings

Introduces the concept of gap sequences induced by any number sequence.

Defines paths, circuits, traces, and lengths to analyze the conjecture.

Provides new insights into the structure of sequences related to the Gilbreath conjecture.

Abstract

Motivated by the Gilbreath conjecture, we develop the notion of the gap sequence induced by any sequence of numbers. We introduce the notion of the path and associated circuits induced by an originator and study the conjecture via the notion of the trace and length of a path.