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On the Gap sequence and the Gilbreath conjecture

Theophilus Agama

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.

On the Gap sequence and 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.

Paper Structure

This paper contains 6 sections, 12 theorems, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. The notion of a path induced by a sequence
  4. The length of a path
  5. The notion of a circuit
  6. Reduction of the Gilbreath conjecture to the language of circuit

Key Result

Proposition 2.2

Let $\{d_j^{k}\}_{j=1}^{t}$ be a path of order $k\geq 1$ with maximal step $t$ with originator $\{a_i\}_{i=1}^{n}$. The path $\{d_i^{k+1}\}_{i\geq 1}$ has exactly $t-1$ maximal steps. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Gilbreath phenomenon with the path of order $k$ highlighted and its length interpreted as the horizontal sum of the terms on that row.
  • Figure 2: Gilbreath phenomenon with the trace of the $s^{\text{th}}$ segment marked as the sum of all its occurrences across the array.
  • Figure 3: Upside-down triangular representation of the Gilbreath framework: originator, paths of successive order, prime segments, trace, and circuit.

Theorems & Definitions (30)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4: Step-order equation
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • ...and 20 more