Working Paper Article Version 3 This version is not peer-reviewed

# Gilbreath Sequences and Proof of Conditions for Gilbreath Conjecture

Version 1 : Received: 6 March 2020 / Approved: 8 March 2020 / Online: 8 March 2020 (17:19:33 CET)
Version 2 : Received: 10 April 2020 / Approved: 12 April 2020 / Online: 12 April 2020 (14:53:47 CEST)
Version 3 : Received: 21 September 2020 / Approved: 22 September 2020 / Online: 22 September 2020 (08:49:55 CEST)

How to cite: Gatti, R. Gilbreath Sequences and Proof of Conditions for Gilbreath Conjecture. Preprints 2020, 2020030145 Gatti, R. Gilbreath Sequences and Proof of Conditions for Gilbreath Conjecture. Preprints 2020, 2020030145

## Abstract

The conjecture attributed to Norman L. Gilbreath, but formulated by Francois Proth in the second half of the 1800s, concerns an interesting property of the ordered sequence of prime numbers $P$. Gilbreath conjecture stated that, if we compute the absolute values of differences of consecutive primes on ordered sequence of prime numbers, and if this calculation is repeated for the terms in the new sequence and so on, every sequence will start with 1. In this paper the concept of Gilbreath sequence, Gilbreath triangle and Gilbreath equation are defined and on the basis of the results obtained from their properties, an inductive proof is produced, which establishes the necessary condition to state that Gilbreath conjecture is true.

## Subject Areas

Gilbreath's conjecture; Gilbreth's sequence; sequence; prime numbers; number theory

Comment 1
Commenter: Riccardo Gatti
Commenter's Conflict of Interests: Author
Comment: Improvement of the layout and references. Correction of proof of theorem (2).
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