Preprint Article Version 1 This version is not peer-reviewed

Gilbreath's Sequences and Proof of Conditions for Gilbreath's 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)

How to cite: Gatti, R. Gilbreath's Sequences and Proof of Conditions for Gilbreath's Conjecture. Preprints 2020, 2020030145 (doi: 10.20944/preprints202003.0145.v1). Gatti, R. Gilbreath's Sequences and Proof of Conditions for Gilbreath's Conjecture. Preprints 2020, 2020030145 (doi: 10.20944/preprints202003.0145.v1).

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’s conjecture stated that, computing the absolute value of differences of consecutive primes on ordered sequence of prime numbers, and if this calculation is done for the terms in the new sequence and so on, every sequence will starts with 1. In this paper is defined the concept of Gilbreath’s sequence, Gilbreath’s triangle and Gilbreath’s equation. On the basis of the results obtained from the proof of properties, an inductive proof is produced thanks to which it is possible to establish the necessary condition to state that the Gilbreath's conjecture is true.

Subject Areas

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