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Preserved in Portico This version is not peer-reviewed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)
Version 3 : Received: 21 September 2020 / Approved: 22 September 2020 / Online: 22 September 2020 (08:49:55 CEST)
Version 4 : Received: 20 February 2023 / Approved: 21 February 2023 / Online: 21 February 2023 (14:30:09 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)
Version 4 : Received: 20 February 2023 / Approved: 21 February 2023 / Online: 21 February 2023 (14:30:09 CET)
A peer-reviewed article of this Preprint also exists.
Gatti, R. Gilbreath Equation, Gilbreath Polynomials, and Upper and Lower Bounds for Gilbreath Conjecture. Mathematics 2023, 11, 4006. Gatti, R. Gilbreath Equation, Gilbreath Polynomials, and Upper and Lower Bounds for Gilbreath Conjecture. Mathematics 2023, 11, 4006.
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.
Keywords
Gilbreath's conjecture; Gilbreth's sequence; sequence; prime numbers; number theory
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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