Working Paper Article Version 2 This version is not peer-reviewed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)
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 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 is defined the concept of Gilbreath sequence, Gilbreath triangle and Gilbreath 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 Gilbreath conjecture is true.

Keywords

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

Subject

Computer Science and Mathematics, Algebra and Number Theory

Comments (1)

Comment 1
Received: 12 April 2020
Commenter: Riccardo Gatti
Commenter's Conflict of Interests: Author
Comment: Improvement of demonstration and clarification of definitions and lemmas
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