Preprint Article Version 4 Preserved in Portico This version is not peer-reviewed

Gilbreath Equation, Gilbreath Polynomials, Upper and Lower Bound 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

Let $S=\left(s_1, \ldots, s_n\right)$ be a finite sequence of integers. Then $S$ is a Gilbreath sequence of length $n$, $S\in\mathbb{G}_n$, iff $s_1$ is even or odd and $s_2, \ldots, s_n$ are respectively odd or even and $\min\mathbb{K}_{\left(s_1, \ldots, s_m\right)}\leq s_{m+1}\leq\max\mathbb{K}_{\left(s_1, \ldots, s_m\right)}\forall m\in\left[\left.1, n\right)\right.$. This, applied to the order sequence of prime number $P$, defines Gilbreath polynomials and two integer sequences A347924 \cite{oeisA347924} and A347925 \cite{oeisA347925} which are used to prove that Gilbreath conjecture $GC$ is implied by $p_n-2^{n-1}\leqslant\mathcal{P}_{n-1}\left(1\right)$ where $\mathcal{P}_{n-1}\left(1\right)$ is the $n-1$-th Gilbreath polynomial at 1.

Keywords

Gilbreath conjecture; Gilbreath sequence; Gilbreath equation; integer sequence; prime numbers; number theory

Subject

Computer Science and Mathematics, Algebra and Number Theory

Comments (1)

Comment 1
Received: 21 February 2023
Commenter: Riccardo Gatti
Commenter's Conflict of Interests: Author
Comment: We have completed the proof of the implication of the conjecture and have provided an explanation to the sequences published by OEIS: A347924 and A347925.
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