preprints.org > computer science and mathematics > algebra and number theory > doi: 10.20944/preprints202003.0145.v4

Preprint Article Version 4 Preserved in Portico This version is not peer-reviewed

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)

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.

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

Computer Science and Mathematics, Algebra and Number Theory

* All users must log in before leaving a comment