Gilbreath Equation, Gilbreath Polynomials, and Upper and Lower Bounds for Gilbreath Conjecture

Author:

Gatti Riccardo1ORCID

Affiliation:

1. National Laboratory of Molecular Biology and Stem Cell Engineering, Istituto Nazionale di Biostrutture e Biosistemi (INBB) c/o Eldor Lab, Via di Corticella 183, 40128 Bologna, Italy

Abstract

Let S=s1,…,sn be a finite sequence of integers. Then, S is a Gilbreath sequence of length n, S∈Gn, iff s1 is even or odd and s2,…,sn are, respectively, odd or even and minKs1,…,sm≤sm+1≤maxKs1,…,sm,∀m∈1,n. This, applied to the order sequence of prime number P, defines Gilbreath polynomials and two integer sequences, A347924 and A347925, which are used to prove that Gilbreath conjecture GC is implied by pn−2n−1⩽Pn−11, where Pn−11 is the n−1-th Gilbreath polynomial at 1.

Publisher

MDPI AG

Subject

General Mathematics,Engineering (miscellaneous),Computer Science (miscellaneous)

Reference10 articles.

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5. Gatti, R. (2020). The On-Line Encyclopedia of Integer Sequences, OEIS Foundation Inc.. Available online: https://oeis.org/A347924.

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