A random analogue of Gilbreath's conjecture

2020/05/01 by Zachary Chase, Chase, Zachary
Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2005.00530

openalex publication_date 2020/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A well-known conjecture of Gilbreath, and independently Proth from the 1800s, states that if a_0,n = p_n denotes the n^th prime number and a_i,n = |a_i-1,n-a_i-1,n+1| for i, n ≥ 1, then a_i,1 = 1 for all i ≥ 1. It has been postulated repeatedly that the property of having a_i,1 = 1 for i large enough should hold for any choice of initial (a_0,n)_{n ≥ 1} provided that the gaps a_0,n+1-a_0,n are not too large and are sufficiently random. We prove (a precise form of) this postulate.

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