A random analogue of Gilbreath’s conjecture

Abstract

A well-known conjecture of Gilbreath, and independently Proth from the 1800s, states that if $$a_{0,n} = p_n$$ a 0 , n = p n denotes the nth prime number and $$a_{i,n} = |a_{i-1,n}-a_{i-1,n+1}|$$ a i , n = | a i - 1 , n - a i - 1 , n + 1 | for $$i, n \ge 1$$ i , n ≥ 1 , then $$a_{i,1} = 1$$ a i , 1 = 1 for all $$i \ge 1$$ i ≥ 1 . It has been postulated repeatedly that the property of having $$a_{i,1} = 1$$ a i , 1 = 1 for i large enough should hold for any choice of initial $$(a_{0,n})_{n \ge 1}$$ ( a 0 , n ) n ≥ 1 provided that the gaps $$a_{0,n+1}-a_{0,n}$$ a 0 , n + 1 - a 0 , n are not too large and are sufficiently random. We prove (a precise form of) this postulate.