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

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.