Prime chains and Pratt trees

TL;DR

This paper investigates the distribution and height of prime chains and Pratt trees, providing estimates, bounds, and a stochastic model suggesting typical heights are near e log log p, under certain conjectures.

Contribution

It introduces new bounds on prime chain distributions, analyzes Pratt tree heights, and proposes a stochastic model based on branching random walks.

Findings

H(p)  c \, ext{log log p} for most p

H(p)  ( ext{log p})^{1-c'} for almost all p

The stochastic model indicates H(p) is close to e log log p for most p

Abstract

We study the distribution of prime chains, which are sequences p_1,...,p_k of primes for which p_{j+1}\equiv 1\pmod{p_j} for each j. We give estimates for the number of chains with p_k\le x (k variable), and the number of chains with p_1=p and p_k \le px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with p_k=p, which is also the height of the Pratt tree for p. We show H(p)\ge c\log\log p and H(p)\le (\log p)^{1-c'} for almost all p, with c,c' explicit positive constants. We can take, for any \epsilon>0, c=e-\epsilon assuming the Elliott-Halberstam conjecture. A stochastic model of the Pratt tree, based on a branching random walk, is introduced and analyzed. The model suggests that for most p, H(p) stays very close to e \log\log p.