Prime chains and Pratt trees
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.
Keywords
Cite
@article{arxiv.0904.0473,
title = {Prime chains and Pratt trees},
author = {Kevin Ford and Sergei V. Konyagin and Florian Luca},
journal= {arXiv preprint arXiv:0904.0473},
year = {2010}
}
Comments
v4. Very minor revision. Small corrections, e.g. sentence preceding Theorem 6, last sentence in the proof of Lemma 5.2. Updated reference [22]