Prime chains and Pratt trees

Details Prime chains and Pratt trees Prime chains and Pratt trees

2009-04-02

arxiv.org/abs/0904.0473v4

Geom. Funct. Anal. 20 (2010), 1231-1258

Mathematics

Number Theory

v4. Very minor revision. Small corrections, e.g. sentence preceding Theorem 6, last sentence in the proof of Lemma 5.2. Update

Scientific paper

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.

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