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- W4300616646 abstract "We study the distribution of prime chains, which are sequences p_1,...,p_k of primes for which p_{j+1}equiv 1pmod{p_j} for each j. We give estimates for the number of chains with p_kle 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 cloglog 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 loglog p." @default.
- W4300616646 created "2022-10-03" @default.
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- W4300616646 date "2009-04-02" @default.
- W4300616646 modified "2023-09-27" @default.
- W4300616646 title "Prime chains and Pratt trees" @default.
- W4300616646 doi "https://doi.org/10.48550/arxiv.0904.0473" @default.
- W4300616646 hasPublicationYear "2009" @default.
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