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Geometric and Functional Analysis

, Volume 20, Issue 5, pp 1231–1258 | Cite as

Prime Chains and Pratt Trees

  • Kevin Ford
  • Sergei V. Konyagin
  • Florian Luca
Article

Abstract

Prime chains are sequences \(p_{1}, \ldots , p_{k}\) of primes for which \({p_{j+1} \equiv 1}\) (mod p j ) for each j. We introduce three new methods for counting long prime chains. The first is used to show that \({N(x; p) = O_{\varepsilon}(x^{1+\varepsilon})}\), where N(x; p) is the number of chains with p 1 = p and \({p_k \leq p_x}\). The second method is used to show that the number of prime chains ending at p is ≍ log p for most p. The third method produces the first nontrivial upper bounds on H(p), the length of the longest chain with p k = p, valid for almost all p. As a consequence, we also settle a conjecture of Erdős, Granville, Pomerance and Spiro from 1990. A probabilistic model of H(p), based on the theory of branching random walks, is introduced and analyzed. The model suggests that for most \({p \leq x}\), H(p) stays very close to e log log x.

Keywords and phrases

Prime chains Pratt trees branching random walks 

2010 Mathematics Subject Classification

Primary 11N05 11N36 Secondary 60J80 

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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-Champaign UrbanaUrbanaUSA
  2. 2.Stecklov Mathematical InstituteMoscowRussia
  3. 3.Instituto de MatemáticasUniversidad Nacional Autonoma de MéxicoMoreliaMéxico

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