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


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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  1. AF.
    L. Addario-Berry, K. Ford, Poisson–Dirichlet branching random walks, preprint.Google Scholar
  2. AR.
    Addario-Berry L., Reed B.: Minima in branching random walks. Ann. Prob. 37, 1044–1079 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  3. B.
    Bachman M.: Limit theorems for the minimal position in a branching random walk with independent logconcave displacements. Adv. Appl. Prob. 32, 159–176 (2000)CrossRefGoogle Scholar
  4. BaH.
    Baker R.C., Harman G.: Shifted primes without large prime factors. Acta Arith. 83, 331–361 (1998)zbMATHMathSciNetGoogle Scholar
  5. Ban et al.
    W.D. Banks, J. Friedlander, F. Luca, F. Pappalardi, I.E. Shparlinski, Coincidences in the values of the Euler and Carmichael functions, Acta Arith. 122 (2006), 207–234.Google Scholar
  6. BanS.
    Banks W.D., Shparlinski I.E.: On values taken by the largest prime factor of shifted primes. J. Aust. Math. Soc. 82(1), 133–147 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  7. BasKW.
    Bassily N.L., Kátai I., Wijsmuller M.: Number of prime divisors of \({\phi{_{k}}(n)}\), where \({\phi_k}\) is the k-fold iterate of \({\phi}\). J. Number Theory 65, 226–239 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  8. Bay.
    Bayless J.: The Lucas–Pratt primality tree. Math. Comp. 77, 495–502 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  9. Bi1.
    Biggins J.D.: The first- and last-birth problems for a multitype age-dependent branching process. Adv. Appl. Prob. 31, 446–459 (1976)CrossRefMathSciNetGoogle Scholar
  10. Bi2.
    Biggins J.D.: Chernoff’s theorem in the branching randon walk. J. Appl. Prob. 14, 630–636 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  11. Bil.
    Billingsly P.: On the distribution of large prime divisors, Collection of Articles Dedicated to the Memory of Alfréd Rényi, I. Period. Math. Hungar. 2, 283–289 (1972)CrossRefMathSciNetGoogle Scholar
  12. Bo.
    E. Bombieri, Le grand crible dans la théorie analytique des nombres, 2nd ed. (in French; English summary) Ast´erisque 18, Société Mathématique de France, Paris (1987).Google Scholar
  13. Bor.
    C.W. Borchardt, Über eine Interpolationsformel für eine Art Symmetrischer Functionen und über Deren Anwendung, Math. Abh. der Akademie der Wissenschaften zu Berlin (1860), 1–20.Google Scholar
  14. C.
    M.A. Cherepnev, Some properties of large prime divisors of numbers of the form p – 1, Mat. Zametki 80 (2006), 920–925 (in Russian); English transl. in Math. Notes 80 (2006), 863–867.Google Scholar
  15. CrP.
    R. Crandall, C. Pomerance, Prime Numbers: A Computational Perspective, 2nd ed., Springer-Verlag, 2005.Google Scholar
  16. D.
    H. Davenport, Multiplicative Number Theory, 3rd ed., Graduate Texts inMathematics 74, Springer-Verlag, New York (2000).Google Scholar
  17. DoG.
    Donnelly P., Grimmett G.: On the asymptotic distribution of large prime factors. J. London Math. Soc. (2) 47, 395–404 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  18. E.
    Erdős P.: On the normal number of prime factors of p - 1 and some related problems concerning Euler’s \({\phi}\)-function. Quart. J. Math. Oxford 6, 205–213 (1935)CrossRefGoogle Scholar
  19. EGPS.
    P. Erdős, A. Granville, C. Pomerance, C. Spiro, On the normal behavior of the iterates of some arithmetic functions, in “Analytic Number Theory, Proceedings of a Conference in Honor of Paul T. Bateman”, Birkhäuser, Boston (1990), 165–204.Google Scholar
  20. EP.
    Erdős P., Pomerance C.: On the normal number of prime factors of \({\phi(n)}\), Number Theory (Winnipeg, Man. 1983). Rocky Mountain J. Math. 15(2), 343–352 (1985)CrossRefMathSciNetGoogle Scholar
  21. FL.
    K. Ford, F. Luca, The number of solutions of λ(x) = n, INTEGERS, Special Volume in Honor of Melvyn Nathanson and Carl Pomerance, to appear.Google Scholar
  22. FLP.
    Ford K., Luca F., Pomerance C.: Common values of the arithmetic functions \({\phi}\) and σ. Bull. London Math. Soc. 42, 478–488 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  23. GT.
    Green B., Tao T.: The primes contain arbitrarily long arithmetic progressions. Ann. Math. 167, 481–547 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  24. HR.
    Halberstam H., Richert H.-E.: Sieve Methods. Academic Press, London (1974)zbMATHGoogle Scholar
  25. HaT.
    R.R. Hall, G. Tenenbaum, Divisors, Cambridge Tracts in Math., 1988.Google Scholar
  26. HiT.
    Hildebrand A., Tenenbaum G.: Integers without large prime factors. J. de Théorie des Nombres de Bordeaux 5, 411–484 (1993)zbMATHMathSciNetGoogle Scholar
  27. K.
    Kátai I.: Some problems on the iteration of multiplicative number-theoretical functions. Acta Math. Acad. Scient. Hung. 19, 441–450 (1968)zbMATHCrossRefGoogle Scholar
  28. L.
    Lamzouri Y.: Smooth values of iterates of the Euler phi-function. Canad. J. Math. 59, 127–147 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  29. LuP.
    F. Luca, C. Pomerance, Irreducible radical extensions and Euler-function chains, in “Combinatorial Number Theory”, de Gruyter, Berlin (2007), 351–361.Google Scholar
  30. MP.
    Martin G., Pomerance C.: The iterated Carmichael λ-function and the number of cycles of the power generator. Acta Arith. 188, 305–335 (2005)CrossRefMathSciNetGoogle Scholar
  31. Mc.
    McDiarmid C.: Minimal positions in a branching random walk. Ann. Appl. Prob. 5(1), 128–139 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  32. P.
    Pomerance C.: Very short primality proofs. Math. Comp. 48, 315–322 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  33. Pr.
    Pratt V.: Every prime has a succinct certificate. SIAM J. Comput. 4(3), 214–220 (1975)zbMATHCrossRefMathSciNetGoogle Scholar

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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