Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 

 

Cullen numbers with the Lehmer property


Authors: José María Grau Ribas and Florian Luca
Journal: Proc. Amer. Math. Soc. 140 (2012), 129-134
MSC (2010): Primary 11A05; Secondary 11N25, 11A07
Published electronically: May 20, 2011
Corrigendum: Proc. Amer. Math. Soc. 141 (2013), 2941-2943
MathSciNet review: 2833524
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Here, we show that there is no positive integer $ n$ such that the $ n$th Cullen number $ C_n=n2^n+1$ has the property that it is composite but $ \phi(C_n)\mid C_n-1$.


References [Enhancements On Off] (What's this?)

  • 1. William D. Banks, Ahmet M. Güloğlu, and C. Wesley Nevans, On the congruence 𝑁≡𝐴\pmod{𝜑(𝑁)}, Integers 8 (2008), A59, 8. MR 2472077
  • 2. William D. Banks and Florian Luca, Composite integers 𝑛 for which 𝜑(𝑛)\mid𝑛-1, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 10, 1915–1918. MR 2352307, 10.1007/s10114-005-0731-1
  • 3. P.  Berrizbeitia and J.  G.  Fernandes, `Observaciones sobre la primalidad de los números de Cullen', short communication in ``Terceras Jornadas de Teoría de Números'' (http://campus.usal.es/$ \sim$tjtn2009/doc/abstracts.pdf).
  • 4. J. Cilleruelo and F. Luca, `Repunit Lehmer numbers', Proc. Edinburgh Math. Soc. 54 (2011), 55-65.
  • 5. G. L. Cohen and P. Hagis Jr., On the number of prime factors of 𝑛 if 𝜙(𝑛)(𝑛-1), Nieuw Arch. Wisk. (3) 28 (1980), no. 2, 177–185. MR 582925
  • 6. J. M. Grau and A. M. Oller-Marcén, `An $ {\widetilde{O}}(\log^2 N)$ time primality test for generalized Cullen numbers', preprint, 2010, to appear in Math. Comp.
  • 7. E. Heppner, Über Primzahlen der Form 𝑛2ⁿ+1 bzw. 𝑝2^{𝑝}+1, Monatsh. Math. 85 (1978), no. 2, 99–103 (German, with English summary). MR 0567136
  • 8. C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge University Press, Cambridge-New York-Melbourne, 1976. Cambridge Tracts in Mathematics, No. 70. MR 0404173
  • 9. Florian Luca, Fibonacci numbers with the Lehmer property, Bull. Pol. Acad. Sci. Math. 55 (2007), no. 1, 7–15. MR 2304295, 10.4064/ba55-1-2
  • 10. F. Luca, On the greatest common divisor of two Cullen numbers, Abh. Math. Sem. Univ. Hamburg 73 (2003), 253–270. MR 2028519, 10.1007/BF02941281
  • 11. F. Luca and C. Pomerance, `On composite integers $ n$ for which $ \phi(n)\mid n-1$', Bol. Soc. Mat. Mexicana, to appear.
  • 12. Florian Luca and Igor E. Shparlinski, Pseudoprime Cullen and Woodall numbers, Colloq. Math. 107 (2007), no. 1, 35–43. MR 2283130, 10.4064/cm107-1-5
  • 13. Carl Pomerance, On the distribution of amicable numbers, J. Reine Angew. Math. 293/294 (1977), 217–222. MR 0447087
  • 14. Carl Pomerance, On composite 𝑛 for which 𝜙(𝑛)\mid𝑛-1. II, Pacific J. Math. 69 (1977), no. 1, 177–186. MR 0434938

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11A05, 11N25, 11A07

Retrieve articles in all journals with MSC (2010): 11A05, 11N25, 11A07


Additional Information

José María Grau Ribas
Affiliation: Departamento de Mátematicas, Universidad de Oviedo, Avenida Calvo Sotelo, s/n, 33007 Oviedo, Spain
Email: grau@uniovi.es

Florian Luca
Affiliation: Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P. 58089, Morelia, Michoacán, México
Email: fluca@matmor.unam.mx

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10899-2
Received by editor(s): October 14, 2010
Received by editor(s) in revised form: November 11, 2010
Published electronically: May 20, 2011
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2011 American Mathematical Society