Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERALIZED CULLEN NUMBERS WITH THE LEHMER PROPERTY

  • Kim, Dae-June (Department of Mathematical Sciences Seoul National University) ;
  • Oh, Byeong-Kweon (Department of Mathematical Sciences and Research Institute of Mathematics Seoul National University)
  • Received : 2012.08.24
  • Published : 2013.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

We say a positive integer n satisfies the Lehmer property if ${\phi}(n)$ divides n - 1, where ${\phi}(n)$ is the Euler's totient function. Clearly, every prime satisfies the Lehmer property. No composite integer satisfying the Lehmer property is known. In this article, we show that every composite integer of the form $D_{p,n}=np^n+1$, for a prime p and a positive integer n, or of the form ${\alpha}2^{\beta}+1$ for ${\alpha}{\leq}{\beta}$ does not satisfy the Lehmer property.

Keywords

References

  1. J. Cilleruelo and F. Luca, Repunit Lehmer numbers, Proc. Edinb. Math. Soc. (2) 54 (2011), no. 1, 55-65.
  2. G. L. Cohen and P. Hagis Jr., On the number of prime factors of n if ${\phi}(n)|(n-1)$, Nieuw.Arch. Wisk. (3) 28 (1980), no. 2, 177-185.
  3. J. M. Grau Ribas and F. Luca, Cullen numbers with the Lehmer property, Proc. Amer. Math. Soc. 140 (2012), no. 1, 129-134.
  4. C. Hooley, Applications of Sieve Methods to the Theory of Numbers, Cambridge Tracts in Mathematics, No. 70. Cambridge University Press, Cambridge-New York-Melbourne, 1976.
  5. D. H. Lehmer, On Euler's totient function, Bull. Amer. Math. Soc. 38 (1932), no. 10, 745-757. https://doi.org/10.1090/S0002-9904-1932-05521-5
  6. F. Luca, Fibonacci numbers with the Lehmer property, Bull. Pol. Acad. Sci. Math. 55 (2007), no. 1, 7-15. https://doi.org/10.4064/ba55-1-2

Cited by

  1. Pell numbers with the Lehmer property vol.28, pp.1-2, 2017, https://doi.org/10.1007/s13370-016-0449-5