Proceedings of the American Mathematical Society

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

 

 

New results on the least common multiple of consecutive integers


Authors: Bakir Farhi and Daniel Kane
Journal: Proc. Amer. Math. Soc. 137 (2009), 1933-1939
MSC (2000): Primary 11A05
Published electronically: December 29, 2008
MathSciNet review: 2480273
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: When studying the least common multiple of some finite sequences of integers, the first author introduced the interesting arithmetic functions $ g_k$ $ (k \in \mathbb{N})$, defined by $ g_k(n) := \frac{n (n + 1) \dots (n + k)} {\operatorname{lcm}(n, n+1, \dots, n + k)}$ $ (\forall n \in \mathbb{N} \setminus \{0\})$. He proved that for each $ k \in \mathbb{N}$, $ g_k$ is periodic and $ k!$ is a period of $ g_k$. He raised the open problem of determining the smallest positive period $ P_k$ of $ g_k$. Very recently, S. Hong and Y. Yang improved the period $ k!$ of $ g_k$ to $ \operatorname{lcm}(1 , 2, \dots , k)$. In addition, they conjectured that $ P_k$ is always a multiple of the positive integer $ \frac{\operatorname{lcm}(1 , 2 , \dots , k , k + 1)}{k + 1}$. An immediate consequence of this conjecture is that if $ (k + 1)$ is prime, then the exact period of $ g_k$ is precisely equal to $ \operatorname{lcm}(1 , 2 , \dots , k)$.

In this paper, we first prove the conjecture of S. Hong and Y. Yang and then we give the exact value of $ P_k$ $ (k \in \mathbb{N})$. We deduce, as a corollary, that $ P_k$ is equal to the part of $ \operatorname{lcm}(1 , 2 , \dots , k)$ not divisible by some prime.


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

  • 1. P. Bateman, J. Kalb, and A. Stenger, A limit involving least common multiples, Amer. Math. Monthly, 109 (2002), 393-394.
  • 2. J. Cilleruelo, The least common multiple of a quadratic sequence, preprint (2008).
  • 3. Bakir Farhi, Minorations non triviales du plus petit commun multiple de certaines suites finies d’entiers, C. R. Math. Acad. Sci. Paris 341 (2005), no. 8, 469–474 (French, with English and French summaries). MR 2180812, 10.1016/j.crma.2005.09.019
  • 4. Bakir Farhi, Nontrivial lower bounds for the least common multiple of some finite sequences of integers, J. Number Theory 125 (2007), no. 2, 393–411. MR 2332595, 10.1016/j.jnt.2006.10.017
  • 5. Denis Hanson, On the product of the primes, Canad. Math. Bull. 15 (1972), 33–37. MR 0313179
  • 6. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
  • 7. Shaofang Hong and Weiduan Feng, Lower bounds for the least common multiple of finite arithmetic progressions, C. R. Math. Acad. Sci. Paris 343 (2006), no. 11-12, 695–698 (English, with English and French summaries). MR 2284695, 10.1016/j.crma.2006.11.002
  • 8. S. Hong and Y. Yang, On the periodicity of an arithmetical function, C. R. Math. Acad. Sci. Paris, 346 (2008), 717-721.
  • 9. -, Improvements of lower bounds for the least common multiple of finite arithmetic progressions, Proc. Amer. Math. Soc., 136 (2008), 4111-4114.
  • 10. M. Nair, On Chebyshev-type inequalities for primes, Amer. Math. Monthly 89 (1982), no. 2, 126–129. MR 643279, 10.2307/2320934

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11A05

Retrieve articles in all journals with MSC (2000): 11A05


Additional Information

Bakir Farhi
Affiliation: Département de Mathématiques, Université du Maine, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France
Email: bakir.farhi@gmail.com

Daniel Kane
Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02139
Email: aladkeenin@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09730-X
Keywords: Least common multiple, arithmetic function, exact period.
Received by editor(s): July 28, 2008
Received by editor(s) in revised form: August 17, 2008
Published electronically: December 29, 2008
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.