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Proceedings of the American Mathematical Society
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
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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.


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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
PII: S 0002-9939(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.