New results on the least common multiple of consecutive integers
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- by Bakir Farhi and Daniel Kane
- Proc. Amer. Math. Soc. 137 (2009), 1933-1939
- DOI: https://doi.org/10.1090/S0002-9939-08-09730-X
- Published electronically: December 29, 2008
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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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Bibliographic 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
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 1933-1939
- MSC (2000): Primary 11A05
- DOI: https://doi.org/10.1090/S0002-9939-08-09730-X
- MathSciNet review: 2480273