New results on the least common multiple of consecutive integers

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dc.creatorFarhi, Bakir
dc.creatorKane, Daniel
dc.date2008-08-11
dc.date.accessioned2026-07-07T09:56:00Z
dc.date.available2026-07-07T09:56:00Z
dc.descriptionWhen 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 $g_k(n) := \frac{n (n + 1) ... (n + k)}{\lcm(n, n + 1, >..., n + k)}$ $(\forall n \in \mathbb{N} \setminus \{0\})$. He proved that $g_k$ $(k \in \mathbb{N})$ is periodic and $k!$ is a period of $g_k$. He raised the open problem consisting to determine the smallest positive period $P_k$ of $g_k$. Very recently, S. Hong and Y. Yang have improved the period $k!$ of $g_k$ to $\lcm(1, 2, ..., k)$. In addition, they have conjectured that $P_k$ is always a multiple of the positive integer $\frac{\lcm(1, 2, >..., k, k + 1)}{k + 1}$. An immediate consequence of this conjecture states that if $(k + 1)$ is prime then the exact period of $g_k$ is precisely equal to $\lcm(1, 2, ..., k)$. 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 $\lcm(1, 2, ..., k)$ not divisible by some prime.
dc.description8 pages, to appear
dc.identifierhttps://arxiv.org/abs/0808.1507
dc.identifierhttp://arxiv.org/abs/0808.1507
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/166829
dc.subjectNumber Theory
dc.subject11A05
dc.titleNew results on the least common multiple of consecutive integers
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