New results on the least common multiple of consecutive integers
New results on the least common multiple of consecutive integers
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DOI：
10.1090/s0002-9939-08-09730-x
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发表时间：
2008-08
期刊：
影响因子：
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通讯作者：
Bakir Farhi;D. Kane
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文献类型：
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作者：
Bakir Farhi;D. Kane
在研究一些整数有限序列的最小公倍数时，第一作者引入了有趣的算术函数\(g_k\)（\(k\in N\)），其定义为\(g_k(n):=\frac{n(n + 1)\cdots(n + k)}{\text{lcm}(n,n + 1,\cdots,n + k)}\)（\(\forall n\in N\setminus\{0\}\)）。他证明了对于每个\(k\in N\)，\(g_k\)是周期函数，且\(k!\)是\(g_k\)的一个周期。他提出了确定\(g_k\)的最小正周期\(P_k\)这一开放问题。最近，S. 洪和Y. 杨将\(g_k\)的周期\(k!\)改进为\(\text{lcm}(1,2,\cdots,k)\)。此外，他们猜想\(P_k\)总是正整数\(\frac{\text{lcm}(1,2,\cdots,k,k + 1)}{k + 1}\)的倍数。这个猜想的一个直接结果是，如果\((k + 1)\)是质数，那么\(g_k\)的精确周期恰好等于\(\text{lcm}(1,2,\cdots,k)\)。在本文中，我们首先证明S. 洪和Y. 杨的猜想，然后给出\(P_k\)（\(k\in N\)）的精确值。作为一个推论，我们推断出\(P_k\)等于\(\text{lcm}(1,2,\cdots,k)\)中不被某个质数整除的部分。
When studying the least common multiple of some finite sequences of integers, the first author introduced the interesting arithmetic functions g k (k ∈ N), defined by g k (n):= n(n+1)...(n+k) lcm(n,n+1,...,n+k) (Vn ∈ N \ {0}). He proved that for each k ∈ 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 lcm(1,2,..., k). In addition, they conjectured that P k is always a multiple of the positive integer lcm(1,2,...,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 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 ∈ N). We deduce, as a corollary, that P k is equal to the part of lcm(1, 2 k) not divisible by some prime.