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arXiv 0903.0530math.NT
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连续等差数列项的最小公倍数

The least common multiple of consecutive arithmetic progression terms

Shaofang Hong, Guoyou Qian

备注：
10 pages. To appear in Proceedings of the Edinburgh Mathematical Society
发表信息：
Proceedings of the Edinburgh Mathematical Society 54 (2011), 431-441
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摘要：

令$k\ge 0,a\ge 1$和$b\ge 0$为整数。通过$g_{k,a,b}(n):=\frac{(b+na)(b+(n+1)a)...(b+(n+k)a)} {{\rm lcm}(b+na,b+(n+1)a,...,b+(n+k)a)}.$设$a=1$和$b=0$为任意正整数$n$定义算术函数$g_{k,a,b}$，则$g_{k,a,b}$成为Farhi以前介绍的算术函数. Farhi证明了$g_{k,1,0}$是周期的，$k!$也是周期的。Hong和Yang将Farhi的周期$k!$改进为${\rm lcm}(1,2,...,k)$，并证明了$\frac{{\rm lcm}(1,2,...,k,k+1)}{k+1}$可整除$g_{k,1,0}$的最小周期。最近，Farhi和Kane证明了这个猜想，并确定了$g_{k,1,0}$的最小周期。对于一般整数$a\ge 1$和$b\ge 0$，很自然地会问一个有趣的问题：$g_{k,a,b}$是周期性的吗？如果是，那么$g_{k,a,b}$的最小周期是多少？我们首先证明了算术函数$g_{k,a,b}$是周期的。随后，我们提供了详细的$p$ -进分析的周期函数$g_{k,a,b}$。最后，我们确定了$g_{k,a,b}$的最小周期。我们的结果将Farhi-Kane定理从正整数集推广到一般的算术级数。

英文摘要：

Let $k\ge 0,a\ge 1$ and $b\ge 0$ be integers. We define the arithmetic function $g_{k,a,b}$ for any positive integer $n$ by $g_{k,a,b}(n):=\frac{(b+na)(b+(n+1)a)...(b+(n+k)a)} {{\rm lcm}(b+na,b+(n+1)a,...,b+(n+k)a)}.$ Letting $a=1$ and $b=0$, then $g_{k,a,b}$ becomes the arithmetic function introduced previously by Farhi. Farhi proved that $g_{k,1,0}$ is periodic and that $k!$ is a period. Hong and Yang improved Farhi's period $k!$ to ${\rm lcm}(1,2,...,k)$ and conjectured that $\frac{{\rm lcm}(1,2,...,k,k+1)}{k+1}$ divides the smallest period of $g_{k,1,0}$. Recently, Farhi and Kane proved this conjecture and determined the smallest period of $g_{k,1,0}$. For the general integers $a\ge 1$ and $b\ge 0$, it is natural to ask the interesting question: Is $g_{k,a,b}$ periodic? If so, then what is the smallest period of $g_{k,a,b}$? We first show that the arithmetic function $g_{k,a,b}$ is periodic. Subsequently, we provide detailed $p$-adic analysis of the periodic function $g_{k,a,b}$. Finally, we determine the smallest period of $g_{k,a,b}$. Our result extends the Farhi-Kane theorem from the set of positive integers to general arithmetic progressions.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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