Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

The least common multiple of consecutive arithmetic progression terms

Shaofang Honga1 and Guoyou Qiana1

a1 Mathematical College, Sichuan University, Chengdu 610064, People's Republic of China, (sfhong@scu.edu.cn; qiangy1230@gmail.com)

Abstract

Let k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetic function gk,a,b for any positive integer n by

$$ g_{k,a,b}(n):=\frac{(b+na)(b+(n+1)a)\cdots(b+(n+k)a)}{\operatorname{lcm}(b+na,b+(n+1)a,\dots,b+(n+k)a)}. $$

If we let a = 1 and b = 0, then gk,a,b becomes the arithmetic function that was previously introduced by Farhi. Farhi proved that gk,1,0 is periodic and that k! is a period. Hong and Yang improved Farhi's period k! to lcm(1, 2, … , k) and conjectured that (lcm(1, 2, … , k, k + 1))/(k + 1) divides the smallest period of gk,1,0. Recently, Farhi and Kane proved this conjecture and determined the smallest period of gk,1,0. For the general integers a ≥ 1 and b ≥ 0, it is natural to ask the following interesting question: is gk,a,b periodic? If so, what is the smallest period of gk,a,b? We first show that the arithmetic function gk,a,b is periodic. Subsequently, we provide detailed p-adic analysis of the periodic function gk,a,b. Finally, we determine the smallest period of gk,a,b. Our result extends the Farhi–Kane Theorem from the set of positive integers to general arithmetic progressions.

(Received March 28 2009)

(Online publication February 25 2011)

Keywords

  • arithmetic progression;
  • least common multiple;
  • p-adic valuation;
  • arithmetic function;
  • smallest period

2010 Mathematics subject classification

  • Primary 11B25;
  • 11N13;
  • 11A05