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The least common multiple of consecutive arithmetic progression terms

Part of: Elementary number theory Multiplicative number theory Sequences and sets

Published online by Cambridge University Press:  25 February 2011

Shaofang Hong  and
Guoyou Qian
Shaofang Hong
Affiliation:
Mathematical College, Sichuan University, Chengdu 610064, People's Republic of China, ([email protected]; [email protected])
Guoyou Qian
Affiliation:
Mathematical College, Sichuan University, Chengdu 610064, People's Republic of China, ([email protected]; [email protected])
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Abstract

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

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.

Keywords

arithmetic progressionleast common multiplep-adic valuationarithmetic functionsmallest period

MSC classification

Secondary: 11B25: Arithmetic progressions 11N13: Primes in progressions 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 2 , June 2011 , pp. 431 - 441
Copyright
Copyright © Edinburgh Mathematical Society 2011

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