Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T04:24:49.795Z Has data issue: false hasContentIssue false
  • English
  • Français

The least common multiple of consecutive arithmetic progression terms

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

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])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1.Apostol, T. M., Introduction to analytic number theory (Springer, 1976).Google Scholar
2.Bachman, G. and Kessler, T., On divisibility properties of certain multinomial coefficients, II, J. Number Theory 106 (2004), 1–12.Google Scholar
3.Bateman, P., Kalb, J. and Stenger, A., A limit involving least common multiples, Am. Math. Mon. 109 (2002), 393–394.CrossRefGoogle Scholar
4.Bennett, M. A., Bruin, N., Györy, K. and Hajdu, L., Powers from products of consecutive terms in arithmetic progression, Proc. Lond. Math. Soc. 92 (2006), 273–306.Google Scholar
5.Cilleruelo, J., The least common multiple of a quadratic sequence, Compositio Math., in press.Google Scholar
6.Farhi, B., Minoration non triviales du plus petit commun multiple de certaines suites finies d'entiers, C. R. Acad. Sci. Paris S'er. I 341 (2005), 469–474.CrossRefGoogle Scholar
7.Farhi, B., Nontrivial lower bounds for the least common multiple of some finite sequences of integers, J. Number Theory 125 (2007), 393–411.CrossRefGoogle Scholar
8.Farhi, B. and Kane, D., New results on the least common multiple of consecutive integers, Proc. Am. Math. Soc. 137 (2009), 1933–1939.Google Scholar
9.Green, B. and Tao, T., The primes contain arbitrarily long arithmetic progressions, Annals Math. (2) 167 (2008), 481–547.CrossRefGoogle Scholar
10.Hanson, D., On the product of the primes, Can. Math. Bull. 15 (1972), 33–37.CrossRefGoogle Scholar
11.Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, 4th edn (Oxford University Press, 1960).Google Scholar
12.Hong, S. and Enoch Lee, K. S., Asymptotic behavior of eigenvalues of reciprocal power LCM matrices, Glasgow Math. J. 50 (2008), 163–174.CrossRefGoogle Scholar
13.Hong, S. and Feng, W., Lower bounds for the least common multiple of finite arithmetic progressions, C. R. Acad. Sci. Paris Sér. I 343 (2006), 695–698. The least common multiple of consecutive arithmetic progression terms 11CrossRefGoogle Scholar
14.Hong, S. and Kominers, S. D., Further improvements of lower bounds for the least common multiple of arithmetic progressions, Proc. Am. Math. Soc. 138 (2010), 809–813.Google Scholar
15.Hong, S. and Loewy, R., Asymptotic behavior of eigenvalues of greatest common divisor matrices, Glasgow Math. J. 46 (2004), 551–569.CrossRefGoogle Scholar
16.Hong, S. and Yang, Y., Improvements of lower bounds for the least common multiple of finite arithmetic progressions, Proc. Am. Math. Soc. 136 (2008), 4111–4114.Google Scholar
17.Hong, S. and Yang, Y., On the periodicity of an arithmetical function, C. R. Acad. Sci. Paris Sér. I 346 (2008), 717–721.Google Scholar
18.Hua, L.-K., Introduction to number theory (Springer, 1982).Google Scholar
19.Myerson, G. and Sander, J., What the least common multiple divides, II, J. Number Theory 61 (1996), 67–84.Google Scholar
20.Nair, M., On Chebyshev-type inequalities for primes, Am. Math. Mon. 89 (1982), 126–129.Google Scholar
21.Saradha, N., Squares in products with terms in an arithmetic progression, Acta Arith. 86 (1998), 27–43.Google Scholar
22.Saradha, N. and Shorey, T. N., Almost squares in arithmetic progression, Compositio Math. 138 (2003), 73–111.CrossRefGoogle Scholar