Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-11T11:37:26.706Z Has data issue: false hasContentIssue false
  • English
  • Français

THE LEAST COMMON MULTIPLE OF CONSECUTIVE TERMS IN A QUADRATIC PROGRESSION

Part of: Elementary number theory Sequences and sets

Published online by Cambridge University Press:  12 April 2012

GUOYOU QIAN ,
QIANRONG TAN  and
SHAOFANG HONG
GUOYOU QIAN
Affiliation:
Mathematical College, Sichuan University, Chengdu 610064, PR China (email: [email protected], [email protected])
QIANRONG TAN
Affiliation:
School of Computer Science and Technology, Panzhihua University, Panzhihua 617000, PR China (email: [email protected])
SHAOFANG HONG*
Affiliation:
Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, PR China (email: [email protected], [email protected], [email protected])
*
✉For correspondence; e-mail: [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 be any given positive integer. We define the arithmetic function gk for any positive integer n by We first show that gk is periodic. Subsequently, we provide a detailed local analysis of the periodic function gk, and determine its smallest period. We also obtain an asymptotic formula for log lcm0≤i≤k {(n+i)2+1}.

Keywords

quadratic progressionleast common multiplep-adic valuationarithmetic functionsmallest period

MSC classification

Secondary: 11B25: Arithmetic progressions 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 3 , December 2012 , pp. 389 - 404
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

S. Hong was supported partially by National Science Foundation of China Grant #10971145 and by the PhD Programs Foundation of Ministry of Education of China Grant #20100181110073.

References

[1]Bateman, P., Kalb, J. and Stenger, A., ‘A limit involving least common multiples’, Amer. Math. Monthly 109 (2002), 393–394.Google Scholar
[2]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.CrossRefGoogle Scholar
[3]Chebyshev, P. L., ‘Memoire sur les nombres premiers’, J. Math. Pures Appl. 17 (1852), 366–390.Google Scholar
[4]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
[5]Farhi, B. and Kane, D., ‘New results on the least common multiple of consecutive integers’, Proc. Amer. Math. Soc. 137 (2009), 1933–1939.CrossRefGoogle Scholar
[6]Green, B. and Tao, T., ‘The primes contain arbitrarily long arithmetic progressions’, Ann. of Math. (2) 167 (2008), 481–547.CrossRefGoogle Scholar
[7]Hanson, D., ‘On the product of the primes’, Canad. Math. Bull. 15 (1972), 33–37.CrossRefGoogle Scholar
[8]Hardy, G. H. and Littlewood, J. E., ‘Some problems of partitio numerorum III: On the expression of a number as a sum of primes’, Acta Math. 44 (1923), 1–70.CrossRefGoogle Scholar
[9]Hong, S. and Qian, G., ‘The least common multiple of consecutive arithmetic progression terms’, Proc. Edinb. Math. Soc. 54 (2011), 431–441.CrossRefGoogle Scholar
[10]Hong, S., Qian, G. and Tan, Q., ‘The least common multiple of a sequence of products of linear polynomials’, Acta Math. Hungar., 135 (2012), 160–167.CrossRefGoogle Scholar
[11]Hong, S. and Yang, Y., ‘On the periodicity of an arithmetical function’, C.R. Acad. Sci. Paris Ser. I 346 (2008), 717–721.CrossRefGoogle Scholar
[12]Hua, L.-K., Introduction to Number Theory (Springer, Berlin, 1982).Google Scholar
[13]Iwaniec, H., ‘Almost-primes represented by quadratic polynomials’, Invent. Math. 47 (1978), 171–188.Google Scholar
[14]Nair, M., ‘On Chebyshev-type inequalities for primes’, Amer. Math. Monthly 89 (1982), 126–129.CrossRefGoogle Scholar
[15]Saradha, N. and Shorey, T. N., ‘Almost squares in arithmetic progression’, Compositio Math. 138 (2003), 73–111.CrossRefGoogle Scholar