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    Hong, Shaofang Qian, Guoyou and Tan, Qianrong 2012. The least common multiple of a sequence of products of linear polynomials. Acta Mathematica Hungarica, Vol. 135, Issue. 1-2, p. 160.


    QIAN, GUOYOU TAN, QIANRONG and HONG, SHAOFANG 2012. THE LEAST COMMON MULTIPLE OF CONSECUTIVE TERMS IN A QUADRATIC PROGRESSION. Bulletin of the Australian Mathematical Society, Vol. 86, Issue. 03, p. 389.


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  • Proceedings of the Edinburgh Mathematical Society, Volume 54, Issue 2
  • June 2011, pp. 431-441

The least common multiple of consecutive arithmetic progression terms

  • Shaofang Hong (a1) and Guoyou Qian (a1)
  • DOI: http://dx.doi.org/10.1017/S0013091509000431
  • Published online: 25 February 2011
Abstract
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

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.

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1.T. M. Apostol , Introduction to analytic number theory (Springer, 1976).

2.G. Bachman and T. Kessler , On divisibility properties of certain multinomial coefficients, II, J. Number Theory 106 (2004), 112.

3.P. Bateman , J. Kalb and A. Stenger , A limit involving least common multiples, Am. Math. Mon. 109 (2002), 393394.

6.B. Farhi , Minoration non triviales du plus petit commun multiple de certaines suites finies d'entiers, C. R. Acad. Sci. Paris S'er. I 341 (2005), 469474.

7.B. Farhi , Nontrivial lower bounds for the least common multiple of some finite sequences of integers, J. Number Theory 125 (2007), 393411.

8.B. Farhi and D. Kane , New results on the least common multiple of consecutive integers, Proc. Am. Math. Soc. 137 (2009), 19331939.

9.B. Green and T. Tao , The primes contain arbitrarily long arithmetic progressions, Annals Math. (2) 167 (2008), 481547.

10.D. Hanson , On the product of the primes, Can. Math. Bull. 15 (1972), 3337.

13.S. Hong and W. Feng , Lower bounds for the least common multiple of finite arithmetic progressions, C. R. Acad. Sci. Paris Sér. I 343 (2006), 695698. The least common multiple of consecutive arithmetic progression terms 11

14.S. Hong and S. D. Kominers , Further improvements of lower bounds for the least common multiple of arithmetic progressions, Proc. Am. Math. Soc. 138 (2010), 809813.

16.S. Hong and Y. Yang , Improvements of lower bounds for the least common multiple of finite arithmetic progressions, Proc. Am. Math. Soc. 136 (2008), 41114114.

17.S. Hong and Y. Yang , On the periodicity of an arithmetical function, C. R. Acad. Sci. Paris Sér. I 346 (2008), 717721.

19.G. Myerson and J. Sander , What the least common multiple divides, II, J. Number Theory 61 (1996), 6784.

20.M. Nair , On Chebyshev-type inequalities for primes, Am. Math. Mon. 89 (1982), 126129.

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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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