Improvements of lower bounds for the least common multiple of finite arithmetic progressions

Hong, Shaofang; Yang, Yujuan

doi:10.1090/s0002-9939-08-09565-8

Cited by 12 publications

(22 citation statements)

References 16 publications

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“…Hong and Feng published [8] a nontrivial lower bound for the least common multiple of finite arithmetic progressions which confirmed Farhi's conjecture. Recently, Hong and Yang [10] improved the lower bounds of Farhi, Hong and Feng.…”

Section: Introductionmentioning

confidence: 99%

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On the periodicity of an arithmetical function

Hong

Yang

2008

Comptes Rendus. Mathématique

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Section: Introductionmentioning

confidence: 99%

“…Récemment, Hong et Yang [10] ont amélioré les minorations de Farhi et Hong et Feng. D'un autre côté, Farhi [3,4], a étudié le plus petit commun multiple d'un nombre fini d'entiers consécutifs. Soit k 0 un entier, il est démontré dans [3] et [4] que ppcm(n, n + 1, .…”

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On the periodicity of an arithmetical function

Hong

Yang

2008

Comptes Rendus. Mathématique

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“…On the other hand, Farhi [2] obtained several nontrivial bounds and posed a conjecture which was later confirmed by Hong and Feng [7]. Hong and Feng [7] also got an improved lower bound for sufficiently long arithmetic progressions; this result was later sharpened further by Hong and Yang [11]. We notice that Hong and Yang [12] and Farhi and Kane [5] obtained some related results regarding the least common multiple of a finite number of consecutive integers.…”

Section: Introductionmentioning

confidence: 56%

“…Following Hong and Yang [11], we denote, for each integer 0 ≤ k ≤ n, C n,k := u k · · · u n (n − k)! , L n,k := lcm(u k , .…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

New lower bounds for the least common multiples of arithmetic progressions

Wu¹,

Tan

Hong³

2013

Chin. Ann. Math. Ser. B

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For relatively prime positive integers u 0 and r and for 0 ≤ k ≤ n, define u k := u 0 + kr. Let Ln := lcm(u 0 , u 1 , ..., un) and let a, l ≥ 2 be any integers. In this paper, we show that, for integers α ≥ a and r ≥ max(a, l − 1) and n ≥ lαr, we have Ln ≥ u 0 r (l−1)α+a−l (r + 1) n .Particularly, letting l = 3 yields an improvement to the best previous lower bound on Ln obtained by Hong and Kominers.2000 Mathematics Subject Classification. Primary 11B25, 11N13, 11A05.

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“…For some other results, see, for example, [4,12,15,21,22]. Meanwhile, the topic of the least common multiple of any given sequence of positive integers has received a lot of attention from many authors: see, for example, [2, 3, 5-7, 10, 11, 13, 14, 16,19,20]. For detailed background information about the least common multiple of finite arithmetic progressions, we refer readers to [17].…”

Section: Introductionmentioning

confidence: 99%

The least common multiple of consecutive arithmetic progression terms

Hong¹,

Gong²

2011

Proceedings of the Edinburgh Mathematical Society

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Let $k\ge 0,a\ge 1$ and $b\ge 0$ be integers. We define the arithmetic function $g_{k,a,b}$ for any positive integer $n$ by $g_{k,a,b}(n):=\frac{(b+na)(b+(n+1)a)...(b+(n+k)a)} {{\rm lcm}(b+na,b+(n+1)a,...,b+(n+k)a)}.$ Letting $a=1$ and $b=0$, then $g_{k,a,b}$ becomes the arithmetic function introduced previously by Farhi. Farhi proved that $g_{k,1,0}$ is periodic and that $k!$ is a period. Hong and Yang improved Farhi's period $k!$ to ${\rm lcm}(1,2,...,k)$ and conjectured that $\frac{{\rm lcm}(1,2,...,k,k+1)}{k+1}$ divides the smallest period of $g_{k,1,0}$. Recently, Farhi and Kane proved this conjecture and determined the smallest period of $g_{k,1,0}$. For the general integers $a\ge 1$ and $b\ge 0$, it is natural to ask the interesting question: Is $g_{k,a,b}$ periodic? If so, then what is the smallest period of $g_{k,a,b}$? We first show that the arithmetic function $g_{k,a,b}$ is periodic. Subsequently, we provide detailed $p$-adic analysis of the periodic function $g_{k,a,b}$. Finally, we determine the smallest period of $g_{k,a,b}$. Our result extends the Farhi-Kane theorem from the set of positive integers to general arithmetic progressions.Comment: 10 pages. To appear in Proceedings of the Edinburgh Mathematical Societ

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Improvements of lower bounds for the least common multiple of finite arithmetic progressions

Abstract: Abstract. Let u 0 , r, α and n be positive integers such that (u 0 , r) = 1. LetThis improves the lower bound of L n obtained previously by Farhi, Hong and Feng.

Cited by 12 publications

References 16 publications

On the periodicity of an arithmetical function

On the periodicity of an arithmetical function

New lower bounds for the least common multiples of arithmetic progressions

The least common multiple of consecutive arithmetic progression terms

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