Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1124.11005
Farhi, Bakir
Nontrivial lower bounds for the least common multiple of some finite sequences of integers. (English)
[J] J. Number Theory 125, No. 2, 393-411 (2007). ISSN 0022-314X; ISSN 1096-1658/e

Given a finite sequence of nonzero integers $u_0,\dots,u_n$, the author gives effective bounds for their least common multiple. For example, Theorem 3 shows that if $u_0,\dots,u_n$ is a strictly increasing arithmetic progression of nonzero integers, then for any non-negative integer $n$, $\text{lcm}\{u_0,\dots,u_n\}$ is a multiple of the rational number ${{u_0u_1\ldots u_n}\over {n!(\text{gcd}\{u_0,u_1\})^n}}$. The author also shows that this lower bound is optimal in some cases. When $u_0$ and the difference of the progression $r$ are coprime he shows that this number is $\ge u_0(r+1)^{n-1}$. He also gives lower bounds for the case when $(u_n)_{n}$ is a quadratic sequence; i.e., is the set of the consecutive values of a quadratic polynomial. For example, he shows that $\text{lcm}\{1^2+1,2^2+1,\dots,n^2+1\}\ge .32(1.442)^n$. The proofs are elementary. Reviewer's remark. An asymptotic formula for $\log \text{lcm}\{u_0,\dots,u_n\}$ when $u_0,\dots,u_n$ is an arithmetic progression is due to [{\it P. Bateman}, A limit involving least common multiples, Am. Math. Mon. 109, 393--394 (2002)]. For the case of quadratic irreducible polynomials $f(X)\in {\Bbb Z}[X]$, J. Cilleruelo has recently shown that $\log \text{lcm}\{f(1),\dots,f(n)\}\sim n\log n$ as $n$ tends to infinity. When $f(X)=X^2+1$, he showed that the next term of the asymptotic expansion is $Bn+o(n)$ and computed the constant $B$. According to these results, the author's lower bounds are `effective' but of a much smaller order than the actual size of these numbers.
[Florian Luca (Morelia)]

MSC 2000:: *11A05 Multiplicative structure of the integers
11B83 Special sequences of integers and polynomials

Keywords: least common multiple; arithmetic progressions; quadratic sequences

Citations: Zbl 1124.11300

Cited in: Zbl 1213.11014

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]