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Nontrivial lower bounds for the least common multiple of some finite sequences of integers. (English)
J. Number Theory 125, No. 2, 393-411 (2007).
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.
Reviewer: Florian Luca (Morelia)
Classification: F65