*(English)*Zbl 1124.11005

Given a finite sequence of nonzero integers ${u}_{0},\cdots ,{u}_{n}$, the author gives effective bounds for their least common multiple. For example, Theorem 3 shows that if ${u}_{0},\cdots ,{u}_{n}$ is a strictly increasing arithmetic progression of nonzero integers, then for any non-negative integer $n$, $\text{lcm}\{{u}_{0},\cdots ,{u}_{n}\}$ is a multiple of the rational number $\frac{{u}_{0}{u}_{1}...{u}_{n}}{n!{\left(\text{gcd}\{{u}_{0},{u}_{1}\}\right)}^{n}}$. 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 ${\left({u}_{n}\right)}_{n}$ is a quadratic sequence; i.e., is the set of the consecutive values of a quadratic polynomial. For example, he shows that $\text{lcm}\{{1}^{2}+1,{2}^{2}+1,\cdots ,{n}^{2}+1\}\ge \xb732{(1\xb7442)}^{n}$. The proofs are elementary.

Reviewer’s remark. An asymptotic formula for $log\text{lcm}\{{u}_{0},\cdots ,{u}_{n}\}$ when ${u}_{0},\cdots ,{u}_{n}$ is an arithmetic progression is due to [*P. Bateman*, A limit involving least common multiples, Am. Math. Mon. 109, 393–394 (2002)]. For the case of quadratic irreducible polynomials $f\left(X\right)\in \mathbb{Z}\left[X\right]$, J. Cilleruelo has recently shown that $log\text{lcm}\left\{f\right(1),\cdots ,f(n\left)\right\}\sim nlogn$ as $n$ tends to infinity. When $f\left(X\right)={X}^{2}+1$, he showed that the next term of the asymptotic expansion is $Bn+o\left(n\right)$ 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.