Nontrivial lower bounds for the least common multiple of some finite sequences of integers (Q996205)

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 \(\geq u_0(r+1)^{n-1}\). 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 \(\text{lcm}\{1^2+1,2^2+1,\dots,n^2+1\}\geq .32(1.442)^n\). 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 [\textit{P. Bateman}, A limit involving least common multiples, Am. Math. Mon. 109, 393--394 (2002; Zbl 1124.11300)]. For the case of quadratic irreducible polynomials \(f(X)\in {\mathbb Z}[X]\), 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.