Given a finite sequence of nonzero integers , the author gives effective bounds for their least common multiple. For example, Theorem 3 shows that if is a strictly increasing arithmetic progression of nonzero integers, then for any non-negative integer , is a multiple of the rational number . The author also shows that this lower bound is optimal in some cases. When and the difference of the progression are coprime he shows that this number is . He also gives lower bounds for the case when is a quadratic sequence; i.e., is the set of the consecutive values of a quadratic polynomial. For example, he shows that . The proofs are elementary.
Reviewer’s remark. An asymptotic formula for when 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 , J. Cilleruelo has recently shown that as tends to infinity. When , he showed that the next term of the asymptotic expansion is and computed the constant . According to these results, the author’s lower bounds are ‘effective’ but of a much smaller order than the actual size of these numbers.