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Nontrivial lower bounds for the least common multiple of some finite sequences of integers. (English) Zbl 1124.11005

Given a finite sequence of nonzero integers u 0 ,,u n , the author gives effective bounds for their least common multiple. For example, Theorem 3 shows that if u 0 ,,u n is a strictly increasing arithmetic progression of nonzero integers, then for any non-negative integer n, lcm{u 0 ,,u n } is a multiple of the rational number u 0 u 1 ...u n n!(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 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 lcm{1 2 +1,2 2 +1,,n 2 +1}·32(1·442) n . The proofs are elementary.

Reviewer’s remark. An asymptotic formula for loglcm{u 0 ,,u n } when u 0 ,,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(X)[X], J. Cilleruelo has recently shown that loglcm{f(1),,f(n)}nlogn 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.


MSC:
11A05Multiplicative structure of the integers
11B83Special sequences of integers and polynomials