Nontrivial lower bounds for the least common multiple of some finite sequences of integers
✍ Scribed by Bakir Farhi
- Publisher
- Elsevier Science
- Year
- 2007
- Tongue
- English
- Weight
- 201 KB
- Volume
- 125
- Category
- Article
- ISSN
- 0022-314X
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✦ Synopsis
We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetic progressions and lower bounds less efficient (but nontrivial) for quadratic sequences whose general term has the form u n = an(n + t) + b with (a, t, b) ∈ Z 3 , a 5, t 0, gcd(a, b) = 1. From this, we deduce for instance the lower bound: lcm{1 2 + 1, 2 2 + 1, . . . , n 2 + 1} 0, 32(1, 442) n (for all n 1). In the last part of this article, we study the integer lcm(n, n + 1, . . . , n + k) (k ∈ N, n ∈ N * ). We show that it has a divisor d n,k simple in its dependence on n and k, and a multiple m n,k also simple in its dependence on n. In addition, we prove that both equalities: lcm(n, n + 1, . . . , n+ k) = d n,k and lcm(n, n + 1, . . . , n+ k) = m n,k hold for an infinitely many pairs (n, k).