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

Abstract

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 $u_n = a n (n + t) + b$ with $(a, t, b) \in {\mathbb{Z}}^3, a \geq 5, t \geq 0, \rm{gcd}(a, b) = 1$. From this, we deduce for instance the lower bound: $\mathrm{lcm}\{1^2 + 1, 2^2 + 1, ..., n^2 + 1\} \geq 0,32 (1,442)^n$ (for all $n \geq 1$). In the last part of this article, we study the integer $\mathrm{lcm}(n, n + 1, ..., n + k)$ $(k \in \mathbb{N}, n \in {\mathbb{N}}^*)$. 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: $\mathrm{lcm}(n, n + 1, ..., n + k) = d_{n, k}$ and $\mathrm{lcm}(n, n + 1, ..., n + k) = m_{n, k}$ hold for an infinitely many pairs $(n, k)$.