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

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 $(a, t, b) \in {$\mathbb{Z}$}^3, a \geq 5, t \geq 0, \rm{gcd}(a, b) = 1$. From this, we deduce for instance the lower bound: $$\mathrm{l}\mathrm{c}\mathrm{m}${1^2 + 1, 2^2 + 1, …, n^2

• 1} \geq 0,32 (1,442)^n$(forall$n \geq 1$). In the last part of this article, we study the integer $$\mathrm{l}\mathrm{c}\mathrm{m}$(n, n + 1, …, n + k)$ $(k\in \mathbb{N},n\in {\mathbb{N}}^{\ast })$. 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: $\mathrm{l}\mathrm{c}\mathrm{m}(n,n+1,\mathrm{.}\mathrm{.}\mathrm{.},n+k)=d_{n,k}$ and $\mathrm{l}\mathrm{c}\mathrm{m}(n,n+1,\mathrm{.}\mathrm{.}\mathrm{.},n+k)=m_{n,k}$ hold for an infinitely many pairs $(n,k)$.