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

16 pages, published in Journal of Number Theory