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 un=an(n+t)+b with (a,t,b)∈Z3, a⩾5, t⩾0, gcd(a,b)=1. From this, we deduce for instance the lower bound: lcm{12+1,22+1,…,n2+1}⩾0,32n(1,442) (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 dn,k simple in its dependence on n and k, and a multiple mn,k also simple in its dependence on n. In addition, we prove that both equalities: lcm(n,n+1,…,n+k)=dn,k and lcm(n,n+1,…,n+k)=mn,k hold for an infinitely many pairs (n,k).