On the least common multiple of several consecutive values of a polynomial

Abstract:

The periodicity is proved for the arithmetic function defined as the quotient of the product of $k+1$ values (where $k \geq 1$) of a polynomial $f\in {\mathbb Z}[x]$ at $k + 1$ consecutive integers ${f(n) f(n + 1) \cdots f(n + k)}$ and the least common multiple of the corresponding integers $f(n)$, $f(n + 1)$, …, $f(n + k)$. It is shown that this function is periodic if and only if no difference between two roots of $f$ is a positive integer smaller than or equal to $k$. This implies an asymptotic formula for the least common multiple of $f(n)$, $f(n+1)$, …, $f(n+k)$ and extends some earlier results in this area from linear and quadratic polynomials $f$ to polynomials of arbitrary degree $d$. A period in terms of the reduced resultants of $f(x)$ and $f(x+\ell )$, where $1 \leq \ell \leq k$, is given explicitly, as well as few examples of $f$ when the smallest period can be established.