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