Polynomials with values which are powers of integers

Rachid Boumahdi and Jesse Larone

Address:
Laboratoire d’Arithmétique, Codage, Combinatoire et Calcul Formel, Université des Sciences et Technologies Houari Boumédiène, 16111, El Alia, Algiers, Algeria
Département de mathématiques et de statistiques, Université Laval, Québec, Canada, G1V 0A6

E-mail:
r_boumehdi@esi.dz
jesse.larone.1@ulaval.ca

Abstract: Let $P$ be a polynomial with integral coefficients. Shapiro showed that if the values of $P$ at infinitely many blocks of consecutive integers are of the form $Q(m)$, where $Q$ is a polynomial with integral coefficients, then $P(x)=Q( R(x))$ for some polynomial $R$. In this paper, we show that if the values of $P$ at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form $m^q$ where $q$ is an integer greater than 1, then $P(x)=( R(x))^q$ for some polynomial $R(x)$.

AMSclassification: primary 13F20.

Keywords: integer-valued polynomial.

DOI: 10.5817/AM2018-2-119