Geodesic
Polynomials with values which are powers of integers
Archivum mathematicum, Tome 54 (2018) no. 2, pp. 119-125 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $R(x)$.
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 $R(x)$.
DOI : 10.5817/AM2018-2-119
Classification : 13F20
Keywords: integer-valued polynomial 
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Boumahdi, Rachid; Larone, Jesse. Polynomials with values which are powers of integers. Archivum mathematicum, Tome 54 (2018) no. 2, pp. 119-125. doi: 10.5817/AM2018-2-119

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