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Keywords:
integer-valued polynomial
integer-valued polynomial
Summary:
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 $R(x)$.
References:
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