Exercise 1.4.16* (Generalization)

Question

Show that if $f(x)$ is a polynomial of degree $n$ with real coefficients, which takes integral values on a certain set of $n+1$ consecutive integers, then $f(x)$ is integer-valued.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

\begin{proof} Assume that $f \in \R_n[x]$ takes integral values on the set $\{p, p+1,\ldots, p+n\}$, where $p \in \N$.

Consider the polynomial $P(x) = f(p+x)$. Then $P(0) = f(p), \ldots,P(n)= f(p+n)$ are integers. By Exercise 1.4.15, $P$ is integer-valued.

Since $f(x) = P(x-p)$, $f(k) = P(k-p) \in \Z$ for all $k \in \Z$. So $f$ is integer-valued.
\end{proof}

Exercise 1.4.16* (Generalization)

Answers

Comments

Exercise 1.4.16* (Generalization)

Answers

Comments

Add answer