Problem

Source: 2017 Taiwan TST Round 1

Tags: algebra, polynomial



Let $n$ be an odd number larger than 1, and $f(x)$ is a polynomial with degree $n$ such that $f(k)=2^k$ for $k=0,1,\cdots,n$. Prove that there is only finite integer $x$ such that $f(x)$ is the power of two.