Problem

Source: China TST 2004 Quiz

Tags: algebra, polynomial, calculus, integration, number theory



Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k+3$, where $ k$ is also an integer.