Problem

Source: ELMO Shortlist 2024/N5

Tags: Elmo, number theory



Let T be a finite set of squarefree integers. (a) Show that there exists an integer polynomial P(x) such that the set of squarefree numbers in the range of P(n) across all nZ is exactly T. (b) Suppose that T is allowed to be infinite. Is it still true that for all choices of T, such an integer polynomial P(x) exists? Allen Wang