Problem

Source: CWMO 2003, Problem 3

Tags: modular arithmetic, pigeonhole principle, number theory unsolved, number theory



Let $ n$ be a given positive integer. Find the smallest positive integer $ u_n$ such that for any positive integer $ d$, in any $ u_n$ consecutive odd positive integers, the number of them that can be divided by $ d$ is not smaller than the number of odd integers among $ 1, 3, 5, \ldots, 2n - 1$ that can be divided by $ d$.