Problem

Source: China TST 2004 Quiz

Tags: number theory unsolved, number theory



The largest one of numbers $ p_1^{\alpha_1}, p_2^{\alpha_2}, \cdots, p_t^{\alpha_t}$ is called a $ \textbf{Good Number}$ of positive integer $ n$, if $ \displaystyle n= p_1^{\alpha_1} \cdot p_2^{\alpha_2} \cdots p_t^{\alpha_t}$, where $ p_1$, $ p_2$, $ \cdots$, $ p_t$ are pairwisely different primes and $ \alpha_1, \alpha_2, \cdots, \alpha_t$ are positive integers. Let $ n_1, n_2, \cdots, n_{10000}$ be $ 10000$ distinct positive integers such that the $ \textbf{Good Numbers}$ of $ n_1, n_2, \cdots, n_{10000}$ are all equal. Prove that there exist integers $ a_1, a_2, \cdots, a_{10000}$ such that any two of the following $ 10000$ arithmetical progressions $ \{ a_i, a_i + n_i, a_i + 2n_i, a_i + 3n_i, \cdots \}$($ i=1,2, \cdots 10000$) have no common terms.