moldovan wrote: Prove that for any integers $ n \ge 2$ there is a set $ A_n$ of $ n$ distinct positive integers such that for any two distinct elements $ i,j \in A_n, |i - j|$ divides $ i^2 + j^2.$ This problem has been posted before. You should notice some ideas : Let A be a set of numbers , we defind $ A + k = \{x + k|x\in A\}$ . 1) If $ A$ be a good set then $ A + d$ also is a good set where $ d = \prod_{i < j}(j - i)$ . 2)If A be a good set ,then $ A + \{0\}$ is a good set . Induction on N will do the rest .

Take $ A_n = \{ n!,2*n!,...,n*n!\}$.