# Product of arithmetic progressions

Let $(a_1,a_2\ldots,a_n)$ and $(b_1,b_2,\ldots,b_n)$ be two permutations of arithmetic progressions of natural numbers. For which $n$ is it possible that $(a_1b_1,a_2b_2,\dots,a_nb_n)$ is an arithmetic progression?

The sequence is (trivially) an arithmetic progression when $n=1$ or $2$. We can notice that $(a_1,a_2,\ldots,a_n)$ and $(b_1,b_2,\ldots,b_n)$ cannot be in identical order, since that would mean they both have to be increasing or decreasing, but then the differences of the resulting product terms cannot be constant.

My computer found these for 4, 5 and 6: $$A: 1×10, 11×4, 6×13, 16×7\\ B: 8×4, 6×9, 4×19, 7×14, 5×24\\ C: 7×35, 31×11,19×23,13×41,37×17,25×29$$

• $4×41,6×31,8×26,5×46,7×36$ – Empy2 Sep 12 '18 at 10:25