Product of arithmetic progressions

Question

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.

Now posted to MO, https://mathoverflow.net/questions/312896/product-of-arithmetic-progressions — Gerry Myerson, Oct 15 '18 at 11:23

Empy2 · Accepted Answer · 2018-09-13T05:38:23.420

0

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

edited Sep 13 '18 at 05:38

answered Sep 10 '18 at 00:35

Empy2

50,853

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

Product of arithmetic progressions

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