3
$\begingroup$

Recall that $29 = \prod\limits_{k=1}^6 (2 + 2 \zeta_{7}^{k} + \zeta_{7}^{2k})$ and $43 = \prod\limits_{k=1}^6 (2 + \zeta_{7}^{k})$ where $\zeta_7=e^{2i\pi/7}$.

Are there any integers $(a_0,a_1,\ldots,a_6)$ such that $2017 = \prod\limits_{k=1}^6 (a_0 + a_1 \zeta_{7}^{k} + \cdots + a_6 \zeta_{7}^{6k})$?

P.S.

$2017 = (3 \zeta_{7} + 3 \zeta_{7}^{3} + 2 \zeta_{7}^{4} + \zeta_{7}^{5}) * (2 \zeta_{7} + 3 \zeta_{7}^{3} + 3 \zeta_{7}^{4} + \zeta_{7}^{5}) * (1 + 3 \zeta_{7} + 2 \zeta_{7}^{2} + 3 \zeta_{7}^{4}) * (\zeta_{7}^2 + 2 \zeta_{7}^{3} + 3 \zeta_{7}^{4} + 3 \zeta_{7}^{6}) * (\zeta_{7}^2 + 3 \zeta_{7}^{3} + 3 \zeta_{7}^{4} + 2 \zeta_{7}^{6}) * (1 + 3 \zeta_{7}^3 + 2 \zeta_{7}^{5} + 3 \zeta_{7}^{6}).$

  • 2
    $a_n = (0,1,2,3,0,3,0)$ – 2017-01-15
  • 1
    Because of your first sentence about $29$ and $43$, I didn't realize until now that $2017$ is prime. Ah, an auspicious year for math. :) – 2017-01-15

1 Answers 1

1

By Winther's comment, we get the formula.

$2017$

$ = (\zeta_{7} + 2 \zeta_{7}^{2} + 3 \zeta_{7}^{3} + 3 \zeta_{7}^{5}) * (\zeta_{7}^2 + 3 \zeta_{7}^{3} + 2 \zeta_{7}^{4} + 3 \zeta_{7}^{6}) * (3 \zeta_{7} + 3 \zeta_{7}^{2} + \zeta_{7}^{3} + 2 \zeta_{7}^{6}) * (2 \zeta_{7} + \zeta_{7}^{4} + 3 \zeta_{7}^{5} + 3 \zeta_{7}^{6}) * (3 \zeta_{7} + 2 \zeta_{7}^{3} + 3 \zeta_{7}^{4} + \zeta_{7}^{5}) * (3 \zeta_{7}^2 + 3 \zeta_{7}^4 + 2 \zeta_{7}^{5} + \zeta_{7}^{6}).$

$ = \prod\limits_{k=1}^6 (\zeta_{7}^{k} + 2 \zeta_{7}^{2k} + 3 \zeta_{7}^{3k} + 3 \zeta_{7}^{5k}).$

I found such $[a0, a1, \cdots, a6]$ by programming.

$[0, 0, 1, 2, 3, 0, 3], [0, 0, 1, 3, 2, 0, 3], [0, 0, 1, 3, 3, 0, 2], [0, 0, 2, 0, 3, 3, 1], [0, 0, 2, 3, 3, 1, 3], [0, 0, 3, 0, 2, 3, 1], [0, 0, 3, 0, 3, 2, 1], [0, 0, 3, 1, 3, 3, 2], [0, 1, 2, 3, 0, 3, 0], [0, 1, 2, 3, 3, 0, 3], [0, 1, 3, 0, 3, 3, 2], [0, 1, 3, 2, 0, 3, 0], [0, 1, 3, 3, 0, 2, 0], [0, 2, 0, 0, 1, 3, 3], [0, 2, 0, 3, 3, 1, 0], [0, 2, 3, 1, 0, 0, 3], [0, 2, 3, 3, 0, 3, 1], [0, 2, 3, 3, 1, 3, 0], [0, 3, 0, 0, 1, 2, 3], [0, 3, 0, 0, 1, 3, 2], [0, 3, 0, 1, 2, 3, 3], [0, 3, 0, 2, 3, 1, 0], [0, 3, 0, 3, 2, 1, 0], [0, 3, 0, 3, 3, 2, 1], [0, 3, 1, 0, 2, 3, 3], [0, 3, 1, 3, 3, 2, 0], [0, 3, 2, 1, 0, 0, 3], [0, 3, 3, 1, 0, 0, 2], [0, 3, 3, 2, 0, 1, 3], [0, 3, 3, 2, 1, 0, 3], [1, 0, 0, 2, 0, 3, 3], [1, 0, 0, 3, 0, 2, 3], [1, 0, 0, 3, 0, 3, 2], [1, 0, 2, 3, 3, 0, 3], [1, 0, 3, 0, 3, 3, 2], [1, 2, 3, 0, 3, 0, 0], [1, 2, 3, 3, 0, 3, 0], [1, 3, 0, 0, 2, 3, 3], [1, 3, 0, 3, 3, 2, 0], [1, 3, 2, 0, 3, 0, 0], [1, 3, 3, 0, 2, 0, 0], [1, 3, 3, 2, 0, 0, 3], [2, 0, 0, 1, 3, 3, 0], [2, 0, 0, 3, 1, 3, 3], [2, 0, 1, 3, 0, 3, 3], [2, 0, 3, 0, 0, 1, 3], [2, 0, 3, 3, 1, 0, 0], [2, 1, 0, 0, 3, 0, 3], [2, 1, 0, 3, 0, 3, 3], [2, 3, 0, 3, 0, 0, 1], [2, 3, 1, 0, 0, 3, 0], [2, 3, 3, 0, 3, 0, 1], [2, 3, 3, 0, 3, 1, 0], [2, 3, 3, 1, 3, 0, 0], [3, 0, 0, 1, 2, 3, 0], [3, 0, 0, 1, 3, 2, 0], [3, 0, 0, 2, 3, 3, 1], [3, 0, 1, 2, 3, 3, 0], [3, 0, 2, 0, 0, 1, 3], [3, 0, 2, 3, 1, 0, 0], [3, 0, 3, 0, 0, 1, 2], [3, 0, 3, 0, 1, 2, 3], [3, 0, 3, 1, 0, 2, 3], [3, 0, 3, 2, 1, 0, 0], [3, 0, 3, 3, 2, 0, 1], [3, 0, 3, 3, 2, 1, 0], [3, 1, 0, 0, 2, 0, 3], [3, 1, 0, 0, 3, 0, 2], [3, 1, 0, 2, 3, 3, 0], [3, 1, 3, 0, 0, 2, 3], [3, 1, 3, 3, 2, 0, 0], [3, 2, 0, 0, 3, 1, 3], [3, 2, 0, 1, 3, 0, 3], [3, 2, 0, 3, 0, 0, 1], [3, 2, 1, 0, 0, 3, 0], [3, 2, 1, 0, 3, 0, 3], [3, 3, 0, 2, 0, 0, 1], [3, 3, 0, 3, 0, 1, 2], [3, 3, 0, 3, 1, 0, 2], [3, 3, 1, 0, 0, 2, 0], [3, 3, 1, 3, 0, 0, 2], [3, 3, 2, 0, 0, 3, 1], [3, 3, 2, 0, 1, 3, 0], [3, 3, 2, 1, 0, 3, 0]$