Prove that $\tan^6 20°+\tan^6 40°+\tan^6 80°$ is an integer

Question

Prove that $\tan^6 20°+\tan^6 40°+\tan^6 80°$ is an integer. Doesn't this problem seem a little out of the box? It seems beautiful, but I don't have an idea on how to start. Calculating the value does give me an integer-$32733$. Thanks.

Answer 1

Hint: consider $$x_{n}=a^n+b^n+c^n$$ $$x_{n+3}=(a+b+c)x_{n+2}-(ab+bc+ac)x_{n+1}+abcx_{n}$$ let $a=\tan^2{20}=\tan^2{\dfrac{\pi}{9}},b=\tan^2{40}=\tan^2{\dfrac{2\pi}{9}},c=\tan^2{80}=\tan^2{\dfrac{4\pi}{9}}$ and you can add $d=\tan^2{\dfrac{3\pi}{9}}=3$

It is easy to find $$a+b+c=30,ab+bc+ac=27, abc=3$$ because you can see here lab bhattacharjee full solution so you induction; $x_{n}\in \mathbb{N^{+}}$

Answer 2

Let $\theta = 20^\circ = \frac{\pi}{9}$ and let $\omega_k = \tan(k\theta)$ for $k = 0,1,\ldots 8$. The sum we want can be rewritten as

$$\tan^6(20^\circ) + \tan^6(40^\circ) + \tan^6(80^\circ) = \omega_1^6 + \omega_2^6 + \omega_4^6$$ Since $(\cos(k\theta) + i\sin(k \theta))^9 = e^{i9k\theta} = (-1)^k$, the $\omega_k$ are the nine roots of the polynomial $$ \Im\left[( 1 + i t )^9\right] = t^9-36t^7+126t^5-84t^3+9t = t(t^2-3)(t^6-33t^4+27t^2-3) $$ Notice $\omega_0 = 0, \omega_3 = -\omega_6 = \sqrt{3}$, the factor $t(t^2 - 3) = (t - \omega_0)(t - \omega_3)(t - \omega_6)$.

Together with the identities $\omega_1 = -\omega_8$, $\omega_2 = -\omega_7$, $\omega_4 = -\omega_5$, we find

$$(x - \omega_1^2)(x - \omega_2^2)(x - \omega_4^2) = x^3 - 33x^2 + 27 x - 3$$

Let $p_n = \omega_1^{2n} + \omega_2^{2n} + \omega_3^{2n}$ for $n \in \mathbb{Z}_{+}$. Apply Newton's identities to above cubic polynomial, we have

$$ \begin{cases} p_1 - 33 = 0\\ p_2 - 33 p_1 + 2\times 27 = 0\\ p_3 - 33 p_2 + 27 p_1 - 3\times 3 = 0 \end{cases} \implies \begin{cases} p_1 = 33\\ p_2 = 33 \times 33 - 2\times 27 = 1035\\ p_3 = 33 \times 1035 - 27 \times 33 + 3\times 3 = 33273 \end{cases} $$ As a result, $$\tan^6(20^\circ) + \tan^6(40^\circ) + \tan^6(80^\circ) = \omega_1^6 + \omega_2^6 + \omega_4^6 = p_3 = 33273$$