$\tan{kπ\over7}$幂和

hbghlyj

Crux3697 任意正整数$n$, 证明$$f(n)=\left(\tan \frac{\pi}{7}\right)^{6 n}+\left(\tan \frac{2 \pi}{7}\right)^{6 n}+\left(\tan \frac{3 \pi}{7}\right)^{6 n}$$为整数.
根据MathWorld $\tan{π\over7},\tan{2π\over7},\tan{3π\over7}$为$x^{6}-21 x^{4}+35 x^{2}-7$的根,
则$a=\tan^2{π\over7},b=\tan^2{2π\over7},c=\tan^2{3π\over7}$为$x^3-21 x^2+35x-7$的3个不同实根.
根据\begin{aligned} a^{n+3}+b^{n+3}+c^{n+3}= & (a+b+c)\left(a^{n+2}+b^{n+2}+c^{n+2}\right) \\ & -(a b+b c+c a)\left(a^{n+1}+b^{n+1}+c^{n+1}\right)+a b c\left(a^{n}+b^{n}+c^{n}\right)\end{aligned}我们有$$f (n + 3) = 21f (n + 2) − 35f (n + 1) + 7f (n) $$
且 $ f (1) = 21, f (2) = 371, f (3) = 7077$ 为整数, 因此 $f (n)$ 为整数, 对任意 $n ∈ ℕ$.

kuing

类似于《撸题集》P.587~588 题目 4.10.8