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Show that $z^n + \dfrac{1}{z^n} ∈ \mathbb{Q}$, when $z + \dfrac{1}{z} = 3$

If it helps, for $n\in\big\{2,3,4,5\big\}$, $\;z^n + \dfrac{1}{z^n} = 7, 18, 47, 123.$

Angelo
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Very Nice
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  • See also https://oeis.org/A005248 – lhf Apr 04 '21 at 11:24
  • The proof to this question is the same as https://math.stackexchange.com/questions/3444893/proof-that-xn-frac1xn-in-mathbb-z-by-complete-induction/3444908#3444908 – Very Nice Apr 04 '21 at 11:38

2 Answers2

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Hint: Expand $$\left(z + \frac1z\right)^n$$

Arthur
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Hint. Let's call $a_n = z^n +\frac{1}{z^n}$. Can you find $a_{n+1}$ from $a_1 \cdot a_n$ ?

Andrea Marino
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