2017 Putnam A2

Let $Q_0(x) = 1$, $Q_1(x) = x$, and

$$

Q_n(x) = \frac{(Q_{n-1}(x))^2 - 1}{Q_{n-2}(x)}

$$

for all $n \geq 2$. Show that, whenever $n$ is a positive integer, $Q_n(x)$ is equal to a polynomial with integer coefficients.

设 $Q_0(x) = 1$, $Q_1(x) = x$, 并且

$$

Q_n(x) = \frac{(Q_{n-1}(x))^2 - 1}{Q_{n-2}(x)}

$$

对于所有 $n \geq 2$。证明，只要 $n$ 是正整数，$Q_n(x)$ 就等于具有整数系数的多项式。