IMC 1997 · Problem 4

Let $\alpha$ be a real number, $1 < \alpha < 2$.

a) Show that $\alpha$ has a unique representation as an infinite product

$$\alpha = \left( 1 + \frac{1}{n_1} \right) \left( 1 + \frac{1}{n_2} \right) \dots$$

where each $n_i$ is a positive integer satisfying

$$n_i^2 \le n_{i+1}.$$

b) Show that $\alpha$ is rational if and only if its infinite product has the following property:

For some $m$ and all $k \ge m$,

$$n_{k+1} = n_k^2.$$