(i) Let $x \geqslant 2$ be a real number and let $P(x)=\prod_{p \leqslant x}\left(1-\frac{1}{p}\right)^{-1}$, where the product is taken over all primes $p \leqslant x$. Prove that $P(x)>\log x$.

(ii) Define the continued fraction of any positive irrational real number $x$. Illustrate your definition by computing the continued fraction of $1+\sqrt{3}$.

Suppose that $a, b, c$ are positive integers with $b=a c$ and that $x$ has the periodic continued fraction $[b, a, b, a, \ldots]$. Prove that $x=\frac{1}{2}\left(b+\sqrt{b^{2}+4 c}\right)$.