Question 1
Let $p_1,p_2,p_3,\ldots$ be the primes in increasing order, and let $0\lt x_0\lt 1$. For every positive integer $k$, define $$x_k=\begin{cases}0,&x_{k-1}=0,\\ \left\{p_k/x_{k-1}\right\},&x_{k-1}\ne0.\end{cases}$$ Here $\{t\}=t-\lfloor t\rfloor$ is the fractional part. Find, with proof, all starting values $x_0$ for which the sequence eventually becomes $0$.