MathDB

Let

p_1, p_2, p_3, \ldots

be the prime numbers listed in increasing order, and let

x_0

be a real number between 0 and 1. For positive integer

k

, define x_k = \begin{cases} 0 & \mbox{if} \; x_{k-1} = 0, \$$.1in] {\displaystyle \left\{ \frac{p_k}{x_{k-1}} \right\}} & \mbox{if} \; x_{k-1} \neq 0, \end{cases} where

\{x\}

denotes the fractional part of

x

. (The fractional part of

x

is given by

x - \lfloor x \rfloor

where

\lfloor x \rfloor

is the greatest integer less than or equal to

x

.) Find, with proof, all

x_0

satisfying

0 < x_0 < 1

for which the sequence

x_0, x_1, x_2, \ldots

eventually becomes 0.

Problem Statement