Solution
= Solution
The denominators of the convergents are
$$
1,1,2,3,14,17,31,48,\ldots.
$$
They increase strictly from $q_2$ onward, and none equals $13$. Equivalently, $34/13=[2;1,1,1,1,2]$, which ceases to agree with $[2;\overline{1,1,1,4}]$ after the third partial quotient. Hence there is no positive index $x$ with $p_x/q_x=34/13$.