= Solution
The <continued fraction>[simple continued fraction] algorithm for $\sqrt{29}$ begins with $a_0=5$. Its complete quotients repeat after
$$
\sqrt{29}
=5+\cfrac1{
2+\cfrac1{
1+\cfrac1{
1+\cfrac1{
2+\cfrac1{
10+\cdots}}}}}.
$$
Thus
$$
\boxed{\sqrt{29}=[5;\overline{2,1,1,2,10}]}.
$$
The convergents through one period are
$$
5,\quad \frac{11}{2},\quad\frac{16}{3},
\quad\frac{27}{5},\quad\frac{70}{13}.
$$
The last one gives the <Pell-type norm from a periodic continued fraction>[negative Pell equation] solution
$$
70^2-29\cdot13^2=4900-4901=-1.
$$
Therefore
$$
\boxed{x=70,\qquad y=13}.
$$
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