= Solution
Apply the reciprocal-and-integer-part algorithm for the <continued fraction>. The successive complete quotients after the integer part two are $(\sqrt7+2)/3$, $(\sqrt7+1)/2$, $(\sqrt7+1)/3$, and $\sqrt7+2$. Their integer parts are $1,1,1,4$, and taking the next reciprocal repeats the first complete quotient. Therefore
$$
\boxed{\sqrt7=[2;\overline{1,1,1,4}].}
$$
The first four <continued fraction convergents> are $2,3,5/2,8/3$. The last gives $8^2-7\cdot3^2=1$. Squaring its unit $8+3\sqrt7$ gives $127+48\sqrt7$, whose <norm> is again one. Thus two positive solutions of the <Pell equation> are
$$
\boxed{(x,y)=(8,3),\quad(127,48).}
$$
Direct verification gives $127^2-7\cdot48^2=16129-16128=1$.
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