Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 3 / 9H / Solution

For a simple continued fraction, defineMultiplying the matrices $\left(\begin{array}{cc}{a}_j& 1\\ 1& 0\end{array}\right)$ givesTheir determinants also give $p_n{q}_{n-1}-p_{n-1}{q}_n=(-1{)}^{n-1}$. If the infinite tail is $\beta =[a_{n+1},a_{n+2},\dots ]>a_{n+1}$, the same formula impliesThe denominators of these continued fraction convergents tend to infinity, since $q_{n+2}\ge q_{n+1}+q_n$. Hence they converge to $\theta$.

The square-root algorithm for the periodic continued fraction of a quadratic irrational, using $M^{\prime }=Da-M$, $D^{\prime }=(53-M^{\prime 2})/D$ and $a^{\prime }=\lfloor (7+M^{\prime })/D^{\prime }\rfloor$, givesThe first four continued fraction convergents are $7/1,22/3,29/4,51/7$. Direct substitution gives positive solutions to the requested Pell-type equations: