《 Elementary Methods in Number Theory 》Exercise 1.3.12

Let $[a_0,a_1,\cdots,a_N]$ be a finite simple continued fraction,and let $p_n$ and $q_n$ be the numbers defined in Exercise 10.Prove that
\begin{equation}\label{eq:2345}
p_nq_{n-1}-p_{n-1}q_n=(-1)^{n-1}
\end{equation}
and for $n=1,\cdots,N$.Prove that if $a_i\in\mathbf{Z}$ for $i=0,1,\cdots,N$,then $(p_n,q_n)=1$ for $n=0,1,\cdots,N$.

Proof:

(1)Please See 数论概论(Joseph H.Silverman) 定理39.2 连分数相邻收敛项之差定理 .

(2)According to \ref{eq:2345},this is simple.