1. 设$b=r_0$.$r_1,r_2,\cdots$是将欧几里德算法应用于$a$与$b$得到的相继余数，证明每两步会缩小余数至少一半，换句话说，验证
\begin{equation}
r_{i+2}<\frac{1}{2}r_i,i=0,1,2,\cdots
\end{equation}由此证明欧几里德算法在至多$2\log_2(b)$步终止.
证明是很容易的.
2.(Lame's theorem)Let $a$ and $b$ be positive integers with $a>b$.The length of the Euclidean algorithm for $a$ and $b$ ,denoted by $E(a,b)$,is the number of divisions required to find the greatest common divisor of $a$ and $b$.Prove that
\begin{equation}
E(a,b)\leq \log_a b+1
\end{equation}
where $a=\frac{1+\sqrt{5}}{2}$.
Proof:
\begin{equation}
\log_ab+1=a\log_ab
\end{equation}And it is easy to verify that
\begin{equation}
a\log_a b<2\log_2(b)
\end{equation}
So according to 1，the theorem holds.