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%(NZM, Problem ) 

\ctln{\bf Math 445 Homework 9}

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\ctln{Due whenever you feel like it (before November 26...)}

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\item{33.} [NZM, p.333, Problem 7.3.3] Show that if the continued fraction
$[a_0,\ldots ,a_n]$ has convergents $h_i/k_i$, then 
$k_i/k_{i-1}=[a_i,a_{i-1},\ldots,a_0]$ for every $i\geq 1$.

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\item{34.} [NZM, p.336, Problem 7.5.3 (sort of)] If $\alpha < \beta < \gamma$ are irrational numbers, 
$\alpha=[a_0,a_1\ldots]$ , 
$\beta=[b_0,b_1,\ldots]$ , 
$\gamma=[c_0,c_1,\ldots...]$ , and $a_i=c_i$ for $0\leq i\leq n$ , then 
$a_i=b_i=c_i$ for $0\leq i\leq n$ .

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\item{} [Hint: Induction! Use $\alpha = [a_0,\ldots ,a_{i-1},a_i+r_i] $, etc. and Problem \#30 to compare 
$\displaystyle a_{i+1}=\lfloor{{1}\over{r_i}}\rfloor$ , etc. 
Note that if $x<y$ then $\lfloor x\rfloor \leq \lfloor y\rfloor$ .]

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\item{35.} [NZM, p. 333, \# 7.3.6] Let $p$ be prime and suppose
$u^2\equiv -1\pmod{p}$ (so $p\equiv 1\pmod{4}$). 
Let $[a_0,\ldots ,a_n]$ be the continued fraction expansion of 
$\dsl {{u}\over{p}}$ , and let $i$ be the largest integer with $k_i\leq \sqrt{p}$ . Show that
$\dsl |{{h_i}\over{k_i}}-{{u}\over{p}}| < {{1}\over{k_i\sqrt{p}}}$, and $|h_ip-k_iu|<\sqrt{p}$ .
Setting $x=k_i$ and $y=h_ip-uk_i$, show that $p|x^2+y^2$ and $x^2+y^2<2p$, so 
$x^2+y^2=p$ .

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\item{36.} [NZM, p. 336, \# 7.4.7] Show that if $x=[a_0,a_1,\ldots]$ is the 
simple continued fraction expansion of (the irrational number) $x$, with convergents 
$h_n/k_n$, then 

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\ctln{$k_n|k_{n-1}x-h_{n-1}|+k_{n-1}|k_nx-h_n|=1$}

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\item{} for all $n$. [Hint: figure out how to (correctly!) remove the absolute value signs.]




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