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%(NZM, Problem ) 

\ctln{\bf Math 445 Homework 5}

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\ctln{Due Friday, October 10}

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\item{17.} Show that if $a^2+2b^2=c^2$ and $p|(a,c)$ for some prime $p$, then $p|b$.

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\item{} (Note: $p=2$ is treated differently...)

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\item{18.} Show that if $a^2+2b^2=c^2$ and $(a,c)=1$, then $a$ and $c$ are odd, and $b$ is even.


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\item{19.} Continuing the analysis: with $(a,c)=1$, writing 

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\ctln{$2b^2=c^2-a^2=(c-a)(c+a)=(2u)(2v)$ ,}

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\item{} show that $(u,v)=1$, one of $u,v$ is odd and the other even, and 
$u=2x^2,v=y^2$ or $u=y^2,v=2x^2$ for some integers $x,y$.

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\item{20.} Finishing the analysis: show that $(2x^2-y^2)^2+2(2xy)^2=(2x^2+y^2)^2$, and every solution
to $a^2+2b^2=c^2$ with $a,b,c\in{\Bbb N}$ has this form: 

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\ctln{$a=|2x^2-y^2|$, $b=2xy$, $c=2x^2+y^2$, with $x,y\geq 0$ and $y$ odd .}





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