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This book, which needs to be returned quite soon, has a problem I don't know where to start. How do I find a 4 parameter solution to the equation

$x^2+axy+by^2=u^2+auv+bv^2$

The title of the section this problem comes from is entitled (as this question is titled) "Numbers of the Form $x^2+axy+by^2$", yet it deals almost exclusively with numbers of the form $x^2+y^2$. It looks like almost an afterthought or a preview of what's to come where it gives the formula

$(m^2+amn+bn^2)(p^2+apq+bq^2)=r^2+ars+bs^2,r=mp-bnq,s=np+mq+anq$

Then 6 of the 7 problems use this form. The first few involve solving the form $z^k=x^2+axy+by^2$, which I quickly figured out are solved by letting $z=u^2+auv+bv^2$, then using the above formula to get higher powers. So for $z^2$ for example, I set $m=p=u$ and $n=q=v$ to get $x$ and $y$ in terms of $u$ and $v$. But for this problem, I'm drawing a blank.

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    You should specify which of the letters $a,b,u,v,x,y$ are constants and which are variables.2012-04-09
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    And specify what book and what pages. Your expressions for $r,s$ are the simplest description of the fact that the Gauss composition of the principal form with itself is again itself. Meanwhile, the formula with $r,s$ is a four parameter solution.2012-04-09
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    The book is one anyone is unlikely to see. Diophantine Analysis by Robert D. Carmichael. It is not mentioned anywhere which are constants and which are variables, but I have assumed a and b to be constant.2012-04-09
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    Forgot the page number. Page 25.2012-04-09
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    @WillJagy While it is a 4 parameter solution, it is not a 4 parameter solution for the equation in the problem I'm asked to solve.2012-04-10
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    Carmichael's book is a well-known classic.2012-04-24
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    @lhf I was just looking online for books on Diophantine analysis at local libraries. There were maybe about 2 results that didn't include "approximation" (on second thought maybe those would have been helpful...). It wasn't until I picked it up that I realized how old it was and wasn't until it was about due to go back I realized who the author was. No book drop for that one. :)2012-04-24

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