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October 8th, 2017, 05:59 AM  #1 
Newbie Joined: Oct 2017 From: netherlands Posts: 4 Thanks: 0  Stuck in problem with level surfaces.
Given is the function f: R^2 > R, with f(x,y)=x^2+y^26xy+8y The level surface f(x,y)=1 contains infinitely much points (x,y) where x and y are integer. How can I prove this? 
October 8th, 2017, 12:41 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,590 Thanks: 1434 
I'd start by rotating the system via $\begin{pmatrix}x' \\ y'\end{pmatrix} = \dfrac{1}{\sqrt{2}}\begin{pmatrix}1 &1\\1 &1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} $ That gets you a standard quadratic form you can play with. 
October 8th, 2017, 10:46 PM  #3  
Newbie Joined: Oct 2017 From: netherlands Posts: 4 Thanks: 0  Quote:
I don't understand why rotating the system helps me prove this. I understand rotation with a matrix, but I don't know how I rotate my function with 2 variables. If I get a standard quadratic function, then I have an idea how I can prove this.  
October 9th, 2017, 02:20 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 21,036 Thanks: 2274 
How far did you get?

October 9th, 2017, 02:49 PM  #5 
Senior Member Joined: Sep 2016 From: USA Posts: 670 Thanks: 440 Math Focus: Dynamical systems, analytic function theory, numerics 
Suppose you have a quadratic equation of the form \[ (xa)^2 + (yb)^2 = c. \] Can you see some relatively simple restrictions on $a,b,c$ which require this equation to have infinitely many integral solutions? 
October 10th, 2017, 08:04 AM  #6 
Newbie Joined: Oct 2017 From: netherlands Posts: 4 Thanks: 0  I am trying to prove it with the formula of Pell. Thank you

October 10th, 2017, 10:04 AM  #7 
Newbie Joined: Oct 2017 From: netherlands Posts: 4 Thanks: 0 
I have this already (x3y)^22*(2y1)^2=1 Call m=x3y and n=2y1. Then I recognize the Pell theorem: n^22p^2=1. This equation has a integer solution for n=p=1. But how can I find the other solutions with the Pell theorem of n and p? (and x and y?) 
October 10th, 2017, 11:31 AM  #8  
Senior Member Joined: Sep 2015 From: USA Posts: 2,590 Thanks: 1434  Quote:
 

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