# x,y equation

• Sep 14th 2013, 08:36 PM
RuyHayabusa
x,y equation
x^2-6xy+3y^2=100 x,y are natural numbers.

Help would be appreciated.
• Sep 15th 2013, 12:44 AM
BobP
Re: x,y equation
Use the usual formula to solve as a quadratic in x.
You need the term under the square root sign to be a perfect square.
• Sep 15th 2013, 01:46 AM
topsquark
Re: x,y equation
To add to bobP's comment, once you get your expression for a perfect square you are likely to need to use a spreadsheet to find the x values. At least I could find no pattern for the first 5 perfect squares.

-Dan
• Sep 16th 2013, 02:16 AM
BobP
Re: x,y equation
Wolfram Alpha gives a complete set of integer solutions. They look quite messy and I haven't checked them out in any way.
The first few though are not difficult following the quadratic/discriminant route.
You need $6y^{2}+100$ to be a square, so check out the sequence $6y^{2}=21, 44, 69, 96,...$ .
$6y^{2}=96$ gets you an early hit and a second comes not much further on.
• Sep 16th 2013, 04:28 PM
topsquark
Re: x,y equation
Quote:

Originally Posted by BobP
Wolfram Alpha gives a complete set of integer solutions. They look quite messy and I haven't checked them out in any way.
The first few though are not difficult following the quadratic/discriminant route.
You need $6y^{2}+100$ to be a square, so check out the sequence $6y^{2}=21, 44, 69, 96,...$ .
$6y^{2}=96$ gets you an early hit and a second comes not much further on.

I may have been using a more complicated system than you. (I had to define a z that was equal to the square root of the discriminant.) This is a rather common difficulty of mine. :)

How did you code your question to Wolfram|Alpha? I tried to do that to check my work but was unable to get it to solve the problem

-Dan
• Sep 17th 2013, 01:32 AM
BobP
Re: x,y equation
I'm not a regular user of Wolfram Alpha, in fact this is the first time that I've actually asked it a question.
I simply typed in the equation and pressed the compute button.
It came back with a whole pile of stuff, including a graph, which was pleasing to see, and a section headed 'Integer Solutions '.