# Find a solution for a Pell's equation

1. Jun 24, 2011

Hi all,

I'm trying to find the integer solutions for

6n^2 -18n +16 = m^2

using pell equation, any idea

thanks

2. Jun 24, 2011

### disregardthat

Try to write 6n^2-18n in a different way to make the equation look like a standard pell equation.

3. Jun 24, 2011

I tryed and found this form:

3/2 (2n-3)^2 -m^2 = -5/2

4. Jun 24, 2011

OR

m^2 - 3/2 (2n-3)^2 = 5/2

5. Jun 24, 2011

and by multiply the second form with 2/5 I get

2/5 m^2 - 3/5 (2n-3)^2 = 1

6. Jun 24, 2011

### epsi00

7. Jun 25, 2011

### RamaWolf

Just a few solutions (n, m): (0, 4), (1, 2), (2, 2), (3, 4), (8,16), (17,38), (66,158), (155, 376), (640, 1564), ...

8. Jun 25, 2011

### RamaWolf

Some solutions with negative n: (-5,16), (-14, 38), (-63,158), (-152,376), (-637,1564)

9. Jun 25, 2011

thanks, great help

but how I know all the +solutions as large as I can.
what steps you follow so you found these solutions?
(no negativs solutions)

10. Jun 25, 2011

### RamaWolf

I got the solutions with the help of a small computer program:
It goes like this:

For n = 0 to 999 do
$\space\space\space$w= 6*n$^{2}$-18*n+16
$\space\space\space$If w is a perfect square i.e. w=m$^{2}$
$\space\space\space\space$ then Return (n,m)
$\space\space\space$end_If
end_For

...sort of experimental math !

11. Jun 26, 2011

### epsi00

You go on Alperton's site ( which I provided above in my first post ) and you enter your parameter to that your equation becomes:
6 x2 - y2 - 18 x + 16 = 0

and you click on solve it and it does it for you. Now under Modes ( below the solve it button ), you have two choices:
1- just solution ( for when you are in a hurry )
2- step by step ( for when you want to in fact learn something )

if you choose step by step, you will be taught how to solve this kind of equation. But if you do not take the time to check every single post ( like mine above ), then you may miss some important stuff.