Find a solution for a Pell's equation

  • Thread starter 76Ahmad
  • Start date
  • #1
48
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Hi all,

I'm trying to find the integer solutions for

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

using pell equation, any idea


please help
thanks
 

Answers and Replies

  • #2
disregardthat
Science Advisor
1,854
33
Try to write 6n^2-18n in a different way to make the equation look like a standard pell equation.
 
  • #3
48
0
I tryed and found this form:

3/2 (2n-3)^2 -m^2 = -5/2
 
  • #4
48
0
OR

m^2 - 3/2 (2n-3)^2 = 5/2
 
  • #5
48
0
and by multiply the second form with 2/5 I get

2/5 m^2 - 3/5 (2n-3)^2 = 1
 
  • #7
95
2
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
95
2
Some solutions with negative n: (-5,16), (-14, 38), (-63,158), (-152,376), (-637,1564)
 
  • #9
48
0
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
95
2
I got the solutions with the help of a small computer program:
It goes like this:

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

...sort of experimental math !
 
  • #11
84
0
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)


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.
 

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