- #1

76Ahmad

- 48

- 0

I'm trying to find the integer solutions for

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

using pell equation, any idea

please help

thanks

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter 76Ahmad
- Start date

In summary, Alperton provides a small computer program that will solve an equation for you. You can also solve it yourself, but if you want step by step help, then you choose this option.

- #1

76Ahmad

- 48

- 0

I'm trying to find the integer solutions for

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

using pell equation, any idea

please help

thanks

Physics news on Phys.org

- #2

disregardthat

Science Advisor

- 1,866

- 34

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

- #3

76Ahmad

- 48

- 0

I tryed and found this form:

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

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

- #4

76Ahmad

- 48

- 0

OR

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

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

- #5

76Ahmad

- 48

- 0

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

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

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

- #6

epsi00

- 84

- 0

- #7

RamaWolf

- 95

- 2

- #8

RamaWolf

- 95

- 2

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

- #9

76Ahmad

- 48

- 0

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

RamaWolf

- 95

- 2

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

epsi00

- 84

- 0

76Ahmad said:

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.

Pell's equation is a type of Diophantine equation in mathematics that takes the form x^2 - Dy^2 = 1, where D is a positive non-square integer.

Solving Pell's equation has many practical applications, such as in number theory, cryptography, and finding rational approximations for square roots. It also has theoretical importance in understanding the properties of integer solutions to polynomial equations.

Yes, there is a well-known algorithm called the Chakravala method, which was developed by Indian mathematician Brahmagupta in the 7th century. This method involves finding continued fractions of the square root of D and using them to generate convergents that are solutions to Pell's equation.

Yes, if D is a perfect square, then the equation reduces to a simple linear equation that can be easily solved. Similarly, if D is a prime number, there are known conditions for which the equation has integer solutions.

Yes, in fact, it is known that if x and y are solutions to Pell's equation, then so are the numbers 2x + Dy and 2y + x. This means that given one solution, an infinite number of other solutions can be generated.

- Replies
- 6

- Views
- 4K

- Replies
- 11

- Views
- 5K

- Replies
- 2

- Views
- 2K

- Replies
- 1

- Views
- 1K

- Replies
- 6

- Views
- 2K

- Replies
- 4

- Views
- 2K

- Replies
- 4

- Views
- 2K

- Replies
- 14

- Views
- 10K

- Replies
- 10

- Views
- 18K

- Replies
- 6

- Views
- 3K

Share: