Find a solution for a Pell's equation

• 76Ahmad
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.
76Ahmad
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

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

I tryed and found this form:

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

OR

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

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

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

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

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

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)

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 !

76Ahmad said:
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.

1. What is a Pell's equation?

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.

2. Why is solving Pell's equation important?

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.

3. Is there a general method for finding solutions to Pell's equation?

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.

4. Are there any special cases of Pell's equation that are easier to solve?

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.

5. Can Pell's equation have infinitely many 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.

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