# Thread: [SOLVED] Method of Ascent for Diophantine Equations

1. ## [SOLVED] Method of Ascent for Diophantine Equations

Hello All, in my text it mentions a "method of ascent" procedure for proving that there are infinitely many solutions to certain Diophantine Equations. As an example, it lists:

(x^2) - 3*(y^2) = 1

For familiarity, the book gives the following description for the "method of ascent" :

Method of Ascent shows that given one solution (u,v), another solution can be computed (w,z) which is in a sense larger. The proof will then involve finding a pair of formulas in forms like:

w=x+y and z=x-y. Please note that neither one of thse formulas work.

Specifically for this problem, (x^2) - 3*(y^2) = 1 , it notes that a pair of second degree formulas do work, where one of them has a cross term and the other formula involves the number 3.

Does anyone know how to do this? I'm stumped! Any help is greatly appreciated!

2. Originally Posted by Samson
Hello All, in my text it mentions a "method of ascent" procedure for proving that there are infinitely many solutions to certain Diophantine Equations. As an example, it lists:

(x^2) - 3*(y^2) = 1

For familiarity, the book gives the following description for the "method of ascent" :

Method of Ascent shows that given one solution (u,v), another solution can be computed (w,z) which is in a sense larger. The proof will then involve finding a pair of formulas in forms like:

w=x+y and z=x-y. Please note that neither one of thse formulas work.

Specifically for this problem, (x^2) - 3*(y^2) = 1 , it notes that a pair of second degree formulas do work, where one of them has a cross term and the other formula involves the number 3.

Does anyone know how to do this? I'm stumped! Any help is greatly appreciated!
Suppose $(x_i,y_i)$ is a solution.

Then $x_{i+1} = x_1 x_i + 3 y_1 y_i$ and $y_{i+1} = x_1 y_i + y_1 x_i$.

3. Originally Posted by chiph588@
Suppose $(x_i,y_i)$ is a solution.

Then $x_{i+1} = x_1 x_i + 3 y_1 y_i$ and $y_{i+1} = x_1 y_i + y_1 x_i$.
Aren't there integer based solutions to this though?

4. Originally Posted by Samson
Aren't there integer based solutions to this though?
The solution I gave you is integer valued...

5. Originally Posted by chiph588@
The solution I gave you is integer valued...
Ah, I see! I missed the constant in front of that first term during my first read through . Thank you!