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Math Help - [SOLVED] Method of Ascent for Diophantine Equations

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    [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!
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Samson View Post
    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 .
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  3. #3
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    Quote Originally Posted by chiph588@ View Post
    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?
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  4. #4
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Samson View Post
    Aren't there integer based solutions to this though?
    The solution I gave you is integer valued...
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    Quote Originally Posted by chiph588@ View Post
    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!
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